SymTFT Fans: The Symmetry Theory of 4d N=4 Super Yang-Mills on spaces with boundaries
Iñaki García Etxebarria Jesús Huertas Angel M. Uranga

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    IFT-UAM/CSIC-24-126[]Department of Mathematical Sciences,
          Durham University,
        
          Durham, DH1 3LE, United Kingdom[]Instituto de Física Teórica IFT-UAM/CSIC,
        
          C/ Nicolás Cabrera 13-15, Campus de Cantoblanco, 28049 Madrid, Spaininaki.garcia-etxebarria@durham.ac.ukj.huertas@csic.esangel.uranga@csic.esThe SymTFT construction is an efficient way of studying the
          symmetries of Quantum Field Theories. In this paper we initiate the
          study of the SymTFT construction for interacting theories on spaces
          with boundaries, in the concrete example of d=4=4 Super
          Yang-Mills theory with Gaiotto-Witten boundary conditions. The
          resulting SymTFT has a number of peculiarities, most prominently a
          fan-like structure with a number of different SymTFT sectors joined
          by gapless interfaces that merge at the Gaiotto-Witten boundary
          SCFT. SymTFT Fans
         The Symmetry Theory of 4d =4 Super Yang-Mills on spaces with boundaries 
            Angel M. Uranga^
            September 9, 2024
        ======================================================================================
        
        
        
        § INTRODUCTION
        
        
        In the Euclidean formulation of Lorentz-invariant Quantum Field
        Theories, internal symmetries give rise to topological operators of
        codimension one. During the last few years many papers, starting with
        <cit.>, have convincingly shown that it is very
        illuminating to think of all the topological operators present
        in any given Quantum Field Theory as symmetries, sometimes referred to
        as categorical symmetries to keep in mind both that we are
        significantly broadening the traditional definition of symmetry, and
        that category theory is a natural tool for talking about the
        topological QFTs describing the topological subsector of the
        theory. Generic topological operators in QFT are not necessarily
        codimension one, and the fusion rules for topological operators are
        not necessarily group-like, so the resulting algebraic structures can
        be very rich, and encode very subtle information about the Quantum
        Field Theory at hand. We refer the reader to any of the excellent
        recent reviews
        <cit.>
        for surveys of different aspects of this rapidly developing field.
        
        We will focus on a recent proposal for capturing the categorical
        symmetries of a theory known (somewhat imprecisely in our case, for
        reasons we will explain momentarily) as the SymTFT construction
        <cit.>. In
        the simplest variant of this approach, the  QFT
        is realised in terms of the interval compactification of a
         topological field theory (the
        SymTFT), with gapped boundary conditions at one end, and
        gapless boundary conditions at the other end. The SymTFT construction
        arises very naturally from geometric engineering in string theory and
        holography, and indeed we will use holography in this work to analyse
        the SymTFT construction in a regime that has not been much studied
        before: d-dimensional theories on manifolds with boundaries. As we
        will see, in this context it is still possible to understand the
        symmetries of the d-dimensional theory on a manifold with boundary
        in terms of a (d+1)-dimensional construction on a space with
        corners, where additional gapless degrees of freedom live.
        
        The resulting bulk is not fully topological, so it would perhaps be
        better to talk about a symmetry theory instead (as emphasised
        in a related context in <cit.>, for instance), but in
        the cases we study in this paper the non-topological nature of the
        bulk ends up being fairly mild: the (d+1)-dimensional theory is
        topological except on non-topological d-dimensional interfaces. So
        by a small abuse of language we will still refer to the bulk as the
        SymTFT.
        
        The specific theories we will focus on are d=4=4 SYM on a
        space with boundary. We choose the boundary conditions preserving half
        of the superconformal symmetry which have been studied by Gaiotto and
        Witten in <cit.>, and which we briefly
        review in section <ref>. Our basic tool for understanding the
        SymTFT for such configurations is the holographic dual, constructed in
        <cit.>,
        and reviewed in section <ref>. We extract the SymTFT from
        the holographic dual in section <ref>. The resulting SymTFT
        is similar to the SymTree construction introduced in
        <cit.>, although with a number of differences that we
        describe in detail in section <ref>. A striking feature of
        our setup is the various gapless domain walls connecting the different
        SymTFTs are arranged in a fan configuration, with the
        three-dimensional Gaiotto-Witten BCFT living at the tip of the
        fan. See figure <ref> for a sketch of the
        SymTFT.
        
        
        
        
        
        A number of expected features of the SymTFT can be understood from
        brane physics in the original string theory setup. We develop various
        aspects of the relation between these two viewpoints in
        section <ref>, in particular the effect of translating
        topological operators between different sectors in the SymTFT
        fan. Section <ref> goes further in this direction,
        studying the SymTFT interpretation of bringing 7-branes from infinity
        — an operation which is fairly natural and well understood from the
        brane point of view, but leads to some puzzles (which we resolve)
        when reinterpreted from the SymTFT point of view. Finally, in
        <ref> we generalise the discussion in the previous
        sections by allowing for orientifolds localised at the boundary.
        
        
        In this work we exploit the supergravity duals of the QFT with
        boundary to learn about its topological sector and extract the SymTFT,
        but our results are also relevant in the reverse direction. Indeed,
        gravitational solutions with End of the World (ETW) boundary
        configurations, dubbed dynamical cobordisms
        <cit.>[For
          related ideas, see
          <cit.>
          for early references, and
          <cit.>
          for recent works.], provide dynamical realisations of the Swampland
        Cobordism Conjecture <cit.>, generalising the concept
        of bubbles of nothing <cit.> (see
        <cit.>
        for recent developments). In this spirit, the ETW configurations we
        will consider were in fact described as dynamical cobordisms of
        AdS_5×^5 in <cit.>. Hence, our in-depth
        understanding of the topological structure of the gravitational bulk
        theory and its interplay with the topology of the ETW configuration
        are a first step towards understanding it for more general dynamical
        cobordisms, providing a valuable tool for the study of topological
        properties of general cobordism defects ending spacetime.
        
        
        
        Note added: As we were finishing this paper we became aware of
        a number of upcoming papers by other groups discussing various aspects
        of SymTFTs for theories on spacetimes with boundaries
        <cit.>. We are very
        thankful to the authors of these papers for agreeing to coordinate
        submission.
        
        
        
        
        
        
        § GAIOTTO-WITTEN CONFIGURATIONS
        
        
        In this section we will quickly review — focusing on the ingredients
        most relevant for the discussion of the SymTFT — the boundary
        conditions studied in <cit.>, which
        preserve half of the superconformal symmetry, namely OSp(4|4), of
        d=4=4(N) SYM. These boundary conditions involve a 3d
        SCFTs (BCFT_3) on the boundary of the 4d theory. In these works the
        CFT_4/BCFT_3 systems were realised as systems of N semi-infinite
        D3-brane ending on a set of NS5- and D5-branes (with finite segments
        of D3-branes suspended between them), thus realising a quiver gauge
        theory of the kind introduced in <cit.>, which under
        suitable conditions flows in the IR to an interacting 3d =4
        SCFT.
        
        
        
         §.§ The brane construction and its field theory
        
        
        We consider N D3-branes along the direction 012, of semi-infinite
        extent in the direction 3 (namely x^3>0, without loss of
        generality), and at the origin in the remaining directions.  These
        D3-branes end on stacks of NS5-branes, which span the directions 012
        456, and at the origin in the direction 3 789, and stacks of D5-branes
        spanning 012 789, and at the origin in 3 456. Actually, in order to
        define the D3-brane boundary on the NS5- and D5-brane system, it is
        more appropriate to consider the NS5- and D5-branes to be slightly
        separated in the direction x^3, and to admit D3-branes suspended
        between them. Once a specific set of constraints, to be reviewed
        below, are satisfied, one can take the limit of coincident 5-branes,
        which implements the flow to the strongly coupled IR BCFT_3 boundary
        conditions for the CFT_4. Because the UV theory, defined by the
        Hanany-Witten brane configuration, preserves 8 supercharges (3d
        =4), the resulting BCFT preserves maximal OSp(4|4)
        superconformal symmetry. Note that the R-symmetry subgroup
        SO(3)× SO(3) of SO(6) is manifest as the rotational symmetry
        in the 3-planes 456 and 789.
        
        A key role in the brane configuration is played by the linking numbers
        of the 5-branes.  These can be defined as the total monopole charge on
        the 5-brane worldvolume theory <cit.>, and can therefore
        be measured at infinity of the 5-brane worldvolume. Hence they are
        invariant under changes in the ordering of branes and the
        corresponding brane creation effects. In practice, given an ordering
        of the 5-branes, the linking number of a 5-brane is given by the net
        number of D3-branes ending on the 5-brane from the right (because
        D3-brane boundaries are (singular) monopoles on the 5-brane
        worldvolume theory) plus the total number of 5-branes of the other
        kind (i.e. NS5 vs D5) to the left of the 5-brane (because they can
        lead to D3-branes ending from the right via Hanany-Witten brane
        creation processes).
        
        There are two conditions for the boundary brane configuration to
        define a supersymmetric BCFT_3: (1) Any D5-brane on which a net
        non-zero number of D3-branes ends from the right is located to the
        right of all the NS5-branes. This constraint ensures that the brane
        configurations have an interpretation in gauge theory, with the
        NS5-branes with suspended D3-branes among them providing a quiver
        gauge theory, and the D5-branes with non-positive net number of
        D3-branes yielding flavours for some of the nodes. (2) For each kind
        of 5-brane (NS5 or D5) the linking numbers are non-decreasing from
        left to right. For D5-branes, this is automatic except for the
        D5-brane to the right of all NS5-branes, and for these the condition
        guarantees the absence of decoupled degrees of freedom in the limit of
        coincident 5-branes; the condition for NS5-branes then follows from
        S-duality.
        
        Related to the above, linking numbers play a role in non-abelian IR enhancement of U(1) global (0-form) flavour symmetries of the 3d UV theory, which are realised as symmetries on the worldvolume of the 5-branes. In the strong coupling regime, n 5-branes of the same kind and with the same linking number lead to monopole operators in the field theory which enhance the symmetry to U(n). Hence linking numbers provide the invariant quantities characterising the BCFT_3.
        
        Let us provide an extra intuition of the interpretation of the linking number in defining the boundary conditions. Consider the case of a single kind of 5-brane, e.g. D5-branes (NS5-branes admit a similar description due to S-duality), organised in m stacks, labelled by an index b, with multiplicities m_b of D5-branes with equal linking number L_b. Since in this case there are no NS5-branes, the linking numbers are realised in terms of D3-branes ending on the D5-branes from the right. Each D5-brane in the b^th stack has L_b D3-branes ending on it, with boundary conditions associated to the L_b-dimensional irreducible representation R_L_b of SU(2) (namely, the SO(3) associated to 789), which is realised by the three (matrix-valued) complex scalars of the 4d =4(N) SYM theory, of the form [X^i,X^j]∼ϵ^ijk X^k. Hence, for the complete system of D5-brane stacks, the boundary condition is associated to a (reducible, in general) representation of SU(2)
        
            R_N=⊕_b=1^m m_b R_L_b  .
        
        The SU(2) commutation relation can be regarded as describing the
        D3-branes puffing up into non-commutative funnels corresponding to the
        D5-branes, whose width is related to the number of D3-branes, i.e. the
        linking numbers. The ordering of linking numbers mentioned above
        ensures that narrow funnels fit inside wider funnels without
        intersections which would introduce additional degrees of freedom.
        
        An efficient way to describe these systems is to focus on the 3d
        =4 theories realised on the D3-branes suspended among the
        5-branes, and to describe the semi-infinite D3-branes as the gauging
        of a global symmetry of the BCFT_3. One prototypical example is that
        of the T[SU(N)] theories <cit.>, which are defined as
        the IR fixed point of the gauge theory on a system with N NS5-branes
        and N D5-branes, realising the gauge theory
        
            U(1) - U(2) - … - U(N-1) -[(N)]  ,
        
        with bifundamental hypermultiplets of consecutive gauge factors, and N hypermultiplets in the fundamental of the U(N-1) group, acted on by the (N) global symmetry. Namely, there are N NS5-branes separated by intervals of (an increasing number of) suspended D3-branes, and N D5-branes between the two rightmost NS5-branes, hence giving flavours to the U(N-1) factor. 
        
        The gauging of the (N) symmetry is implemented by moving the D5-branes infinitely to the right, introducing N semi-infinite D3-branes in crossing the last NS5-brane. We thus get a semi-infinite stack of N D3-branes ending on a set of N NS5-branes, all with linking number 1 (hence with an enhanced U(n) flavour symmetry). A similar configuration (obtained by S-duality and exchange or 345 and 789 (i.e. 3d mirror symmetry) exists for N D3-branes ending on a set of N D5-branes with linking number 1. Hence this provides a boundary condition of the kind (<ref>), with the N dimensional representation being trivial (i.e. N copies of the trivial 1-dimensional representation).
        
        Let us pause for an interesting observation nicely illustrated by this example. The symmetries of fairly general 3d =4 theories (including those from from NS5- and D5-brane configurations) were considered in <cit.>, in particular the just discussed enhanced 0-form flavour symmetries, as well as the 1-form symmetries, so we may apply their results to our setup. In particular, considering the above T[SU(N)] theories, the 1-form symmetry obtained upon gauging the flavour SU(N) (meaning (N) with global structure of SU(N)) was identified in <cit.> to be _N. Hence, despite the fact that the D5-branes introduce flavours in the fundamental of the 4d SU(N), the electric 1-form symmetry is not fully broken, but rather there survives a diagonal combination of it with a _N in the enhanced flavour SU(N) on the D5-branes. Similarly, for other gaugings SU(N)/_p with N=pk, the surviving electric 1-form symmetry is _k<cit.>. 
        
        The generalisation of the T[SU(N)] theories to the general configurations of D5-branes described in (<ref>) is known as the T_ρ[SU(N)], where ρ is a partition of N given by
        
            N= 1+…+1_m_1  times+2+…+2_m_2  times+…
        
        i.e. with m_b parts equal to L_b, according to the decomposition of the representation (<ref>).
        
        As a concrete example, we may consider the theory corresponding to the
        trivial partition ρ of N into 1 part equal to N. We thus have
        1 D5-brane with linking number N, namely the N D3-branes end on 1
        D5-brane exploiting the N-dimensional irreducible representation of
        SU(2). By extending the arguments in <cit.>, one
        can check that upon gauging of SU(N) the electric 1-form symmetry of
        the theory is _N. Analogously the 1-form symmetry for
        SU(N)/_p gauging is _k with k=N/p. We will recover this
        pattern from the SymTFT in section <ref>.
        
        
        There is a generalisation of the above class to the case with both
        NS5- and D5-branes. The starting point is to also introduce stacks of
        n_a NS5-branes with linking numbers K_a, in addition to the
        earlier stacks of D5-branes, obeying the rules provided above, to
        define a 3d =4 theory with no semi-infinite D3-branes. Then one
        can rearrange the configuration by locating all the D5-branes to the
        left of the NS5-branes. We end up with a set of N D3-branes on a
        segment, ending on D5-branes from the left (resp. NS5-branes from the
        right) according to a partition ρ (resp. ρ̂) of N,
        equivalently the multiplicities and linking numbers of the
        5-branes. The data (ranks and fundamental flavours of all the unitary
        gauge factors in the quiver) of the original gauge theories can be
        obtained from the partitions. These gauge theories, under the
        conditions already explained, flow to IR 3d SCFTs known as
        T_ρ^ρ̂[SU(N)], and provide BCFT_3 boundary conditions
        by gauging their (N) global symmetry.
        
        Let us be more explicit on this last point. In addition to the above defined linking number, it is useful to introduce an equivalent set of linking numbers K̃_a, L̃_b as the net number of D3-branes ending on the 5-brane from the right  minus the total number of 5-branes of the other kind to the  right of the 5-brane. These new versions differ from the old ones just in an overall shift by the total numbers of D5- and NS5-branes, respectively. In the 3d T_ρ^ρ̂[SU(N)] theories, we have N D3-branes ending on the NS5-branes from the right and on the D5-branes on the left, hence the total sum of NS5-brane linking numbers ∑_a n_aK_a is equal to N, and similarly for the total sum of D5-brane linking numbers ∑ m_bL̃_b=-N. Upon gauging the (N) flavour symmetry, we have an extra set of N semi-infinite D3-branes ending on the 5-brane set from the right, hence the condition that the BCFT_3 provides a boundary condition for the 4d =4su(N) SYM theory is
        
        
            N=∑_a n_aK_a + ∑ m_bL̃_b  ,
        
        These ingredients will be manifest in the gravitational dual description in the next section, and will be inherited as key ingredients in the SymTFT.
        
        Before that, we would like to mention that the Hanany-Witten brane configurations can be used not only to provide boundary conditions for 4d =4(N), but also more general configurations. These include configurations of NS5- and D5-branes separating two stacks of semi-infinite D3-branes (a particular case of Janus configurations <cit.>), which we will exploit in the discussion of orientifold theories in section <ref>. In addition, one can also construct theories in which there are several configurations of NS5- and D5-branes separated by large distances, with sets of D3-branes suspended between consecutive configurations. These have gravitational duals including multiple AdS_5×^5 asymptotic regions, and will be briefly mentioned in section <ref>.
        
        
        
         §.§ The holographic dual: End of the World branes
        
        
        In this section, we will briefly review the gravitational duals of the brane configurations described in the last section. They are given by a particular class of explicit 10d supergravity backgrounds studied in <cit.> (see also <cit.> for recent applications).
        
        
        
        
        The most general supergravity solutions preserving 16 supersymmetries
        and SO(2,3)× SO(3)× SO(3) symmetry were explicitly
        constructed in <cit.>. They are
        gravitational duals of Hanany-Witten brane configurations of NS5- and
        D5-branes with stacks of D3-branes as in the previous section
        <cit.>. The geometries in general have a “bagpipe”
        structure <cit.> (see figure <ref>), namely
        they look like AdS_4×^6, with ^6 a 6d compact manifold
        (the “bag”), save for a number of AdS_5×^5 throats (the
        “pipes”) sticking out of it. We will eventually focus on the
        specific case with only one AdS_5×^5 ending on the bag,
        which provides the gravity dual of 4d =4(N) SYM on a 4d
        spacetime with boundary.
        
        The most general supergravity solutions preserving 16 supersymmetries
        and SO(2,3)× SO(3)× SO(3) symmetry have the structure of a
        fibration of AdS_4×_1^2×_2^2 over an oriented
        Riemann surface Σ. The ansatz for the 10d metric is
        
            ds^2= f_4^2 ds^2_AdS_4+f_1^2 ds^2__1^2+f_2^2 ds^2__2^2+ds^2_Σ  .
        
        Here f_1, f_2, f_3 are functions of a complex coordinate w of Σ, in terms of which 
        
            ds^2_Σ=4ρ^2 |dw|^2  ,
        
        for some real function ρ. There are also non-trivial backgrounds for the NSNS and RR 2-forms and the RR 4-form; we will provide only the necessary results, referring the reader to the references for further details. 
        
        There are closed expressions for the different functions in the above metric, describing the BPS solution. As an intermediate step, we define the real functions
        
            W∂_w h_1∂_w̅h_2+∂_w h_2∂_w̅h_1 
              ,   N_1 2h_1h_2|∂_w h_1|^2-h_1^2 W    ,  N_2 2h_1h_2|∂_w h_2|^2-h_2^2 W
        
        in terms of the functions h_1,h_2 to be specified below. The dilaton is given by
        
            e^2Φ=N_2/N_1  ,
        
        and the functions are given by
        
            ρ^2=e^-1/2Φ√(N_2|W|)/h_1h_2  ,   f_1^2=2e^1/2Φ h_1^2√(|W|/N_1)  ,   f_2^2=2e^-1/2Φ h_2^2√(|W|/N_2)  ,  f_4^2=2e^-1/2Φ√(N_2/|W|)  .
        
        
        In the following we focus on the solutions describing a single asymptotic AdS_5×^5 region, as befits the gravitational dual of a stack of semi-infinite D3-branes ending on a configuration of 5-branes, i.e. a 4d =4(N) SYM theory with a boundary on which it couples to a Gaiotto-Witten BCFT_3.
        
        The Riemann surface Σ can be taken to correspond a quadrant w=r e^iφ, with r∈ (0,∞) and φ∈[π/2,π]. At each point of the quadrant we have AdS_4×_1^2×_2^2, fibered such that ^2_1 shrinks to zero size over φ=π (negative real axis) and ^2_2 shrinks to zero size over φ=π/2 (positive imaginary axis). 
        Hence the boundaries of the quadrant Σ are actually not boundaries of the full geometry, which closes off smoothly over those edges. The general solution includes a number of 5-brane sources, describing the NS5- and D5-branes, and spikes corresponding to asymptotic regions AdS_5×^5, which describe 4d =4 SYM sectors on D3-branes. 
        
        All these features are encoded in the quantitative expression for the functions h_1, h_2, which for this class of solutions, have the structure 
        
            h_1   = 4 Im(w)+2∑_b=1^md̃_b log(|w+il_b|^2/|w-il_b|^2)  ,
            
            h_2   =-4 Re(w)-2∑_a=1^n d_a log(|w+k_a|^2/|w-k_a|^2)  .
        
        Here the supergravity solution includes 5-brane sources, which are
        located at specific points in the two edges of the quadrant. Concretely the NS5-branes
        are along the φ=π axis, with stacks of multiplicity n_a at
        positions w=-k_a, and the D5-branes are along the φ=π/2
        axis, with stacks of multiplicity m_b at positions w=il_b. The
        multiplicities of 5-branes are related to the parameters d_a,
        d̃_b via
        
            n_a= 32π^2 d_a∈   ,   m_b= 32 π^2 d̃_b∈  .
        
        Specifically, the supergravity solution near one of this punctures is locally of the form of a NS5-brane spanning AdS_4×^2_2 (respectively a D5-brane
        spanning AdS_4×^2_1), with n_a units of NSNS 3-form flux
        (respectively m_b units of RR 3-form flux) on the ^3
        surrounding the 5-brane source. The latter is easily visualised, by
        simply taking a segment in Σ forming a half-circle around the
        5-brane puncture, and fibering over it the ^2 fibre shrinking at
        the endpoints, so that we get a topological ^3, see figure
        <ref>. Hence, near each 5-brane puncture, the metric factorises as
        AdS_4×^2×^3 fibered along a local radial
        coordinate r̃ parametrising the distance to the puncture.
        
        
        
        
        Asymptotically, at r→∞ far away from the 5-branes, the two
        ^2's combine with the coordinate φ to form a ^5, so
        in this limit we have AdS_5×^5, with N units of RR 5-form
        flux.  The parameter N will shortly be related to other quantities
        in the solution. The ^5 is manifest in figure <ref>,
        by fibering _1^2×_2^2 over an arc φ∈[π/2,π]
        at fixed r>|k_a|,|l_b|, so that each of the two ^2 shrinks to
        zero size at one of the endpoints of the arc, leading to a topological
        ^5.
        
        The ^5 can be deformed to smaller values of r, and the 5-form
        flux is preserved, except when the arc crosses one of the 5-brane
        punctures. At this point, the 5-form flux can escape through the
        5-brane and decreases, leaving less flux in the inner ^5 shell;
        this is the gravitational manifestation that some number of D3-branes
        ends on the 5-brane. Hence, the change in the 5-form flux encodes the
        linking number of the 5-branes in the corresponding stack. Following
        <cit.>, in the ^2×^3 geometry around the
        stacks of n_a NS5-branes and of m_b D5-branes, there are
        non-trivial integrals
        
            ∫_^2_2 C_2=K_a  ,  ∫_^3_1 H_3=n_a   ;  ∫__1^2 B_2=L̃_b  ,  ∫_^3_2 F_3=m_b  .
         
        Note that here we have used the usual linking numbers K_a for the NS5-branes, but the modified linking numbers L̃_b, introduced in section <ref> for the D5-branes. We recall that the linking numbers K_a, L_b (respectively K̃_a, L̃_b) of 5-branes are defined as the net number of D3-branes ending from the right on the 5-brane plus (respectively, minus) the number of 5-branes of opposite kind to the left (respectively, to the right) of the 5-brane.
        
        With this proviso, the change in the flux 
        
            F̃_5=F_5+B_2F_3-C_2H_3  ,
        
        with F_5=dC_4, upon crossing an NS5-brane (resp. D5-brane) is given by n_aK_a (resp. m_bL̃_b), with no sum over indices. Hence the stacks gather the 5-branes with same linking number, and their (n_a), (m_b) gauge symmetry is the holographic dual of the enhanced non-abelian flavour symmetries (or their duals) for the BCFT_3 mentioned in <ref>.
        
        An important point about (<ref>) is that the integrals of NSNS and RR 2-form potentials on the ^2's are essentially integers, related to linking numbers of the brane configuration. This may seem striking, because periods of p-form potentials are not topological quantities, and can be changed continuously. Hence, we clarify that the values in (<ref>) correspond to a particular choice for B_2, C_2 and C_4, specifically leading to a vanishing F_5=dC_4 on the asymptotic ^3×^2's associated to the 5-branes, see <cit.>. Any topological deformation of the configuration, such that the periods of B_2, C_2 are no longer integer, implies a change in C_4 such that F_5 is no longer vanishing on ^3×^2, and compensates thing off, so that we recover the same flux of F̃_5 in (<ref>). Thus, the invariant statement is that the 5-form flux jumps by the corresponding integer quantity upon crossing the 5-branes.
        
        The flux over the ^5's decreases in such crossings until we reach
        the region r<|k_a|,|l_b|, in which there is no leftover flux, so
        that the ^5 shrinks and spacetime ends in an smooth way at r=0,
        see figure <ref>. This implies that the relation between
        the asymptotic 5-form flux N and the 5-brane parameters is
        
            N=∑_a n_a K_a+∑_b m_bL̃_b  ,
        
        nicely dovetailing (<ref>). 
        
        A final point we had skipped is the relation between the location of the 5-brane punctures k_a, l_b and the linking numbers K_a, L̃_b, i.e. the 2-form fluxes (<ref>). The positions of the 5-brane punctures are determined by the expressions
        
        
            K_a=32π(k_a +2∑_bd̃_b arctan(k_a/l_b))   ,   L̃_b=32π(l_b +2∑_a d_a arctan(k_a/l_b))  .
        
        We refer the reader to the references for a derivation of this result and other related details.
        
        As we had anticipated, and as emphasised in <cit.>, the above supergravity background describes a bordism to nothing of the asymptotic AdS_5×^5 via an End of the World (ETW) configuration <cit.>. We review this perspective in what follows, also as a recap of the fundamental parameters determining the structure of the solution, and their interplay with the field theory parameters. 
        
        Let us describe the asymptotic AdS_5(×^5) region in the Poincaré patch, so the the holographic boundary (which is a half-space semi-infinite in the direction 3) is manifest. The AdS_4 geometry in the supergravity solution corresponds to an AdS_4 slicing of AdS_5, where the slices of the foliations correspond to radial lines sticking out of the holographic 3d boundary of AdS_4, see figure <ref>. The slices of this foliation are labelled by the coordinate r=|w| in the Riemann surface Σ of the supergravity solution. The asymptotic region r→∞ in the 10d supergravity solution corresponds to the radial lines closer to the 4d holographic boundary, and correspond to an AdS_5 (×^5) with N units of 5-form flux. 
        
        As one moves inward in the Riemann surface to values of r corresponding to the 5-brane puncture locations, the 5d geometry hits 5-brane sources spanning AdS_4 slices. In the Poincaré picture they correspond to lines sticking out radially from the 3d boundary of the 4d holographic boundary in a fan-like structure. The 5-brane sources actually correspond to local internal geometries ^2×^3, with background fields (<ref>), namely 3-form fluxes n_a or m_b (encoding the 5-brane multiplicity for NS5- and D5-branes, respectively), and 2-form backgrounds K_a, L̃_b. The latter determine the explicit angular locations of the 5-brane stacks in Poincaré coordinates (equivalently, the radial positions of the 5-brane punctures in Σ in the 10d solution) via (<ref>). The 5-form flux over the ^5 decreases upon crossing the 5-branes by amounts n_aK_a or m_bL̃_b, respectively. Eventually, the 5-form flux is completely peeled off and at a critical angle in  Poincaré coordinates the ^5 shrinks and spacetime ends. Hence, the solution describes and End of the World (ETW) configuration.
        
        A final word of caution about the above 5d picture. As is familiar,
        the scale of the internal geometries are comparable to those of the
        corresponding AdS geometries [This is known as `no scale
          separation' in the swampland literature <cit.> (see
          <cit.> for a review and further references).]. In
        our setup this holds both for the asymptotic AdS_5×^5 and
        the AdS_4×_6“bag” dual of the BCFT_3. As usual in
        holography, one may often still use a lower dimensional perspective,
        by removing the internal space via a dimensional reduction, rather
        than a physical effective theory below a decoupled KK scale. This is
        extremely useful in cases where the dimensional reduction is a
        consistent truncation, as in the case of the asymptotic
        AdS_5×^5, where the 5d perspective has been usefully
        exploited for over two decades. To some extent, as discussed in detail
        in <cit.>, this does not apply to the configuration
        with the ETW configuration, due to its reduced symmetry, and the
        lower-dimensional perspective has a more limited applicability. This
        will however not pose any problem for our analysis, which is focused
        on topological aspects related to the generalised symmetries of the
        QFT and its SymTFT as obtained from the topological sector of the
        holographic realisation. We turn to this study in the next section.
        
        
        
        
        § THE SYMTFT
        
        
        A natural way to obtain the SymTFT in d-dimensional quantum field
        theories with a (d+1)-dimensional holographic dual is by extracting
        the topological sector of the dual
        <cit.>. (More
        generally, it is possible to perform a similar analysis for
        geometrically engineered QFTs without requiring the existence of a
        tractable large N gravitational dual, see for instance
        <cit.>
        for works in this direction.)
        
        In our present context of the 4d =4(N) SYM with
        Gaiotto-Witten BCFT_3 boundary conditions, we may use this same
        strategy by analysing the holographic dual reviewed in section
        <ref>. Indeed, in the asymptotic AdS_5×^5
        region the recipe amounts to reducing the 10d Chern-Simons coupling of
        type IIB string theory on an ^5 with N units of RR 5-form flux,
        which, as studied originally in <cit.> leads to the 5d
        SymTFT[In this paper it will be sufficient for us to think of
          B_2 and C_2 as differential forms, as opposed to cochains in
          differential cohomology or K-theory.]
        
            S_5d=N/2π∫__5 B_2 dC_2  ,
        
        where we have allowed for a general 5d manifold , and we work in
        the convention where the supergravity fields B_2, C_2 are
        2π-periodic. (In a number of places below we will just highlight
        the parametric dependence on N or similar parameters fixing the
        order of the discrete symmetries.)
        
        In the case of the holographic dual of the 4d SCFT with BCFT_3
        boundary conditions, we encounter a crucial new structure, which we
        dub the SymTFT Fan, as we approach the ETW configuration. Indeed, the
        holographic 5d bulk is split into several pieces by the presence of
        the 5-branes, which moreover carry non-topological field theories on
        their worldvolume. Each piece of the 5d bulk is still associated to a
        compactification of the 10d theory on ^5, albeit with decreasing
        number of units of RR 5-form flux, until the flux is eventually peeled
        off and the ^5 can safely shrink, see figure <ref>,
        in the spirit of the cobordism to nothing explained in the previous
        section. Hence, we expect that the dimensional reduction on ^5,
        and the SymTFT (<ref>) for different effective values of N
        still plays and important role, but new structures are required to
        describe the 5-brane theories and how they glue the SymTFTs.
        
        
        
         
        Since our Symmetry Theory at hand has several boundaries, we fix some
        terminology for them to avoid confusions, see figure
        <ref>: we refer to the holographic 4d half-space as
        holographic boundary or physical boundary, as in standard SymTFTs,
        because it supports the local degrees of freedom of the d=4=4
        theory; the same applies to the holographic 3d space supporting the
        BCFT_3, which we will also refer to as the 3d boundary of the 4d
        physical theory. The topological boundary is that at which topological
        boundary conditions are set for the SymTFTs. Finally, the boundary of
        the 5d bulk spacetime, at which the ^5 shrinks and spacetime
        ends, is referred to as ETW boundary. We also sometimes use ETW brane
        or ETW configuration for the complete systems of 5-branes and ETW
        boundary providing the bordism to nothing for the asymptotic 5d bulk,
        in practice the AdS_5(×^5) region.
        
        
        
        
         §.§ The SymTFT Fan: Gluing SymTFTs via U(1)s
         
        
        In this section we discuss the structures underlying the gluing of the
        different SymTFT wedges via the 5-brane theories into the SymTFT
        Fan. This structure of the Symmetry Theory connects very nicely with
        several recently introduced tools in the generalisation of the concept
        of SymTFTs.
        
        
        
          §.§.§ Relation to SymTrees
        
        
        We now address the key point about how the SymTFTs in the different ^5 regions of the geometry are related. The crucial observation is that, as explained in section <ref>, in the full supergravity description, the regions near the 5-branes correspond to spikes locally corresponding to AdS_4×^2 times a cone over ^3, as befits the local geometry around a backreacted stack of 5-branes. For simplicity, in what follows we focus on a single stack of D5-branes (NS5-branes work similarly due to S-duality), and we denote by n the number of 5-branes in the stack and P its linking number.
        This implies that there are n units of RR 3-form flux over the ^3 (NSNS 3-form flux for NS5-branes), and there is a non-trivial background of the NSNS 2-form B_2 over the ^2 (RR 2-form for NS5-branes)
        
            ∫_^2B_2=P  .
        
        
        
        
        This implies that when the solution approaches a 5-brane, the geometry
        corresponding to the compactification on ^5 with N units of RR
        5-form flux splits into a region corresponding to ^5 with N-nP
        units of RR 5-form flux (the 5d region “across” the 5-brane in
        figure <ref>), and a region corresponding to a
        compactification on ^2×^3 with nP units of 5-form flux.
        The latter arises from
        
            ∫_^2×^3F̃_5=∫_^2×^3(F_5+B_2F_3)=nP  ,
        
        where we have used the fact that there are no D3-brane sources and hence no F_5 flux in the ^2×^3 regions. In other words, the ^2×^3 is homologically the difference of the two ^5's in both sides of the 5-brane, see figure <ref>, hence the total flux of F̃_5 must be conserved.
        
        
        
        
        The splitting of the geometry implies a splitting of the SymTFT, as follows. As suggested above, the two ^5 regions (on both sides of the 5-brane) lead to two SymTFTs of the kind (<ref>) with coefficients N and M N-nP, which we denote by SymTFT_N and SymTFT_M, respectively. On the other hand, out of their meeting locus there sticks out another SymTFT (which we denote by SymTFT_^2×^3), obtained from the reduction of the topological sector of the 10d theory on the compact ^2×^3 geometry with the above mentioned fluxes and background fields. The structure of this SymTFT_^2×^3 will be discussed later on, but we anticipate it contains a coupling (<ref>) with coefficient nP. The three SymTFTs meet at a juntion, which supports non-topological degrees of freedom, which will be discussed later on.  The complete picture of the Symmetry Theory is hence a modification of figure <ref> in which each 5-brane is replaced by a branch SymTFT, as depicted in figure <ref>.
        
        This local structure (local in the sense that it is away from the physical and topological boundaries of the SymTFTs) is identical to the SymTrees introduced in <cit.>, in which a SymTFT can split into multiple decoupled SymTFTs at a (non-topological) junction theory. However, it is crucial to notice that there are the following two important (and related) differences between our setup and the SymTree constructions in <cit.>:
        
        ∙ The first is that in the SymTrees in <cit.> the splitting of the `trunk' SymTFT into the branch SymTFTs happens as one moves from the topological boundary to the physical boundary of the complete Symmetry Theory; in our present setup, the splitting of the SymTFTs happens along a direction  parallel to the boundaries of the complete Symmetry Theory, namely each of the SymTFTs in the tree (as well as the junction theory) extend across the Symmetry Theory sandwich between the topological and the physical boundaries. On the topological boundary, this implies that each of the SymTFTs after the splitting has its own set of topological boundary conditions, rather than being determined by the junction theory as in <cit.>. On the physical boundary, the parent SymTFT_N ends up on the 4d SCFT on the half-space, whereas interestingly  all the SymTFTs after the splitting end up on the  same 3d BCFT. This is related to the next point.
        
        ∙ The second main difference, related to the previous one, is that in <cit.> the splitting of the SymTFT is a consequence of the existence of a energy scale below which the full theory is separated into different decoupled sectors. In our context, both the 4d physical theory and its 3d boundary theory enjoy superconformal symmetry, so there are no scales in the system, hence the splitting of the SymTFT is not controlled by the criteria explained in <cit.>. The difference is clearly displayed in the holographic dual, where, in the SymTrees in <cit.>, the splitting occurs along the holographic direction, hence is controlled by a physical energy scale, whereas in our setup the splitting occurs along a direction parallel to the holographic boundary and takes places at all scales. In fact, in the supergravity solution the amount of separation between the branches after splitting increases with the distance to the holographic boundary in a characteristic fan-like structure; this nicely dovetails the behaviour in a setup with conformal invariance, where the physical separation is related to the energy scale as dictated by dimensional analysis, due to the absence of fundamental energy scales. 
        
        
        
        From the above argument, we expect on general grounds that the fan-like structure will occur for the holographic dual of any CFT with BCFT boundary conditions. This is a particularly useful observation since explicit supergravity solutions of gravitational duals of CFTs on spacetimes with boundaries are scarce. We thus expect that our observation of the role of SymTFT Fans in 4d =4 SYM with boundaries extends to more general classes of CFTs with boundaries, and are useful to uncover their Symmetry Theory structures, even when the supergravity solution is now known. 
        
        
        
        
          §.§.§ The 5-brane SymTFT 
        
        
        Let us now go into more detail into the SymTFT_^2×^3 corresponding to the reduction of the topological sector of the 10d theory on ^2×^3. The first observation is that, in a sense that will be made more precise later on in the context of nesting, it corresponds to the 5d SymTFT of the 4d physical theory of the stack of n 5-branes compactified on ^2 (and propagating on AdS_4 or whatever replaces it in more general context). This has a (n)≃(n)⊕(1) gauge symmetry. Leaving the (1) factor for later on, we focus on the non-abelian factor, which leads to a 4d supersymmetric (n) pure SYM theory. The configuration is actually closely related to the realisation in <cit.> of this gauge theory as the context of the Klebanov-Strassler holographic dual <cit.>. In fact, this holographic dual is based on a cone over T^1,1^2×^3 with n units of RR 3-form flux over the ^3 and nP units of 5-form flux over ^2×^3. Therefore the derivation of the SymTFT of the 5-brane theory essentially follows <cit.>. We will not need its detailed structure and simply note that it includes a coupling (<ref>) with coefficient nP due to (<ref>); we refer the reader to appendix A in <cit.> for additional details.
        
        
        
        
          §.§.§ The junction theory 
        
        
        In addition to the (n) theory, there is the (1) degree of freedom. As explained for SymTrees in <cit.>, the (1) is actually not localised on the 5-brane physical theory, but rather belongs to the junction theory. In fact, from our earlier discussion about the general structure of the splitting of SymTFTs and the matching of the B_2,C_2 sectors with action (<ref>) with coefficients N, M, nP, the local structure of the junction is identical to that in the SymTree corresponding to an adjoint Higgsing (N)→  [ (N_1)⊕(N_2) ], with N_1=nP and N_2=M in our case. Following <cit.>, the (1) dynamics can be isolated as the decomposition (N)→(N_1)⊕(N_2)⊕(1) inherited from SU(N)→ [SU(N_1)× SU(N_2) × U(1)]/_L with L= lcm (N_1,N_2) the least common multiple of N_1, N_2.
        
        Let us provide a simple heuristic argument that there is a (1) theory supported at the junction theory describing the gluing of two ^5's and a ^2×^3 geometry (which 5-form fluxes related by their homology relation). The argument is in the spirit of the SymTree structure in <cit.> for a Higgsing, but, as explained, the local structure of the SymTrees applies analogously to our case. Consider a stack of m D3-branes located at a conifold singularity, and split it into two stacks of m_1, m_2 D3-branes (with m_1+m_2=m) located at slightly different positions away from the singular point. We may take the distance between the two points much smaller than the (already small) distance from them to the singular point. This corresponds to performing a Higgsing of the (m)⊕(m) Klebanov-Witten theory <cit.> to the diagonal =4(m) SYM theory, which is subsequently Higgsed down to two decoupled =4(m_1) and (m_2) SYM theories. At the level of the Symmetry Theory, the above system is described by a SymTree, where one branch corresponds to a SymTFT_^2×^3 (i.e. reduction over the base space of the conifold singularity T^1,1^2×^3), which turns into a SymTFT_m (i.e. reduction on the ^5 around the joint stack of m D3-branes) to reproduce the Higgsing to the diagonal SU(m), which then splits in a junction into a SymTFT_m_1 and SymTFT_m_2 (corresponding to the reductions on the ^5's around the individual stacks of m_1 and m_2 D3-branes). Overall, we have a  SymTree with asymptotic branches given by a SymTFT_^2×^3 and the two ^5 theories, SymTFT_m_1 and SymTFT_m_2. The junction theory is obtained by smashing together the transition SymTFT_^2×^3 → SymTFT_m and the junction of the SymTFT_m, SymTFT_m_1 and SymTFT_m2. The latter junction was studied explicitly in <cit.>, where it was shown that it supports a (1) gauge theory, arising from the group theory analysis of the Higgsing 
        
            (m)→[(m_1)⊕(m_2)]=(m_1)⊕(m_2)⊕(1)_L  ,
        
        with L= lcm (m_1,m_2). On the other hand, the former transition, associated to the Higgsing (m)⊕(m)→(m) does not support any (1)'s, because the Higgsing by the bifundamental matter lowers the rank; hence we do not expect additional non-trivial degrees of freedom. The complete junction theory is therefore a 4d (1) gauge theory identical to that associated to the Higgsing (<ref>). 
        
        As explained, this applies to the local structure of the trivalent SymTree describing the crossing of the 5-brane in the Symmetry Theory of the 4d =4(N) SYM on a half-space coupled to BCFT_3 degrees of freedom. We may even follow the analysis in <cit.> to write down the junction conditions between the background fields of the 1-form symmetries in the different SymTFTs. In our case, for a SymTree corresponding to a D5-brane SymTFT_^2×^3 branch, focusing on the coupling of the junction theory to the RR 2-form background field C_2, at the location of the junction we have
        
            .m_1/gC_2^(m_1)|_ jnct.=.m_2/gC_2^(m_2)|_ jnct.=.m_1/gC_2^(m)|_ jnct.  .
        
        Here the relation is in _g, with g= gcd(m_1,m_2), and in
        our case[Note that in our system, we have m_2+m=m_1,
          i.e. m=m_1-m_2 as opposed to the above Higgsing, in which
          m=m_1+m_2. The sign flip is however irrelevant for the formal
          arguments to derive the presence of the (1) in the junction
          theory and the derivation of the junction conditions. It is formally
          equivalent to considering that m_2 describes a stack of
          anti-D3-branes.]m_1=N, m_2=nP, m=M. The superindex in the
        fields C_2 corresponds to its value in each of the SymTFTs,
        evaluated at the location of the junction. There is a similar
        gluing condition for B_2.
        
        
        An interesting implication of the above junction conditions is that there is a local 1-form symmetry _g. In fact, this reproduces the electric 1-form symmetries studied in particular cases in section <ref>. For instance, for the boundary conditions corresponding to the T[SU(N)] theory, the Symmetry Theory contains one stack of N D5-branes, each with linking number 1, joining the SymTFT_N of the asymptotic 5d theory, and a trivial SymTFT corresponding to the region with no 5-form flux near the ETW boundary. The jump in 5-form flux is hence from N units directly to zero, and the junction conditions above lead to an unbroken _N symmetry, matching the expected electric 1-form symmetry. Actually, the specific symmetry realised in the physical theory is determined by the choice of boundary conditions; this can be exploited to reproduce the electric 1-form symmetries for other gaugings SU(N)/_p, although we leave the details as an exercise for the interested reader.
        
        Similarly, for the configuration corresponding to a single D5-brane with linking number N, the junction conditions indicate an unbroken _N, in agreement with the field theory result in section <ref>. We will provide a different argument for the _g symmetry in the general case in section <ref>.
        
        In the following section we provide a rederivation of the above results from a different perspective, which exploits the worldvolume fields and couplings in explicit D5-branes.
        
        
        
        
        
          §.§.§ Retraction of the 5-brane SymTFT 
        
        
        An interesting operation introduced in <cit.> for
        SymTrees is the retraction of one or a subset of the branches, which
        in their context corresponded to smashing together the physical
        boundary of those branches with the junction theory. In our present
        setup, we can also consider the retraction of branches in the SymTree,
        and in particular it is interesting to consider the retraction of the
        branches corresponding to the 5-branes. This now corresponds to simply
        collapsing the corresponding SymTFT_^2×^3, bringing the
        (n) physical theory on top of the junction theory. In terms of
        figure <ref>, we collapse the blue/green lines of each
        physical 5-brane theory with the blue/green line of its corresponding
        junction theory. The result of this retraction should precisely
        correspond to figure <ref>, with a fan of SymTFTs of the
        kind (<ref>) separated by the non-topological theories on
        the worldvolume of explicit 5-branes.
        
        
        In fact it is possible to recover the main properties derived from the geometric realisation of the SymTree junction by using the properties of explicit 5-branes in the system, as follows. Since the interesting structure is related to the center of mass U(1), we simply take a single D5-brane, i.e. n=1 (general n simply introduces some multiplicities in the coefficients; also NS5-branes lead to similar results up to S-duality). Let us consider the topological couplings arising from the D5-brane worldvolume, in particular
        
        
            ∫_D5 (  C_4  F_D5+ C_2  F_D5 F_D5 )  .
        
        where 
        
            F_D5=B_2 + F_D5  .
        
        Here B_2 is the pullback of the bulk B_2 on the D5-brane worldvolume, and F_D5 is the  field strength for the D5-brane worldvolume U(1) 1-form gauge connection F_D5=dA_D5. In (<ref>) and similar forthcoming expressions we will skip numerical factors in the coefficients, and just highlight the parametric dependence on the flux or brane multiplicities.
        
        We now use the linking number is given by the total worldvolume monopole charge, namely
        
            ∫_^2 F_D5=P  .
        
        This is the D5-brane probe version of the supergravity background (<ref>), because the NSNS 2-form B_2 in the 5-brane backreacted geometry encodes the 2-form B_2 in the probe approximation together with the 5-brane worldvolume gauge field A_D5 (which is just a trivialisation of the latter). Hence we get the 4d couplings
        
            P∫_4d (  C_4 + C_2  F_D5 )  .
        
        The first term in (<ref>) describes the coupling of the D5-brane to the RR 4-form, and implies that the RR 5-form flux on ^5 jumps by P units as one crosses the D5-brane. Regarding the second term, the contribution of B_2 in F_D5 leads to a coupling
        
        
            P∫_4d C_2 ∧ B_2  .
        
        This nicely reproduces the topological coupling of the 5-brane branch SymTFT_^2×^3 in the former SymTree before the retraction. It accounts for the jump in the coefficient of the trunk SymTFT (<ref>) by P as one crosses the 5-brane,  nicely dovetailing the change in RR 5-form flux discussed in the previous paragraph.
        
        Finally, the remaining term has the structure
        
        
            P∫_4d C_2 ∧ F_D5  .
        
        This should be interpreted as the coupling of the U(1) in the junction theory with the background field C_2, as anticipated in the previous section. Note that in the case n=1, the (n) piece of the 5-brane theory is trivial, so the 5-brane theory is just the (1) junction theory.
        
        Note that for a D5-brane, the junction theory couples to the background field C_2, ultimately related to the 1-form symmetry of the holographic dual 4d SCFT. Clearly NS5-branes lead to similar couplings for B_2, the background field ultimately related to the dual 1-form symmetry of the holographic dual 4d SCFT. In later sections this will become manifest in the behaviour of diverse charged operators upon crossing the 5-branes.
        
        
        
          §.§.§ A remark on Nesting of the 5-brane SymTFT 
        
        
        We would now like to remark on an interesting point. We have emphasised that the supergravity solution for the ETW configuration with the 5-brane sources in section <ref> is the gravity dual of the 3d Gaiotto-Witten BCFT_3. It is thus reasonable to expect that at the topological level the structure of the Symmetry Theory in the ETW configuration, with the SymTree-like structure in figure <ref> corresponds to the Symmetry Theory of the 3d BCFT. 
        
        This however leads to a seemingly striking feature: The 3d BCFT is actually coupled to a set of non-topological 4d gauge theories, i.e. the physical 4d theory on the 5-brane worldvolumes. Each of the latter provides the physical boundary of a 5d SymTFT_^2×^3, while the 3d BCFT is the physical boundary of a fan of SymTFTs separated by junction theories. 
        In other words, the 3d BCFT lies in a corner of a  5d SymTFT (or rather, a set of them) bounded on one side by a 4d physical theory describing the gauged version of the flavour symmetry (or a set of them), and on the other side by a SymTFT (or rather, a junction theory at the intersection of several SymTFTs).
        
        Although this structure may seem exotic, it falls in the recent generalised structure of Nesting of Symmetry Theories introduced in <cit.>. Specifically, this structure provides the Symmetry Theory for d-dimensional physical theories whose flavour symmetries are described in terms of a (d+1)-dimensional physical theory sticking out of the original one. This whole configuration is regarded as the physical boundary of a Symmetry Theory, given by the (d+2)-dimensional SymTFT of the (d+1)-dimensional flavour physical theory, bounded by the (d+1)-dimensional SymTFT of the original d-dimensional physical theory.  In other words, the d-dimensional physical theory sits at a corner of a (d+2) SymTFT, at the intersection of a boundary given by the (d+1) flavour physical theory and another boundary given by the (d+1)-dimensional SymTFT.
        
        In our case, we have a similar (but even more complex) setup. The BCFT_3 is sitting at the corner of the 5d SymTFT_^2×^3, at the intersection of a 4d boundary given by the physical 5-brane theory and a 4d boundary given by the junction theory of the SymTFT_^2×^3 with the ^5 SymTFTs. In fact, the situation is even more complex because there is one such structure per 5-brane, so the BCFT_3 is sitting at the 3d corner of several such nestings.
        
        Although the details of this picture are is not essential for our purposes, we will refer to this setup at various points in later sections.
        
        
        
        
         §.§ Topological operators and crossing the SymTFT Fan
        
        
        We have discussed the behaviour of the different topological fields in the Symmetry Theory of the 4d =4(N) theory on spacetime with a boundary coupled to a Gaiotto-Witten BCFT_3. In this section we discuss the behaviour of various topological operators, in particular as they move across 5-branes between different SymTFT wedges. 
        
        
        
          §.§.§ Generalities
        
        
        
        Since our SymTFT Fans have a local SymTree structure, studying the motion of operators across 5-branes in our setup is similar to studying the motion of topological operators among the trunk and the branch SymTFTs in a SymTree in <cit.>. The only differences arise from the different orientation of the SymTree with respect to the topological and physical boundaries of the SymTFTs. 
        
        For instance, consider topological operators stretching between the two boundaries of a SymTFT in the SymTree in our setup (i.e. describing charged operators), and localised on the SymTree branches; they behave similarly to symmetry operators in <cit.>, i.e. topological operators in the bulk of the SymTFT not ending on its boundaries. Hence, in our context, charged operators in the trunk SymTFT of the SymTree can be moved across the junction theory and become charged operators of one of the branch SymTFTs, or of several of them simultaneously, with the possibility that they leave some operator of the junction theory along the way, see figure <ref>. 
        
        In section <ref> we had introduced the complementary representation of our Symmetry Theories by retracting the SymTFT branches corresponding to the 5-branes, as a set of wedges of SymTFTs (<ref>) with different coefficients, separated by explicit D5- and NS5-branes carrying induced D3-brane charge to ensure conservation of the 5-form flux. 
        It is easy to follow this retraction in the configurations in figure <ref>, which basically amounts to smashing together the two blue lines corresponding to the junction theory and the physical boundary of the 5-brane branch SymTFT, and stacking the possibly present defects of the latter (labelled V^(2)) on top of the possibly present defect of the junction theory. The set of possibilities thus reduces, as cases (ii) and (iv) become of the same kind, and so do the cases (v) and (vi).
        
        In the discussion of topological operators below we will find that the same crossing transitions admits different brane interpretations in terms of the full SymTree configuration or its retraction. This will be discussed in the explicit examples below.
        
        Regarding topological symmetry operators, they are characterised by the fact that they link the charged defect operators above, hence, in our setup they are localised in the interval between the physical and the topological boundaries. A first kind is those which are  localised also in the SymTree. When moved across the 5-branes, their behaviour is similar to that of topological symmetry operators in the SymTree in <cit.>. For illustration of several possibilities, one may take those in figure <ref>, but considering them localised, rather than stretched, in the SymTFT interval. Their behaviour upon retraction of the 5-brane SymTFTs also follows that indicated above. A second class of topological symmetry operator is obtained when they stretch in the direction parallel to the physical and topological boundary of the Symmetry Theory, and hence hit the junction theory (or several of them). The possible behaviours of such operators at the junction are similar to the behaviour of topological defect operators in <cit.>, and some particular examples are shown in figure <ref>.
        
        
        
        
        We will study some particular examples of operators and their behaviour with respect to the junctions in the coming sections.
        
        
        
          §.§.§ Line operators, F1s and D1s
        
        
        We start the discussion in the SymTFT_N which is dual to the bulk of the 4d =4(N) SYM theory. As is familiar since <cit.>, the basic set of topological operators are realised in string theory in terms of F1 or D1 strings stretching from the topological to the physical boundary (representing Wilson and 't Hooft line operators, respectively), and D1 or F1 string worldsheets linking them (representing the symmetry generating operators, respectively). 
        
        Consider now the SymTFT_M theory at the other side of e.g. a D5-brane (with M=N-nP in earlier notation). In this SymTFT we have similar sets of topological operators realised in terms of F1 and D1 strings. This seemingly leads to the following puzzle. Consider for instance the F1 stretched between the topological and physical boundaries in the SymTFT_N, and act on it with a D1 linking it, so it transforms by a phase e^2π i/N. We can consider the same kind of configuration, but in the SymTFT_M, in which case the charged operator transforms by a phase e^2π i /M. The puzzle arises because seemingly both the F1 and the D1 can be moved from the SymTFT_N to the SymTFT_M without any discontinuity, i.e. without crossing the D5-brane, by simply avoiding it in the internal geometry. Yet the phase transformation seems to jump discontinuously in the transition.
        
        An equivalent way to phrase the above puzzle is to consider the homological difference between the F1s in the two SymTFTs; this corresponds to an F1 which extends from the SymTFT_N to the SymTFT_M across the 5-brane (although it may avoid it in the internal space, so no actual crossing occurs). Considering a D1 which links this F1, it seemingly acts by introducing different phases in the different SymTFTs.
        
        This puzzle was already addressed in <cit.>, in the SymTree which describes the adjoint Higgsing (N)→(M) ⊕(N-M)⊕(1). The configuration of an F1 starting in the SymTFT_N, crossing the junction and entering the SymTFT_M (equivalently the SymTFT_N-M) describes the relation between the Wilson line operators in the different theories. As explained in <cit.>, when translating the stringy realisation in terms of a F1 into field theory operators, the F1 in the SymTFT_M corresponds to an (M) Wilson line, but with the extra dressing of a Wilson line of the (1)⊂(M), which ultimately is recast as an operator of the (1) junction theory (and similarly for the SymTFT_N-M). This accounts for the extra phases required to compensate for the jump in the phase when one acts on the object with a D1 linking it. 
        
        Clearly, the same explanation holds in our context, with the SymTFT of
        the 5-brane (with its induced N-M units of D3-brane charge) playing
        the role of the SymTFT_N-M. In terms of the diagrams in figure
        <ref>, the topological operators crossing the junction
        are dressed by a junction theory operator, as in diagram (iii). Coming
        back to the original picture with the two F1s describing charged
        operators in the SymTFT_N and SymTFT_M, the above argument means
        that the same microscopic object (the F1) corresponds to two different
        operators in the two theories, and when transforming one into the
        other, there arises an additional dressing by an operator of the
        U(1) junction theory. In terms of the diagrams in figure
        <ref>, the topological operators picks up a junction theory
        operator upon crossing the 5-brane, as in diagram (ii). Clearly a
        similar discussion holds for D1 line operators.
        
        A complementary microscopic viewpoint on the above discussion is as
        follows. From the string theory perspective, both the F1 and the D1
        charges are initially -valued. But the presence of e.g. N units
        of 5-form flux in the compact space implies there are physical
        processes which violate their charge in N units. These are the
        baryonic vertex given by D5-brane wrapped on the 5d compact space,
        which emits N F1 strings <cit.>, and its S-dual
        wrapped NS5-brane, which emits N D1 strings.  This allows to recast
        the 5d action (<ref>) in terms of _N-valued fields in a
        5d topological _N theory. Our statements mean that these discrete
        fields for the different SymTFTs are to be identified but up to an
        overall normalisation, which in fact corresponds to the junction
        conditions (<ref>).
        
        
        
        
          §.§.§ The baryon vertex
        
        
        The change in the order of the discrete symmetry between the SymTFTs is hence encoded in the existence of the baryon vertex and the dual baryon vertex, and how they behave when moved across the 5-branes, as we study in this section.
        
        In short, as we will see in explicit examples below,  in the full SymTree configuration the crossing transitions can be understood in terms of geometric relations and Freed-Witten consistency conditions [As usual we use this term to refer to incompatibilities of branes and fluxes in general for non-torsion classes <cit.>, i.e. beyond the original Freed-Witten study of the torsion case <cit.>.]. On the other hand, in the Symmetry Theory after the retraction, the crossing transitions can be understood in terms of Hanany-Witten brane creation effects <cit.> involving the explicit 5-branes. This is expected given the general relation between Freed-Witten consistency conditions and Hanany-Witten effects (see <cit.> for discussion).
        
        For simplicity, we focus on the setup of a single D5-brane stack, as has been implicitly done in the previous pictures, with a single D5-brane with linking number P. Hence, we consider a SymTree with a trivalent junction in which the SymTFT_N (corresponding to reduction on ^5 with N units of 5-form flux) splits into a SymTFT_M (with M units of flux on ^5) and SymTFT_^2×^3 with N-M=P units of 5-form flux (from 1 unit of RR 3-form flux over ^3 and a NSNS 2-form period P on the ^2). Equivalently, after the retraction, we have the SymTFT_N and SymTFT_M separated by one explicit D5-brane wrapped on a maximal ^2⊂^5 and carrying P units of induced D3-brane charge due to the non-trivial integral (<ref>) of F_D5 over ^2. 
        
        Consider the baryonic vertex of the SymTFT_N, namely a D5-brane wrapped on the ^5 and emitting N fundamental strings <cit.> ending in the physical boundary corresponding to the 4d QFT on the half-space. As explained above, although the baryonic vertex corresponds to the identity operator in the SymTFT_N, it has a non-trivial meaning in the microscopic string theory realisation, as encoding the actual reduction of -valued charges to _N.
        
        
        
        
        Let us consider the picture of the retracted SymTree, and consider moving this baryonic D5-brane (and its N attached F1s) across the 4d subspace spanned by the D5-brane wrapped on ^2 (which we refer to as 4d D5-branes to avoid confusion). In the process of crossing to the SymTFT_M, there is a Hanany-Witten effect between the baryonic D5-brane and the P units of induced D3-brane charge on the 4d D5-branes, resulting in the creation of P fundamental strings stretching between them. One can now recombine them with P of the original N F1 strings and detach them from the baryonic D5-brane, leaving a D5-brane on ^5 with just M F1 strings, see figure <ref>. This is precisely the baryonic vertex appropriate to represent the identity in the in the SymTFT_M. 
        
        Note however that we still have the P detached strings, which now
        stretch from the 4d D5-brane to the physical boundary of the
        BCFT_3. In holographic language, they correspond to P Wilson
        line operators, and correspond to the dynamical quarks localised at
        the 3d boundary of the 4d CFT. In the language of the Symmetry Theory,
        they define an operator of the junction theory left over after the
        crossing, as explained above.
        
        Let us rederive the above crossing of the baryonic operator but in terms of the full SymTree configuration, rather than its retraction. In this picture, the baryonic operator in the SymTFT_N, namely the D5-brane on the ^5 with N units of 5-form flux (hence emitting N F1 strings), is homologically equivalent to the baryonic operator in the SymTFT_M, i.e. a D5-brane on the ^5 with M units of flux (hence emitting M F1 strings) and a D5-brane on the ^2×^3 of the SymTFT_^2×^3 emitting P F1 strings due to the 5-form flux on this branch. It is easy to check that the latter are due to the induced D3-brane charge, and connect directly with the  Hanany-Witten picture above: the D5-brane on ^2×^3 has P units of induced D3-brane charge due to the NSNS 2-form background (<ref>); the P induced D3-branes wrap on the ^3, which has 1 unit of RR 3-form flux (encoding the 4d D5-brane charge), hence have a Freed-Witten anomaly and must emit P F1 strings. The picture of emission of F1 strings due to 5-form flux or RR 3-form flux on the induced D3-branes is just a reflection of the fact that the 5-form flux in this branch arises from the Chern-Simons couplings, c.f. (<ref>). 
        
        
        
        
        We have thus rederived the crossing process in a simple geometric
        fashion, and recovered the appearance of the operator on the SymTFT of
        the 5-brane theory, see figure <ref> (analogous to the
        transition (i)→ (ii) in figure <ref>, but for symmetry
        operators). We note that, although the geometric picture of the
        crossing would seem to suggest that there is no junction operator
        created in the process, the (1) degree of freedom is delocalised,
        hence we depict the presence of a junction operator, as discussed with
        the explicit 4d D5-brane in the Hanany-Witten description. We also
        emphasise that, although the operators we are discussing correspond to
        the identity operator in each of the SymTFTs, there is non-trivial
        information about how they are related upon crossing the
        junction. This is the reflection of the junction conditions of the
        underlying fields.
        
        We finish by emphasising that the baryon vertices of the different SymTFTs can be used to derive the local _g 1-form symmetry unbroken by the junction conditions. Near the junction of the SymTFT_N, the SymTFT_M and the SymTFT_^2×^3, we can consider processes annihilating F1 strings in sets of N, or in sets of M (or equivalently, of N-M) or integer linear combinations thereof. So by Bezout's theorem, F1 strings are conserved mod g with g= gcd(N,M). For instance, in the particular case of the SymTFT of the T[SU(N)] boundary conditions, we have one stack of n D5-branes with linking number 1, so M=0, and hence we get a _N 1-form symmetry, as discussed in the field theory setup in section <ref>, and in section <ref> from the viewpoint of the junction conditions.
        
        
        
          §.§.§ The dual baryon vertex
        
        
        Let us now consider the dual baryon vertex, which is given by an NS5-brane wrapped on the ^5, emitting N D1 strings, ending on the boundary of the physical 4d theory on the half-space. In the absence of boundary for the 4d QFT, S-duality of 4d =4(N) SYM relates this to the baryonic vertex in the previous section; however, in the presence of boundaries for the 4d QFT, the Symmetry Theory contains explicit 4d D5- or NS5-branes (or their SymTFTs in the SymTree picture), which are not invariant under S-duality. Hence the dual baryon operator must the studied on its own.
        
        We start with a dual baryon vertex of the SymTFT_N, given by one NS5-brane wrapped on the ^5 and emitting N D1 strings due to the 5-form flux. Let us first consider the picture of the retracted SymTree, namely with an explicit 4d D5-brane with P units of induced D3-brane charge, separating the SymTFT_N and SymTFT_M regions. If we now move the NS5-brane across the 4d D5-brane, there is a Hanany-Witten effect between the 4d D5-brane and the NS5-brane, leading to the creation of one D3-brane wrapped on the ^2 and stretched between the NS5- and the D5-brane (for a stack of a number n of 4d D5-branes, there would be n D3-branes created). In addition, there is a non-trivial Hanany-Witten effect of the NS5-brane with the P induced D3-brane charge on the 4d D5-brane, which creates P D1 strings stretched between them. Note that they can be equivalently regarded as P units of D1-brane charge induced on the previous D3-brane due to the non-trivial NSNS 2-form background on the ^2; for clarity we prefer to keep the discussion of the created D3- and D1-branes separate.
        
        As in the discussion above for the F1 strings of the baryonic D5-brane, one can recombine the newly created P D1 strings with P of the D1 strings emitted by the NS5-branes, to obtain an NS5-brane emitting M D1 strings, precisely as required by the change in the 5-form flux on the ^5. In the process we are also left with P D1 strings stretching from the D5-brane to the boundary corresponding to the physical theory of the BCFT_3.
        
        
        
        
        In addition, we have the single D3-brane, wrapped on the ^2 and stretched between the NS5-brane and the 4d D5-brane. An important point is now that the ^2 is topologically trivial on the ^5, but not on the worldvolume of the D5-brane (as it wraps precisely this ^2). This means that the boundary of the D3-brane on the NS5-brane is actually trivial, and the D3-brane can unwind and snap away. On the other hand, the boundary of the D3-brane on the D5-brane is non-trivial, so the D3-brane cannot unwind completely. The result is that the D3-brane wraps a 3-chain B_3 whose boundary is the ^2 wrapped by the D5-brane, ∂ B_3=^2. A minimal-volume representative of this 3-chain is a half-^3 in the ^5 at the location of the 4d D5-brane. In the picture of the geometry as an ^2×^2 fibration over a Riemann surface (the quadrant), the process and the structure of the 3-chain is shown in figure <ref>. In analogy with the previous section, we expect the P D1 strings and the D3-brane stretching out should correspond to an operator in the D5-brane theory. This can be made more clear using the full SymTree picture, as we do next.
        
        Consider the crossing of the NS5-brane in the full SymTree, i.e. growing back the SymTFT_^2×^3 associated to the 5-brane theory. We start with the dual baryon vertex of the SymTFT_N, namely the NS5-brane wrapped on the ^5 with N units of 5-form flux and emitting N D1 strings. To move it across the junction, we simply use the homology relation among the ^5's and ^2×^3. We obtain an NS5-brane wrapped on the ^5 with M units of 5-form flux, emitting M D1 strings, and an NS5-brane wrapped on ^2×^3, with P units of 5-form flux, emitting P D1 strings. The former is simply the dual baryon vertex of the SymTFT_M, as expected. The NS5-brane on ^2×^3, in addition to the Freed-Witten anomaly enforcing the emission of the P D1 strings, has a Freed-Witten anomaly due to the single unit of RR 3-form flux on ^3, which enforces the emission of a D3-brane wrapped on ^2 (as above, we keep their discussion of the emitted D3- and D1-branes separate). The situation is very similar to the above Hanany-Witten derivation of the crossing, but with an interesting novelty. The ^2 wrapped by the emitted D3-brane is  non-trivial in the ^2×^3 geometry of the SymTFT_^2×^3, although it is trivial in the global geometry. Namely, the D3-brane can unwind but only in the region of the junction, in which the ^2×^3 geometry blends with the ^5 in which the ^2 is trivial. This means that the 3-chain B_3 wrapped by the D3-brane stretches from the 4d D5-brane on the physical boundary of the SymTFT_^2×^3 to the junction theory, and ends there.
        
        
        
        
        In formal terms, the dual baryon operator in the
        SymTFT_^2×^3 emitting P D1 strings is non-genuine, so
        it must be dressed by a operator which stretches to the junction
        theory, as sketched in figure <ref>.
        
        It is satisfying to recover the by now familiar phenomenon that in
        SymTFTs arising from holography subtle effects involving topological
        operators in the SymTFT description follow easily from long-understood
        string theory physics. We will encounter similar phenomena in even
        richer setups in the next sections.
        
        
        
        § INCLUSION OF 7-BRANES
        
        
        In the previous section we have emphasised that the Symmetry Theory of our system has the local structure of a SymTree, albeit with a different location, as compared with <cit.>, of the topological and physical boundaries. This implies that the junction theory reaches the “topological” boundary, and requires the introduction of physical, non-topological, boundary conditions there. In a string theory context, a natural choice is to use string theory objects to provide boundary conditions. In this section we explore the introduction of 7-branes on which the 4d 5-branes end.
        
        
        
         §.§ Generalities
        
        
        We will phrase our discussion in terms of the retracted SymTree picture, with explicit 5-branes in the configuration. This facilitates the discussion of 5-branes ending on 7-branes. We note that in order to carry out a similar picture in the full SymTree picture, one should consider introducing a SymTFT for the 7-brane theory, and consider it as the boundary of the SymTFT_^2×^3, in the spirit of the nesting of SymTFTs in <cit.>.
        
        
        
          §.§.§ The 7-brane boundary conditions
        
        
        As in previous sections, we focus on the case of 4d D5-branes separating the SymTFT_N and SymTFT_M (by S-duality, similar consideration follow for NS5-branes ending on the S-dual 7-branes). We also focus on the case of a single D5-brane, with P=N-M units of induced D3-brane charge. As is familiar (already from <cit.> in a T-dual setup), D5-branes can end on NS5-branes or on D7-branes. The two choices differ in what boundary condition they impose on the D5-brane worldvolume gauge fields. The choice of D7-branes is natural in our setup since it allows to nicely translate the flavour symmetries realised on the D5-branes to the D7-branes worldvolume theory.
        
        Hence, we consider D7-branes wrapped on ^5, and spanning a 3d Minkowski subspace on the AdS_4 slice, initially at a location pushed onto the topological boundary, so that they provide boundary conditions for the D5-branes ending on them. Since the D7-branes are located at a boundary, there are no closed paths in the Symmetry Theory which go around the D7-branes, and the axion C_0 can be defined globally, i.e. there is no need to specify a branch cut (equivalently, it is sticking out outwards, away from the Symmetry Theory).
        
        It is interesting to notice that the D7-branes just introduced are of
        the kind considered in <cit.>. Following it, we can
        consider moving the D7-branes into the 5d bulk of the Symmetry
        Theory. In doing that, it is necessary to introduce the branch cut
        implementing the SL(2,) monodromy of the 7-brane. From our above
        discussion, it is clear it stretches from the location of the 7-brane
        in the bulk to the topological boundary. Hence, the endpoint of the
        branch cut at the topological boundary still encodes the relevant
        information about the boundary condition of the D5-brane (or junction)
        theory. From <cit.>, this choice of branch cut
        orientation corresponds to the 7-branes creating a duality
        interface. Namely, the topological boundary conditions of the
        SymTFT_N and SymTFT_M are related by a non-trivial SL(2,)
        duality transformation due to crossing of the branch cut. As
        emphasized in <cit.>, although the particular shape of
        a D7-brane branch cut is not physically observable, its endpoint at
        the topological boundary is, and leads to observable effects.
        
        
        
        
        Hence the topological boundary of the SymTFT Fan is a set of gapped boundary conditions for the different SymTFTs, separated by 7-branes on which the 5-branes are ending, see figure <ref>a. If the 7-branes are pushed into the bulk, to make their effects more manifest, the SymTFT gapped boundary conditions are separated by the endpoints of the 7-brane branch cuts, see figure <ref>b.
        
        Since we are focusing on the case of a single D5-brane, it is clear that it suffices to consider a single D7-brane on which it ends. We would like however to make a comment regarding the general case of a stack of n>1 D5-branes. Recall that the worldvolume gauge fields on the D5-brane have Neumann boundary condition on the physical boundary, so as to couple to the currents of the (n) enhanced global symmetry of the BCFT_3, as required by the holographic interpretation. On the other hand, they have Dirichlet boundary conditions on the D7-brane. This implies that there is a version of the s-rule in <cit.>, which requires that each D7-brane can be the endpoint of at most one D5-brane, so that one needs n D7-branes. A complementary heuristic explanation is that the (n) global symmetry of the physical theory is  morally realised as the worldvolume (n) gauge symmetry on a stack of n D7-branes (more precisely, on the bound system of the stack of n D5-branes ending on n D7-branes)[This is similar to the realisation of flavour symmetries in Hanany-Witten setups when the flavour branes are moved off the interval between the NS5-branes, a process which creates new D-branes ending on the flavour branes.]. 
        
        
        
          §.§.§ The 5-form flux and the need for explicit D3-branes
        
        
        We return to the setup with a single D5-brane separating the SymTFT_N and the SymTFT_M with P=N-M, and ending on a D7-brane in the bulk of the Symmetry Theory, with a branch cut stretching to the topological boundary.
        
        
        
        
        As explained earlier, the jump in flux across the D5-brane, implicit in the separation between the SymTFT_N and the SymTFT_M, is accounted for by the P units of induced D3-brane charge. Since the D5-brane ends on the D7-brane, which is localised in the 5d bulk, it would now be possible to cross from the SymTFT_N to the SymTFT_M without encountering any discontinuity, other than crossing the D7-brane branch cut. But the D7-brane branch cut simply implements the SL(2,) transformation, which does not act on the 5-form flux. Hence, we are forced to introduce an explicit stack of P D3-branes stretching from the D7-brane to the topological  boundary of the SymTFT, see figure <ref>. 
        
        The D3-brane charge is actually not ending on the 7-brane, but rather
        it continues along the corresponding 5-brane in the form of induced
        D3-brane charge. This is as it should be, since D3-branes cannot end
        on 7-branes. This is also clear when looking at the 4d D3-brane
        worldvolume coupling
        
            P∫_4d C_2 B_2     ,
        
        which is the continuation of the D5-brane coupling (<ref>) and explains the jump 
        in the coefficient of the 5d topological coupling B_2F_3 from the SymTFT_N to the SymTFT_M. Hence the structure of the SymTFT Fan is maintained even when the 5-branes end on the 7-branes.
        
        
        
        
        Although we will not use it, we would like to remark on the corresponding picture in the full SymTree. As explained before, the 5d SymTFT_^2×^3 is now bounded by the 4d Symmetry Theory of the 3d 7-brane theory, but which should now correspond to a junction theory beyond which we have the SymTFT_P of the stack of P D3-branes, see figure <ref>. Hence we again have a combination of SymTrees and Nesting of SymTFTs. 
        
        We conclude by noting that the effect of the D3-brane charge was crucial for the Hanany-Witten effect for the baryonic D5- and NS5-branes across the 4d D5-brane in sections <ref> and <ref>. In the configuration where 4d D5-brane ends on a D7-brane, the extra P explicit D3-branes from the D7-brane to the topological boundary guarantee that the same effect takes place if the baryon vertex is moved from the SymTFT_N to the SymTFT_M across the D7-brane branch cut.
        
        In the following section we discuss in more detail the effects related to moving objects between the SymTFTs across the branch cut. As expected from <cit.>, this implements the SL(2,) monodromies corresponding to a duality interface in the theory.
        
        
        
        
         §.§ The SL(2,) monodromy
        
        
        In this section we discuss the effects experienced in the Symmetry
        Theory and its topological operators as one crosses from the
        SymTFT_N to the SymTFT_N across the D7-brane branch cut (and the
        P explicit D3-branes). This is a particular case of the phenomena
        discussed in <cit.>, and implements the SL(2,)
        monodromies corresponding to a duality interface in the theory.
        
         As above, we focus on the simple setup of a single D5-brane ending on a single D7-brane. In our discussions below, the effect of the D3-brane charge (either explicit or induced on the 4d D5-brane), which has already been mentioned, will we included implicitly, except for a few explicit mentions.
        
        
        
        
          §.§.§ Stacking of TQFT
        
        
        We now derive the effect of the D7-brane branch cut on the Symmetry Theory. Recall that the physical effect of the D7-brane branch cut in string theory is that the 10d RR axion shifts C_0→ C_0+1. From the perspective of the holographic dual 4d =4(N) SYM theory, this corresponds to a shift of the θ angle. There is a mixed anomaly between the shift of the θ angle and the electric 1-form symmetry, encoded in the 5d TFT
        
            S=2π i N-1/N∫_Ydθ/2π∪ P( B)/2  ,
        
        where B is the background coupling to the electric 1-form symmetry and P is the Pontryagin square.
        For the geometries of our interest (5d spaces foliated with 4d slices given by spin manifolds over which θ is constant) P( B) is even, so the only relevant piece is the 1/N term. We will also be sensitive only to the de Rham version of P∼ B^2.
        
        We now follow the derivation in <cit.> of this 5d TQFT from the holographic setup. We use the well-defined 3-form field strength
        
            F̃_3= dC_2-C_0dB_2  ,
        
        to rewrite the 5d topological coupling (<ref>) as
        
            N∫_5d B_2F_3=N∫_5dB_2 F̃_3-N∫_5d dC_0 B_2 B_2   .
        
        The last term reproduces (with the usual redefinitions B_2∼ B/N) the 1/N term in (<ref>).
        
        Hence, if we take an interval I across the D7-brane branch cut (so that C_0→ C_0+1 along it),  and integrate the 5d coupling, its variation is given by
        
            Δ∫_5d(- NdC_0 B_2 B_2 ) =
            -N∫_4d B_2B_2  .
        
        This is precisely the kind of variation required, corresponding to the stacking of a TQFT in the SymTFT (i.e. of a counterterm in the dual QFT), as befits the shift of the θ angle. One may worry that, on the two sides of the D5/D7 system, the 5-form flux changes by P units, and this requires the coefficient of the above coupling to jump as N→ M. Actually, this is solved because of the additional crossing of the P explicit D3-branes, which have a worldvolume coupling
        
            P∫_4dC_0 B_2B_2  .
        
        This amounts to P units of a boundary contribution of the coupling ∫_5d dC_0 B_2B_2, so that on the SymTFT_M side the coefficient is effectively reduced from M to N.
        
        In short, the effect of the D7-brane branch cut is the stacking of a TQFT, according to the effect of the SL(2,) as a shift of the θ angle corresponding to the duality interface.
        
        
        
          §.§.§ Line operators and (p,q) string webs
        
        
        Let us consider the fate of line operators described in terms of F1 and D1 strings in the SymTFT_N as they cross the branch cut. As explored in section <ref>, this can be efficiently discussed by considering F1 or D1 strings which stretch between the SymTFT_N and the SymTFT_M across the junction theory. The crossing over the P explicit D3-branes leads to the effects already discussed in section <ref>, which account for the change in the order of the discrete valuedness of the background fields, and hence of the topological operators[Incidentally, note that in this context, the D3-branes separating the SymTFT_N and the SymTFT_M is the retracted version of a SymTree corresponding precisely to adjoint Higgsing. Namely, the SymTFT corresponding to the D3-branes is a SymTFT_P associated to reduction on ^5 with P units of 5-form flux.]. Hence we only need to consider their transformation across the branch cut of the 7-brane, i.e. the SL(2,) transformation [ 1 1; 0 1 ]. The F1 is a (1,0) string, so it is invariant under this action. On the other hand, the D1 is a (0,1) string, and hence turns into a (1,1) string, as shown in figure <ref>a. For charged line operators, this is just the manifestation of the transformation of monopoles into dyons upon a shift of the θ angle.
        
        
        
        
        It is interesting to consider the above transformation of the D1 strings when they are moved across the D7-brane, so that they cross over the D5-brane, see figure <ref>b. In this crossing there is a Hanany-Witten effect creating an F1 string stretching from the D7-brane to the D1 string, and turning the latter into a (1,1) string due to charge conservation in the resulting string junction. Hence we recover the same overall result, even though the D1 string has not crossed over the D7-brane branch cut. 
        
        Hence we recover the interpretation of D7-branes (with branch cuts stretched to the topological boundary) as duality interfaces in <cit.>, via their action on F1 and D1 strings.
        
        
        
          §.§.§ Baryon and dual baryon vertices
        
        
        In the discussion in the previous section we have briefly mentioned the jump in the order of topological operators when they cross from the SymTFT_N to the SymTFT_M. In order to get a more direct insight we quickly derive the transformation of baryon and dual baryon vertices as they move across the D7-brane branch cut (and the P D3-branes), extending the discussion in sections <ref> and <ref>.
        
        We start with the baryon vertex in the SymTFT_N region, namely a D5-brane wrapped on the ^5 and emitting N F1 strings. Both the baryonic D5-brane and the F1 strings are left invariant when moved across the D7-brane branch cut. On the other hand, when moved across the P D3-branes, there are P newly created F1 strings. They combine with P of the original F1 strings emitted by the baryonic D5-brane, leaving the latter with M F1 strings. This corresponds to the baryon vertex of the SymTFT_M region. In addition, there remain P F1 strings being emitted by the D3-branes. In the full SymTree picture, these would correspond to the baryon vertex of the SymTFT_P region, as well as an operator of the (1) junction theory. The result is as in section <ref>, with the explicit D3-branes here playing the role of the induced D3-brane charge there.
        
        
        
        
        Consider now the dual baryon vertex in the SymTFT_N region, namely an NS5-brane wrapped on the ^5 and emitting N D1s, see figure <ref>a. When moved across the P D3-branes, there is a Hanany-Witten creation of P suspended D1 strings, see figure <ref>b. Upon recombining, we have an NS5-brane emitting M D1 strings, and P D1 strings stretching from the P D3-branes, see figure <ref>c. These P D1s will be pretty much spectators in what follows, so we do not mention them explicitly, and focus on the NS5-brane. We now move the NS5-brane across the D7-brane branch cut, dragging its M D1 strings with it. Upon crossing, the NS5-brane turns into a (1,1) 5-brane, which emits M(1,1) strings, which turn into D1s when they cross the branch cut, see figure <ref>d. This reproduces the expected effect of the SL(2,) monodromy on the dual baryonic vertex.
        
        It is interesting to continue manipulating the above object in order to relate it to a simple dual baryonic vertex of the SymTFT_M. To do so, we move across the 4d D5-brane the M D1 strings, which necessarily cross the D7-brane without possibly avoiding it. This crossing produces P F1 strings which combine with the (1,1) and D1 strings forming a (p,q) string web, see figure <ref>e, in analogy with section <ref>. At the topological level, we can split the (1,1) 5-brane, and the string web including its M attached (1,1) strings, as 1 NS5-brane emitting M D1 strings, and 1 D5-brane emitting M F1 strings which end on the D7-brane, see figure <ref>f. 
        
        The NS5-brane emitting M D1 strings is just the dual baryon vertex of the SymTFT_M. On the other hand, the baryonic D5-brane emitting M F1s ending on the D7-brane can be pushed onto the latter and regarded as a defect associated to the junction theory.
        
        We hope these examples suffice to illustrate the behaviour of different objects in the configuration, and their interplay with the SL(2,) duality interface introduced by the D7-branes.
        
        
        
        § ORIENTIFOLD AND ORBIFOLD BOUNDARIES
        
        
         §.§ Generalities on orientifold and orbifold 5-planes
        
        
        In this section we discuss a generalisation of the above boundary
        configuration by allowing for the introduction of orientifold planes,
        and orbifolds related to them. We will focus on orientifold/orbifolds
        which are localised at the boundary in the direction 3. Namely, we are
        not considering for instance the well-studied case of O3-planes
        parallel to D3-branes, leading to =4(n) or (n)
        theories (see
        <cit.>
        for discussions of the holographic duals of such systems). Moreover,
        we are also interested in configurations that preserve the same
        symmetries and supersymmetries preserved by D5- and NS5-brane boundary
        conditions, so as to stay in the set of boundary conditions studied in
        <cit.>. This essentially restricts us
        to consider the introduction of O5-planes in the same directions as
        the D5-brane, namely 012 789, or orbifold 5-planes in the same
        directions as the NS5-branes, namely 012 456.
        
        Orientifold 5-planes (O5-planes for short) are defined by quotienting
        by the orientifold action Ω R, where Ω is worldsheet
        parity and R is a geometric _2 involution acting as a
        coordinate flip in the directions 3456,
        (x^3,x^4,x^5,x^6)→ (-x^3,-x^4,-x^5,-x^6). As discussed, this
        preserves the same supersymmetries as a D5-brane along 012789. The 6d
        plane fixed under R is an O5-plane, and there are several discrete
        choices for its properties (in analogy with O3-planes in
        <cit.>), classified by discrete RR and NSNS backgrounds
        on the _3 geometry around it <cit.>. The
        O5^--plane and O5^+-plane differ in the value of the NSNS 2-form
        on an _2⊂_3, carry RR charge ∓ 2 (as measured
        in D5-units in the double cover), and project the symmetry (n) on
        a stack of D5-branes on top of them down to (n) or (n),
        respectively.
        
        There are variants of these, dubbed O5^±, which
        arise when the non-trivial _2 RR background C_0 is turned on at
        the O5-plane location. The O5^- has RR charge -1
        and can be described as the O5^- with one stuck D5-brane on top, so
        it leads to a gauge symmetry (n+1) when n additional D5-branes
        are located on top of it. The O5^+ is an exotic
        version of the O5^+-plane, has RR charge +2 and also leads to a
        group (n) when n D5-branes (counting before orientifolding,
        here n must be even) are located on top of it.
        
        Let us briefly discuss orbifold 5-planes. They are obtained by
        quotienting by the orbifold action R'(-1)^F_L, where R' is a
        geometric _2 involution flipping the coordinates
        (x^3,x^7,x^8,x^9)→(-x^3,-x^7,-x^8,-x^9). As discussed, this
        preserves the same supersymmetries as an NS5-brane along 012456. The
        6d plane fixed under R' is the orbifold 5-plane, and there are also
        several variants <cit.>. Interestingly, the orbifold
        5-planes are related by S-duality to O5-planes <cit.>; in
        particular the perturbative one, which has a localised twisted sector
        spectrum given by the 6d (1,1)U(1) vector multiplet, is S-dual to
        the O5^- with 2 D5-branes on top, which realises the same localised
        field content as an (2) vector multiplet in the open string
        sector. Because of this kind of S-duality relations, we will focus our
        discussion on the introduction of O5-planes, with additional NS5- and
        D5-branes, and will not consider orbifold 5-planes any further.
        
        
        
        
         §.§ Brane configurations
        
        
        In order to describe Hanany-Witten brane configurations for systems of D3-branes ending on a set of NS5- and D5-branes in the presence of O5-planes, it is useful to consider the configuration in the covering space. There we have a _2 invariant system, with one stack of N semi-infinite D3-branes in x^3→∞ and its image stack in x^3→ -∞, ending on a (_2-invariant) `middle' configuration of NS5-, D5-branes, and D3-branes suspended among them, with the O5-plane sitting at x^3=0. 
        
        Thus the configuration is morally a back-to-back double copy of the kind of systems considered in previous sections. One key difference is that at the ETW boundary, which is given by the O5-plane (and possible additional 5-branes on top of it), the number of D3-branes need not vanish. This will be nicely reproduced in the gravity dual picture, as discussed later on. In general the number of D3-branes at the O5-plane location is different from the asymptotic value N, so we denote it by n in the following.
        
        Given the above, most of the discussion regarding NS5- and D5-branes away from the O5-plane behave locally just like the configurations of NS5- and D5-branes in the previous sections. The only position at which the presence of the orientifold quotient is felt  locally is precisely the O5-plane. Hence, we next describe different possible configurations of 5-branes on top of the O5-plane. Each of them can then be used to construct a general class of 3d theories defining boundary conditions, by simply sprinkling additional NS5- and D5-branes and their _2 images. We hence turn to study the different configurations of 5-branes on top of the O5-plane.
        
        
        
          §.§.§ Boundary conditions for O5-planes with no NS5-branes
        
        
        The simplest possibility is that the O5-plane does not have additional
        NS5-branes on top. This possibility corresponds to the boundary
        conditions reducing the gauge symmetry considered in
        <cit.>. Hence, we locally have a stack
        of n D3-branes in the double cover, intersected by an O5-plane, and
        we can then use standard orientifold rules to read out the local
        breaking of the (n) symmetry due to the orientifold
        projection. In this respect, recall that the orientifold projection of
        the O5-plane on the D3-branes is of the opposite kind (namely 
        vs ) as compared with the action on D5-branes, because the
        corresponding mixed open string sector has 4 DN directions
        <cit.>.
        
        For instance, the boundary conditions defined by an O5^+-plane reduce the symmetry (n) to (n), which corresponds to Class I in <cit.>. In more detail, splitting the 4d =4 vector multiplet in terms of a local 3d =4 vector multiplet and a 3d =4 hypermultiplet in the adjoint of (n), the (n) part of the vector multiplet is even under the O5^+-plane action (and the remaining generators are odd), and the  part (plus a singlet) of the hypermultiplet is even (while the  part is odd). This provides the definition of Dirichlet or Neumann boundary conditions for the different fields.
        
        Similarly, the O5^--plane boundary condition project the (n)
        gauge symmetry down to (n) (recall that n is necessarily even
        for this orientifold projection), which corresponds to Class II in
        <cit.>. In more detail, in the covering space the
        (n) part of the 3d =4 vector multiplet is even (and the
        rest is odd), while the  part of the hypermultiplet is even
        (and the + 1 is odd).
        
        For the O5^--plane, the configuration is
        equivalent to the O5^--plane with one stuck D5-brane on top. Hence,
        to the above boundary condition we must add one localised
        half-hypermultiplet flavour (in the fundamental of the local (n)
        symmetry and charged under the _2≃ O(1) on the stuck
        D5-brane) arising from the D3-D5 open string sector. Finally, the
        O5^+-plane behaves similarly to the O5^+-plane,
        with minor modifications due to the non-trivial value of the RR axion
        C_0=1/2 on top.
        
        Clearly, the above configurations can be enriched by addition additional D5-brane pairs on top of the O5-plane. At the level of the gauge theory, the main effect is the addition of extra localised flavours of the D3-brane gauge theory. We will not discuss these possibilities explicitly, but they are implicitly included in our analysis. For instance, as already mentioned, by allowing for a pair of D5-branes on top of an O5^--plane we reproduce the physics of the S-dual orbifold 5-planes. In fact, such orbifolds provide a realisation  of Class III in <cit.>; in terms of the O5^--plane with two D5-branes stack on them, one splits the stack of D3-branes as n=p+q, with the two stacks of p, q D3-branes ending on the two possible stuck D5-branes (equivalently, carrying opposite charges under the D5-brane (2)≃(1) worldvolume symmetry).
        
        In addition, recall that the configurations can be completed by adding general configurations of NS5- and D5-branes obeying the rules of the configurations in previous sections (and their _2 images), to define general classes of quiver gauge theories providing  Gaiotto-Witten BCFT_3 orientifold boundary conditions.
        
        
        
          §.§.§ Boundary conditions for O5-planes with one stuck NS5-brane
        
        
        A second possible local configuration corresponds to including one stuck NS5-brane on top of the O5-plane[This is possible for the O5^±-planes and the O5^--plane, but not for the O5^+-plane, because it requires C_0=1/2, and the orientifold image of an NS5-brane is not just an NS5-brane, but a (1,1) 5-brane.] (this was introduced in brane configurations in <cit.>, see <cit.> for review). 
        In the covering space, we have two _2 image D3-brane stacks ending on the NS5-brane from opposite sides in x^3, and the O5-plane crossing through their intersection. The D3-brane gauge symmetry in the covering space is (n)⊕(n) and it projects down to (n) after the orientifold. Hence, interestingly, in the quotient the (n) gauge symmetry is not reduced by the boundary conditions. We again can consider the different possible kinds of O5-plane in turn. They differ in how the orientifold acts on the 3d =4 hypermultiplet in the bifundamental (,) of the parent theory.
        
        For the O5^+-plane, the bifundamental matter is projected down to a 2-index symmetric representation  (plus a singlet) of the (n) symmetry. The simplest way to check this is to realise that giving a vev to this field corresponds to recombining the half D3-branes and moving the recombined infinite D3-brane away from the NS5-brane and along the O5-plane (i.e. the directions 789); for fields in the , the vev breaks the symmetry down to (n), in agreement with the  expected symmetry on D3-branes intersecting an O5^+ with no NS5-brane at the intersection, see section <ref>.
        
        Similarly, for the O5^--plane the bifundamental matter is projected down to a . For the O5^- (equivalently an O5^--plane with one stuck D5-brane), in addition to the  we get an additional full hypermultiplet flavour in the fundamental. We get a full, rather than a half, hypermultiplet because in this case the D3-branes are split in two halves by the NS5-branes, and the orientifold swaps the the D5-D3 open strings for the two halves, in contrast with the previous section, where there was a single D3-D5 open string sector which was mapped to itself under the quotient, enforcing the orientifold projection down to a half-hypermultiplet. Of course this matches the group theoretical fact that the fundamental of (n), even when charged under the D5-brane O(1)≃_2, is not a pseudoreal representation, so it is not possible to have half-hypermultiplets. For the O5^+ we get a variant of the result of the O5^+, namely matter in the  plus a singlet.
        
        In conclusion, the brane configurations are therefore very similar to those considered in previous sections, with the only novelty of the new object in the game. We now turn to a quick discussion of their role in the 1-form symmetries of the system, and then turn to the discussion of the supergravity dual, and the SymTFT picture it produces.
        
        
        
          §.§.§ The 1-form symmetries
        
        
        In this section we quickly carry out the discussion of the impact of the O5-plane orientifold quotient in the 1-form symmetries of the system. Recall that the configurations with one stuck NS5-brane in section <ref> or without it in section <ref> are related by a Higgs mechanism by giving vevs to the localised hypermultiplets in the former. Thus, since 1-form symmetries are insensitive to vevs for local operators, the results are essentially identical and  will thus depend mainly on the kind of O5-plane in the configuration. We can then study the 1-form symmetries in either of the two pictures, or carry out the discussion simultaneously.
        
        The O5^--plane configurations and 1-form symmetries
        
        Let us start discussing the configurations with an O5^--plane. In
        the configuration with no stuck NS5-brane, we have seen in section
        <ref> that the n D3-branes intersecting the
        O5^--plane have boundary condition projecting the vector multiplets
        down to (n) and the hypermultiplets down to the
        . Considering the possible global structures of an (n)
        theory[This can be made more precise by considering the
          semi-infinite D3-branes to end on some faraway NS5-branes, so that
          it describes a 3d genuine (n) theory. We will implicitly work
          similarly in the discussion of 1-form symmetry in forthcoming
          examples.]<cit.>, we can have the (n) or
        ((n)/_2)_± theories. The former admits Wilson lines in the
        fundamental representation, while the latter do not (and admit
        magnetic or dyonic lines, respectively for the ± choices). In
        short, we have potential _2 electric or magnetic (or the dyonic
        diagonal combination) 1-form symmetries, which are jointly described
        by _2-valued 2-form background fields C_2, B_2. We will
        recover this picture in the SymTFT arising from the holographic dual
        in section <ref>.
        
        We recover the same result for the configuration with one stuck NS5-brane, in which we have an (n) theory with a localised hypermultiplet in the , recall section <ref>. For simplicity, we start with the case of even n. The presence of the dynamical hypermultiplet implies that the potentially present electric 1-form symmetry of the (n) theory is broken to at most _2. Therefore the global structure of the gauge theory is either SU(n) with a _2 electric 1-form symmetry, admitting _2 valued Wilson lines, or (SU(n)/_n/2)_0,1 in two variants, admitting magnetic or dyonic Wilson lines. This matches the picture discussed above.
        
        With one stuck NS5-brane, it is possible to have odd n. This is compatible with the Higgsing realised by moving the NS5-brane off the D3-brane stack because one of the D3-branes is dragged along with the NS5-brane; equivalently, the Higgsing with a hypermultiplet in the  for odd n necessarily has one vev entry equal to zero. Hence, only in the limit of infinite separation we really connect with the case with no stuck NS5-brane and even n.
        For finite separation, the presence of this extra D3-brane may modify the structure of symmetries in the theory. In fact, it is possible to tensor the dynamical matter in the  to build dynamical objects in the fundamental representation, so that no leftover 1-form symmetry is present.
        
        The O5^+-plane configurations and the non-BPS spinor
        
        Let us now consider the  configurations with the O5^+-plane. In principle one could expect this case to be fairly similar to the above, but there are two main (and related) differences. The first is that the orientifold projection on a set of n D3-branes intersecting the O5^+-plane leads to an (n) symmetry, whose pattern of possible global structures is more involved than for (n), due to the role of line operators in the spinor representations. The second is that, as we discuss next, on top of matter in two-index tensor representations, the O5^+-plane supports new dynamical degrees of freedom, precisely in the spinor representation. In the following we discuss the appearance of this spinor states.
        
        Building on the characterisation in <cit.> of D-brane charges via K-theory, <cit.> proposed the 
        classification of D-brane charges localised on Op-planes based on equivariant K-theory on ^p+1× (^9-p/Ω I_9-p), where I_9-p flips the coordinates or ^9-p. In particular, localised (d+1)-dimensional branes on the (p+1)-dimensional Op^±-planes are classified by the real or symplectic K-theory groups KR(^9-p, ^p-d) and KH(^9-p, ^p-d). These are easily  obtained by the isomorphisms
        
        
            KR(^9-p, ^p-d)≃ KO(^2p-d-1)   ,   KH(^9-p, ^p-d)≃ KO(^2p-d+3)  .
        
        In particular, upon using mod-8 Bott periodicity of KO groups, for an O5^+-plane the relevant K-theory groups are  KO(^5-d). The non-trivial groups are
        
            KO(^0)=KO(^4)=   ,   KO(^1)= KO(^2)=_2  ,
        
        which describe the BPS D5- and D1-brane, and a _2-charged non-BPS 4-brane and a 3-brane. 
        
        We are interested in the 3-brane, for which we now provide a
        microscopic construction. To describe a _2 charged 3-brane in the
        directions 0 789 (hence along the O5^+-plane), we consider a
        D7-brane along 03 456 789 and at the origin in the directions 12, and
        its orientifold image, namely an anti-D7-brane (which we denote by
        D7) spanning 03 456 789 and also at the origin in the
        directions 12. Away from the O5^+-plane they can both annihilate via
        tachyon condensation, but the tachyon is odd under the projection and
        must vanish at the O5^+-plane location, leading to a localised
        3-brane charge. A simple way to check the above picture is to
        (formally) perform T-duality along the directions 3456, so that the
        O5^+-plane maps to an O9^+-plane, and the D7-D7 pair
        maps into a D3-D3 pair, whose tachyon is projected out,
        so it provides a stable non-BPS state. This non-BPS D3-brane in the
        non-supersymmetric (32) theory in <cit.> was
        identified in <cit.> with the class KSp(^6)=_2
        and built explicitly with precisely the above microscopic
        construction.
        
        Let us now introduce a stack of n D3-branes intersecting the O5^+-plane. Using the above microscopic construction, it is easy to see that the open strings between the D7-D7 and the n D3-branes have 8 DN directions, and lead to fermion zero modes in the fundamental representation, so the resulting state transforms in the spinor representation of the (n) in the n D3-branes. We also observe that for even n one in fact gets spinors of both chiralities; this is in contrast with other instances of non-BPS spinor states (such as the type D0-brane) for which a worldvolume _2 gauge symmetry enforces a projection eliminating one of the chiralities <cit.>.
        
        One may worry that the state is higher-dimensional, as it additionally extends in the directions 456 789, and would see to play no role in the discussion of symmetries of the 4d theory. However the relevant point is that it has a 1d intersection with the D3-branes, so it effectively defines a line operator in the spinor representation for most purposes[In the holographic dual in section <ref> and its SymTFT sector in section <ref> this will be more manifest, as the additional dimensions are actually compact in the near horizon geometry of the D3-branes.]. In particular, the global structure of the symmetry group must be compatible with the existence of objects transforming in the spinor representation. In addition, one may consider configurations where the non-BPS brane is compactified in the extra directions 789, and form the analogue of  dynamical finite energy brane-antibrane pairs, which can be nucleated and break line operators in the spinor representation. It is in this sense that we refer to them as spinor states.
        
        The presence of these spinor states will be key in our identification below of the 1-form symmetries of this system when regarded as a boundary configuration of the 4d theory. Incidentally, we point out that such spinor states are absent for the O5^- configurations, so that our identification of the 1-form symmetry structure in the earlier discussion is not modified.
        
        The O5^+-plane configurations and 1-form symmetries: odd n
        
        Consider now the configurations with the O5^+-plane, for which n may be even or odd. We start with the case of n odd and look first at the case with no stuck NS5-brane.  As shown in section <ref>, the n D3-branes intersecting the O5^--plane have boundary condition projecting the vector multiplets down to (n) and the hypermultiplets down to . For odd n, the center of (n) is _2, with the _2 charge carried by the spinor representation. The possible global structures are Spin(n), admitting spinor Wilson line operators, and SO(n)_±, without them but admitting monopole or dyon line operators, respectively. Because of the existence of the non-BPS spinor discussed above, we are led to a Spin(n) global structure. This would seemingly lead to a _2 electric 1-form symmetry, but would be problematic, because the bulk (n) has no _2 subgroup of its center _n for odd n. Happily, the fact that the spinor is  dynamical (i.e. can be used to break line operators in the spinor representation) saves the day, because it breaks the _2 1-form symmetry. Hence, there is no left-over 1-form symmetry in this case.
        
        If there is one stuck NS5-brane, recalling section <ref>, the boundary conditions preserve an (n) symmetry but there is a localised hypermultiplet in the . Hence, it would seem that there is a potential _2 electric 1-form symmetry, with the _2 charge carried by  line operators in the fundamental representation. This would however be at odds with the Higgsing to the (n) theory, where the _2 1-form symmetry is ultimately broken by the spinor state. However, since the latter is a stable state, it must survive in the unHiggsing to the (n) theory and be present even when the NS5-brane sits on top of the O5-plane. In this situation, although the presence of the NS5-brane prevents a fully microscopic description, it is possible to guess its (n) quantum numbers. In particular, two of them can be combined into a fundamental representation of (n) (just as two spinors can combine into a vector in the (n) subgroup), hence the _2 1-form symmetry is broken, matching the above result.
        
        The O5^+-plane configurations and 1-form symmetries:
          even n
        
        We now quickly consider the case of even n, starting with the configuration with no stuck NS5-brane. As shown in section <ref>, the n D3-branes intersecting the O5^+-plane have boundary condition projecting the vector multiplets down to (n) and the hypermultiplets down to . The center C of (n) is _4 for n=4k+2 or _2×_2 for n=4k, so the global structures are either Spin(n) or Spin(n)/H with H a non-trivial subgroup of C. The former is the only structure admitting spinors of both chiralities, hence is the appropriate global structure in our case. The _4 or _2×_2 electric 1-form symmetry is however broken by the  dynamical presence of the spinors themselves, so no non-trivial 1-form symmetry remains.
        
        In the configuration with a stuck NS5-brane, as discussed in section <ref>, we have (n) boundary conditions with a localised hypermultiplet in the . As in the case of odd n, the would-be present _2 1-form symmetry is broken by the _2-charged non-BPS states, so no 1-form symmetry is preserved in this case.
        
        The O5^±-plane configurations
        
        Let us now consider the configurations with the O5^--plane, which can be regarded as the previous ones with the addition of one stuck D5-brane on top of the O5-plane. Clearly, the present of the extra fundamental flavours from the D3-D5 open string sector, the electric 1-form symmetry is completely broken. For the case of the O5^+-plane, we have a situation similar to the O5^+-plane, and we will not discuss it further.
        
        
        
         §.§ Holographic dual
        
        
        In this section we describe the gravitational dual of the configurations of 4d =4(N) SYM on half-space with orientifold boundary conditions of the kind considered in section <ref>. The discussion of the corresponding supergravity solutions require a slight generalisation beyond those in section <ref>, as we explain next.
        
        
        
          §.§.§ Supergravity solutions with multiple AdS_5×^5 asymptotic regions
        
        
        The supergravity solutions considered in <cit.> (see also <cit.> for further works) are actually the most general compatible with SO(2,3)× SO(3)× SO(3) symmetry and 16 supersymmetries. In fact, they describe the near horizon solution of stacks of D3-branes with general Hanany-Witten configurations of NS5- and D5-branes. These include configurations defining BCFT_3 boundary conditions for 4d =4(N) SYM, but also configurations in which several 4d =4(N_i) SYM sectors are separated by configurations of NS5- and D5-branes (with extra D3-branes suspended among them). We quickly review their basic structure, emphasising only the points necessary for the generalisation we need, and refer the reader to the literature for further details.
        
        The corresponding gravity duals are given by
        AdS_4×^2_1×^2_2 fibered over a Riemann surface,
        which can be conveniently described as a disk. Its boundary is divided
        into segments of two kinds, in which either ^2_1 or ^2_2
        shrink. Segments of different kinds are separated by punctures,
        which in general can correspond to asymptotic AdS_5×_5
        regions, where the ^5 is realised by fibering
        _1^2×_2^2 over an arc in the Riemann surface with one
        endpoint in each of the two boundary segments, as we discussed in the
        example in figure <ref>. We denote by N_i the RR 5-form
        flux on the ^5 in the i^th puncture. In case this flux is
        zero, the geometry at this point is actually smooth, and describes the
        shrinking of ^5 to zero size, as in the case of the origin in
        figure <ref>.
        
        Within each segment, there may be additional punctures, which describe NS5- or D5-branes, according to the kind of segment on which it is located. These punctures describe local regions with geometry AdS_4×^2×^3, where the ^2×^3 is obtained by fibering _1^2×_2^2 over an arc in the Riemann surface with the two endpoints on both sides of the puncture in the boundary segment, as we discussed in the example in figure <ref>. There is an NSNS or RR 3-form flux over ^3 encoding the 5-brane charge, and an integral of the 2-form field (of the opposite kind) over ^2, encoding the 5-brane linking number (or worldvolume monopole charge). We note that the argument below (<ref>) applies also here, essentially relating the integer value of the 2-form field background to the integer value of the RR 5-form F̃_5 flux on ^2×^3.
        
        A particular class is that of solutions with only one asymptotic AdS_5×^5 region, which we have studied in earlier sections. In this case, in our discussion the disk of the Riemann surface was distorted into the quadrant in figure <ref>, with the boundary corresponding to the horizontal and vertical semi-infinite lines, plus the point at infinity. The two axes correspond to segments of the two kinds, and they are separated by a puncture at the origin, with no 5-form flux on the ^5, hence describing a smooth point, and by a puncture at the point at infinity, with N units of 5-form flux, describing the asymptotic AdS_5×^5 dual to 4d (N) SYM theory on half-space. Along each segment/axis, there are 5-brane punctures which describe the BCFT_3 and which have been the focus of most of our study.
        
        The previous paragraphs show that most of the ingredients of the supergravity solution in the general case are essentially already present in the case of an ETW configuration with a single AdS_5×^5 asymptotic region. Hence, most of our discussion of the SymTFT structure of the topological sector of the gravitational background extend to the general case. 
        The general structure is that of a collection of SymTrees, analogous to those introduced in <cit.> to discuss compactifications with several sectors associated to isolated singular points. In our case, we have a set of SymTFT_N_i's associated to the AdS_5×^5 asymptotic regions, which are separated by U(1) junction theories connecting them with the different SymTFT_^2×^3's associated to the 5-branes. We also have a retracted SymTree picture in which the SymTFT_^2×^3's are collapsed onto the junction theories to yield the explicit 5-brane probe theories.  The coefficients of the topological couplings (<ref>) jump across the junctions, as dictated by the induced D3-brane charge on the 5-branes in the retracted SymTree picture, or equivalently by the coefficient of the sector (<ref>) of the corresponding SymTFT_^2×^3 in the full SymTree picture. We will not consider this general  setup any further, but simply adapt it to the class of solutions necessary to describe the gravity dual of the orientifold boundary conditions in section <ref>, and to extract their Symmetry Theories. 
        
        
        
          §.§.§ Orientifold ETW configurations
        
        
        The general class of solutions in the previous section includes the gravitational duals of the orientifolded brane configurations in section <ref>, when described in the double cover, as we explain in this section. 
        
        The appropriate class of solutions is that with two identical asymptotic AdS_5×^5 regions, with the same number N of RR 5-form flux units (and same value of other fields, e.g. the 10d axio-dilaton), and an intermediate region with a similarly _2 symmetric distribution of NS5- and D5-branes. For convenience we depict the Riemann surface Σ as a strip, parametrised by a coordinate z with Im  z∈ [0,π/2], with the two points at infinity Re   z→±∞ corresponding to the two asymptotic AdS_5×^5 regions, and with NS5-brane punctures (resp. D5-brane punctures) in the lower (resp. upper) boundary[The quadrant in the ETW configurations in the previous sections can be described as a strip by simply taking w=-e^-z, see (<ref>) later. The point w=0, i.e. z→∞, corresponds to a closed off puncture, with vanishing 5-form flux, hence no actual asymptotic AdS_5×^5 region at that point <cit.>.]. This strip corresponds to the disk mentioned in section <ref>, with the two boundaries of the strip corresponding to the two kinds of segment in the disk boundary, and the two points at infinity corresponding to punctures describing two asymptotic AdS_5×^5.
        
        The geometry is of the kind explained in section <ref>, with metric (<ref>) and functions (<ref>) defined in terms of h_1,h_2 given by (see e.g. <cit.> for explicit expressions of this form)
        
        h_1    =   [  -iα̃sinh(z-β̃)  - ∑_b γ̃_b  log(tanh( iπ/4- z-δ̃_b/2) )] +  c.c.
        
        h_2    =   [ αcosh(z-β)  - ∑_a  γ_a  log(tanh(z-δ_a/2) )] +  c.c.
        
        These are analogous to (<ref>), but in terms of the strip coordinate z instead of the quadrant coordinate w, and with a slightly different asymptotic behaviour to include the second AdS_5×^5 region. Specifically, they expressions are related by
        
            w=-e^-z   ,   γ_a≡ 2 d_a   ,   γ̃_b≡ 2d̃_b    ,   k_a≡ e^-δ_a   ,   l_b≡ e^-δ̃_b
        
        
        
        The configuration is quotiented by Ω R, with R being inherited[This description of the orientifolding ignores the backreaction of the O5-plane on the geometry. The latter could be accounted for (at least slightly far away from the O5-plane location) by introducing an additional D5-brane source in the expressions of h_1,h_2. In our discussion we will ignore this backreaction, but its topological content will be included in our discussion of the SymTFT in section <ref>.] from its flat space avatar R:(x^3,x^4,x^5,x^6)→ (-x^3,-x^4,-x^5,-x^6). It can be expressed in terms of the ^5 in the internal space, by using its embedding in ^6 as
        
            (x^4)^2+(x^5)^2+(x^6)^2+(x^7)^2+(x^8)^2+(x^9)^2=r^2  .
        
        The fixed point set is hence AdS_4×_1^2. In the AdS_4×_1^2×_2^2 fibration over the strip, it acts as antipodal identification on _2^2 together with a reflection z→ - z on the complex coordinate on the strip. The fixed point set in the strip corresponds to the segment in the imaginary axis, but the only fixed point set in the whole geometry corresponds to the _1^2 sitting at the upper endpoint of this segment (times AdS_4). We thus have an O5-plane on AdS_4×_1^2, and located at z=iπ in the strip; this is precisely the geometry required to preserve the same supersymmetries as the D5-branes.
        
        
        
        
        In order to admit this orientifold quotient, the 5-brane configurations have to be distributed on the corresponding boundaries of Σ symmetrically under this reflection with respect to the imaginary axis, see figure <ref>. For notational convenience, let us label the 5-brane stacks with indices a and b taking both positive and negative values, with the a^th and b^th stacks being orientifold images of the -a^th and -b^th (In case there are 5-branes on top of the O5-plane (for instance for the O5^--plane), we simply include them by using a label a=0 or b=0). Hence, for (<ref>) to be symmetric under z→- z, we must have 
        
            δ_a=-δ_-a   ,   δ̃_b=-δ̃_-b   ,   γ_a=γ_-a   ,  γ̃_b=γ̃_-b  ,
        
        as well as  β=0, β̃=0. Namely, the positions and multiplicities of 5-branes on the different stacks must respect the symmetry. In particular, in our ETW brane notation in previous sections, n_-a=n_a, m_-b=m_b.
        
        This also ensures, although we will not be explicit about it, that the NSNS 2-form and RR 2- and 4-form backgrounds also respect the symmetry, implying that the assignment of 5-form flux on the asymptotic ^5's and the ^2×^3 on the 5-brane throats are _2 symmetric.  Finally, noting that the NSNS 2-form is intrinsically odd under the orientifold action, and recalling (<ref>) we have L̃_-b=-L̃_b; on the other hand, the RR 2-form is intrinsically even under the orientifold, but the orientation of _1^2 flips, so recalling (<ref>) we have K_-a=-K_a. In case there are 5-branes on top of the O5-plane, we simply have K_0=L_0=0, in agreement with the fact that there is no flux jump due to the _2 symmetry.
        
        From the perspective of the configuration after the quotient, we have a configuration that away from the location of the orientifold plane has the same structure as the solutions in section <ref>. The main new ingredient is thus the presence of the orientifold plane and it action on local fields and localised degrees of freedom. This will also be the only main new ingredient in our study of the SymTFT.
        
        
        
         §.§ The SymTFT
        
        
          §.§.§ General structure
        
        
        
        The most efficient way to extract the structure of the Symmetry Theory in this class of models is to describe the supergravity solutions in the previous section in Poincaré coordinates of the 5d solution. As shown in figure <ref>, in the covering space we have a full 4d Poincaré holographic boundary, with a _2 symmetric fan of wedges separated by 5-branes spanning AdS_4 slices, and the O5-plane (possibly with 5-branes on top of it) sitting in the AdS_4 orthogonal to the 4d holographic boundary. As in previous sections, the Symmetry Theory picture can be extracted by taking the 4d holographic boundary to describe the 4d physical boundary, and the asymptotic infinity in the Poincaré radial direction to describe the topological boundary. Also, although we have described the configuration in the covering space, we can make the orientifold identification manifest, by simply folding the configuration along the vertical axis, and reach a configuration exhibiting an ETW boundary, which corresponds to the O5-plane (possibly with 5-branes).
        
        The picture clearly shows that the structure of the Symmetry Theory is a variant of the SymTFT Fan encountered in previous sections. In fact, away from the O5-plane it is locally exactly of that kind. Namely, we have a SymTFT Fan built by putting together wedges containing branches of SymTFTs of the kind (<ref>) for different values of their coefficients, joined via junction theories to SymTFT_^2×^3's corresponding to the 5-branes, with fluxes dictated by flux conservation. 
        
        New features are however encountered in the local region around the
        O5-plane, i.e. the ETW boundary. First, the 5-form flux does not go to
        zero at the location of the O5-plane, we rather have a non-zero number
        n of flux quanta, and correspondingly the ^5 does not shrink to
        zero size. Hence, spacetime is ending due to the presence of the
        O5-plane; this is similar to how type IIA theory can end on O8-planes,
        as in each boundary of type I' theory. The second is the presence of a
        new object, the O5-plane itself, whose role we discuss in the
        following.
        
        In order to understand the role of the O5-plane in the Symmetry Theory, the presence of additional 5-branes in the SymTFT Fan away from it is not relevant, so we may simply focus on the configuration in a small neighbourhood around the O5-plane. This sector of the Symmetry Theory in the covering space is therefore a 5d SymTFT_n spanning the whole non-compact 4d spacetime, times the interval between the physical and topological boundaries, but cut in two halves by an O5-plane (possibly with 5-branes on top). The integer n is the amount of flux remaining after the asymptotic value N has been peeled of by the 5-branes away from the O5-plane.
        
        The above picture describes the retracted version of the full Symmetry Theory. A more detailed picture is obtained by activating the full SymTree structure, considering the two orientifold images of the SymTFT_n joining at a junction theory with the SymTFT_^2×^3's associated to the O5-plane (and the possible D5-branes on top of it). One may think that in the presence of an NS5-brane stuck on the O5-plane there should be an additional SymTFT_^2×^3 branch in the SymTree. However since the linking number of such NS5-brane is zero, as discussed above, or equivalently the integral of the RR 2-form over the ^2 vanishes, so does the total 5-form flux over the corresponding ^2×^3. Therefore this would-be SymTFT_^2×^3 is actually trivial. This is the SymTFT manifestation of the fact that the theories with a stuck NS5-brane can be Higgsed to those without it, with  no modification of their 1-form symmetry structure. 
        
        These ingredients will be important in the discussion of the 1-form symmetries, to which we turn next.
        
        
        
          §.§.§ The 1-form symmetries in the SymTFT
        
        
        In this section we carry out the discussion of the 1-form symmetries in the SymTFT of the different O5-plane configurations. The discussion will parallel, and match, the discussion in section <ref>. For simplicity we restrict to configurations without stuck NS5-brane, since they simply add a topologically trivial sector to the SymTFT structure, as we have just explained.
        
        The O5^--plane configuration and 1-form symmetries
        
        Consider the configuration with an O5^--plane, hence we need n even. In the configuration with no stuck NS5-brane, we have two SymTFT_n theories joining at a junction with the O5^--plane. The latter may be regarded as an extra branch corresponding to a SymTFT_^2×^3 with -2 units of 5-form flux. The relevant observation is that, because gcd(n,2)=2, there is an unbroken _2 in the full theory. This is precisely the structure required to match the unbroken _2 1-form symmetry discussed in section <ref>.
        
        One may argue that the factor of 2 arises because we are working in
        the covering space, and wonder about a possible intrinsic description
        in the orientifold quotient. Indeed, such a description is obtained by
        simply realising that the orientifold makes some of the 2-form fields
        _2-valued. For B_2, this simply follows from the fact that it
        is odd under the orientifold action, hence it is projected down to a
        _2-valued gauge field. For C_2, this can be seen by noticing
        that two stuck D1 strings can become orientifold images of each other
        and move away from the orientifold plane, showing the field under
        which they are charged is _2-valued.
        
        The O5^+-plane configuration and 1-form symmetries: odd n
        
        Consider now the O5^+-plane configuration, starting with the case of an odd number n of RR 5-form flux units in the bulk SymTFT. In this case the SymTree connects two SymTFT_n theories with the SymTFT_^2×^3 associated to the O5-plane, which contains a term of the form (<ref>) with coefficient +2 in the covering space. Since now gcd(n,2)=1, there is no left-over 1-form symmetry, in agreement with the field theory analysis in section <ref>.
        
        From the perspective of the theory in the quotient, the action of the
        O5-plane on the 2-form fields is as in the case of the O5^--plane
        above, hence it would seem that there is a _2 symmetry. However,
        for odd n this is broken by the baryonic vertex of the SymTFT_n
        theories. Alternatively, there is a way more intrinsic to the
        orientifold configuration to explain the breaking of the _2
        symmetry. This is given by a vertex corresponding to the gravitational
        realisation of the (n) spinor introduced in section
        <ref>. The microscopic construction is the
        near horizon version of the flat space construction mentioned there,
        i.e. we consider a D7-D7 pair spanning a 3d subspace
        along the O5-plane volume and wrapped on the RP^3×^2
        around the O5-plane (with a nontrivial _2 Wilson exchanging the
        two objects along non-trivial 1-cycles H_1( RP^3,)=_2, to
        account for the orientifold action). This describes a spinor state in
        the theory, which breaks the above _2 symmetry as explained in
        the field theory approach.
        
        The O5^+-plane configuration and 1-form symmetries: even n
        
        Consider now the case of an O5^+-plane with an even number n of RR 5-form flux units in the bulk SymTFT. In this case we have a SymTree with an unbroken _2 symmetry because gcd(n,2)=2. This is however broken by the presence of the spinor mentioned above, so no non-trivial 1-form symmetry remains, in agreement with the field theory analysis. 
        
        The O5^±-plane configurations
        
        Let us quickly consider the O5^±-plane configurations. The O5^--plane corresponds to the O5^--plane with an extra D5-brane on top, hence its charge is +1 in the covering space. This means that, even if n is even, there is no _2 symmetry preserved in the SymTree because gcd(n,1)=1. This agrees with the field theory description. Equivalently, from the perspective of the orientifold quotient theory, the projection on 2-form fields is as in the O5^--plane, so one might think that there is a _2 symmetry. However, the electric 1-form symmetry is broken by the presence of explicit flavours in the fundamental due to the stuck D5-brane, in agreement with the above result. Either way, there is no non-trivial 1-form symmetry in this case.
        
        For the case of the O5^+-plane, we have a situation similar to the O5^+-plane, and we will not discuss it further.
        
        
        
        
        
        § CONCLUSIONS
        
        
        In this work we have disclosed the structure of the 5d SymTFT of 4d
        =4(N) SYM on a space with boundary coupled to a
        Gaiotto-Witten BCFT_3, by extracting the key topological information
        from the gravitational dual of configurations of D3-branes ending on a
        system of NS5- and D5-branes. The Symmetry Theory turns out to display
        an extremely rich structure, consisting of a fan of SymTFTs
        (<ref>) with different levels, coupled via (1) junction
        theories to the SymTFTs of the 5-branes, in a SymTree-like
        structure. The SymTFT Fan realises 0-form flavour symmetries and their
        enhancement, and allows for the identification of unbroken 1-form
        symmetries of the SYM theory with boundaries.
        
        
        
        We have performed a systematic characterisation of the physical
        effects undergone by different topological operators when moved across
        different SymTFTs of the Symmetry Theory, by using the holographic
        string theory realisation, truncated at the topological level. Subtle
        field theory effects, such as the variation of the order of the
        discrete symmetries, are very simply realized in terms of familiar
        brane dynamics, such as Hanany-Witten brane creation effects and
        Freed-Witten consistency conditions.
        
        
        
        We have further discussed the introduction of 7-branes to provide
        boundary conditions for the 5-brane theories, and studied their role
        as duality interfaces of the physical gauge theory, implementing
        SL(2,) duality transformation on topological operators moved
        across the 7-brane branch cuts.
        
        
        
        We have carried out a similar analysis for orientifold boundary conditions. By extracting the topological information of configurations including the different kinds of O5-planes in the gravitational duals,  we have shown that the corresponding Symmetry Theories are obtained by enriching the SymTFT Fan with the local physics of the O5-plane neighbourhood. We have used string theory techniques to characterized the structure of 1-form symmetries for different kinds of orientifold boundary conditions. 
        
        
        
        Our work opens several interesting directions, some of which are:
        
        
          *  The use of 5-brane configurations to define boundary conditions allows to construct holographic dual pairs for boundaries of 4d theories with less supersymmetry. We expect our techniques to be helpful in extracting the corresponding SymTFTs for such constructions. For instance, as discussed in <cit.> the holographic duals of =3 S-fold <cit.> or =2 orbifold theories with boundaries can be obtained by considering quotients of the parent ETW configurations with symmetric sets of 5-branes. It is straightforward to apply our techniques to formulate the SymTFT Fans for these configurations, and it would be interesting to explore possible new features arising from the quotients.
        
        
          *  It is possible to construct 5-brane configurations with smaller supersymmetry but still providing BCFT_3 boundary conditions for the 4d (N) SYM theories, for instance by using rotated configurations of NS5- and D5-branes preserving 4 supersymmetries <cit.>. Such BCFT_3 5-brane configurations were considered in <cit.> and share many features of those in <cit.>. Hence, although they do not have a known supergravity dual, our techniques may well allow for the construction of their corresponding SymTFTs, hopefully uncovering new classes of SymTFT Fans, thus enlarging the classification of symmetry structures for supersymmetric boundary conditions of 4d (N) SYM theories.
        
        
          *  Several features of the Symmetry Theories we have considered seem related to the fact that we are dealing with conformal boundary conditions for a conformal field theory. In particular, the fan-like structure of junction theories in the holographic dual is  directly related to conformal invariance. This suggests that the Symmetry Theories of general conformal field theories with conformal boundary conditions are given in terms of a suitable SymTFT Fan combining wedges of bulk 5d SymTFTs and flavour SymTFTs into a local SymTree structure. It would be interesting to explore a general formulation of this SymTFT Fans for other classes of CFTs.
        
        
          *  The characterisation of the topological sector of supergravity backgrounds with ETW branes is also relevant from the perspective of the Cobordism Conjecture <cit.>, as it provides a direct link between the topological sector in the bulk and that on the ETW branes. This provides a powerful tool in the characterisation of cobordism defects in general theories of Quantum Gravity. Our work is an important first step in explicitly developing this program in the context of holographic dual pairs, and we expect it to seed further developments relevant to swampland studies.
        
        We hope to come back to these and other interesting questions in the future.
        
        
        
        
        
        We thank Roberta Angius, Andrés Collinucci, Matilda Delgado, Miguel
        Montero and Xingyang Yu for helpful discussions.  I.G.E. and
        J.H. thank Harvard University for hospitality during part of the
        development of this work. I.G.E. is partially supported by STFC grants
        ST/T000708/1 and ST/X000591/1 and by the Simons Foundation
        collaboration grant 888990 on Global Categorical Symmetries. The work
        by J.H. and A.U is supported through the grants CEX2020-001007-S and
        PID2021-123017NB-I00, funded by MCIN/AEI/10.13039/ 501100011033 and by
        ERDF A way of making Europe. The work by J. H. is also supported by
        the FPU grant FPU20/01495 from the Spanish Ministry of Education and
        Universities.
        
        
        
          
        Data access statement. There is no additional research
        data associated with this work.
        
        JHEP
