aainstitutetext: Physics Department, University of Cincinnati, PO Box 210011,
Cincinnati OH 45221 USAbbinstitutetext: Department of Physics and Astronomy &
Center for Theoretical Physics, Seoul National University, Seoul 08826,
Republic of Korea

# Vertex algebra of extended operators in 4d N=2 superconformal field theories

Philip C. Argyres b Matteo Lotito a Mitch Weaver<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We construct a class of extended operators in the cohomology of a pair of
twisted Schur supercharges of 4d ${\mathcal{N}}{=}2$ SCFTs. The extended
operators are constructed from the local operators in this cohomology — the
Schur operators — by a version of topological descent; in particular, they are
world volume integrals of certain super descendants of Schur operators. Their
world volumes extend in directions transverse to a spatial plane in Minkowski
space-time and their correlators are meromorphic functions only of the
positions where they intersect this plane. These operators enlarge the vertex
operator algebra of the Schur operators. We illustrate this enlarged vertex
algebra by computing some extended-operator product expansions for the free
hypermultiplet SCFT.

##  1 Extended operators in twisted Schur cohomology

Extended operators, such as Wilson-’t Hooft line operators, play a central
role in our understanding of gauge field theories. Progress on analytic
control of correlation functions of extended operators in strongly-coupled
gauge theories has been made over the past 30 years by examining extended
operators in supersymmetric gauge theories and the closely related
constructions of topological extended operators in topologically twisted
supersymmetric gauge theories. But the spectrum and correlators of
supersymmetric extended operators in strongly coupled non-lagrangian
supersymmetric field theories — e.g., in isolated superconformal field
theories (SCFTs) — is much less well understood.

In the last decade much insight into the structure of the local operator
algebra of 4d ${\mathcal{N}}{=}2$ SCFTs has come from the realization
Beem_2015  that the operator algebra of certain BPS local operators — the
twist-translated Schur operators — have the structure of a vertex operator
algebra (VOA), which is more or less the chiral half of a 2d CFT. Twist-
translated Schur operators are the local operators in the cohomology of a pair
of nilpotent supercharges in the superconformal algebra,
${\mathbbmtt{T}}_{\pm}$, called twisted Schur supercharges.

It is natural to ask, in this context, whether there are extended operators in
twisted Schur cohomology, and if so, what is the structure of their operator
algebra. We will show that there are, indeed, line, surface, and domain wall
operators in twisted Schur cohomology. They are constructed from the local
Schur operators in a way analogous to how extended topological operators are
constructed from local operators by topological descent Witten_1988 . The line
operators and one of the surface operators are supported only on light-like
world volumes (i.e., on sections of a light cone) and so only exist in
Lorentzian signature space-time. Furthermore, we will show that their
correlators, with each other and with twist-translated Schur operators, depend
meromorphically only on a single complex coordinate, $z$, entering into the
parameterization of the space-time position of each extended operator
insertion. The $(z,\overline{z})$ coordinates parameterize a spatial 2-plane
in space-time which we will call the VOA plane. Thus the extended operator
algebra forms a vertex algebra extending the VOA of the twist-translated Schur
operators.

This note provides a concise introduction to and summary of these results. A
more extensive derivation, including many technical details, is provided in
Argyres_2022xx .

VOAs corresponding to 4d ${\mathcal{N}}{=}2$ SCFTs arise when one studies the
protected sector of Schur operators annihilated by the twisted Schur
supercharges introduced in Beem_2015 . We will consider the pair of nilpotent
twisted Schur supercharges

$\displaystyle{\mathbbmtt{T}}_{+}$ $\displaystyle\doteq
Q^{1}_{~{}2}+{\widetilde{S}}^{2\dot{2}},$ $\displaystyle{\mathbbmtt{T}}_{-}$
$\displaystyle\doteq S_{1}^{~{}2}-{\widetilde{Q}}_{2\dot{2}}.$ (1)

We follow the notation and conventions of Lemos_2020  unless otherwise
specified.111We use $1$ and $2$ for spinor indices in place of $+$ and $-$
used in Beem_2015 ; Lemos_2020 . We reserve $\pm$ for lightcone indices.
${\mathbbmtt{T}}_{\pm}$ are the same as ${\mathbbmtt{Q}}_{1}$ and
${\mathbbmtt{Q}}_{2}$ in Beem_2015  and are those of Lemos_2020  with phase
${\zeta}=1$. Local operators in ${\mathbbmtt{T}}_{+}$-cohomology are
automatically also in ${\mathbbmtt{T}}_{-}$-cohomology in a unitary SCFT.

The set of extended operators that we analyze in this work are of a special
type, chosen in order to guarantee that they are non trivial objects in the
cohomology of the twisted Schur supercharges. Starting from a local Schur
operator, we perform a version of topological descent Witten_1988  by
${\mathbbmtt{T}}_{\pm}$-exact generators of the conformal algebra. This gives
rise to a web of descent operators obtained from a single local Schur
operator, ${\mathcal{O}}$,

$\displaystyle{\mathcal{O}}\to{\mathbf{D}}^{X_{1}}[{\mathcal{O}}]\to{\mathbf{D}}^{X_{2}}{\mathbf{D}}^{X_{1}}[{\mathcal{O}}]\to\cdots.$
(2)

Here ${\mathbf{D}}^{X}$ denotes the descent operation corresponding to the
exact generator $X$.

While there is a continuum of exact generators with which to perform descent,
a combination of ${\mathbbmtt{T}}_{\pm}$-cohomology equivalence of various
descent operations together with requiring finiteness of the descent operator
reduces this web to a finite set in which each successive descent increases
the world volume dimension of the descent operator by one. Each
${\mathbf{D}}^{X}$ acts on the proceeding operator by a (fermionic)
supercharge, so flips the statistics of the descent operator; each
${\mathbf{D}}^{X}$ changes the ${\rm U}(1)_{r}$ charge by $\pm 1$;
${\mathbf{D}}^{X}$ preserves the chiral weight, $h_{\mathcal{O}}$, under the
holomorphic conformal $\mathfrak{sl}_{2}$ symmetry of the VOA plane; and
${\mathbf{D}}^{X}$ commutes with twist translations in the sense that
${\mathbf{D}}^{X}[{\partial}_{z}{\mathcal{O}}]~{}=~{}{\partial}_{z}{\mathbf{D}}^{X^{\prime}}[{\mathcal{O}}]$,
so ${\partial}_{z}$ still acts as the translation generator of the extended
vertex algebra. Thus, descent operators, though extended operators in 4d,
behave like vertex operators inserted in the VOA plane.

pointlinessurfaceswalls${\mathcal{O}}$${\bf L^{-}}[{\mathcal{O}}]$${\bf
L^{+}}[{\mathcal{O}}]$${\bf S}[{\mathcal{O}}]$${\widehat{\bf
S}}[{\mathcal{O}}]$${\widehat{\bf W}}^{-}[{\mathcal{O}}]$${\widehat{\bf
W}}^{+}[{\mathcal{O}}]$$P_{-}$$P_{+}$$M_{+3}$$M_{-3}$$P_{+}$$P_{-}$$P_{-}$$M_{+3}$$M_{-3}$$P_{+}$
Figure 1:  The descent web obtained from a scalar Schur operator
${\mathcal{O}}$. Descent operations ${\mathbf{D}}^{X}$ are shown as arrows
labelled by the generator $X$. ${\bf L}^{\pm}$ ${\bf S}$, ${\widehat{\bf S}}$
and ${\widehat{\bf W}}^{\pm}$ denote the possible inequivalent line, surface,
and domain wall operators that can be reached by descent. For example, ${\bf
S}[{\mathcal{O}}]\doteq{\mathbf{D}}^{P_{+}}{\mathbf{D}}^{P_{-}}[{\mathcal{O}}]={\mathbf{D}}^{P_{-}}{\mathbf{D}}^{P_{+}}[{\mathcal{O}}]$.
The operators and descent operations shown in red are special to the case of a
scalar Schur operator ${\mathcal{O}}$. They must be modified in the case of a
spinning Schur operator; see section 3 for a discussion.

The resulting descent web is shown in figure 1. The arrows are the descent
operations labelled by their corresponding generators. $P_{\pm}$ are certain
light-like translations and $M_{\pm 3}$ are light-like combinations of boosts
and rotations, defined below. These generators also determine the space-time
world volumes of the descent operators; they are illustrated in figure 2
below. We have introduced shorthand names, ${\bf L}^{\pm}$, ${\bf S}$,
${\widehat{\bf S}}$, ${\widehat{\bf W}}^{\pm}$, for the iterated descent
operations. The $\pm$ superscripts are correlated with the ${\rm U}(1)_{r}$
charges. The statistics of the descent operators alternate between the rows of
figure 1, while their chiral weights are all the same. This is reminiscent of
the structure of the chiral half of a 2d topological SCFT. But, as we will
discuss below, it is not at all clear whether the enlarged vertex algebra has
the structure of a superconformal vertex algebra.

The ${\widehat{\bf S}}$ and ${\widehat{\bf W}}^{\pm}$ extended surface and
wall operators, shown in red in figure 1, can only be reached via descent with
an $M_{\pm 3}$ generator. Unlike $P_{\pm}$ descent, $M_{\pm 3}$ descent does
not give extended operators which are in ${\mathbbmtt{T}}_{\pm}$ cohomology
for all Schur primaries, ${\mathcal{O}}$. Also, the ${\widehat{\bf W}}^{\pm}$
wall operators intersect the VOA plane in a straight line, so behave in some
respects like a boundary operator in the vertex algebra. However, the
dependence of correlators on the position of this line in the VOA plane is
quasi-topological: correlators do not depend on the position of the line until
it crosses the position of another vertex operator insertion on the VOA plane.
In this note we largely sidestep these interesting complications of the hatted
descent operators by focusing primarily on the subalgebra of
${\mathcal{O}}$-${\bf L}^{\pm}$-${\bf S}$ operators, shown in black in figure
1.

In the next section, we define the twisted Schur descent operation and derive
the descent web, figure 1. In section 3 we discuss questions of convergence
and regularization of operator products. In section 4 we compute some operator
product expansions in the ${\mathcal{O}}$-${\bf L}^{\pm}$-${\bf S}$ vertex
subalgebra for the free hypermultiplet SCFT. We finish in section 5 with
remarks on future directions and open problems.

##  2 Twisted Schur descent

### Topological descent.

Recall the topological descent procedure Witten_1988  for constructing
extended operators from point operators in topological field theories (TFTs)
formed by twisting supersymmetric theories by a nilpotent supercharge
${\mathbbmtt{Q}}$. In a TFT the translation generators $P_{\mu}$ are
${\mathbbmtt{Q}}$-exact, so there exist fermionic generators $Q_{\mu}$ such
that222We use $\circ$ to denote the adjoint action for superalgebra and group
elements. Thus $\circ$ is the (anti)commutator between algebra generators, and
also denotes conjugation of a generator by a group element. We use the
convention that when there are no parentheses $\circ$ acts on everything to
its right.

$\displaystyle{\mathbbmtt{Q}}\circ P_{\mu}$ $\displaystyle=0,$ and
$\displaystyle P_{\mu}$ $\displaystyle={\mathbbmtt{Q}}\circ Q_{\mu}.$ (3)

If ${\mathcal{O}}(0)$ is a ${\mathbbmtt{Q}}$-closed local operator, then so is
${\mathcal{O}}(x)\doteq e^{x^{\mu}P_{\mu}}\circ{\mathcal{O}}(0)$. Define the
extended operator

$\displaystyle{\bf{\Sigma}}[{\mathcal{O}}]\doteq\int_{\Sigma}(Q\cdot
dx)\circ\cdots\circ(Q\cdot dx)\circ{\mathcal{O}}(x),$ (4)

where there are $p$ factors of $Q\cdot dx\doteq Q_{\mu}dx^{\mu}$ in the
integrand, and ${\Sigma}$ is some closed $p$-manifold.
${\bf{\Sigma}}[{\mathcal{O}}]$ is ${\mathbbmtt{Q}}$-closed since the
${\mathbbmtt{Q}}$-variation of the integrand is a total derivative by (3).
Furthermore, the integrand is a closed $p$-form in
${\mathbbmtt{Q}}$-cohomology, so the ${\mathbbmtt{Q}}$/̄cohomology class of
${\bf{\Sigma}}[{\mathcal{O}}]$ only depends on the homology class of
${\Sigma}$. These topological properties of descent operators follow from

$\displaystyle Q_{\mu}\circ Q_{\nu}=0,\qquad\forall\ {\mu},{\nu},$ (5)

which is an additional property of ${\mathbbmtt{Q}}$ in a TFT.

### Generalized descent.

Recast topological descent as an iterative process, where, starting with a
perhaps extended operator ${\mathscr{E}}$ in ${\mathbbmtt{Q}}$-cohomology, we
define its descent with respect to the exact generator $n\cdot P\doteq
n^{\mu}P_{\mu}$, where $n^{\mu}$ is a unit vector, as the new operator
extended in the $n^{\mu}$ direction by an operation

$\displaystyle{\mathbf{D}}^{n\cdot P}[{\mathscr{E}}]\doteq\int
d{\alpha}\,(n\cdot Q)\circ e^{{\alpha}\,n\cdot P}{\mathscr{E}}.$ (6)

That ${\mathbf{D}}^{n\cdot P}[{\mathscr{E}}]$ is in
${\mathbbmtt{Q}}$-cohomology and that $[{\mathbf{D}}^{n\cdot
P},{\mathbf{D}}^{m\cdot P}]=0$ follow from (5).

Let us generalize this construction of extended operators in cohomology to any
nilpotent supercharge ${\mathbbmtt{T}}$ in a superconformal algebra. For any
${\mathbbmtt{T}}$-exact real bosonic generator $X$, there is a supercharge
${\xi}$ such that

$\displaystyle X={\mathbbmtt{T}}\circ{\xi}.$ (7)

Define the $X$-descent of any (perhaps extended) ${\mathbbmtt{T}}$-closed
operator ${\mathscr{E}}$ as

$\displaystyle{\mathbf{D}}^{X}[{\mathscr{E}}]\doteq\int\\!d{\alpha}\,{\xi}\circ
e^{{\alpha}X}\circ{\mathscr{E}},$ (8)

where the integral is over the real line or a circle if the 1-parameter
subgroup generated by $X$ is non-compact or compact, respectively. In all the
cases of interest to us it will be non-compact, so from now on $\int
d{\alpha}=\int_{-\infty}^{\infty}d{\alpha}$. Since $X$ is real,
$e^{{\alpha}X}\circ{\mathscr{E}}$ is a conformally transformed operator
supported at the conformal transform of its original support. Then
${\mathbf{D}}^{X}[{\mathscr{E}}]$ is ${\mathbbmtt{T}}$-closed up to boundary
terms,

$\displaystyle{\mathbbmtt{T}}\circ{\mathbf{D}}^{X}[{\mathscr{E}}]$
$\displaystyle=\int\\!d{\alpha}\,{\mathbbmtt{T}}\circ{\xi}\circ
e^{{\alpha}X}\circ{\mathscr{E}}=\int
d{\alpha}\left\\{({\mathbbmtt{T}}\circ{\xi})\circ
e^{{\alpha}X}\circ{\mathscr{E}}-{\xi}\circ
e^{{\alpha}X}\circ(e^{-{\alpha}X}\circ{\mathbbmtt{T}})\circ{\mathscr{E}}\right\\}$
$\displaystyle=\int d{\alpha}\left\\{X\circ
e^{{\alpha}X}\circ{\mathscr{E}}-{\xi}\circ
e^{{\alpha}X}\circ{\mathbbmtt{T}}\circ{\mathscr{E}}\right\\}=\int
d{\alpha}\frac{d}{d{\alpha}}\left\\{e^{{\alpha}X}\circ{\mathscr{E}}\right\\}$
$\displaystyle=0\qquad\text{if}\qquad\lim_{{\alpha}\to\pm\infty}e^{{\alpha}X}\circ{\mathscr{E}}=0.$
(9)

Using the ${\mathbbmtt{T}}$-closedness of $X$ and ${\mathscr{E}}$, it is then
easy to see that the ${\mathbbmtt{T}}$-cohomology class of
${\mathbf{D}}^{X}[{\mathscr{E}}]$ does not depend on the specific
representative ${\xi}$ chosen to solve (7).

We test whether
“$\lim_{{\alpha}\to\pm\infty}e^{{\alpha}X}\circ{\mathscr{E}}=0$” by whether
the left side gives zero when inserted in an any correlation function. In case
the ${\mathscr{E}}$ are integrals of operators of sufficiently positive
conformal dimension, and to the extent that
$\lim_{{\alpha}\to\pm\infty}e^{{\alpha}X}\circ{\mathscr{E}}$ moves the
operator support to space-time infinity, then convergence in correlators is
ensured in euclidean space. But it is not immediately clear that these are
sufficient conditions for ${\mathbf{D}}^{X}[{\mathscr{E}}]$ to be well-defined
and in ${\mathbbmtt{T}}$-cohomology in Minkowski space-time because of
divergences in integrated correlators for distant light-like separated
insertions. We discuss these issues in some detail in section 3 below.

If ${\mathcal{O}}$ is a point operator, say at the origin, then
${\mathbf{D}}^{X}[{\mathcal{O}}]$ is a line operator if $e^{{\alpha}X}$ moves
the origin; otherwise it will be a new point operator. Similarly, when
${\mathscr{E}}$ is an extended operator localized on some $p$-dimensional
variety ${\Sigma}$, ${\mathbf{D}}^{X}[{\mathscr{E}}]$ is either a
$(p+1)$-dimensional extended operator if $e^{{\alpha}X}$ does not preserve
${\Sigma}$ as a set, or is another $p$-dimensional extended operator if it
does. In our case, descents which preserve ${\Sigma}$ all either vanish or
diverge, so only descents which increase the dimensionality of the operator
support are interesting. Note that the analog of the relations (5) which
ensure the topological nature of the dependence of descent operators on their
${\Sigma}$ world volumes in TFTs need no longer hold in the case of
generalized descent.

The number and nature of the resulting ${\mathbbmtt{T}}$-cohomology classes of
descent operators depends on the details of the algebra of the real bosonic
${\mathbbmtt{T}}$-exact generators.

### Twisted Schur cohomology of the superconformal algebra.

We now specialize to descent with respect to the pair ${\mathbbmtt{T}}_{\pm}$
of twisted Schur supercharges defined in (1). Unitarity ensures that point
operators in ${\mathbbmtt{T}}_{+}$-cohomology are automatically also in
${\mathbbmtt{T}}_{-}$-cohomology; these are the Schur operators of the SCFT.
It is a non-obvious fact that this continues to hold for the extended
operators constructed from the Schur operators by ${\mathbbmtt{T}}_{\pm}$
descent. Thus, if ${\mathcal{O}}$ is a Schur, and
${\mathbf{D}}^{+}[{\mathcal{O}}]$ is a descent operator by some
${\mathbbmtt{T}}_{+}$-exact generator, then by (2)
${\mathbf{D}}^{+}[{\mathcal{O}}]$ will be ${\mathbbmtt{T}}_{+}$-closed (up to
boundary terms which must, of course, be shown to vanish). The non-obvious
fact is that ${\mathbf{D}}^{+}[{\mathcal{O}}]$ is also
${\mathbbmtt{T}}_{-}$-closed, even though it is not found by descent with
respect to any ${\mathbbmtt{T}}_{-}$-exact generator.

To show this and other special properties of twisted Schur descent, we compute
the subalgebras of ${\mathbbmtt{T}}_{\pm}$-closed or exact bosonic generators
of the ${\mathcal{N}}{=}2$ superconformal algebra. First, it is convenient to
fix a space-time coordinate system. Let $x^{\mu}$, ${\mu}~{}=~{}1,2,3,4$, be
the usual flat Minkowski coordinates with $x^{1}$ the time, and define
lightcone coordinates in the $x^{1}$-$x^{2}$ and complex coordinates in the
$x^{3}$-$x^{4}$ planes by

$\displaystyle x^{\pm}$ $\displaystyle\doteq\pm x^{1}+x^{2},$ $\displaystyle
z$ $\displaystyle\doteq x^{3}-ix^{4}.$ (10)

Points at $x^{\pm}=0$ with coordinates $(z,{\overline{z}})$ comprise the “VOA
plane”. The tensor generators of the conformal algebra, $P_{\mu}$,
$M_{{\mu}{\nu}}$, and $K_{\mu}$, are related to their spinor components
by333This differs slightly from the convention in Lemos_2020  because we are
working in Minkowski space-time.

$\displaystyle\begin{pmatrix}P_{1\dot{1}}&P_{1\dot{2}}\\\
P_{2\dot{1}}&P_{2\dot{2}}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}2P_{z}&-2iP_{-}\\\
2iP_{+}&-2P_{\overline{z}}\end{pmatrix},$
$\displaystyle\begin{pmatrix}K^{\dot{1}1}&K^{\dot{1}2}\\\
K^{\dot{2}1}&K^{\dot{2}2}\end{pmatrix}$
$\displaystyle=2\begin{pmatrix}K_{\overline{z}}&-iK_{-}\\\
iK_{+}&-K_{z}\end{pmatrix},$ (11)
$\displaystyle\begin{pmatrix}{\mathcal{M}}_{1}^{\ 1}&{\mathcal{M}}_{1}^{\
2}\\\ {\mathcal{M}}_{2}^{\ 1}&{\mathcal{M}}_{2}^{\ 2}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}-{\mathcal{M}}_{2}^{\ 2}&2iM_{-z}\\\
2iM_{+{\overline{z}}}&M_{+-}{-}M_{z{\overline{z}}}\end{pmatrix},$
$\displaystyle\begin{pmatrix}{\widetilde{\mathcal{M}}}^{\dot{1}}_{\
\dot{1}}&{\widetilde{\mathcal{M}}}^{\dot{1}}_{\ \dot{2}}\\\
{\widetilde{\mathcal{M}}}^{\dot{2}}_{\
\dot{1}}&{\widetilde{\mathcal{M}}}^{\dot{2}}_{\ \dot{2}}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}-{\widetilde{\mathcal{M}}}^{\dot{2}}_{\
\dot{2}}&2iM_{-{\overline{z}}}\\\
2iM_{+z}&M_{+-}{+}M_{z{\overline{z}}}\end{pmatrix}.$

Denote the subalgebras of the ${\mathcal{N}}{=}2$ superconformal algebra
${\mathfrak{S}}$ which are closed and exact under twisted Schur supercharges
${\mathbbmtt{T}}_{\pm}$ by ${\mathfrak{Z}}_{\pm}\doteq\\{X\in{\mathfrak{S}}\
|\ {\mathbbmtt{T}}_{\pm}\circ X=0\\}$ and
${\mathfrak{B}}_{\pm}\doteq\\{X\in{\mathfrak{S}}\ |\
X={\mathbbmtt{T}}_{\pm}\circ Y\text{ for some }Y\in{\mathfrak{S}}\\}$,
respectively. Then the even (bosonic) subalgebras are

$\displaystyle{\mathfrak{Z}}_{\pm}$
$\displaystyle=\mathfrak{sl}_{2}\oplus\widehat{\mathfrak{sl}}_{2}\oplus
Z\niplus{\mathfrak{m}}_{\pm},$ $\displaystyle{\mathfrak{B}}_{\pm}$
$\displaystyle=\widehat{\mathfrak{sl}}_{2}\oplus
Z\niplus{\mathfrak{m}}_{\pm},$ (12)

where

$\displaystyle\mathfrak{sl}_{2}$ $\displaystyle\doteq\big{\langle}\
L_{-1}\doteq P_{z},\qquad\quad
L_{0}\doteq\tfrac{1}{2}D+M_{z{\overline{z}}},\qquad\quad L_{1}\doteq
K_{\overline{z}}\ \big{\rangle},$ (13)
$\displaystyle\widehat{\mathfrak{sl}}_{2}$ $\displaystyle\doteq\big{\langle}\
{\widehat{L}}_{-1}\doteq P_{\overline{z}}-R^{-},\
{\widehat{L}}_{0}\doteq\tfrac{1}{2}D-M_{z{\overline{z}}}-\tfrac{1}{2}R,\
{\widehat{L}}_{1}\doteq K_{z}+R^{+}\ \big{\rangle},$
$\displaystyle{\mathcal{Z}}$ $\displaystyle\doteq\big{\langle}\
{\mathcal{Z}}\doteq 2M_{+-}+\tfrac{1}{2}r\ \big{\rangle},$
$\displaystyle{\mathfrak{m}}_{+}$ $\displaystyle\doteq\big{\langle}\ P_{+},\
M_{+z},\ M_{+{\overline{z}}},\ K_{+}\
\big{\rangle},\qquad{\mathfrak{m}}_{-}\doteq\big{\langle}\ P_{-},\ M_{-z},\
M_{-{\overline{z}}},\ K_{-}\ \big{\rangle}.$

Here $r$ is the ${\rm U}(1)_{r}$ generator and $R$, $R^{\pm}$ are ${\rm
SU}(2)_{R}$ generators normalized as in Cordova_2016  so that $r(Q)=-1$ and
$R\in\mathbb{Z}$.444This differs from Lemos_2020  only for the ${\rm
SU}(2)_{R}$ Cartan generator, which is $R\doteq R^{1}_{\ 1}-R^{2}_{\ 2}$ for
us. $\mathfrak{sl}_{2}\oplus\widehat{\mathfrak{sl}}_{2}$ is the algebra of
holomorphic plus antiholomorphic conformal transformations of the VOA plane,
with the antiholomorphic half twisted by ${\rm SU}(2)_{R}$ rotations.
${\mathcal{Z}}$ generates boosts in the $x^{\pm}$ plane combined with a ${\rm
U}(1)_{r}$ rotation.555Our ${\mathcal{Z}}$ is the negative of the
${\mathcal{Z}}$ of Beem_2015 ; Lemos_2020 . Finally, ${\mathfrak{m}}_{\pm}$
are nilpotent ideals of the subalgebras in (12),

$\displaystyle(\mathfrak{sl}_{2}\oplus\widehat{\mathfrak{sl}}_{2}\oplus{\mathcal{Z}})\circ{\mathfrak{m}}_{\pm}\subset{\mathfrak{m}}_{\pm}.$
(14)

The ${\mathfrak{m}}_{\pm}$ ideals play a central role in twisted Schur
descent.

### The algebra of the twisted Schur descent operation.

${\mathbbmtt{T}}_{\pm}$ descent can be performed with respect to
${\mathbbmtt{T}}_{\pm}$-exact generators with a real action in space-time.
From (13) these are real combinations of ${\mathcal{Z}}$ and
${\mathfrak{m}}_{\pm}$ for ${\mathbbmtt{T}}_{\pm}$, respectively. Though the
$\widehat{\mathfrak{sl}}_{2}$ generators are ${\mathbbmtt{T}}_{\pm}$-exact,
they all have complex action on space-time; real combinations with the
$\mathfrak{sl}_{2}$ generators are necessary to generate a real action on
space-time. But since the $\mathfrak{sl}_{2}$ generators are not exact, they
are not suitable for performing twisted Schur descent. We discuss the effect
of real $\mathfrak{sl}_{2}\oplus\widehat{\mathfrak{sl}}_{2}$ generators on
descent operators below.

We now show that twisted Schur descent starting from a primary Schur operator
${\mathcal{O}}$ by a sequence of ${\mathbbmtt{T}}_{+}$ and/or
${\mathbbmtt{T}}_{-}$-exact generators produces a series of extended operators
${\mathscr{E}}$ which share many of the same properties as their Schur
“parent” ${\mathcal{O}}$. Schur operators at the origin satisfy

$\displaystyle{\mathbbmtt{T}}_{\pm}\circ{\mathcal{O}}(0)$ $\displaystyle=0,$
$\displaystyle{\widehat{L}}_{0}\circ{\mathcal{O}}(0)$
$\displaystyle={\mathcal{Z}}\circ{\mathcal{O}}(0)=0,$ (15)

which from (12) immediately imply
$\widehat{\mathfrak{sl}}_{2}\circ{\mathcal{O}}(0)=_{\mathbbmtt{T}}0$, where
“$=_{\mathbbmtt{T}}$” means the equality holds in both ${\mathbbmtt{T}}_{+}$
and ${\mathbbmtt{T}}_{-}$ cohomologies. Schur primaries at the origin
additionally satisfy

$\displaystyle L_{-1}\circ{\mathcal{O}}(0)$
$\displaystyle={\partial}_{z}{\mathcal{O}}(0),$ $\displaystyle
L_{0}\circ{\mathcal{O}}(0)$ $\displaystyle=h_{\mathcal{O}}{\mathcal{O}}(0),$
$\displaystyle L_{1}\circ{\mathcal{O}}(0)$ $\displaystyle=0,$ (16)

where the chiral weights are positive half-integers,
$h_{\mathcal{O}}\in\mathbb{N}/2$.

We will show that extended operators ${\mathscr{E}}$ formed by twisted Schur
descent from a Schur primary ${\mathcal{O}}$ at the origin are automatically
in _both_ ${\mathbbmtt{T}}_{+}$ and ${\mathbbmtt{T}}_{-}$-cohomology, and
satisfy

$\displaystyle{\widehat{L}}_{0}\circ{\mathscr{E}}={\mathcal{Z}}\circ{\mathscr{E}}$
$\displaystyle=_{\mathbbmtt{T}}0,$ $\displaystyle L_{-1}\circ{\mathscr{E}}$
$\displaystyle=_{\mathbbmtt{T}}{\partial}_{z}{\mathscr{E}},$ $\displaystyle
L_{0}\circ{\mathscr{E}}$
$\displaystyle=_{\mathbbmtt{T}}h_{\mathcal{O}}{\mathscr{E}},$ $\displaystyle
L_{1}\circ{\mathscr{E}}$ $\displaystyle=_{\mathbbmtt{T}}0.$ (17)

Let $N_{\pm}\in{\mathfrak{m}}_{\pm}$ be any real vector. Then, since $N_{\pm}$
is ${\mathbbmtt{T}}_{\pm}$-exact, there is a fermionic super generator
${\nu}_{\pm}$ such that

$\displaystyle{\mathbbmtt{T}}_{\pm}\circ{\nu}_{\pm}=N_{\pm}.$ (18)

One then checks that

$\displaystyle{\mathbbmtt{T}}_{\mp}\circ{\nu}_{\pm}$ $\displaystyle=0,$
$\displaystyle{\mathbbmtt{T}}_{\mp}\circ N_{\pm}$
$\displaystyle=\pm{\nu}_{\pm},$ (19)

which are related to

$\displaystyle{\mathbbmtt{T}}_{\pm}\circ{\mathbbmtt{T}}_{\mp}$
$\displaystyle={\mathcal{Z}},$ $\displaystyle{\mathcal{Z}}\circ N_{\pm}$
$\displaystyle=\pm N_{\pm},$ $\displaystyle{\mathcal{Z}}\circ{\nu}_{\pm}$
$\displaystyle=\pm{\nu}_{\pm},$ (20)

and that

$\displaystyle{\nu}_{\pm}\circ{\nu}_{\pm}$ $\displaystyle=0,$ $\displaystyle
N_{\pm}\circ{\nu}_{\pm}$ $\displaystyle=0.$ (21)

Given the definition (18), the relations (19), (20), and (21) follow from the
detailed form of the ${\mathcal{N}}{=}2$ superconformal algebra, i.e., they do
not follow in any obvious way from the automorphisms of that algebra (though
half of them follow from the other half by virtue of the anti-linear
involution which exchanges
${\mathbbmtt{T}}_{+}\leftrightarrow{\mathbbmtt{T}}_{-}$).

Denote by ${\bm{\mu}}_{\pm}$ the fermionic subalgebras satisfying
${\mathbbmtt{T}}_{\pm}\circ{\bm{\mu}}_{\pm}={\mathfrak{m}}_{\pm}$. These
algebras are nilpotent,

$\displaystyle{\mathfrak{m}}_{\pm}\circ{\mathfrak{m}}_{\pm}={\mathfrak{m}}_{\pm}\circ{\bm{\mu}}_{\pm}={\bm{\mu}}_{\pm}\circ{\bm{\mu}}_{\pm}=0,$
(22)

(signs correlated), generalizing (21). An immediate consequence is that

$\displaystyle{\mathbf{D}}^{N_{\pm}}{\mathbf{D}}^{N^{\prime}_{\pm}}[{\mathscr{E}}]=-{\mathbf{D}}^{N^{\prime}_{\pm}}{\mathbf{D}}^{N_{\pm}}[{\mathscr{E}}],$
(23)

which follows from the definition of descent (8).

Assume we have a descent operator ${\mathscr{E}}$ in both
${\mathbbmtt{T}}_{\pm}$ cohomologies and satisfying (17). We will show that
the most general ${\mathbbmtt{T}}_{+}$ descent operation acting on
${\mathscr{E}}$ produces a new operator, ${\mathbf{D}}^{X}[{\mathscr{E}}]$ as
in (8), which is also in both ${\mathbbmtt{T}}_{\pm}$ cohomologies and also
satisfies (17). (The same argument works for ${\mathbbmtt{T}}_{-}$ descent.)
We can choose as the descent generator $X$ any real
${\mathbbmtt{T}}_{+}$-exact generator, which, up to an overall irrelevant
rescaling, therefore has one of the three possible forms ${\mathcal{Z}}$,
${\mathcal{Z}}+N_{+}$, or $N_{+}$, for some non-zero
$N_{+}\in{\mathfrak{m}}_{+}$. First,

$\displaystyle{\mathbf{D}}^{\mathcal{Z}}[{\mathscr{E}}]$
$\displaystyle=\int\\!\\!d{\alpha}\,{\mathbbmtt{T}}_{-}\circ
e^{{\alpha}{\mathcal{Z}}}\circ{\mathscr{E}}=\int\\!\\!d{\alpha}\,{\mathbbmtt{T}}_{-}\circ{\mathscr{E}}=0,$
(24)

where we used (20), the induction hypotheses (17), and that ${\mathscr{E}}$ is
${\mathbbmtt{T}}_{-}$-closed. So descent by ${\mathcal{Z}}$ gives zero. Next,

$\displaystyle{\mathbf{D}}^{{\mathcal{Z}}+N_{+}}[{\mathscr{E}}]$
$\displaystyle=\int_{-\infty}^{+\infty}\\!\\!d{\alpha}\,({\mathbbmtt{T}}_{-}+{\nu}_{+})\circ
e^{{\alpha}({\mathcal{Z}}+N_{+})}\circ{\mathscr{E}}=\int_{-1}^{+\infty}\\!\\!dy\,{\nu}_{+}\circ
e^{yN_{+}}\circ{\mathscr{E}},$ (25)

using that
$e^{{\alpha}({\mathcal{Z}}+N_{+})}=e^{yN_{+}}e^{{\alpha}{\mathcal{Z}}}$ with
$y=e^{\alpha}-1$ which follows from ${\mathcal{Z}}\circ N_{+}=N_{+}$, and we
also used that
$e^{-yN_{+}}\circ{\mathbbmtt{T}}_{-}={\mathbbmtt{T}}_{-}+y{\nu}_{+}$ which
follows from ${\mathbbmtt{T}}_{-}\circ N_{+}={\nu}_{+}$ and ${\nu}_{+}\circ
N_{+}=0$. The final expression in (25) is like $N_{+}$ descent except with a
boundary at finite points in space-time. We therefore expect it not to be
${\mathbbmtt{T}}_{+}$-closed. Indeed, by the same argument as in (2),
${\mathbbmtt{T}}_{+}\circ{\mathbf{D}}^{{\mathcal{Z}}+N_{+}}[{\mathscr{E}}]=e^{yN_{+}}\circ{\mathscr{E}}|^{y=\infty}_{y=-1}\neq
0$. So descent by ${\mathcal{Z}}+N_{+}$ fails to give an operator in
cohomology.

We thus only ever need to consider descent operators of the form

$\displaystyle{\mathbf{D}}^{N_{+}}[{\mathscr{E}}]$
$\displaystyle=\int\\!\\!d{\alpha}\,{\nu}_{+}\circ
e^{{\alpha}N_{+}}\circ{\mathscr{E}},$ $\displaystyle N_{+}$
$\displaystyle\in{\mathfrak{m}}_{+}.$ (26)

A similar formula defines ${\mathbf{D}}^{N_{-}}[{\mathscr{E}}]$ for
$N_{-}\in{\mathfrak{m}}_{-}$.

We now show ${\mathbf{D}}^{N_{+}}[{\mathscr{E}}]$ is in both
${\mathbbmtt{T}}_{\pm}$ cohomologies. It is in
${\mathbbmtt{T}}_{+}$-cohomology by the generalized descent construction up to
boundary terms as in (2). These boundary terms need to be evaluated for
particular choices of $N_{+}$ and ${\mathscr{E}}$. Assume for the moment that
these boundary terms vanish. Then we compute

$\displaystyle{\mathbbmtt{T}}_{-}\circ{\mathbf{D}}^{N_{+}}[{\mathscr{E}}]$
$\displaystyle=-\int_{-\infty}^{+\infty}\\!\\!d{\alpha}\,{\nu}_{+}\circ
e^{{\alpha}N_{+}}\circ({\mathbbmtt{T}}_{-}+{\alpha}{\nu}_{+})\circ{\mathscr{E}}=0,$
(27)

where in the first step we used the descent algebra identities (18) and (19)
to commute ${\mathbbmtt{T}}_{-}$ to the right, and the vanishing follows from
the induction hypothesis that ${\mathscr{E}}$ is ${\mathbbmtt{T}}_{-}$-closed
and from the nilpotency relations (21). The identity (27) — which does not
have an analog in topological descent — is crucial in allowing twisted Schur
descent to extend the vertex operator algebra of twist-translated Schur
operators.

Note that the ${\rm U}(1)_{r}$ charges of the twisted Schur supercharges are
$r({\mathbbmtt{T}}_{\pm})=\mp 1$, so by (18), $r({\nu}_{\pm})=\pm 1$ since
bosonic generators have zero $r$-charge. Then it follows from (26) that

$\displaystyle r({\mathbf{D}}^{N_{\pm}}[{\mathscr{E}}])=r({\mathscr{E}})\pm
1.$ (28)

Note also that, using the relation ${\nu}_{+}={\mathbbmtt{T}}_{-}\circ N_{+}$
in the definition of ${\mathbf{D}}^{N_{+}}[{\mathscr{E}}]$ and commuting a
${\mathbbmtt{T}}_{-}$ to the right to annihilate ${\mathscr{E}}$, one derives
an identity of the form
${\mathbf{D}}^{N_{+}}[{\mathscr{E}}]={\mathbf{D}}^{N_{+}}[{\mathscr{E}}]+$boundary
terms. So these boundary terms must vanish if
${\mathbf{D}}^{N_{+}}[{\mathscr{E}}]$ is well-defined, giving

$\displaystyle{\alpha}{\nu}_{+}\circ
e^{{\alpha}N_{+}}\circ{\mathscr{E}}\big{|}^{{\alpha}=+\infty}_{{\alpha}=-\infty}$
$\displaystyle={\mathbbmtt{T}}_{-}\circ\left(e^{{\alpha}N_{+}}\circ{\mathscr{E}}\big{|}^{{\alpha}=+\infty}_{{\alpha}=-\infty}\right).$
(29)

Since the boundary terms on the right vanish by (2) for
${\mathbf{D}}^{N_{+}}[{\mathscr{E}}]$ to be ${\mathbbmtt{T}}_{+}$-closed, it
follows that the apparently less convergent boundary terms on the left must
also vanish.

We now show that ${\mathbf{D}}^{N_{+}}[{\mathscr{E}}]$ satisfies the induction
hypotheses (17) if ${\mathscr{E}}$ does. It immediately follows that
${\mathcal{Z}}\circ{\mathbf{D}}^{N_{+}}[{\mathscr{E}}]=\widehat{\mathfrak{sl}}_{2}\circ{\mathbf{D}}^{N_{+}}[{\mathscr{E}}]=_{\mathbbmtt{T}}0$
since ${\mathcal{Z}}$ and $\widehat{\mathfrak{sl}}_{2}$ are both
${\mathbbmtt{T}}_{\pm}$-exact. To show the
$L_{0}\circ{\mathbf{D}}^{N_{+}}[{\mathscr{E}}]$ and
$L_{1}\circ{\mathbf{D}}^{N_{+}}[{\mathscr{E}}]$ relations we need the action
of $\mathfrak{sl}_{2}$ on the ${\mathfrak{m}}_{\pm}$ and ${\bm{\mu}}_{\pm}$
subalgebras. Because $\mathfrak{sl}_{2}$ is ${\mathbbmtt{T}}_{\pm}$-closed, if
$N_{\pm}\in{\mathfrak{m}}_{\pm}$ and ${\nu}_{\pm}\in{\bm{\mu}}_{\pm}$ are
related by ${\mathbbmtt{T}}_{\pm}\circ{\nu}_{\pm}=N_{\pm}$, then for any
element $X\in\mathfrak{sl}_{2}$, $N^{\prime}_{\pm}\doteq X\circ N_{\pm}$ and
${\nu}^{\prime}_{\pm}\doteq X\circ{\nu}_{\pm}$ are related in the same way.

Consider $X\in\mathfrak{sl}_{2}$. By commuting $X$ to the right one finds666We
are violating our associativity convention in writing the last term which is
meant to be $N^{\prime}_{+}\circ{\nu}_{+}\circ
e^{{\alpha}N_{+}}\circ{\mathscr{E}}$, and not
$(N^{\prime}_{+}\circ{\nu}_{+})\circ e^{{\alpha}N_{+}}\circ{\mathscr{E}}$
which vanishes identically.

$\displaystyle X\circ{\mathbf{D}}^{N_{+}}[{\mathscr{E}}]$
$\displaystyle=_{\mathbbmtt{T}}{\mathbf{D}}^{N_{+}}[X\circ{\mathscr{E}}]+\int\\!\\!d{\alpha}({\nu}^{\prime}_{+}{+}{\alpha}N^{\prime}_{+}{\circ}{\nu}_{+}){\circ}e^{{\alpha}N_{+}}{\circ}{\mathscr{E}}.$
(30)

Using ${\nu}^{\prime}_{+}={\mathbbmtt{T}}_{-}\circ N^{\prime}_{+}$, the
integrated term in (30) is

$\displaystyle\int\\!\\!d{\alpha}\cdots$
$\displaystyle=_{{\mathbbmtt{T}}_{-}}\int\\!\\!d{\alpha}N^{\prime}_{+}{\circ}(-{\mathbbmtt{T}}_{-}{+}{\alpha}N^{\prime}_{+}{\circ}{\nu}_{+}){\circ}e^{{\alpha}N_{+}}{\circ}{\mathscr{E}}=-\int\\!\\!d{\alpha}N^{\prime}_{+}{\circ}e^{{\alpha}N_{+}}{\circ}({\mathbbmtt{T}}_{-}{+}{\alpha}{\nu}_{+}{-}{\alpha}{\nu}_{+}){\circ}{\mathscr{E}}$

which vanishes in ${\mathbbmtt{T}}_{-}$-cohomology. Similarly, using
$N^{\prime}_{+}={\mathbbmtt{T}}_{+}\circ{\nu}^{\prime}_{+}$,
$N_{+}={\mathbbmtt{T}}_{+}\circ{\nu}_{+}$, and ${\mathbbmtt{T}}_{+}\circ
e^{{\alpha}N_{+}}\circ{\mathscr{E}}=0$, this term is, in
${\mathbbmtt{T}}_{+}$-cohomology, the boundary term

$\displaystyle\int\\!\\!d{\alpha}\cdots$
$\displaystyle=_{{\mathbbmtt{T}}_{+}}\int\\!\\!d{\alpha}{\nu}^{\prime}_{+}{\circ}(1{+}{\alpha}N_{+}){\circ}e^{{\alpha}N_{+}}{\circ}{\mathscr{E}}=\int\\!\\!d{\alpha}\frac{d}{d{\alpha}}\left\\{{\alpha}{\nu}^{\prime}_{+}{\circ}e^{{\alpha}N_{+}}{\circ}{\mathscr{E}}\right\\}={\alpha}{\nu}^{\prime}_{+}{\circ}e^{{\alpha}N_{+}}{\circ}{\mathscr{E}}\Big{|}^{+\infty}_{-\infty}.$

This boundary term is closely related to the boundary term in (29) that
vanishes for ${\mathbf{D}}^{N_{+}}[{\mathscr{E}}]$ to be a well-defined
operator in ${\mathbbmtt{T}}_{\pm}$-cohomology, making it plausible that it,
too, vanishes. But we do not have a general argument proving this, and resort
to checking it on a case-by-case basis.

The upshot is that, by taking $X\in\\{L_{-1},L_{0},L_{1}\\}$ in (30), we have
shown that the ${\mathbf{D}}^{N_{+}}[{\mathscr{E}}]$ descent operator
satisfies (17) with ${\mathbf{D}}^{N_{+}}[{\mathscr{E}}]$ in place of
${\mathscr{E}}$.

### Cohomological equivalence of descent operators.

We have now reduced each step of the twisted Schur descent procedure (26) to
choosing an element $N_{+}\in{\mathfrak{m}}_{+}$ or an
$N_{-}\in{\mathfrak{m}}_{-}$ with which to perform descent. Since
${\mathfrak{m}}_{\pm}$ are 4-dimensional vector spaces, there is an
$\mathbb{R}\mathbb{P}^{3}$ of inequivalent choices of $N_{\pm}$, since the
overall scale of $N_{\pm}$ can be absorbed into the integration measure of the
descent operation. We will now show that almost all these choices are, in
fact, equivalent in ${\mathbbmtt{T}}_{\pm}$-cohomology, and only a finite set
of choices of $N_{\pm}\in{\mathfrak{m}}_{\pm}$ needs to be considered.

The key is to consider the action of the $\widehat{\rm SO}(2,2)$ group of
twisted conformal transformations of the VOA plane on descent operators.
$\widehat{\rm SO}(2,2)$ is generated by elements of
$\mathfrak{sl}_{2}\oplus\widehat{\mathfrak{sl}}_{2}$ with real space-time
action. “Twisted” refers to the accompanying ${\rm SU}(2)_{R}$ transformation
that is part of $\widehat{\mathfrak{sl}}_{2}$. To this end, define the
superconformal group elements

$\displaystyle{\widehat{g}}(z)$
$\displaystyle\doteq\exp\\{zL_{-1}+{\overline{z}}{\widehat{L}}_{-1}\\},$
$\displaystyle\text{twist translations},$ $\displaystyle{\widehat{g}}(\ell)$
$\displaystyle\doteq\exp\\{\ell L_{0}+{\overline{\ell}}{\widehat{L}}_{0}\\},$
$\displaystyle\text{twist rotation and dilatation},$ (31)
$\displaystyle{\widehat{g}}({\beta})$
$\displaystyle\doteq\exp\\{{\beta}L_{1}+{\overline{\beta}}{\widehat{L}}_{1}\\},$
$\displaystyle\text{twist special conformal transformations},$

for $z$, $\ell$, and ${\beta}$ arbitrary complex numbers, which act on the VOA
plane as indicated above. Consider, say,
${\widehat{g}}({\beta})\circ{\mathscr{E}}=e^{{\beta}L_{1}}e^{{\overline{\beta}}{\widehat{L}}_{1}}\circ{\mathscr{E}}$.
Since ${\mathscr{E}}$ is ${\mathbbmtt{T}}_{\pm}$-closed and
${\widehat{L}}_{1}$ is ${\mathbbmtt{T}}_{\pm}$-exact, we have
${\widehat{g}}({\beta})\circ{\mathscr{E}}=_{\mathbbmtt{T}}e^{{\beta}L_{1}}\circ{\mathscr{E}}$.
Furthermore, since $L_{1}\circ{\mathscr{E}}=_{\mathbbmtt{T}}0$ if
${\mathscr{E}}$ is built from a Schur primary, we have
${\widehat{g}}({\beta})\circ{\mathscr{E}}=_{\mathbbmtt{T}}{\mathscr{E}}$. A
similar argument for ${\widehat{g}}(\ell)$ and ${\widehat{g}}(z)$ implies

$\displaystyle{\widehat{g}}({\beta})\circ{\mathscr{E}}$
$\displaystyle=_{\mathbbmtt{T}}{\mathscr{E}},$
$\displaystyle{\widehat{g}}(\ell)\circ{\mathscr{E}}$
$\displaystyle=_{\mathbbmtt{T}}e^{\ell h_{\mathcal{O}}}{\mathscr{E}},$
$\displaystyle{\widehat{g}}(z)\circ{\mathscr{E}}$
$\displaystyle=_{\mathbbmtt{T}}e^{zP_{z}}\circ{\mathscr{E}},$ (32)

for ${\mathscr{E}}$ a Schur primary descent. The ${\widehat{g}}(z)$ relation
shows, just as for Schur operators, that their twisted Schur cohomology
classes only depend on the holomorphic $z$ coordinate on the VOA plane. The
${\widehat{g}}(\ell)$ and ${\widehat{g}}({\beta})$ relations are much
stronger, implying essentially that their cohomology classes are independent
of both $\ell$ and ${\overline{\ell}}$ as well as ${\beta}$ and
${\overline{\beta}}$.

Since

$\displaystyle{\widehat{g}}\circ{\mathbf{D}}^{N_{+}}[{\mathscr{E}}]$
$\displaystyle=\int\\!\\!d{\alpha}\,({\widehat{g}}\circ{\nu}_{+})\circ
e^{{\alpha}\,{\widehat{g}}\circ
N_{+}}\circ({\widehat{g}}\circ{\mathscr{E}})={\mathbf{D}}^{{\widehat{g}}\circ
N_{+}}[{\widehat{g}}\circ{\mathscr{E}}],$ (33)

where we have used that ${\mathfrak{m}}_{\pm}$ are ideals (14) of the
superconformal algebra cohomology subalgebras, the relations (32) imply

$\displaystyle{\mathbf{D}}^{N_{+}}[{\mathscr{E}}]$
$\displaystyle=_{\mathbbmtt{T}}{\mathbf{D}}^{{\widehat{g}}({\beta})\circ
N_{+}}[{\mathscr{E}}],$ $\displaystyle{\mathbf{D}}^{N_{+}}[{\mathscr{E}}]$
$\displaystyle=_{\mathbbmtt{T}}{\mathbf{D}}^{{\widehat{g}}(\ell)\circ
N_{+}}[{\mathscr{E}}],$ (34) $\displaystyle
e^{zP_{z}}\circ{\mathbf{D}}^{N_{+}}[{\mathscr{E}}]$
$\displaystyle=_{\mathbbmtt{T}}{\mathbf{D}}^{{\widehat{g}}(z)\circ
N_{+}}[e^{zP_{z}}\circ{\mathscr{E}}].$

Now compute the ${\widehat{g}}(z),{\widehat{g}}(\ell),{\widehat{g}}({\beta})$
actions on ${\mathfrak{m}}_{\pm}$ explicitly. Parameterize $N_{+}$ as

$\displaystyle{\mathfrak{m}}_{+}\ni N_{+}$ $\displaystyle\doteq
pP_{+}+mM_{+z}+\overline{m}M_{+{\overline{z}}}+kK_{+},$ $\displaystyle p,k$
$\displaystyle\in\mathbb{R},$ $\displaystyle m$ $\displaystyle\in\mathbb{C}.$
(35)

A short calculation gives

$\displaystyle{\widehat{g}}({\beta}):\big{(}p,m,k\big{)}$
$\displaystyle\to\big{(}\ p\,,\,m{+}2p{\overline{\beta}}\,,\,k{-}{\rm
Re}(m{\beta}){-}p{\beta}{\overline{\beta}}\ \big{)},$ $\displaystyle
e^{-(\ell+{\overline{\ell}})/2}\,{\widehat{g}}(\ell):\big{(}p,m,k\big{)}$
$\displaystyle\to\big{(}\
p\,,\,e^{-{\overline{\ell}}}m\,,\,e^{-\ell-{\overline{\ell}}}k\ \big{)},$ (36)
$\displaystyle{\widehat{g}}(z):\big{(}p,m,k\big{)}$ $\displaystyle\to\big{(}\
p{-}{\rm Re}(m{\overline{z}}){-}kz{\overline{z}}\,,\,m{+}2kz\,,\,k\ \big{)},$

where we have used the freedom to rescale by an overall factor in the second
line. One can show that if $p\neq 0$ there is a sequence of
${\widehat{g}}({\beta})$ and ${\widehat{g}}(\ell)$ transformations which takes
$(p,m,k)\to(1,0,0)$ or $(1,0,\pm 1)$. Likewise, if $p=0$ and $m\neq 0$ one can
set $(p,m,k)\to(0,1,0)$, and if $p=m=0$ one can set $(p,m,k)\to(0,0,1)$. There
is a similar result for ${\mathfrak{m}}_{-}$. Thus at any stage there are just
10 possible inequivalent descent operations: ${\mathbf{D}}^{P_{\pm}}$,
${\mathbf{D}}^{M_{\pm 3}}$, ${\mathbf{D}}^{K_{\pm}}$,
${\mathbf{D}}^{P_{\pm}+K_{\pm}}$, and ${\mathbf{D}}^{P_{\pm}-K_{\pm}}$.

### The descent web.

Starting from a Schur primary, ${\mathcal{O}}$, at the origin, we now map out
its possible inequivalent descent operators. Since the conformal
transformations $e^{{\alpha}K_{\pm}}$ and $e^{{\alpha}M_{\pm 3}}$ do not move
the origin, we expect ${\mathbf{D}}^{K_{\pm}}[{\mathcal{O}}]$ and
${\mathbf{D}}^{M_{\pm 3}}[{\mathcal{O}}]$ to diverge. This is because, by
(26), these descents are $\int_{-\infty}^{+\infty}d{\alpha}$ integrals of some
local operators at the origin. Since $K_{\pm}\circ{\mathcal{O}}=0$ for
${\mathcal{O}}$ primary, the local operator is just ${\mathcal{O}}$, so
${\mathbf{D}}^{K_{+}}[{\mathcal{O}}]=(\infty)\cdot{\mathcal{O}}$.
$e^{{\alpha}M_{\pm 3}}\circ{\mathcal{O}}$ is a combination, polynomial in
${\alpha}$, of spin components of the Lorentz multiplet that ${\mathcal{O}}$
belongs to, so is at least as divergent.

Since $P_{\pm}$ translate the origin along the $x^{\pm}$ axes, the
${\mathbf{D}}^{P_{\pm}}[{\mathcal{O}}]$ descent operators are light-like line
operators. Furthermore, correlation functions of these operators converge and
the boundary terms appearing in (2) vanish. We discuss the convergence of
these line operators in more detail in the next section. Furthermore

$\displaystyle{\mathbf{D}}^{P_{+}\pm K_{+}}[{\mathcal{O}}]$
$\displaystyle=\int\\!\\!d{\alpha}(\pi_{+}\pm{\kappa}_{+})\circ
e^{{\alpha}(P_{+}\pm
K_{+})}\circ{\mathcal{O}}=\int\\!\\!d{\alpha}(\pi_{+}\pm{\kappa}_{+})\circ
e^{{\alpha}P_{+}}\circ{\mathcal{O}}$
$\displaystyle=\int\\!\\!d{\alpha}e^{{\alpha}P_{+}}\circ(\pi_{+}\pm{\kappa}_{+})\circ{\mathcal{O}}=\int\\!\\!d{\alpha}e^{{\alpha}P_{+}}\circ\pi_{+}\circ{\mathcal{O}}={\mathbf{D}}^{P_{+}}[{\mathcal{O}}],$
(37)

where ${\mathbbmtt{T}}_{+}\circ\pi_{+}=P_{+}$ and similarly for
${\kappa}_{+}$, and we have used the fact that $K_{+}\circ{\mathcal{O}}=0$ for
a Schur primary, the nilpotency relations (22), and that
${\kappa}_{+}\circ{\mathcal{O}}=0$ for a Schur primary. The last relation
follows from ${\kappa}_{+}={\mathbbmtt{T}}_{-}\circ K_{+}$ by (19), implying
${\kappa}_{+}\circ{\mathcal{O}}=({\mathbbmtt{T}}_{-}\circ
K_{+})\circ{\mathcal{O}}={\mathbbmtt{T}}_{-}\circ
K_{+}\circ{\mathcal{O}}-K_{+}\circ{\mathbbmtt{T}}_{-}\circ{\mathcal{O}}=0$
since a Schur primary is annihilated by $K_{+}$ and by ${\mathbbmtt{T}}_{-}$.
A similar argument holds for $P_{-}\pm K_{-}$ descent. Thus, at the first
level of the descent web, the ony independent convergent descent operators are
two lightlike line operators ${\mathbf{D}}^{P_{\pm}}[{\mathcal{O}}]$.

A similar pattern applies to each successive descent operation
${\mathbf{D}}^{X}[{\mathscr{E}}]$: if $X$ does not translate the world volume
of ${\mathscr{E}}$ then the descent diverges, and if it does, it potentially
generates a convergent extended operator of one greater world volume
dimension; and the $P_{\pm}\pm K_{\pm}$ descent operators coincide with the
$P_{\pm}$ descent operators in ${\mathbbmtt{T}}_{\pm}$ cohomology. Accepting
this for the moment, it is easy to map out the possible descent world volumes.

$e^{{\alpha}P_{\pm}}$ acting on the origin generates the lightlike lines

$\displaystyle L^{\pm}\doteq\\{x^{\mp}=z={\overline{z}}=0\\}.$ (38)

$e^{{\alpha}P_{\mp}}$ acting on $L^{\pm}$ (signs correlated) both generate the
plane

$\displaystyle S\doteq\\{z={\overline{z}}=0\\}.$ (39)

$e^{{\alpha}M_{\mp 3}}$ acting on $L^{\pm}$ both generate the cone

$\displaystyle{\widehat{S}}\doteq\\{x^{+}x^{-}+z{\overline{z}}=0,\,z={\overline{z}}\\}.$
(40)

This is the $z=$ real ($x^{4}=0$) slice of the light cone in Minkowski space-
time. The remaining generators, $P_{\pm}$, $M_{\pm 3}$, $K_{\pm}$, and
$K_{\mp}$, do not move $L^{\pm}$. $e^{{\alpha}P_{\pm}}$ acting on
${\widehat{S}}$ and $e^{{\alpha}M_{\pm 3}}$ acting on $S$ generate the wall
(3d hyperplane)

$\displaystyle\widehat{W}\doteq\\{z={\overline{z}}\\}.$ (41)

The remaining generators do not move $S$ or ${\widehat{S}}$. Finally, none of
the generators move $\widehat{W}$. These world volumes are illustrated in
figure 2 which shows the $x^{4}=0$ ($z={\overline{z}}$) slice of Minkowski
space-time.

$x^{1}$$x^{2}$$x^{3}\subset$ VOA plane${\mathcal{O}}$${\bf L^{-}}$${\bf
L^{+}}$${\bf S}$${\widehat{\bf S}}$ Figure 2: The possible world volumes of
twisted Schur descent operators. The figure shows the $x^{4}=0$ 3d slice of
Minkowski space-time; $x^{1}$ is the time coordinate. The $x^{3}$ axis is the
$z={\overline{z}}$ line in the VOA plane. $L^{\pm}$ and ${\widehat{S}}$ are in
the light cone. The $\widehat{W}$ 3d world volume fills the 3d slice shown in
the figure. The extension of the ${\widehat{S}}$ and $\widehat{W}$ world
volumes in the $x^{3}$-direction of the VOA plane is conventional: this
direction can be rotated while keeping the corresponding descent operators in
the same cohomology class.

Acing with these descent operators in succession generates extended operators
with these world volumes, which we denote by the corresponding boldfaced
symbol. For instance, ${\bf
S}[{\mathcal{O}}]\doteq{\mathbf{D}}^{P_{+}}{\mathbf{D}}^{P_{-}}[{\mathcal{O}}]$.
The result is the descent web shown in figure 1. Note that although there is
only one $\widehat{W}$ 3d world volume, figure 1 shows two distinct 3d descent
operators, ${\widehat{\bf W}}^{+}[{\mathcal{O}}]$ and ${\widehat{\bf
W}}^{-}[{\mathcal{O}}]$. This is because they are reached by sequences of
descent operations which imply by (28) that their ${\rm U}(1)_{r}$ charges
necessarily differ by 2, and so they must be distinct operators even though
they share the same world volume.

What is less obvious about the descent web shown in figure 1 is why following
the different descent paths leading to ${\bf S}[{\mathcal{O}}]$,
${\widehat{\bf S}}[{\mathcal{O}}]$, and ${\widehat{\bf
W}}^{\pm}[{\mathcal{O}}]$ give the same descent operator. The anti-
commutativity of the ${\mathbf{D}}^{P_{+}}$ and ${\mathbf{D}}^{P_{-}}$
operations is easy to see. Let $\pi_{\pm}$ be the fermionic supercharges
satisfying ${\mathbbmtt{T}}_{\pm}\circ\pi_{\pm}=P_{\pm}$. Then by (19)
$\pi_{\pm}=\pm{\mathbbmtt{T}}_{\mp}\circ P_{\pm}$. Substituting these in
$\pi_{+}\circ\pi_{-}$, using the Jacobi identities, that
${\mathbbmtt{T}}_{\pm}\circ P_{\pm}=0$, and that $P_{+}\circ P_{-}=0$, one
finds that $\pi_{+}\circ\pi_{-}=-P_{+}\circ{\mathcal{Z}}\circ P_{-}=P_{+}\circ
P_{-}=0$ using (20) in the last steps. Then
${\mathbf{D}}^{P_{+}}{\mathbf{D}}^{P_{-}}[{\mathcal{O}}]=-{\mathbf{D}}^{P_{-}}{\mathbf{D}}^{P_{+}}[{\mathcal{O}}]$
follows immediately from the definition (26) of the descent operation. The
${\mathbf{D}}^{M_{+3}}{\mathbf{D}}^{P_{-}}[{\mathcal{O}}]=_{\mathbbmtt{T}}-{\mathbf{D}}^{M_{-3}}{\mathbf{D}}^{P_{+}}[{\mathcal{O}}]$
identity is more intricate, and is discussed in Argyres_2022xx . Finally,
${\mathbf{D}}^{M_{\pm
3}}{\mathbf{D}}^{P_{\pm}}[{\mathscr{E}}]=_{\mathbbmtt{T}}-{\mathbf{D}}^{P_{\pm}}{\mathbf{D}}^{M_{\pm
3}}[{\mathscr{E}}]$ follows from (23).

In (2) we showed that ${\mathbf{D}}^{P_{+}\pm
K_{+}}[{\mathcal{O}}]={\mathbf{D}}^{P_{+}}[{\mathcal{O}}]$. This relation is
much harder to show at higher levels of descent. In particular,
${\mathbf{D}}^{P_{+}\pm K_{+}}{\bf
L^{-}}[{\mathcal{O}}]=_{\mathbbmtt{T}}{\mathbf{D}}^{P_{+}}{\bf
L^{-}}[{\mathcal{O}}]$ and a similar relationship for ${\mathbf{D}}^{P_{-}\pm
K_{-}}{\bf L^{+}}$ follows from a lengthy calculation and holds only in
cohomology; the details will be reported in Argyres_2022xx . Using this result
and the anti-commutativity (23) of ${\mathfrak{m}}_{+}$ descent, we have

$\displaystyle{\mathbf{D}}^{P_{+}\pm K_{+}}{\bf S}$
$\displaystyle={\mathbf{D}}^{P_{+}\pm K_{+}}{\mathbf{D}}^{P_{+}}{\bf
L^{-}}=-{\mathbf{D}}^{P_{+}}{\mathbf{D}}^{P_{+}\pm K_{+}}{\bf
L^{-}}=_{\mathbbmtt{T}}-{\mathbf{D}}^{P_{+}}{\mathbf{D}}^{P_{+}}{\bf
L^{-}}=0,$ (42) $\displaystyle{\mathbf{D}}^{P_{+}\pm K_{+}}{\widehat{\bf S}}$
$\displaystyle={\mathbf{D}}^{P_{+}\pm K_{+}}{\mathbf{D}}^{M_{+3}}{\bf
L^{-}}=-{\mathbf{D}}^{M_{+3}}{\mathbf{D}}^{P_{+}\pm K_{+}}{\bf
L^{-}}=_{\mathbbmtt{T}}-{\mathbf{D}}^{M_{+3}}{\mathbf{D}}^{P_{+}}{\bf
L^{-}}=-{\widehat{\bf W}}^{+},$

and similarly for $P_{-}\pm K_{-}$ descent.

### VOA plane twist-translations.

So far we have acted with descent on Schur primaries, ${\mathcal{O}}(0)$, at
the origin of the VOA plane. These are twist-translated to other points on the
VOA plane by

$\displaystyle{\mathcal{O}}(z)\doteq{\widehat{g}}(z)\circ{\mathcal{O}}(0)=_{\mathbbmtt{T}}e^{zP_{z}}\circ{\mathcal{O}}(0),$
(43)

where ${\widehat{g}}(z)$ is the twist-translation operator (2). The last
equality in cohomology indicates that the correlators of twist-translated
Schur operators depend only on their holomorphic $z$ coordinates. In the same
way, we twist translate arbitrary twisted Schur descent operators to other
points in the VOA plane by acting with ${\widehat{g}}(z)$,

$\displaystyle{\mathscr{E}}(z)\doteq{\widehat{g}}(z)\circ{\mathscr{E}}(0)=_{\mathbbmtt{T}}e^{zP_{z}}\circ{\mathscr{E}}(0).$
(44)

In the case of ${\mathbf{D}}^{P_{\pm}}$ descent, since ${\widehat{g}}(z)\circ
P_{\pm}=P_{\pm}$ by (2), we find
${\mathbf{D}}^{P_{\pm}}[{\mathscr{E}}](z)=_{\mathbbmtt{T}}{\mathbf{D}}^{P_{\pm}}[{\mathscr{E}}(z)]$.
This means twist-translation commutes with the $P_{\pm}$-descent operation.
Since the ${\bf L}^{\pm}[{\mathcal{O}}]$ and ${\bf S}[{\mathcal{O}}]$ descent
operators only involve $P_{\pm}$ descent, it follows that

$\displaystyle{\bf L}^{\pm}[{\mathcal{O}}](z)$ $\displaystyle={\bf
L}^{\pm}[{\mathcal{O}}(z)],$ $\displaystyle{\bf S}[{\mathcal{O}}](z)$
$\displaystyle={\bf S}[{\mathcal{O}}(z)].$ (45)

By contrast, ${\mathbf{D}}^{M_{+3}}$ descent does not commute with twist-
translations since ${\widehat{g}}(z)\circ M_{+3}=M_{+3}-{\rm Re}(z)P_{+}$ by
(2).

##  3 Boundaries, intersections, and regularizations

Boundaries refer to the boundary terms that must vanish for descent operators
to be ${\mathbbmtt{T}}_{\pm}$-closed as in (2). Note that although these
boundary terms occur at descent integration parameter ${\alpha}=\pm\infty$,
they do not necessarily correspond to points at space-time infinity, as we
will see in the case of $M_{\pm 3}$ descent.

Intersections refer to the fact that correlators (or operator products) need
only be finite if the world volumes of their operator insertions avoid all
space-time intersections. For local (Schur) operators this means simply that
their VOA plane coordinates do not coincide, $z_{1}\neq z_{2}$, but for
extended operators this may not be sufficient to avoid intersections.

Regularization of operator products is needed in Minkowski space-time because
propagators (2-point functions) have poles at light-like separations. We are
interested in time-ordered operator products in Minkowski space-time since
those are the ones for which the twisted Schur cohomology identities (32)
hold. Light cone poles of time-ordered propagators are regulated by the usual
$i{\epsilon}$ prescription, reviewed below.

Because the twisted Schur descent operator world volumes extend to space-time
infinity, the evaluation of their boundary terms at infinity, the avoidance of
their intersections at infinity, and the choice of regularization interact in
an interesting way. We will explain this interaction in this section to some
extent, even though we only have a partial understanding for ${\widehat{\bf
S}}$ and ${\widehat{\bf W}}^{\pm}$ descent operators.

### The ${\mathcal{O}}$-${\bf L}^{\pm}$-${\bf S}$ subalgebra.

We start by explaining these issues in the vertex operator subalgebra
generated by Schur operators and their ${\bf L}^{\pm}$ and ${\bf S}$ descent
operators. The resulting set of vertex operators (at the origin of the VOA
plane, say) is characterized by having support in the $z={\overline{z}}=0$
plane in space-time, and so, in particular, does not contain the
${\widehat{\bf S}}$ or ${\widehat{\bf W}}^{\pm}$ descent operators.

Since the fermionic super generators $\pi_{\pm}$ satisfying
${\mathbbmtt{T}}_{\pm}\circ\pi_{\pm}=P_{\pm}$ are777 Here $Q^{a}_{\
{\alpha}}$, ${\widetilde{Q}}_{a{\dot{\alpha}}}$, $S_{a}^{\ {\alpha}}$, and
${\widetilde{S}}^{a{\dot{\alpha}}}$ are the supertranslation and
superconformal generators of the ${\mathcal{N}}{=}2$ superconformal algebra in
the notation of Beem_2015 ; Lemos_2020 . In particular, ${\alpha}$,
${\dot{\alpha}}$, $a$ are Lorentz left and right spinor indices and ${\rm
SU}(2)_{R}$ doublet index, respectively.

$\displaystyle\pi_{+}$ $\displaystyle=-i{\widetilde{Q}}_{1\dot{1}},$
$\displaystyle\pi_{-}$ $\displaystyle=-iQ^{2}_{\ 1},$ (46)

it follows from the definition (26) of the descent operation that

$\displaystyle{\bf L^{+}}[{\mathcal{O}}]$
$\displaystyle=-iN\int\\!\\!dx^{+}\,{\widetilde{Q}}_{1\dot{1}}\circ
e^{x^{+}P_{+}}\circ{\mathcal{O}}(0),\hskip 28.45274pt{\bf
L^{-}}[{\mathcal{O}}]=-iN\int\\!\\!dx^{-}\,Q^{2}_{\ 1}\circ
e^{x^{-}P_{-}}\circ{\mathcal{O}}(0),$ $\displaystyle{\bf S}[{\mathcal{O}}]$
$\displaystyle=-N^{2}\int\\!\\!dx^{+}dx^{-}\,Q^{2}_{\
1}\circ{\widetilde{Q}}_{1\dot{1}}\circ
e^{x^{+}P_{+}+x^{-}P_{-}}\circ{\mathcal{O}}(0),$ (47)

Their normalizations are a matter of choice; we have put in the factors of $N$
for later convenience. These descent operators are especially easy to work
with since $\pi_{\pm}$ and $P_{\pm}$ all commute. ${\bf
L}^{\pm}[{\mathcal{O}}]$ are reminiscent888The ${\bf L}^{\pm}[{\mathcal{O}}]$
descent lines are not quite light-ray transforms. The natural generalization
of the light-ray transform to a conformal primary with general Lorentz spins
$(j,{\widetilde{\jmath}})$ is ${\bf
L}[{\mathcal{O}}][{\zeta},\widetilde{\zeta}]=\int
ds\,{\mathcal{O}}_{{\alpha}_{1}\cdots{\alpha}_{j}\,{\dot{\alpha}}_{1}\cdots{\dot{\alpha}}_{\widetilde{\jmath}}}(s{\zeta}^{\alpha}\widetilde{\zeta}^{\dot{\alpha}})\
{\zeta}^{{\alpha}_{1}}\cdots{\zeta}^{{\alpha}_{j}}\,\widetilde{\zeta}^{{\dot{\alpha}}_{1}}\cdots\widetilde{\zeta}^{{\dot{\alpha}}_{\widetilde{\jmath}}}$,
where ${\mathcal{O}}$ is integrated over the lightline
$x^{\mu}{\sigma}_{\mu}^{{\alpha}{\dot{\alpha}}}=s{\zeta}^{\alpha}\widetilde{\zeta}^{\dot{\alpha}}$
where ${\zeta}^{\alpha}$ and $\widetilde{\zeta}^{\dot{\alpha}}$ are
polarization spinors such that $x^{\mu}$ is real. The ${\bf
L}^{\pm}[{\mathcal{O}}]$ light-like line operators are not of this form: their
spinor polarizations do not line up with the $L^{\pm}$ world lines. of the
light-ray transform operators introduced in Kravchuk_2018:1805.00098 , and
some of the discussion of Kravchuk_2018:1805.00098 ; Kologlu:2019 ; Chang:2020
on the computation of their correlators and OPEs applies.

Time-ordered correlators of descent operators are simply the world volume
integrals of the time-ordered correlators of the local operators appearing in
their integrands, e.g., $\langle\int
d{\alpha}_{1}{\mathcal{O}}_{1}(x_{1}({\alpha}_{1}))\,\int
d{\alpha}_{2}{\mathcal{O}}_{2}(x_{2}({\alpha}_{2}))\rangle\doteq\int
d{\alpha}_{1}d{\alpha}_{2}\langle{\mathcal{O}}_{1}(x_{1}({\alpha}_{1}))\,{\mathcal{O}}_{2}(x_{2}({\alpha}_{2}))\rangle$.
The light-cone singularities of time-ordered correlators are regulated by the
usual causal $i{\epsilon}$ prescription where the time-difference coordinate
$x_{12}^{{\mu}=1}\doteq t_{12}$ is replaced by the complex value
$(1+i{\epsilon})t_{12}$ and the limit ${\epsilon}\to 0^{+}$ is taken.999Other
$i{\epsilon}$ prescriptions, such as $t_{12}\to t_{12}+i{\epsilon}\,{\rm
sgn}(t_{12})$ also work, though ones which in which the imaginary shift falls
off as $|t_{12}|\to\infty$ do not.

The boundary terms (2) that contribute to ${\mathbbmtt{T}}_{+}\circ{\bf
L^{+}}[{\mathcal{O}}]$ are
$\lim_{|{\alpha}|\to\infty}e^{{\alpha}P_{+}}\circ{\mathcal{O}}(0)$. We
evaluate this limit by inserting it into an arbitrary correlation function,
where it can only give a non-vanishing contribution if there is an OPE channel
involving its conformal conjugate field. The large-separation limit of the
2-point functions of conjugate conformal fields has the form

$\displaystyle\lim_{|x_{12}|\to\infty}\langle{\mathcal{O}}(x_{1})\overline{\mathcal{O}}(x_{2})\rangle\sim|x_{12}|^{-2D_{\mathcal{O}}},$
(48)

where $|x_{12}|^{2}=(x_{1}-x_{2})^{\mu}(x_{1}-x_{2})_{\mu}$ and
$D_{\mathcal{O}}$ is the conformal dimension of ${\mathcal{O}}$. It follows
from unitarity that all operators (except the identity operator) obey
$D_{\mathcal{O}}\geq 1$.

If ${\mathcal{O}}(x_{1})=e^{{\alpha}P_{+}}\circ{\mathcal{O}}(0)$, then
$(x_{1}^{+},x_{1}^{-},z_{1})=({\alpha},0,0)$, and
$|x_{12}|^{2}=({\alpha}-x_{2}^{+})x_{2}^{-}+|z_{2}|^{2}$. Thus the right side
of (48) vanishes as $|{\alpha}|\to\infty$ except perhaps when $x_{2}^{-}=0$,
in which case it may, naively, have a non-zero limit. However, with the
$i{\epsilon}$ regulator turned on,
$|x_{12}|^{2}\approx|x_{12}|^{2}_{{\epsilon}=0}-2i{\epsilon}t_{12}^{2}\approx|x_{12}|^{2}_{{\epsilon}=0}-\frac{i}{2}{\epsilon}{\alpha}^{2}$
as $|{\alpha}|\to\infty$, so the right side of (48) vanishes for all
$x^{\mu}_{2}$. Note that this conclusion would not hold if the imaginary shift
to the time coordinate falls off too fast as the time coordinate gets large;
hence the comment in footnote 9. The “Wick rotation” $t\to(1+i{\epsilon})t$
regularization also seems more consonant with scale invariance.

Thus, in correlators evaluated using the $i{\epsilon}$ prescription, the
boundary terms contributing to ${\mathbbmtt{T}}_{+}\circ{\bf
L^{+}}[{\mathcal{O}}]$ vanish, and so ${\bf L^{+}}[{\mathcal{O}}]$ is in
${\mathbbmtt{T}}_{\pm}$ cohomology. The same argument applies to ${\bf
L^{-}}[{\mathcal{O}}]$. ${\bf S}[{\mathcal{O}}]$ is also
${\mathbbmtt{T}}_{\pm}$-closed with the $i{\epsilon}$ prescription, but in a
less delicate way; see Argyres_2022xx .

### Operator products and intersections in conformally compactified space-
time.

We compute operator products by inserting operators separated in space-time,
and taking the limit as they approach one another. In the case of
${\mathcal{O}}(z)$, ${\bf L}^{\pm}[{\mathcal{O}}](z)$, and ${\bf
S}[{\mathcal{O}}](z)$, if their $z$ coordinates are different their world
volumes do not intersect at any finite points in space-time. However, since
their world volumes extend to infinity, we must be careful also about their
intersections at points at infinity. Here it is useful to work in conformally
compactified space-time, which is the familiar Penrose diamond with boundaries
the future and past light-like infinities ${\mathscr{I}}^{\pm}\simeq I\times
S^{2}$ together with future and past time-like infinity points $i^{\pm}$ and a
space-like infinity point $i^{0}$. Then ${\bf L}^{\pm}(z)$ and ${\bf
L}^{\mp}(0)$ do not intersect at infinity, while ${\bf L}^{\pm}(z)$ and ${\bf
L}^{\pm}(0)$ or ${\bf S}(0)$ intersect at 2 points on ${\mathscr{I}}^{\pm}$,
and ${\bf S}(z)$ and ${\bf S}(0)$ intersect at 4 points on
${\mathscr{I}}^{\pm}$ and at $i^{0}$. The intersections at light-like infinity
of ${\bf L}^{\pm}$ with ${\bf L}^{\pm}$ or ${\bf S}$ turn out to be innocuous:
with the $i{\epsilon}$ prescription they give no divergences. This follows, in
part, because none of these products can close on the identity operator, so no
world volume integrals of the corresponding propagators occur.

By contrast, the ${\bf S}$ with ${\bf S}$ intersections at
${\mathscr{I}}^{\pm}$ and at $i^{0}$ give rise to behaviors which must be
further regulated. We start with the intersection at the space-like point at
infinity, $i^{0}$. This point is also the point at infinity on the VOA plane.
Just as in previous discussions of VOA modules associated to surface operators
Cordova:2017 ; Bianchi_2019 , a surface operator insertion twist-translated to
$z$ on the VOA plane should also be thought of as providing an insertion at
the point at infinity. The point at infinity is moved by the twist special
conformal transformations of the VOA plane, ${\widehat{g}}({\beta})$ in (2).
Thus the ${\bf S}[{\mathcal{O}}](z)\,{\bf S}[{\mathcal{O}}](0)$ operator
product has an intersection at $i^{0}$, and the ${\bf S}[{\mathcal{O}}](z)\
{\widehat{g}}({\beta}){\circ}{\bf S}[{\mathcal{O}}](0)$ product does not for
${\beta}\neq 0$. The first cohomology equivalence of (2) indicates that the
operator product should not depend on ${\beta}$, but the topological
distinction between an intersection at $i^{0}$ when ${\beta}=0$ and no
intersection when ${\beta}\neq 0$ makes it possible that there are two
different (${\beta}$-independent) answers in these two cases. For example, for
the scalar $D_{\mathcal{O}}=1$ Schur primaries ${\mathcal{O}}=q_{I}$ of the
free hypermultiplet SCFT discussed in the next section, we find that
$\langle{\bf S}[q_{I}](z)\ {\widehat{g}}({\beta}){\circ}{\bf
S}[q_{J}](0)\rangle=-{\epsilon}_{IJ}z^{-1}$ for ${\beta}\neq 0$, but vanishes
for ${\beta}=0$. We compute all ${\bf S}$-${\bf S}$ products regulated in this
way to remove the intersection at space-like infinity.

In computing these ${\beta}$-regulated products, one must take into account
that the ${\widehat{g}}({\beta})$ transformations act discontinuously on
Minkowski space-time since they are special conformal transformations. The
natural way to compute these products is on the Lorentzian cylinder, which is
covered by an infinite number of copies of conformally compactified Minkowski
space-time, and on which conformal transformations act continuously
Penrose:1964 . Also, time-ordered Lorentzian cylinder correlators of unitary
CFTs are the analytic continuation of correlators in the CFT on the euclidean
4-sphere Luscher:1974 .

But these correlators suffer from a second intersection problem. Turning on a
relative twist conformal transformation, ${\beta}\neq 0$, does not remove the
intersections at the four points on ${\mathscr{I}}^{\pm}$ at light-like
infinity. This is because ${\widehat{g}}({\beta})$ transformations fix the
$L^{\pm}$ lines. Indeed, there are no conformal transformations that both
preserve twisted Schur cohomology and also remove these intersection points.
In the $\langle{\bf S}[{\mathcal{O}}]{\bf S}[{\mathcal{O}}]\rangle$ 2-point
function, the world volume integral over the relative positions of the two
insertions is regulated by the $i{\epsilon}$ prescription, but the remaining
integral over the world volume of one of the insertions is not absolutely
convergent. One must therefore separately regulate this integral.

The natural prescription is to “Wick rotate” these correlators to euclidean
space. There are no light-like infinities in Euclidean space: there is just a
single point, $i^{0}$, at (spatial) infinity compactifying Euclidean space to
$S^{4}$. A relative (euclidean) ${\widehat{g}}({\beta})$ special conformal
transformation separates the intersection of two 2-surfaces at $i^{0}$, and
completely regulates their 2-point function.

We do not currently know the precise prescription corresponding to a small
“Wick rotation” of a 2-surface in the Lorentzian cylinder which regulates both
light-like separations and space-time intersections and which is an analytic
continuation of time-ordered correlators. A guess is simply to complexify the
${\widehat{g}}({\beta})$ transformation by adding in an additional small
relative $e^{i{\delta}}$ phase between ${\beta}$ and ${\overline{\beta}}$,
making them not be complex conjugates, while treating (perhaps naively) the
resulting correlators as being time-ordered. This, indeed, regulates the
2-point function, but also violates the cohomology Ward identity (2), giving a
${\beta}$\- and ${\overline{\beta}}$-dependent answer. (This cohomology
violation happens in a relatively mild way: the regulated expression is given
as a sum of finite terms which is the same as the Euclidean expression — which
does give a ${\beta}$\- and ${\overline{\beta}}$-independent sum — but with
some relative minus signs.) Another guess is to perform the usual
$\exp{i{\delta}P_{t}}$ small Wick rotation on the ${\bf S}$ world volume
before performing the ${\widehat{g}}({\beta})$ transformation. This also
regulates the integrals, but still does not give a cohomologically consistent
result, presumably because it violates the time-ordered nature of the
correlator.

In light of this difficulty, we therefore resort in the calculations of the
next section to the simple expedient of using the euclidean result for the
${\bf S}$-${\bf S}$ 2-point function. Mathematically, this Euclidean
regularization is easy, but comes at the cost of not having a physically
satisfactory interpretation in terms of extended operator world volumes in
Minkowski space-time or analytically continued Wightman functions.

### Convergence of the ${\widehat{\bf S}}$ and ${\widehat{\bf W}}^{\pm}$
descent operators.

These operators are reached by descent involving the $M_{\pm 3}$ generators.
The infinite-parameter boundaries of the space-time orbits of these generators
include finite points in space-time, unlike the orbits of the $P_{\pm}$
translation generators. This fact makes the ${\widehat{\bf S}}$ and
${\widehat{\bf W}}^{\pm}$ operators less convergent than the ${\bf L}^{\pm}$
and ${\bf S}$ operators.

Consider, for example, the ${\widehat{\bf
S}}[{\mathcal{O}}]_{-}\doteq{\mathbf{D}}^{M_{-3}}{\bf L^{+}}[{\mathcal{O}}]$
operator. (We put the “$-$” subscript to indicate that this is the
${\widehat{\bf S}}$ operator found by $M_{-3}$ descent; we will discuss its
equivalence to the one found by $M_{+3}$ descent shortly.) The $M_{-3}$ light-
like combination of a rotation and a boost generates a subgroup with elements
$\\{g({\xi})\doteq e^{-{\xi}M_{-3}},\,{\xi}\in(-\infty,\infty)\\}$, whose
space-time orbits are not compact. Acting on the $L^{+}$ line $g({\xi})$ has
the effect of rotating $L^{+}$ to other light-like lines in the light cone
${\widehat{S}}$ shown in figure 2, reaching the $L^{-}$ line only in the
${\xi}\to\pm\infty$ limit. Thus the ${\xi}\to\pm\infty$ boundaries of the
${\mathbf{D}}^{M_{-3}}$ descent operation include finite points in space-time,
namely, all the points on $L^{-}$. Parameterize the ${\widehat{S}}$ light cone
by $x^{\mu}(t,{\theta})$ with
$(x^{1},x^{2},x^{3},x^{4})=t(1,\cos{\theta},\sin{\theta},0)$ with
$t\in\mathbb{R}$ and ${\theta}\in[-\pi,\pi]$. Thus ${\theta}=0$ is $L^{+}$ and
${\theta}=\pm\pi$ is $L^{-}$. One then shows that
${\mathbbmtt{T}}_{-}\circ{\widehat{\bf S}}[{\mathcal{O}}]_{-}$ is a sum of
boundary terms, one at $t=\pm\infty$ and one at ${\theta}=\pm\pi$.

The ${\theta}$ boundary term gives

$\displaystyle{\mathbbmtt{T}}_{-}\circ{\widehat{\bf S}}[{\mathcal{O}}]_{-}$
$\displaystyle\supset-\frac{i}{2}\Big{[}\big{\\{}(1{+}\cos{\theta}){\widetilde{Q}}_{1\dot{1}}+\sin{\theta}{\widetilde{Q}}_{2\dot{1}}\big{\\}}\circ
e^{x^{\mu}(t,{\theta})P_{\mu}}\circ
e^{i{\xi}{\widetilde{\mathcal{M}}}^{\dot{1}}_{\
\dot{2}}}\circ{\mathcal{O}}(0)\Big{]}^{{\theta}=+\pi}_{{\theta}=-\pi}.$ (49)

If the Schur ${\mathcal{O}}$ is a component of a field with Lorentz
spins101010We normalize our Lorentz spins to be integers, following the
conventions of Cordova_2016 . $(j,{\widetilde{\jmath}})$ then

$\displaystyle e^{i{\xi}{\widetilde{\mathcal{M}}}^{\dot{1}}_{\
\dot{2}}}\circ{\mathcal{O}}(0)\sim{\mathcal{O}}_{\dot{1}\cdots\dot{1}}(0)+\cdots+{\xi}^{\widetilde{\jmath}}{\mathcal{O}}_{\dot{2}\cdots\dot{2}}(0)$
(50)

where we show only the ${\widetilde{\jmath}}$ dotted spinor indices of
${\mathcal{O}}$. Since ${\xi}=\tan({\theta}/2)$ it follows that the boundary
terms are non-zero for ${\widetilde{\jmath}}>0$. Note that these non-zero
boundary terms cannot cancel against possible boundary terms at $t=\pm\infty$
because they are local operators at finite points in space-time. Thus
${\widehat{\bf S}}[{\mathcal{O}}]_{-}$ is not in twisted Schur cohomology
unless ${\widetilde{\jmath}}=0$. An analogous argument shows that
${\widehat{\bf S}}[{\mathcal{O}}]_{+}\doteq{\mathbf{D}}^{M_{+3}}{\bf
L^{-}}[{\mathcal{O}}]$ is only in cohomology if $j=0$.

The $t=\pm\infty$ boundary term is (taking ${\widetilde{\jmath}}=0$)

$\displaystyle{\mathbbmtt{T}}_{-}\circ{\widehat{\bf S}}[{\mathcal{O}}]_{-}$
$\displaystyle\supset-\frac{i}{2}\int_{-\pi}^{+\pi}\\!\\!d{\theta}\Big{[}t\big{\\{}\sin{\theta}{\widetilde{Q}}_{1\dot{1}}+(1{-}\cos{\theta}){\widetilde{Q}}_{2\dot{1}}\big{\\}}\circ{\mathcal{O}}(x^{\mu}(t,{\theta}))\Big{]}^{t=+\infty}_{t=-\infty}.$
(51)

Just as in the discussion of the boundary terms contributing to
${\mathbbmtt{T}}_{\pm}\circ{\bf L}^{\pm}[{\mathcal{O}}]$, these boundary terms
vanish when inserted in correlation functions because even though the
$|t|\to\infty$ limit leads to light-like separations, the $i{\epsilon}$
regularization implies the propagator still falls off as $|t|^{-2D}$. In this
case $D=D_{\mathcal{O}}+\frac{1}{2}$ because of the ${\widetilde{Q}}$’s in
(51), but this shift is compensated for by the explicit factor of $t$ in (51)
giving the overall large-$|t|$ behavior $\sim|t|^{-2D_{\mathcal{O}}}$. This
converges since $D_{\mathcal{O}}\geq 1$.

Thus we expect that ${\widehat{\bf S}}[{\mathcal{O}}]$ descent operators exist
in twisted Schur cohomology when
$j_{\mathcal{O}}{\widetilde{\jmath}}_{\mathcal{O}}=0$. In the case of scalar
Schur primaries (i.e., ${\mathcal{O}}$’s with $j={\widetilde{\jmath}}=0$)
there are two descent paths to ${\widehat{S}}$, leading to the ${\widehat{\bf
S}}[{\mathcal{O}}]_{+}$ and ${\widehat{\bf S}}[{\mathcal{O}}]_{-}$ operators.
An intricate calculation shows that they differ by only by boundary terms at
infinity which, again, can be shown to vanish in correlators. A similar
analysis implies ${\widehat{\bf W}}^{+}[{\mathcal{O}}]$ exists when
$j_{\mathcal{O}}=0$, and ${\widehat{\bf W}}^{-}[{\mathcal{O}}]$ exists when
${\widetilde{\jmath}}_{\mathcal{O}}=0$.

This result implies that the descent web pictured in figure 1 only applies to
scalar Schur primaries. For Schur primaries with $j>0$ the $M_{+3}$ descent
arrows disappear, and in cases with ${\widetilde{\jmath}}>0$ the $M_{-3}$
arrows disappear. In particular, if ${\mathcal{O}}$ has
$j{\widetilde{\jmath}}>0$ — as is the case, for example, of the Virasoro
operator, $T$, of the VOA — then the ${\widehat{\bf S}}[T]$ and ${\widehat{\bf
W}}^{\pm}[T]$ descent vertex operators do not exist.

Note that the ${\widehat{S}}$ and $\widehat{W}$ world volumes break the
rotational invariance of the VOA plane by picking out the $z={\overline{z}}$
direction. Denote rotated world volumes by ${\widehat{S}}_{\phi}$ and
${\widehat{W}}_{\phi}$ where ${\phi}$ is the angle from the real $z$-axis in
the VOA plane. Thus the ${\widehat{S}}_{\phi}$ world volume is thus extended
in the $z=e^{2i{\phi}}{\overline{z}}$ direction. The corresponding descent
operators, ${\widehat{\bf S}}_{\phi}$ and ${\widehat{\bf W}}_{\phi}^{\pm}$,
are generated by descent with $M_{\pm{\phi}}\doteq\cos{\phi}M_{\pm
3}+\sin{\phi}M_{\pm 4}$. Since ${\widehat{g}}(\ell)$ for imaginary
$\ell=i{\phi}$ has the effect of rotating these world volumes in the VOA plane
in just this way, by (35) and (2), the ${\widehat{g}}(\ell)$ cohomology
equivalence of (2) shows that these are equivalent operators in
${\mathbbmtt{T}}_{\pm}$ cohomology:

$\displaystyle{\widehat{\bf S}}_{\phi}[{\mathcal{O}}]$
$\displaystyle=_{\mathbbmtt{T}}{\widehat{\bf S}}[{\mathcal{O}}],$
$\displaystyle{\widehat{\bf W}}_{\phi}^{\pm}[{\mathcal{O}}]$
$\displaystyle=_{\mathbbmtt{T}}{\widehat{\bf W}}^{\pm}[{\mathcal{O}}].$ (52)

Thus there is nothing special about the $z={\overline{z}}$ direction in the
VOA plane in the extended vertex algebra.

However, once we insert a ${\widehat{\bf W}}_{\phi}^{\pm}$ operator in a
correlator, then any other ${\widehat{\bf S}}_{{\phi}^{\prime}}$ or
${\widehat{\bf W}}_{{\phi}^{\prime}}^{\pm}$ insertion must have parallel world
volume, i.e., all must have ${\phi}^{\prime}={\phi}+n\pi$. This is simply
because if their world volumes are not parallel in this sense, then they will
intersect the ${\widehat{\bf W}}_{\phi}^{\pm}$ insertion at finite points in
space-time.

Generally, the world volumes of an ${\widehat{\bf S}}_{\phi}(0)$ operator and
a twist-translated ${\widehat{\bf S}}_{{\phi}^{\prime}}(z)$ operator do not
intersect at any finite points in space-time. The exception is the specific
arrangement where ${\rm arg}(z)=({\phi}+{\phi}^{\prime}+\pi)/2$. Thus
correlators $\langle{\widehat{\bf S}}_{{\phi}_{1}}(z_{1}){\widehat{\bf
S}}_{{\phi}_{2}}(z_{2})\cdots\rangle$ will have regions of cohomological
equivalence in the configuration space of the $({\phi}_{i},z_{i})$’s. The
boundaries of these regions are the excluded configurations where pairs of
${\widehat{\bf S}}$ world volumes intersect in space-time. The geometry and
topology of these regions seems complicated.

Lastly, the ${\widehat{\bf W}}_{\phi}^{\pm}$ world volume intersects the VOA
plane along the $z=e^{2i{\phi}}{\overline{z}}$ line. Thus correlators of a
${\widehat{\bf W}}_{\phi}^{\pm}$ operator with any other vertex algebra
operators is only well-defined if those other insertions avoid this line. In
this way ${\widehat{\bf W}}^{\pm}$ operators behave like boundary (or line)
operators in the VOA plane, and so are not, strictly speaking, vertex
operators. But, this vertex-boundary operator is topological in the sense that
${\widehat{g}}(\ell)$ rotation of the boundary line in the VOA plane will not
change the correlator in cohomology until the rotation moves the boundary line
across the insertion point of another vertex operator in the VOA plane.

##  4 Extended vertex algebra of the free hypermultiplet theory

We now calculate some operator products in the ${\mathcal{O}}$-${\bf
L}^{\pm}$-${\bf S}$ sub-vertex algebra of the free hypermultiplet SCFT. In
particular, we will calculate the OPEs among the free dimension-1 scalar
Schur, $q$, the dimension-3 vector Schur operator, $T$, and their ${\bf
L}^{\pm}$ and ${\bf S}$ descent operators. $T$ is the Virasoro operator in the
VOA which is the chiral theory of a free symplectic boson.

The free hypermultiplet superconformal primaries are the free complex scalars,
${\rho}^{a}_{I}$, satisfying the reality conditions
$({\rho}^{a}_{I})^{\dagger}={\epsilon}^{IJ}{\epsilon}_{ab}{\rho}^{b}_{J}$,
where $a,b\in\\{1,2\\}$ are fundamental $\mathfrak{su}(2)_{R}$ indices and
$I,J\in\\{1,2\\}$ are fundamental $\mathfrak{su}(2)_{F}$ flavor
indices.111111All ${\epsilon}$ symbols satisfy
${\epsilon}_{12}=-{\epsilon}^{12}=1$. There are also free fermions given by
${\psi}_{{\alpha}I}\doteq\frac{1}{2}{\epsilon}_{ba}Q^{a}_{\
{\alpha}}{\rho}^{b}_{I}$ and
${\widetilde{\psi}}_{{\dot{\alpha}}I}\doteq\frac{1}{2}{\widetilde{Q}}_{a{\dot{\alpha}}}{\rho}^{a}_{I}$.
Their 2-point functions are

$\displaystyle\langle{\rho}^{a}_{I}(x_{1}){\rho}^{b}_{J}(x_{2})\rangle$
$\displaystyle=\frac{{\epsilon}^{ab}{\epsilon}_{IJ}}{|x_{12}|^{2}},$
$\displaystyle\langle{\psi}_{{\alpha}I}(x_{1}){\widetilde{\psi}}_{{\dot{\alpha}}J}(x_{2})\rangle$
$\displaystyle=\frac{{\epsilon}_{IJ}(x_{12})_{{\alpha}{\dot{\alpha}}}}{|x_{12}|^{4}}.$
(53)

where

$\displaystyle x_{{\alpha}{\dot{\alpha}}}\doteq
x^{\mu}{\sigma}_{{\mu}{\alpha}{\dot{\alpha}}}$
$\displaystyle=\begin{pmatrix}{\overline{z}}&-ix^{+}\\\
ix^{-}&-z\end{pmatrix}.$ (54)

We consider the twist-translated Schur operators

$\displaystyle q_{I}(z)$ $\displaystyle\doteq
u_{a}{\rho}^{a}_{I}(z,{\overline{z}}),$ $\displaystyle T(z)$
$\displaystyle\doteq u_{a}u_{b}\tfrac{1}{2}{\epsilon}^{IJ}{:}\mkern
1.0mu{\rho}^{a}_{I}{\partial}_{z}{\rho}^{b}_{J}\mkern
1.6mu{:}(z,{\overline{z}}),$ (55)

where $u_{a}=(1,-{\overline{z}})$. In the VOA, $q_{I}$ is a free symplectic
boson of chiral weight $h_{q}=1/2$, and $T$ is the Virasoro operator of weight
$h_{T}=2$.

###  ${\bf L}^{\pm}$ and ${\bf S}$ descent operators.

Using the definitions of the descent operators given in section 2, the twist
translated descent operators for $q_{I}$ are

$\displaystyle{\bf L^{+}}[q_{I}](z)$
$\displaystyle=-iN\int_{L^{+}}dx^{+}\,{\widetilde{\psi}}_{\dot{1}I}(x^{+},0,z,{\overline{z}}),$
$\displaystyle{\bf L^{-}}[q_{I}](z)$
$\displaystyle=-iN\int_{L^{-}}dx^{-}\,{\psi}_{1I}(0,x^{-},z,{\overline{z}}),$
(56) $\displaystyle{\bf S}[q_{I}](z)$
$\displaystyle=-N^{2}\int_{S}dx^{+}dx^{-}\,{\partial}_{z}{\rho}^{2}_{I}(x^{+},x^{-},z,{\overline{z}})$

where we choose the normalization factor

$\displaystyle N=(-2\pi i)^{1/2},$ (57)

to simplify the form of the OPEs. The twist translations act trivially on
these descent operators because the local operators being integrated are all
$\mathfrak{su}(2)_{R}$ singlets or of lowest $\mathfrak{su}(2)_{R}$ weight.
This is not the case for the twist translated descent operators of $T$ which
are

$\displaystyle{\bf L^{+}}[T](z)$
$\displaystyle=\frac{-i}{2}{\epsilon}^{IJ}N\int_{L^{+}}dx^{+}\,u_{a}\left({:}\mkern
1.0mu{\widetilde{\psi}}_{\dot{1}I}{\partial}_{z}{\rho}^{a}_{J}\mkern
1.6mu{:}+{:}\mkern
1.0mu{\rho}^{a}_{I}{\partial}_{z}{\widetilde{\psi}}_{\dot{1}J}\mkern
1.6mu{:}\right)(x^{+},0,z,{\overline{z}}),$ $\displaystyle{\bf L^{-}}[T](z)$
$\displaystyle=\frac{-i}{2}{\epsilon}^{IJ}N\int_{L^{-}}dx^{-}\,u_{a}\left({:}\mkern
1.0mu{\psi}_{1I}{\partial}_{z}{\rho}^{a}_{J}\mkern 1.6mu{:}+{:}\mkern
1.0mu{\rho}^{a}_{I}{\partial}_{z}{\psi}_{1J}\mkern
1.6mu{:}\right)(0,x^{-},z,{\overline{z}}),$ (58) $\displaystyle{\bf S}[T](z)$
$\displaystyle=-\frac{{\epsilon}^{IJ}}{2}N^{2}\int_{S}dx^{+}dx^{-}\,\Big{\\{}u_{a}({:}\mkern
1.0mu{\partial}_{z}{\rho}^{2}_{I}{\partial}_{z}{\rho}^{a}_{J}\mkern
1.6mu{:}+{:}\mkern
1.0mu{\rho}^{a}_{I}{\partial}_{z}{\partial}_{z}{\rho}^{2}_{J}\mkern
1.6mu{:})(x^{+},x^{-},z,{\overline{z}})$ $\displaystyle\hskip
128.0374pt\mbox{}+({:}\mkern
1.0mu{\psi}_{1I}{\partial}_{z}{\widetilde{\psi}}_{\dot{1}J}\mkern
1.6mu{:}-{:}\mkern
1.0mu{\widetilde{\psi}}_{\dot{1}I}{\partial}_{z}{\psi}_{1J}\mkern
1.6mu{:})(x^{+},x^{-},z,{\overline{z}})\Big{\\}}.$

For the OPEs involving a descent operator, one must often compute integrals of
propagator-like expressions to simplify the result. Whenever possible we
perform such integrals in Minkowski space-time using the standard
$i{\epsilon}$ prescription but, as is the case with 2-point functions of ${\bf
S}$, this is not always sufficient. Curiously, in cases where there are space-
time intersections and the computation of integrals are involved, we have
checked that the $i{\epsilon}$ prescription is sufficient to compute them, as
long as these integrals come from terms not closing on the identity.

An important property of these OPEs is that they are meromorphic functions of
$z$ as required by cohomology Ward identities. This is a check that we are
computing descent operator products correctly.

A priori, additional ${\mathbbmtt{T}}_{\pm}$-closed extended operators might
appear on the right sides of the OPEs which are not equivalent in cohomology
to the descent operators of $q_{I}$ and $T$. This would indicate we were
“missing” distinct extended operators in the ${\mathcal{O}}$-${\bf
L}^{\pm}$-${\bf S}$ subset which are not necessarily captured by descent. In
practice, seemingly distinct ${\mathbbmtt{T}}_{\pm}$-closed surface/line
operators indeed appear in the OPEs involving a descent operator. However,
whenever this occurs, we have always found that subtle integral cohomology
identities end up showing these seemingly new extended operators are in fact
${\mathbbmtt{T}}_{\pm}$-equivalent to the descent operators of $q_{I}$ and $T$
or their normal-ordered products. We detail these cohomology identities in
Argyres_2022xx .

In the end, we find that the OPEs among these Schur operators and their
descent operators can all be put into canonical forms which we specify in a
condensed notation with the following tables.

OPEs among $q_{I},\,{\bf L}^{\pm}[q_{I}]$ and ${\bf S}[q_{I}]$:

$\displaystyle
X_{I}(z)Y_{J}(0)\sim\,{\epsilon}_{IJ}\frac{{\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{a}}}{z}\mbox{\quad
for\quad}{\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{a}}\in\mathbb{R}$
$\displaystyle\begin{array}[]{c|cccc}X_{I}\backslash Y_{J}&q_{J}&{\bf
S}[q_{J}]&{\bf L^{+}}[q_{J}]&{\bf L^{-}}[q_{J}]\\\ \hline\cr
q_{I}&-1&-1&0&0\\\ {\bf S}[q_{I}]&-1&-1&0&0\\\ {\bf L^{+}}[q_{I}]&0&0&0&-1\\\
{\bf L^{-}}[q_{I}]&0&0&1&0\end{array}$ (64)

The coefficient of the ${\bf S}[q_{I}]{\bf S}[q_{J}]$ OPE comes from
performing the integrals involved in Euclidean space; see the discussion in
section 3.

OPEs among $T,\,{\bf L}^{\pm}[T],{\bf S}[T]$ and $q_{I},\,{\bf
L}^{\pm}[q_{I}],{\bf S}[q_{I}]$:

$\displaystyle
X_{T}(z)Y_{I}(0)\sim\,\left(\frac{1}{2z^{2}}+\frac{1}{z}{\partial}_{z}\right){\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{V_{I}}}(0)\mbox{\quad
for\quad}{\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{V_{I}}}\in\\{q_{I},\,{\bf
L}^{\pm}[q_{I}],\,{\bf S}[q_{I}]\\}$
$\displaystyle\begin{array}[]{c|cccc}X_{T}\backslash Y_{I}&q_{I}&{\bf
S}[q_{I}]&{\bf L^{+}}[q_{I}]&{\bf L^{-}}[q_{I}]\\\ \hline\cr
T&q_{I}&q_{I}&0&0\\\ {\bf S}[T]&{\bf S}[q_{I}]&{\bf S}[q_{I}]&{\bf
L^{+}}[q_{I}]&{\bf L^{-}}[q_{I}]\\\ {\bf L^{+}}[T]&{\bf L^{+}}[q_{I}]&{\bf
L^{+}}[q_{I}]&0&q_{I}\\\ {\bf L^{-}}[T]&{\bf L^{-}}[q_{I}]&{\bf
L^{-}}[q_{I}]&q_{I}&0\end{array}$ (70)

OPEs among $T,{\bf L}^{\pm}[T],{\bf S}[T]$:

$\displaystyle X_{T}(z)Y_{T}(0)$
$\displaystyle\sim-\frac{{\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{a}}}{2z^{4}}+\frac{2({\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{U}}{\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{V}})}{z^{3}}+\frac{{\partial}_{z}({\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{U}}{\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{V}})}{z^{2}}\
\,+\frac{{\partial}_{z}^{2}({\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{U}}{\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{V}})}{z}$
$\displaystyle\hskip
99.58464pt+\frac{2({\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{U}}{\wedge}{\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{V}}^{\prime})}{z^{2}}+\frac{{\partial}_{z}({\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{U}}{\wedge}{\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{V}}^{\prime})}{z}$
$\displaystyle\hskip
165.02597pt+\frac{({\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{U}}^{\prime}{\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{V}}^{\prime})}{z}$
(71)
$\displaystyle\equiv\,({\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{a}},{\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{U}},{\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{V}})\mbox{\quad
for\quad}{\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{a}}\in\mathbb{R}\mbox{\quad
and\quad}{\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{U}},{\color[rgb]{0.87890625,0.00390625,0.1328125}\definecolor[named]{pgfstrokecolor}{rgb}{0.87890625,0.00390625,0.1328125}{V}}\in\\{q,\,{\bf
L}^{\pm}[q],\,{\bf S}[q]\\},$

where we have defined the normal ordered products

$\displaystyle(UV)$ $\displaystyle\doteq-\frac{{\epsilon}^{IJ}}{4}{:}\mkern
1.0muU_{I}V_{J}\mkern 1.6mu{:}(0)\ ,$ $\displaystyle(U{\wedge}V^{\prime})$
$\displaystyle\doteq+\frac{{\epsilon}^{IJ}}{4}{:}\mkern
1.0mu(U_{I}\,{\partial}_{z}V_{J}-{\partial}_{z}U_{I}\,V_{J})\mkern
1.6mu{:}(0)\ ,$ (72) $\displaystyle(U^{\prime}V^{\prime})$
$\displaystyle\doteq{\epsilon}^{IJ}{:}\mkern
1.0mu{\partial}_{z}U_{I}\,{\partial}_{z}V_{J}\mkern 1.6mu{:}(0)\ .$

Then

$\displaystyle\begin{array}[]{c|cccc}X_{T}\backslash Y_{T}&T&{\bf S}[T]&{\bf
L^{+}}[T]&{\bf L^{-}}[T]\\\ \hline\cr T&(1,q,q)&(1,q,{\bf S}[q])&(0,q,{\bf
L^{+}}[q])&(0,q,{\bf L^{-}}[q])\\\ {\bf S}[T]&(1,{\bf S}[q],q)&(1,{\bf
S}[q],{\bf S}[q])&(0,{\bf S}[q],{\bf L^{+}}[q])&(0,{\bf S}[q],{\bf
L^{-}}[q])\\\ {\bf L^{+}}[T]&(0,{\bf L^{+}}[q],q)&(0,{\bf L^{+}}[q],{\bf
S}[q])&(0,{\bf L^{+}}[q],{\bf L^{+}}[q])&(1,{\bf L^{+}}[q],{\bf L^{-}}[q])\\\
{\bf L^{-}}[T]&(0,{\bf L^{-}}[q],q)&(0,{\bf L^{-}}[q],{\bf S}[q])&(-1,{\bf
L^{-}}[q],{\bf L^{+}}[q])&(0,{\bf L^{-}}[q],{\bf L^{-}}[q])\end{array}$ (78)

Note that

$\displaystyle(UV)$ $\displaystyle=-(-)^{|U|\cdot|V|}(VU),$
$\displaystyle(U{\wedge}V^{\prime})$
$\displaystyle=+(-)^{|U|\cdot|V|}(V{\wedge}U^{\prime}),$
$\displaystyle(U^{\prime}V^{\prime})$
$\displaystyle=-(-)^{|U|\cdot|V|}(V^{\prime}U^{\prime}),$

where $|U|=0$ for a boson, and $|U|=1$ for a fermion.

We now make a few preliminary observations on the structure of this operator
algebra. The most striking observation is that while the ${\bf L}^{\pm}[q]$
OPEs in (64) describe a pair of conjugate free symplectic fermions, upon
diagonalizing the bosonic 2-point functions (64) by defining the new basis
(suppressing flavor indices)

$\displaystyle q_{\pm}\doteq\frac{1}{2}(q\pm{\bf S}[q]),$ (79)

their OPEs, in the notation of (64), become

$\displaystyle\begin{array}[]{c|cc}X\backslash Y&q_{-}&q_{+}\\\ \hline\cr
q_{-}&0&0\\\ q_{+}&0&-1\end{array}\qquad.$ (83)

This means that $q_{+}$ is a free symplectic boson. But $q_{-}$ has a
completely regular OPE, analogous to a chiral ring element. In fact, it is
immediate from (64) and (70) that $q_{-}$ has completely regular operator
products with _all_ the fields we have constructed.

This raises the question of whether $q_{-}$ vanishes in cohomology: does

$\displaystyle{\bf S}[q_{I}]=_{\mathbbmtt{T}}q_{I}\qquad?$ (84)

We do not see a direct way of proving such a relation just on the basis of
their expressions as operator representatives of twisted Schur cohomology
classes in the 4d SCFT. An indirect way is to show that all their operator
products coincide with a generating set of operators for the extended vertex
algebra. The VOA of a free symplectic boson is (strongly) generated by
$q_{I}$, but we do not know of a generating set of operators for the extended
vertex algebra, even though the underlying 4d SCFT is free. Indeed,
determining a generating set requires understanding the cohomological
equivalences among descent operators and their normal-ordered products. Note
that even if the equivalence in (84) between a local operator and its surface
descent operator were true, this would have to be a property particular to
$q_{I}$. For instance, we must have $T\neq_{\mathbbmtt{T}}{\bf S}[T]$ since
their operator products with ${\bf L}^{\pm}[q_{I}]$ shown in (70) are
different.

Another observation is that, as expected, the $T$ vertex operator has the
usual Virasoro OPE with central charge $-1$,

$\displaystyle
T(z)T(0)\sim\frac{-1}{2z^{4}}+\frac{2T(0)}{z^{2}}+\frac{{\partial}_{z}T(0)}{z},$
(85)

by virtue of the fact that
$T=(q{\wedge}q^{\prime})=\frac{1}{2}{\epsilon}^{IJ}q_{I}{\partial}_{z}q_{J}$
by (55). But although $T$ is a Virasoro operator, it is not the stress-energy
operator of the algebra. This is because to be a stress-energy operator of a
vertex algebra, the algebra must be decomposable into a basis of primary
operators, $\Phi_{i}(z)$, and their descendants,
${\partial}_{z}^{n}\Phi_{i}(z)$, such that

$\displaystyle
T(z)\Phi_{i}(0)\sim\frac{h_{i}\Phi_{i}(0)}{z^{2}}+\frac{{\partial}_{z}\Phi_{i}(0)}{z},$
(86)

for some $h_{i}\in\mathbb{N}/2$. If a stress-energy operator exists obeying
(85) and (86), then the vertex algebra is called a _conformal vertex algebra_
kac1998vertex  or, somewhat confusingly, a _vertex operator algebra_. Define
the modes of $T$ by the expansion $T(z)=\sum_{n}T_{n}z^{-n-2}$ in the usual
way. The conformal vertex algebra conditions imply that the action of these
modes on vertex operators coincides with that of the $L_{n}$
$\mathfrak{sl}_{2}$ generators, i.e., $T_{n}=L_{n}$ for $n\in\\{-1,0,1\\}$. In
particular, $T_{-1}={\partial}_{z}$ and the eigenvalues of $T_{0}$ are the
chiral weights $h_{i}$. The chiral weight of $T(z)$ is $h_{T}=2$.

In our case $T$ does not obey (86), so it is not a stress-energy operator for
the extended vertex algebra. This is clear from (70) since $T(z){\bf
S}[q](0)\sim(\cdots)q(0)$. A key question is then whether there exists a
stress-energy operator for our extended vertex algebra. We look for one by
searching for suitable bosonic operators with chiral weight 2. Recalling that
$q$ and its descent operators ${\bf L}^{\pm}[q]$ and ${\bf S}[q]$ all have
weight $h_{q}=1/2$, while $T$ and its descent operators all have weight
$h_{T}=2$, the possible weight-2 bosonic operators appearing in the OPEs we
have calculated are

$\displaystyle(q{\wedge}q^{\prime}),\quad(q{\wedge}{\bf
S}[q]^{\prime}),\quad({\bf S}[q]{\wedge}{\bf S}[q]^{\prime}),\quad T,\quad{\bf
S}[T],\quad({\bf L^{+}}[q]{\wedge}{\bf L^{-}}[q]^{\prime}),$ (87)

in the notation of (4). We have already remarked that
$T=(q{\wedge}q^{\prime})$. If (84) were true, then the first four operators of
(87) would be equivalent. Equivalences in ${\mathbbmtt{T}}_{\pm}$ cohomology
among the remaining weight-2 operators, though possible, are not apparent. We
find no linear combination of these operators which satisfies both (85) and
(86) for the subset of operators we are examining, even if we assume (84) is
true. This, unfortunately, does not settle the question of whether the
extended vertex algebra is a conformal vertex algebra since we have only
looked at a limited number of operators and their OPEs.

##  5 Future directions

We have shown how a version of topological descent applied to twist-translated
Schur operators gives rise to a large set of new protected line, surface, and
wall operators, and enlarges the vertex operator algebra of any 4d
${\mathcal{N}}{=}2$ SCFT. Some details of this construction were deferred to
Argyres_2022xx .

A basic open question about this construction is to formulate an appropriate
regularization of correlation functions of surface or wall operators whose
world volumes meet at light-like infinity. This was discussed in section 3,
where we sidestepped the problem by computing ${\bf S}$-${\bf S}$ correlators
in euclidean signature. It is not clear how this can work for ${\widehat{\bf
S}}$-${\widehat{\bf S}}$ correlators.

Another task is to understand the analytic structure of correlators involving
the ${\widehat{\bf W}}^{\pm}$ quasi-topological boundary operators.

A number of open questions were also raised in section 4 concerning the
structure of the ${\mathcal{O}}$-${\bf L}^{\pm}$-${\bf S}$ subalgebra. A key
question is whether the extended vertex algebra is a conformal vertex algebra
(a.k.a., a vertex operator algebra). The computation of the OPEs recorded in
section 4 relied crucially on various series of cohomological operator
identities, whose derivation is given in Argyres_2022xx . To sort out these
central structural questions it seems necessary to find a way of
systematically organizing and generating these identities.

Extended vertex algebra OPE computations in the free vector multiplet SCFT is
a next obvious step. By combining free hypermultiplet and vector multiplet
results, weak coupling extended vertex algebras in conformal gauge theories
can also be obtained. It would be interesting to study extended vertex algebra
modules built on light-like Wilson-’t Hooft-type line operators in twisted
Schur cohomology. These should also be calculationally accessible in conformal
gauge theories.

###### Acknowledgements.

It is a pleasure to thank C. Beem, L. Bianchi, A. Bourget, M. Bullimore, M.
Dedushenko, C. Elliott, A. Ferrari, I. Garcia-Etxebarria, J. Grimminger, O.
Gwilliam, P. Kravchuk, M. Lemos, M. Martone, L. Rastelli, and D. Simmons-
Duffin for helpful discussions and suggestions. PCA and MW are supported in
part by DOE grant DE-SC0011784. ML is supported in part by the National
Research Foundation of Korea (NRF) Grant 2021R1A2C2012350.

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