Nadav Drukker,1 Ziwen Kong,1 Malte Probst,1

Marcia Tenser,2,3 and Diego Trancanelli2,4

1Department of Mathematics, King’s College London,
The Strand, WC2R 2LS London, United-Kingdom

2Institute of Physics, University of São Paulo, 05314-970 São Paulo, Brazil

3Dipartimento di Fisica, Università degli Studi di Milano-Bicocca,
Piazza della Scienza 3, 20126 Milano, Italy

4Dipartimento di Scienze Fisiche, Informatiche e Matematiche,
Università di Modena e Reggio Emilia, via G. Campi 213/A, 41125 Modena, Italy
&
INFN Sezione di Bologna, via Irnerio 46, 40126 Bologna, Italy

{nadav.drukker, mltprbst, marciatenser<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
# Conformal and non-conformal hyperloop deformations of the 1/2 BPS circle

###### Contents

  1. 1 Introduction and summary
  2. 2 ${\cal N}=4$ Chern-Simons-matter theories on $S^{3}$
  3. 3 The 1/2 BPS Wilson loop and its symmetries
  4. 4 Two-node hyperloops
    1. 4.1 Deformations with $\Pi\neq 0$
    2. 4.2 Deformations with $\Pi=0$
  5. 5 Longer quivers and twisted hypers
    1. 5.1 Deformations with $\Pi\neq 0$
    2. 5.2 Deformations with $\Pi=0$
  6. 6 Special cases
    1. 6.1 Single node bosonic loops
    2. 6.2 Two-node hyperloops with $\Pi\neq 0$
    3. 6.3 Two-node hyperloops with $\Pi=0$
    4. 6.4 Hyperloops with twisted hypers and $\Pi\neq 0$
    5. 6.5 Hyperloops with twisted hypers and $\Pi=0$
  7. A Symmetries of the 1/2 BPS Wilson loop
  8. B The covariant derivative
  9. C Extra fermionic terms

##  1 Introduction and summary

A distinguishing feature of three-dimensional supersymmetric conformal field
theories are the vast moduli spaces of BPS line operators annihilated by some
supercharges. For operators that are conformal, this was understood from an
algebraic point of view in [1], but many examples of conformally invariant
circular line operators, including continuous families of them, were found
before, see for example [2, 3, 4, 5, 6, 7, 8, 9, 10] and [11] for a review.

In the absence of an approach allowing for a full classification, we continue
here to develop and employ constructive methods of identifying BPS Wilson loop
operators called _hyperloops_ , finding a plethora of new observables, some of
which are conformally invariant and some of which are not, greatly enlarging
the known moduli spaces.

The theories we study are ${\mathcal{N}}=4$ supersymmetric Chern-Simons-matter
with either linear or circular quiver structure, characterized by the coupling
of the gauge multiplet to hypermultiplets and twisted hypermultiplets [12, 13,
14, 15]. The 2-node circular quiver has ${\mathcal{N}}=6$ supersymmetry and is
the ABJ(M) theory [16, 17], so most of what we say applies there as well. For
concreteness, we consider theories on $S^{3}$ and focus on operators supported
along a great circle.111Of course, it would be interesting to consider other
contours, such as latitudes, or generic curves on an $S^{2}\subset S^{3}$,
along the lines of what has been done in [18, 19, 20] for ${\cal N}=4$ super
Yang-Mills in four dimensions and in [21] for the ABJ(M) theory.

In a recent paper [10], some of us already studied Wilson loops in this same
setting. Those hyperloops were written as deformations of bosonic Wilson loops
that preserve 2 or 4 supercharges (so they are 1/8 or 1/4 BPS). Starting with
particular block-diagonal combinations of bosonic connections
${\mathcal{L}}_{\text{bos}}$ annihilated by a supercharge $Q$, it was found
that one can deform them as follows

${\mathcal{L}}_{\text{bos}}\to{\mathcal{L}}={\mathcal{L}}_{\text{bos}}-iQG+G^{2},$
(1.1)

where $G$ is a matrix constructed out of bosonic fields in the
hypermultiplets. The resulting operator is still supersymmetric, by
construction, and is defined in terms of a superconnection containing the
fermionic fields, which is something typical of supersymmetric Chern-Simons
theories [22]. Another peculiarity of three-dimensional theories is that the
$Q$ variation of ${\mathcal{L}}$ does not vanish per se, as it happens in the
four-dimensional counterpart of these objects, but it is instead a total
covariant derivative, so the entire Wilson loop, which is a gauge invariant
object, is still annihilated by $Q$.

In the current work we apply a similar philosophy to [10], but we employ as
the starting point of the deformation the 1/2 BPS Wilson loop found in [4]
(see also [3]), rather than a bosonic loop:

${\mathcal{L}}_{1/2}\to{\mathcal{L}}={\mathcal{L}}_{1/2}+\textrm{deformation},$
(1.2)

with the details of the deformation given after (4.1) below. The 1/2 BPS loop
is also a particular deformation of the bosonic loop as in (1.1), so our
current construction includes all of those found previously.

Moreover, unlike the construction in [10], where a single choice of
supercharge based on the original Wilson loops was employed, here we consider
any supercharge annihilating the 1/2 BPS loop, so any linear combination of a
basis of 8 supercharges. In particular, in cases when the supercharge $Q$ has
an appropriate kernel, we find infinite-dimensional moduli spaces, since
(roughly speaking) we can insert any of the operators in the kernel any number
of times at any point along the loop.

This new procedure allows us to uncover new families of supersymmetric line
operators. For example, we have discovered:

  * •

Previously unrecognized bosonic loops preserving 2 and 3 supercharges, which
are therefore 1/8 and 3/16 BPS, in addition to the known ones preserving 2 or
4 supercharges, see Section 6.1.

  * •

New 1/8 and 1/4 BPS loops that do not share supercharges with any known
bosonic Wilson loops, so could not have been found by relying on (1.1). Of
particular note is a subclass of these loops, which depends on one parameter
(after fixing 4 supercharges), for which the variation of the superconnection
under conformal transformations of the circle is a total derivative, see
Section 6.3.2.

This forms a new class of previously unrecognized line operators that are
classically conformally invariant. Unlike the 1/2 BPS or 1/4 BPS bosonic
loops, the one-dimensional conformal algebra is not generated by the
supercharges that they preserve, but is an outer automorphism of it. As we
cannot rely on supersymmetry to guarantee conformality, it would be extremely
interesting to examine them at the quantum level and verify whether they are
truly conformally invariant.

There are various natural directions that could be pursued starting from these
results. The most obvious one is to try to compute the expectation value of
these operators, using localization for example. This typically starts with
determining to which cohomological class the various operators belong. In
previous examples [10] based on (1.1), as well as in the original papers [2,
4], it was found that the bosonic operators and their fermionic deformations
are cohomologically equivalent. In this context we know however that this does
not hold, as we find loops, such as the latitudes, that are known to have
different expectation values from the 1/2 BPS circle [23, 24, 25, 26]. This of
course makes these new classes of operators even more interesting.

The next natural question is about the holographic duals. While the
holographic duals of 1/2 BPS loops in some ${\cal N}=4$ Chern-Simons-matter
theories have been identified [3, 4, 27], the question of what is dual to less
supersymmetric (and/or higher representation) operators has not been addressed
yet.222A first examination of a possible moduli space of 1/6 BPS loops in
ABJ(M) theory was done in [28, 29].

Finally, it would be interesting to study the moduli spaces of conformal loops
as defect conformal manifolds and analyze the defect conformal field theory
they define, along the lines of what has been done for the ABJ(M) theory in
[30] and see also [31]. For non-conformal loops it would be interesting to
understand their renormalisation group flows [32, 33].

This paper is organised as follows. In the next section we present the
notation for the theories and the supersymmetry variations of the fields. In
Section 3 we present the simplest 1/2 BPS Wilson loop in these theories, which
is the starting point of the deformations. The bulk of the calculations is in
Sections 4 and 5, focusing respectively on loops involving only two nodes of
the quiver and those involving more, respectively. For the benefit of the
casual reader we collect the main results and present a detailed analysis of
special interesting examples in Section 6. Some details are presented in the
appendices.

##  2 ${\cal N}=4$ Chern-Simons-matter theories on $S^{3}$

The theories we study are ${\mathcal{N}}=4$ Chern-Simons-matter theories,
which can be represented in terms of either circular or linear quiver diagrams
[12, 13, 14, 15]. For the most part we focus on a node labeled by $I$ with
gauge field $A_{I}$ and its adjacent node with $A_{I+1}$, but in Section 5 we
also consider more nodes. The edges of the diagram represent hypermultiplets
and twisted hypermultiplets. The hypermultiplet $(q_{I}^{a},\psi_{I\dot{a}})$
couples to $A_{I}$ and $A_{I+1}$, while the twisted hypermultiplet
$(\tilde{q}_{I-1\,\dot{a}},\tilde{\psi}_{I-1}^{a})$ couples to $A_{I}$ and
$A_{I-1}$, and so on in an alternate fashion. The field content is summarized
in the quiver diagram of Figure 1, where the solid lines between nodes
represent the matter fields.

$A_{I}$$A_{I+1}$$-k$$k$$\tilde{q}_{I-1\,\dot{a}},\
\tilde{\psi}^{a}_{I-1}$$\bar{\tilde{q}}^{\dot{a}}_{I-1},\bar{\tilde{\psi}}_{I-1\,a}$$\bar{q}_{I\,a},\
\bar{\psi}^{\dot{a}}_{I}$$q^{a}_{I},\
\psi_{I\,\dot{a}}$$\tilde{q}_{I+1\,\dot{a}},\
\tilde{\psi}^{a}_{I+1}$$\bar{\tilde{q}}^{\dot{a}}_{I+1},\
\bar{\tilde{\psi}}_{I+1\,a}$ Figure 1: The quiver and field content of the
${\cal N}=4$ theory.

The scalar fields in the hypermultiplet have indices $a,b=1,2$ and are
doublets of the $SU(2)_{L}$ R-symmetry. The fermions with indices
$\dot{a},\dot{b}=\dot{1},\dot{2}$ are charged instead under $SU(2)_{R}$. This
is reversed in the twisted hypermultiplets. These indices are raised and
lowered using the appropriate epsilon symbols: $v^{a}=\epsilon^{ab}v_{b}$ and
$v_{a}=\epsilon_{ab}v^{b}$ with $\epsilon^{12}=\epsilon_{21}=1$, and similarly
for the dotted indices.

To write down the Wilson loops and the supersymmetry variations, it is useful
to define moment maps and currents, following [4, 10]

$\displaystyle\mu_{I}{}^{a}_{\ b}$
$\displaystyle=q_{I}^{a}\bar{q}_{I\,b}-\frac{1}{2}\delta^{a}_{b}q_{I}^{c}\bar{q}_{I\,c}\,,\qquad$
$\displaystyle j_{I}^{a\dot{b}}$
$\displaystyle=q_{I}^{a}\bar{\psi}_{I}^{\dot{b}}-\epsilon^{ac}\epsilon^{\dot{b}\dot{c}}\psi_{I\,\dot{c}}\bar{q}_{I\,c}\,,$
(2.1) $\displaystyle\tilde{\mu}_{I}{}^{\dot{a}}_{\ \dot{b}}$
$\displaystyle=\bar{\tilde{q}}_{I-1}^{\,\dot{a}}\tilde{q}_{I-1\,\dot{b}}-\frac{1}{2}\delta^{\dot{a}}_{\dot{b}}\bar{\tilde{q}}_{I-1}^{\,\dot{c}}\tilde{q}_{I-1\,\dot{c}}\,,\qquad$
$\displaystyle\tilde{\jmath}_{I}^{\,\dot{b}a}$
$\displaystyle=\bar{\tilde{q}}_{I-1}^{\,\dot{b}}\tilde{\psi}_{I-1}^{a}-\epsilon^{\dot{b}\dot{c}}\epsilon^{ac}\bar{\tilde{\psi}}_{I-1\,c}\tilde{q}_{I-1\,\dot{c}}\,,$
$\displaystyle\nu_{I}$ $\displaystyle=q_{I}^{a}\bar{q}_{I\,a}\,,\qquad$
$\displaystyle\tilde{\nu}_{I}$
$\displaystyle=\bar{\tilde{q}}_{I-1}^{\,\dot{a}}\tilde{q}_{I-1\,\dot{a}}\,.$

These are bilinears of the matter fields and transform in the adjoint
representation of $U(N_{I})$. Note that other bilinears of the same matter
fields can transform in the adjoint of $U(N_{I\pm 1})$. For example,
$\nu_{I+1}=\bar{q}_{Ia}q_{I}^{a}$ is built out of the same fields as
$\nu_{I}$, but it transforms in the adjoint of $U(N_{I+1})$ because of the
reversed order.

As stated in the Introduction, we define the theory on $S^{3}$ and the
hyperloops we construct are supported along the equator of this sphere. The
corresponding on-shell ${\cal N}=4$ supersymmetry transformations were derived
in [10] by relying on the decomposition of ${\mathcal{N}}=4$ to
${\mathcal{N}}=2$ theories and the transformation rules of the latter in [15,
34]. They are

$\displaystyle\delta A_{\mu\,I}$
$\displaystyle=\frac{i}{k}\xi_{a\dot{b}}\gamma_{\mu}(j_{I}^{a\dot{b}}-\tilde{\jmath}_{I}^{\,\dot{b}a})\,,\hskip-85.35826pt$
(2.2) $\displaystyle\delta q_{I}^{a}$
$\displaystyle=\xi^{a\dot{b}}\psi_{I\,\dot{b}}\,,\qquad$
$\displaystyle\delta\bar{q}_{I\,a}$
$\displaystyle=\xi_{a\dot{b}}\bar{\psi}_{I}^{\dot{b}}\,,$
$\displaystyle\delta\tilde{q}_{I-1\,\dot{b}}$
$\displaystyle=-\xi_{a\dot{b}}\tilde{\psi}_{I-1}^{a}\,,\qquad$
$\displaystyle\delta\bar{\tilde{q}}_{I-1}^{\,\dot{b}}$
$\displaystyle=-\xi^{a\dot{b}}\bar{\tilde{\psi}}_{I-1\,a}\,,\hskip
227.62204pt$ $\displaystyle\delta\psi_{I\,\dot{a}}$
$\displaystyle=i\gamma^{\mu}\xi_{b\dot{a}}D_{\mu}q_{I}^{b}+i\zeta_{b\dot{a}}q_{I}^{b}-\frac{i}{k}\xi_{b\dot{a}}(\nu_{I}q_{I}^{b}-q_{I}^{b}\nu_{I+1})+\frac{2i}{k}\xi_{b\dot{c}}\left(\tilde{\mu}_{I}{}_{\
\dot{a}}^{\dot{c}}q_{I}^{b}-q_{I}^{b}\tilde{\mu}_{I+1}{}_{\dot{a}}^{\
\dot{c}}\right),\hskip-284.52756pt$
$\displaystyle\delta\bar{\psi}_{I}^{\dot{a}}$
$\displaystyle=i\gamma^{\mu}\xi^{b\dot{a}}D_{\mu}\bar{q}_{I\,b}+i\zeta^{b\dot{a}}\bar{q}_{I\,b}-\frac{i}{k}\xi^{b\dot{a}}(\bar{q}_{I\,b}\nu_{I}-\nu_{I+1}\bar{q}_{I\,b})+\frac{2i}{k}\xi^{b\dot{c}}\left(\bar{q}_{I\,b\,}\tilde{\mu}_{I}{}_{\
\dot{c}}^{\dot{a}}-\tilde{\mu}_{I+1}{}_{\dot{c}}^{\
\dot{a}}\bar{q}_{I\,b}\right),\hskip-312.9803pt$
$\displaystyle\delta\tilde{\psi}_{I-1}^{a}$
$\displaystyle=-i\gamma^{\mu}\xi^{a\dot{b}}D_{\mu}\tilde{q}_{I-1\,\dot{b}}-i\zeta^{a\dot{b}}\tilde{q}_{I-1\,\dot{b}}+\frac{i}{k}\xi^{a\dot{b}}(\tilde{q}_{I-1\,\dot{b}}\tilde{\nu}_{I}-\tilde{\nu}_{I-1}\tilde{q}_{I-1\,\dot{b}})\hskip-284.52756pt$
$\displaystyle\quad-\frac{2i}{k}\xi^{b\dot{c}}\left(\tilde{q}_{I-1\,\dot{c}\,}\mu_{I}{}_{\
b}^{a}-\mu_{I-1}{}_{b}^{\ a}\tilde{q}_{I-1\,\dot{c}}\right),\hskip-28.45274pt$
$\displaystyle\delta\bar{\tilde{\psi}}_{I-1\,a}$
$\displaystyle=-i\gamma^{\mu}\xi_{a\dot{b}}D_{\mu}\bar{\tilde{q}}_{I-1}^{\,\dot{b}}-i\zeta_{a\dot{b}}\bar{\tilde{q}}_{I-1}^{\,\dot{b}}+\frac{i}{k}\xi_{a\dot{b}}(\tilde{\nu}_{I}\bar{\tilde{q}}_{I-1}^{\,\dot{b}}-\bar{\tilde{q}}_{I-1}^{\,\dot{b}}\tilde{\nu}_{I-1})\hskip-284.52756pt$
$\displaystyle\quad-\frac{2i}{k}\xi_{b\dot{c}}\left(\mu_{I}{}_{\
a}^{b}\bar{\tilde{q}}_{I-1}^{\,\dot{c}}-\bar{\tilde{q}}_{I-1}^{\,\dot{c}}\mu_{I-1}{}_{a}^{\
b}\right),$

where $\xi_{a\dot{b}}$ are the Killing spinors and
$\zeta_{a\dot{b}}=\frac{1}{3}\gamma^{\mu}\nabla_{\mu}\xi_{a\dot{b}}$. The
covariant derivative acts as, for instance,
$D_{\mu}q^{a}_{I}=\partial_{\mu}q^{a}_{I}-iA_{\mu,I}q^{a}_{I}+iq^{a}_{I}A_{\mu,I}$.

Specifically, each supersymmetry parameter $\xi^{a\dot{b}}$ is a linear
combination of four (conformal) Killing spinors on $S^{3}$ denoted
$\\{\xi^{l}$, $\xi^{\bar{l}}$, $\xi^{r}$, $\xi^{\bar{r}}\\}$, i.e.

$\xi_{\alpha}^{a\dot{b}}=\xi_{\imath}^{a\dot{b}}\xi^{\imath}_{\alpha}+\xi_{\bar{\imath}}^{a\dot{b}}\xi^{\bar{\imath}}_{\alpha}\,,$
(2.3)

where $\imath=l,r$ and $\bar{\imath}=\bar{l},\bar{r}$ label doublets of the
$SO(2,1)$ conformal symmetry along the circle. All together they form a
quartet of the $SO(4,1)$ symmetry of $S^{3}$. The index $\alpha=\pm$ is the
spinor index.

The Killing spinors obey

$\nabla_{\mu}\xi^{l,\bar{l}}=\frac{i}{2}\gamma_{\mu}\xi^{l,\bar{l}}\,,\qquad\nabla_{\mu}\xi^{r,\bar{r}}=-\frac{i}{2}\gamma_{\mu}\xi^{r,\bar{r}}\,.$
(2.4)

Along the circle we take $\gamma_{\varphi}=\sigma_{3}$ and the Killing spinors
reduce to [35]

$\xi^{l}_{\alpha}=\begin{pmatrix}1\\\
0\end{pmatrix},\qquad\xi^{\bar{l}}_{\alpha}=\begin{pmatrix}0\\\
1\end{pmatrix},\qquad\xi^{r}_{\alpha}=\begin{pmatrix}e^{-i\varphi}\\\
0\end{pmatrix},\qquad\xi^{\bar{r}}_{\alpha}=\begin{pmatrix}0\\\
e^{i\varphi}\end{pmatrix},$ (2.5)

whence one finds
$\zeta^{l,\bar{l}}_{a\dot{b}}=\frac{i}{2}\xi^{l,\bar{l}}_{a\dot{b}}$ and
$\zeta^{r,\bar{r}}_{a\dot{b}}=-\frac{i}{2}\xi^{r,\bar{r}}_{a\dot{b}}$.

We work in Euclidean signature and take the gamma-matrices,
$(\gamma^{\mu})_{\alpha}^{\;\;\beta}$, to be given by the Pauli matrices. As
usual, the spinor contractions are such that

$\xi_{1}\xi_{2}\equiv\xi_{1}^{\alpha}\xi_{2,\alpha}=+\xi_{2}\xi_{1}\,,\qquad\xi_{1}\gamma^{\mu}\xi_{2}\equiv\xi_{1}^{\alpha}(\gamma^{\mu})_{\alpha}^{\;\;\beta}\xi_{2,\beta}=-\xi_{2}\gamma^{\mu}\xi_{1}\,.$
(2.6)

It follows that the Killing spinors in (2.5) satisfy
$\xi^{\bar{l}}\xi^{l}=\xi^{l}\xi^{\bar{l}}=1$ and
$\xi^{\bar{l}}\gamma^{\mu}\xi^{l}=-\xi^{l}\gamma^{\mu}\xi^{\bar{l}}=\delta^{\mu}_{\varphi}$,
and similarly for the contractions involving $\xi^{r}$ and $\xi^{\bar{r}}$.

##  3 The 1/2 BPS Wilson loop and its symmetries

The starting point of the deformation (1.2) considered in this paper is a
particular 1/2 BPS loop of the theory. As shown originally for the ABJ(M)
theory in [22] and for ${\mathcal{N}}=4$ theories in [4] (see also [3]), such
a Wilson loop must couple to at least two vector fields, as well as to the
matter fields charged under them. We take the loop built around the $I$ and
$I+1$ nodes of the particular form

$W_{1/2}=\operatorname{sTr}{\cal P}\exp
i\oint{\mathcal{L}}_{1/2}\,d\varphi\,,\qquad{\mathcal{L}}_{1/2}=\begin{pmatrix}{\mathcal{A}}_{I}&-i\bar{\alpha}\psi_{I\dot{1}-}\\\
i\alpha\bar{\psi}^{\dot{1}}_{I+}&{\mathcal{A}}_{I+1}-\frac{1}{2}\end{pmatrix},$
(3.1)

with

${\mathcal{A}}_{I}=A_{\varphi,I}+\frac{i}{k}\big{(}{\nu_{I}}-\tilde{\mu}\indices{{}_{I}^{\dot{1}}{}_{\dot{1}}}+\tilde{\mu}\indices{{}_{I}^{\dot{2}}{}_{\dot{2}}}\big{)}\,,\qquad{\mathcal{A}}_{I+1}=A_{\varphi,I+1}+\frac{i}{k}\big{(}{\nu_{I+1}}-\tilde{\mu}\indices{{}_{I+1}{}_{\dot{1}}^{\dot{1}}}+\tilde{\mu}\indices{{}_{I+1}{}_{\dot{2}}^{\dot{2}}}\big{)}\,.$
(3.2)

The constants $\alpha$ and $\bar{\alpha}$ (which are not complex conjugate to
each other) satisfy $\alpha\bar{\alpha}=2i/k$ and the Wilson loop does not
depend on their actual value, so we could fix them to be equal, but we leave
them instead as a constant gauge parameter. We could allow for them to depend
on $\varphi$ at the expense of a $U(1)$ gauge transformation at the bottom
right entry:
${\mathcal{A}}_{I+1}-\frac{1}{2}\to{\mathcal{A}}_{I+1}-\frac{1}{2}-i\alpha^{-1}\partial_{\varphi}\alpha$.
The origin of the shift $-1/2$ in the connection (and the resulting appearance
of the supertrace if compared with the original definition in [4] in terms of
the trace) is explained in [11].

As we verify below, the eight supercharges preserved by this loop are

$Q^{\dot{2}a+}_{\imath}\,,\qquad Q^{\dot{1}a-}_{\bar{\imath}}\,.$ (3.3)

The spinor indices $\alpha=\pm$ are taken upstairs, to contract with the
downstairs indices of the Killing spinors in (2.5). To relate to the notation
in (2.2), we can represent the supersymmetry transformation as
$\delta=-\xi^{\imath}_{a\dot{b}\alpha}Q^{\dot{b}a\alpha}_{\imath}-\xi^{\bar{\imath}}_{a\dot{b}\alpha}Q^{\dot{b}a\alpha}_{\bar{\imath}}$.

Looking at the form of the Killing spinors along the circle (2.5), one can
write a general superposition of the preserved supercharges (3.3) as

$Q=\eta^{\imath}_{a}Q^{\dot{2}a+}_{\imath}+\bar{\eta}^{\imath}_{a}(\sigma^{1})_{\imath}^{\
\bar{\imath}}Q^{\dot{1}a-}_{\bar{\imath}}=\eta^{\imath}_{a}\bar{v}_{\imath}Q^{\dot{2}a+}+\bar{\eta}^{\imath}_{a}v_{\imath}Q^{\dot{1}a-}\,,$
(3.4)

with Grassmann-even parameters $\eta^{\imath}_{a}$, $\bar{\eta}^{\imath}_{a}$
(which, again, are not complex conjugate) and auxiliary $SO(2,1)$ spinors

$v_{\imath}=\begin{pmatrix}e^{+i\varphi}\\\
1\end{pmatrix}_{\imath}\,,\qquad\bar{v}_{\imath}=\begin{pmatrix}1\\\
e^{-i\varphi}\end{pmatrix}_{\imath}\,.$ (3.5)

The supersymmetry variations generated by a supercharge parameterised in such
fashion can then be computed by reading off

$\xi_{a{\dot{1}}}=\begin{pmatrix}(\eta\bar{v})_{a}\\\
0\end{pmatrix},\qquad\xi_{a{\dot{2}}}=\begin{pmatrix}0\\\
(\bar{\eta}v)_{a}\end{pmatrix}.$ (3.6)

In the right-most expression in (3.4), $Q^{\dot{a}a}$ acts in the same way as
$Q^{\dot{a}a}_{l}$, that is without the extra phases $e^{\pm i\varphi}$, which
have been absorbed in the definition of $v_{\imath}$ and $\bar{v}_{\imath}$.
There are four $\eta^{\imath}_{a}$ and four $\bar{\eta}^{\imath}_{a}$
parameters, but the supercharges are identified up to rescalings, so the space
of real supercharges is in fact $\mathbb{R}\mathbb{P}^{7}$.

As noted already in [4], there exists another Wilson loop with the same gauge
fields and preserving the exact same symmetries, but coupling instead to other
fields in the hypermultiplets. This other operator has the superconnection

${\mathcal{L}}_{1/2}^{\prime}=\begin{pmatrix}{\mathcal{A}}_{I}&-i\bar{\alpha}\psi_{I\dot{2}+}\\\
i\alpha\bar{\psi}^{\dot{2}}_{I-}&{\mathcal{A}}_{I+1}+\frac{1}{2}\end{pmatrix},$
(3.7)

with the opposite sign for the $\nu$’s compared to the ones appearing in
(3.2). All the moduli spaces that we find include in them also this operator
as a special point of enhanced supersymmetry. It is then just a matter of
choice to do the analysis around (3.1), rather than around this one.

Before examining in detail the supersymmetries preserved by the loop defined
in (3.1), let us compute its bosonic symmetries. Our notation and further
details on the algebra can be found in Appendix A. Firstly, notice that the
superconnection (3.1) contains only singlets of the
$\mathfrak{s}\mathfrak{u}(2)_{L}$ R-symmetry, which is clearly preserved. The
bosonic part of ${\cal{L}}_{1/2}$ is also annihilated by transverse rotations
$T_{\perp}$, but it acts on the fermions by the Pauli matrix $\sigma_{3}$, see
(A.6). Since spinor indices appear in ${\cal{L}}_{1/2}$ accompanied by
opposite R-symmetry indices, we can cancel the action of $T_{\perp}$ by an
appropriate multiple of the $\bar{R}_{3}$ generator of the unbroken
${\mathfrak{u}}(1)_{R}$ R-symmetry, and, indeed, the combination
$L_{\perp}\equiv-i\left(T_{\perp}+i\bar{R}_{3}/2\right)$ annihilates
${\cal{L}}_{1/2}$. As for the action of the conformal generators $J_{0}$ and
$J_{\pm}$ on the $1/2$ BPS loop, using the expressions (A.4) and (A.5)

$\displaystyle
iJ_{0}\,{\cal{L}}_{1/2}=\frac{d{\cal{L}}_{1/2}}{d\varphi}-\frac{\partial{\cal{L}}_{1/2}}{\partial\varphi}-[\sigma_{3},{\cal{L}}_{1/2}]\,.$
(3.8)

Since the ${\cal{L}}_{1/2}$ does not contain any explicit
$\varphi$-dependence, we may bring this into the form333The precise definition
of the covariant derivative ${\cal D}_{\varphi}^{{\cal L}_{1/2}}$ is in
Appendix B.

$\displaystyle
iJ_{0}\,{\cal{L}}_{1/2}={\cal{D}}^{{\cal{L}}_{1/2}}_{\varphi}\left({\cal{L}}_{1/2}+\sigma_{3}\right).$
(3.9)

Total covariant derivatives can be integrated away, so this guarantees
invariance of the $1/2$ BPS loop under $J_{0}$. Similar arguments show that
$J_{\pm}$ are preserved as well. Finally, note that while acting on
${\cal{L}}_{1/2}$ with $T_{\perp}$ (or equivalently $\bar{R}_{3}$) gives a
non-zero result, it still takes the form of a covariant derivative

$\displaystyle
T_{\perp}{\cal{L}}_{1/2}\propto[\sigma_{3},{\cal{L}}_{1/2}]={\cal{D}}^{{\cal{L}}_{1/2}}_{\varphi}\sigma_{3}\,.$
(3.10)

Consequently, $\bar{R}_{3}$ and $T_{\perp}$ are preserved separately.

We now proceed to evaluate the action of the supercharge $Q$ in (3.4) on the
superconnection ${\mathcal{L}}_{1/2}$ (3.1) and to verify that it is equal to
a total derivative. This also introduces a lot of the notation required in the
rest of the paper.

First, to write the action of $Q$ on the hypermultiplet fields it is useful to
define rotated scalar fields

$r^{1}\equiv(\eta\bar{v})_{a}q^{a}\,,\quad
r^{2}\equiv(\bar{\eta}v)_{a}q^{a}\,,\quad\bar{r}_{1}\equiv\epsilon^{ab}(\bar{\eta}v)_{a}\bar{q}_{b}\,,\quad\bar{r}_{2}\equiv-\epsilon^{ab}(\eta\bar{v})_{a}\bar{q}_{b}\,,$
(3.11)

where $(\eta\bar{v})_{a}=\eta^{\imath}_{a}\bar{v}_{\imath}$ and likewise for
$(\bar{\eta}v)_{a}$. Now

$Qr^{1}=-\Pi\psi_{\dot{2}+}\,,\quad Qr^{2}=\Pi\psi_{\dot{1}-}\,,\quad
Q\bar{r}_{1}=\Pi\bar{\psi}^{\dot{2}}_{-}\,,\quad
Q\bar{r}_{2}=-\Pi\bar{\psi}^{\dot{1}}_{+}\,,$ (3.12)

where the $\pm$ subscripts are spinor indices and

$\displaystyle\Pi\equiv\epsilon^{ab}(\bar{\eta}v)_{a}(\eta\bar{v})_{b}$ (3.13)

is a quantity that plays a central role in our analysis.

It is not too hard to show, using (2.2), that the second variation of the
rotated scalars is

$\displaystyle Q^{2}r^{1}$
$\displaystyle=\Pi\left(i(\eta\bar{v})_{a}\partial_{\varphi}q^{a}-\frac{1}{2}(\eta\sigma^{3}\bar{v})_{a}q^{a}+{\mathcal{A}}_{I}r^{1}-\frac{2i}{k}\nu_{I}r^{1}-r^{1}{\mathcal{A}}_{I+1}+\frac{2i}{k}r^{1}\nu_{I+1}\right),$
(3.14) $\displaystyle Q^{2}r^{2}$
$\displaystyle=\Pi\left(i(\bar{\eta}v)_{a}\partial_{\varphi}q^{a}-\frac{1}{2}(\bar{\eta}\sigma^{3}v)_{a}q^{a}+{\mathcal{A}}_{I}r^{2}-r^{2}{\mathcal{A}}_{I+1}\right).$

Now, using

$\displaystyle
2i\partial_{\varphi}(\eta\bar{v})_{a}=(\eta\bar{v})_{a}-(\eta\sigma^{3}\bar{v})_{a}\,,\qquad-2i\partial_{\varphi}(\bar{\eta}v)_{a}=(\bar{\eta}v)_{a}+(\bar{\eta}\sigma^{3}v)_{a}\,,$
(3.15)

and

$r^{1}\bar{r}_{1}+r^{2}\bar{r}_{2}=\Pi(q^{1}\bar{q}_{1}+q^{2}\bar{q}_{2})=\Pi\nu\,,$
(3.16)

these second variations can be written as

$\displaystyle Q^{2}r^{1}$
$\displaystyle=\Pi\left(i\partial_{\varphi}r^{1}-\frac{1}{2}r^{1}+{\mathcal{A}}_{I}r^{1}-r^{1}{\mathcal{A}}_{I+1}\right)-\frac{2i}{k}(r^{2}\bar{r}_{2}r^{1}-r^{1}\bar{r}_{2}r^{2})\,,$
(3.17) $\displaystyle Q^{2}r^{2}$
$\displaystyle=\Pi\left(i\partial_{\varphi}r^{2}+\frac{1}{2}r^{2}+{\mathcal{A}}_{I}r^{2}-r^{2}{\mathcal{A}}_{I+1}\right).$

Likewise, the anti-chiral components have double variations given by

$\displaystyle Q^{2}\bar{r}_{1}$
$\displaystyle=\Pi\left(i\partial_{\varphi}\bar{r}_{1}+\frac{1}{2}\bar{r}_{1}+\mathcal{A}_{I+1}\bar{r}_{1}-\bar{r}_{1}\mathcal{A}_{I}\right)-\frac{2i}{k}(\bar{r}_{2}r^{2}\bar{r}_{1}-\bar{r}_{1}r^{2}\bar{r}_{2})\,,$
(3.18) $\displaystyle Q^{2}\bar{r}_{2}$
$\displaystyle=\Pi\left(i\partial_{\varphi}\bar{r}_{2}-\frac{1}{2}\bar{r}_{2}+\mathcal{A}_{I+1}\bar{r}_{2}-\bar{r}_{2}\mathcal{A}_{I}\right).$

It is now straightforward to check that, when $\Pi\neq 0$, the off-diagonal
entries in ${\mathcal{L}}_{1/2}$ are equal to $-i\Pi^{-1}QH$, with

$H=\begin{pmatrix}0&\bar{\alpha}r^{2}\\\ \alpha\bar{r}_{2}&0\end{pmatrix}.$
(3.19)

One can combine this with the results above to find that the supersymmetry
variation of the 1/2 BPS connection is

$Q{\mathcal{L}}_{1/2}=\mathcal{D}_{\varphi}^{{\mathcal{L}}_{1/2}}H\,.$ (3.20)

The covariant derivative used here includes a commutator with the diagonal
part of ${\mathcal{L}}_{1/2}$ and an anticommutator with the off-diagonal
part, as explained in detail in Appendix B.

For the purpose of this calculation it was not needed to evaluate the action
of $Q^{2}$ on $r^{1}$, but only on $r^{2}$. The former is included here as it
is of relevance for the rest of the paper. Also, if one wanted to repeat the
calculation for the other 1/2 BPS loop in (3.7), one would need to replace
$r^{2}$ and $\bar{r}_{2}$ in $H$ in (3.19) with $r^{1}$ and $\bar{r}_{1}$.

##  4 Two-node hyperloops

Here we systematically study continuous deformations of the
${\mathcal{L}}_{1/2}$ in (3.1) preserving the supercharge $Q$ defined in
(3.4). Again, the strategy is not to find a superconnection which is strictly
annihilated by $Q$, but that rather transforms as a total covariant
derivative, precisely as ${\cal L}_{1/2}$ in (3.20) above. For the moment, we
focus on the case in which the hyperloop couples to only two nodes of the
quiver of the theory, but in the next section we generalize this to longer
quivers.444The representation of the hyperloops in terms of quiver diagrams,
which may include some or all of the nodes and edges of the original quiver
defining the gauge theory, is explained in detail in [9, 10].

Following [10], we take a deformation of the form

${\mathcal{L}}={\mathcal{L}}_{1/2}+F+B+C\,,$ (4.1)

where $F$ is off-diagonal and Grassmann-odd, $B$ is a diagonal bilinear of the
scalar fields and $C$ is annihilated by $Q$. This is the most general form
consistent with the gauge group representations, the supermatrix structure and
with all dimensions being equal to one. BPS non-conformal loops with higher
dimension insertions are also possible, but are not considered here.

The condition $QC=0$ distinguishes two cases: when the supercharge annhilates
some of the matter fields and when it does not. Nontrivial solutions include
any BPS bosonic loop where the supersymmetry variation should be simply zero,
rather than a total derivative. We exclude that case at the moment, because
for a compact gauge group the coefficient of the gauge field in the Wilson
loop is the identity (or more precisely $i$). As the gauge field already
appears in the appropriate form in ${\mathcal{L}}_{1/2}$, we should not allow
for extra gauge field terms in $C$. An exception to this would arise if $B$
also has gauge fields, a possibility discussed in Appendix C.

The other possibility is that $Q$ annihilates fields from the hypermultiplet.
Note that the action of $Q$ on the scalars in (3.12) is always proportional to
the bilinear of the parameters $\eta^{\imath}_{a}$ and
$\bar{\eta}^{\imath}_{a}$ that we called $\Pi$. When $\Pi$ is identically
zero, we see that $Q$ has a nontrivial kernel (in this case $r^{1}\propto
r^{2}$, so they do not form a basis of the scalar fields). One has therefore
to distinguish the cases when $\Pi$ does not vanish (or has isolated zeros)
and the case when $\Pi=0$, studied later in Section 4.2.

###  4.1 Deformations with $\Pi\neq 0$

Starting from the ansatz (4.1), we want to determine the most general $F$, $B$
and $C$ giving BPS loops, under the assumption that $\Pi\neq 0$.

The simplest term to address is $C$. The only solutions to $QC=0$ which is at
most bilinear in the fields and excluding the gauge field is
$C=\operatorname{diag}(c_{I},c_{I+1})$, a numerical matrix not containing the
fields. Note that we set the radius $R$ of $S^{3}$ to 1, otherwise this should
scale with $1/R$ on dimensional grounds. The term proportional to the identity
is completely trivial, so we remove it and take $C=\operatorname{diag}(0,c)$.

Moving on to $F$, in order for $QF$ to involve a derivative in the $\varphi$
direction, $F$ is restricted to have the fermions in (3.12). Therefore, if
$\Pi\neq 0$, one can take

$F=-iQG\,,\qquad G=\begin{pmatrix}0&\bar{b}_{a}r^{a}\\\
b^{a}\bar{r}_{a}&0\end{pmatrix}.$ (4.2)

Here the $b^{a}$, $\bar{b}_{a}$ parameters may be functions of $\varphi$.

In terms of $G$, we can combine (3.17) and (3.18) into

${-i}Q^{2}G=\partial_{\varphi}(\Pi G)-i[{\mathcal{L}}_{1/2}^{B},\Pi
G]+i[H^{2},G]-\Pi\hat{\mathcal{G}}\,,$ (4.3)

with the remainder

$\Pi\hat{\mathcal{G}}=\begin{pmatrix}0&\partial_{\varphi}(\Pi\bar{b}_{a})r^{a}-i\Pi\bar{b}_{1}r^{1}\\\
\partial_{\varphi}(\Pi b^{a})\bar{r}_{a}+i\Pi
b^{1}\bar{r}_{1}&0\end{pmatrix}.$ (4.4)

To evaluate the supersymmetry variation, it is sometimes useful to split the
connection into the diagonal (bosonic) and off-diagonal (fermionic) part:
${\mathcal{L}}={\mathcal{L}}^{B}+{\mathcal{L}}^{F}$, and likewise for
${\mathcal{L}}_{1/2}$. One can then write

$\displaystyle Q{\mathcal{L}}$ $\displaystyle=Q{\mathcal{L}}_{1/2}-iQ^{2}G+QB$
(4.5)
$\displaystyle={\mathcal{D}}^{{\mathcal{L}}_{1/2}}H+\partial_{\varphi}(\Pi
G)-i[{\mathcal{L}}_{1/2}^{B},\Pi G]+i[H^{2},G]+QB-\Pi\hat{\mathcal{G}}$
$\displaystyle={\mathcal{D}}^{{\mathcal{L}}_{1/2}}(H+\Pi
G)-i\\{{\mathcal{L}}_{1/2}^{F},\Pi G\\}+i[H^{2},G]+QB-\Pi\hat{\mathcal{G}}$
$\displaystyle={\mathcal{D}}^{{\mathcal{L}}}(H+\Pi G)+i[B,H+\Pi G]+i[C,H+\Pi
G]+i[H^{2},G]-\Pi\hat{\mathcal{G}}$ $\displaystyle\quad-\\{QG,H+\Pi
G\\}-\\{QH,G\\}+QB\,.$

The terms on the last line are all diagonal and vanish by setting
$B=\\{G,H\\}+\Pi G^{2}$. With this form for $B$, also the second and fourth
terms on the previous line (which are cubic in the scalar fields) vanish.
Another way to write these equations is in terms of the variations of the
extra terms in ${\mathcal{L}}$ in (4.1)

$\displaystyle{Q}B$
$\displaystyle=i\\{F,H\\}+\\{\mathcal{L}_{1/2}^{F}+F,\Delta H\\}\,,$ (4.6)
$\displaystyle{Q}F$ $\displaystyle=\partial_{\varphi}\Delta
H-i[\mathcal{L}_{1/2}^{B},\Delta H]-i[B+C,H+\Delta H]\,.$

We see that this is indeed satisfied with $\Delta H=\Pi G$.

The deformed connection can then be written as

${\mathcal{L}}={\mathcal{L}}_{1/2}-iQG+\\{G,H\\}+\Pi G^{2}+C\,,$ (4.7)

and it is a total derivative if we further impose that the remainders in the
last equality of (4.5) cancel

$i[C,H+\Pi G]-\Pi\hat{\mathcal{G}}=0\,.$ (4.8)

These are four differential equations for $b^{a}$ and $\bar{b}_{a}$

$\displaystyle\partial_{\varphi}(\Pi b^{1})-i(c-1)\Pi b^{1}$
$\displaystyle=0\,,$ (4.9) $\displaystyle\partial_{\varphi}(\Pi
b^{2})-ic(\alpha+\Pi b^{2})$ $\displaystyle=0\,,$
$\displaystyle\partial_{\varphi}(\Pi\bar{b}_{1})+i(c-1)\Pi\bar{b}_{1}$
$\displaystyle=0\,,$
$\displaystyle\partial_{\varphi}(\Pi\bar{b}_{2})+ic(\bar{\alpha}+\Pi\bar{b}_{2})$
$\displaystyle=0\,.$

Taking $\hat{c}(\varphi)$ to be the primitive of $c$, the general solution can
be written as

$\Pi b^{1}=e^{-i\varphi+i\hat{c}}\beta^{1}\,,\quad\Pi
b^{2}=e^{i\hat{c}}\beta^{2}-\alpha\,,\quad\Pi\bar{b}_{1}=e^{i\varphi-i\hat{c}}\bar{\beta}_{1}\,,\quad\Pi\bar{b}_{2}=e^{-i\hat{c}}\bar{\beta}_{2}-\bar{\alpha}\,,$
(4.10)

with constant $\beta^{1}$, $\beta^{2}$, $\bar{\beta}_{1}$, $\bar{\beta}_{2}$.

There is a lot of freedom in choosing $c$. It can in principle be an arbitrary
function of $\varphi$, but this is a gauge symmetry, which is absorbed in
${\mathcal{A}}_{I+1}$. We can always fix to the same gauge as in (3.1) by
setting $c=0$. Note that in generic gauges, when $\hat{c}$ is not periodic,
the parameters $b$ and $\bar{b}$ are also not periodic (as it was in the
original paper [22]).

In the gauge $c=0$, the deformed connection (4.7) is

${\mathcal{L}}=\begin{pmatrix}A_{\varphi,I}+M_{a}^{\
b}r^{a}\bar{r}_{b}-\frac{i}{k}(\tilde{\mu}_{I\ \dot{1}}^{\
\dot{1}}-\tilde{\mu}_{I\ \dot{2}}^{\
\dot{2}})&-i\bar{\beta}_{2}\psi_{I\dot{1}-}+ie^{i\varphi}\bar{\beta}_{1}\psi_{I\dot{2}+}\\\
i\beta^{2}\bar{\psi}_{I+}^{\dot{1}}-ie^{-i\varphi}\beta^{1}\bar{\psi}_{I-}^{\dot{2}}&A_{\varphi,I+1}+M_{a}^{\
b}\bar{r}_{b}r^{a}-\frac{i}{k}(\tilde{\mu}_{I+1\,\dot{1}}{}^{\dot{1}}-\tilde{\mu}_{I+1\,\dot{2}}{}^{\dot{2}})-\frac{1}{2}\end{pmatrix},$
(4.11)

where

$M=\Pi^{-1}\begin{pmatrix}\bar{\beta}_{1}\beta^{1}+\frac{i}{k}&e^{i\varphi}\bar{\beta}_{1}\beta^{2}\\\
e^{-i\varphi}\bar{\beta}_{2}\beta^{1}&\bar{\beta}_{2}\beta^{2}-\frac{i}{k}\end{pmatrix}.$
(4.12)

After fixing a supercharge $Q$, the possible space of hyperloops it generates
can be represented by the matrix $M$ in (4.12). It is given by 4 complex
parameters $\beta^{a}$ and $\bar{\beta}_{a}$, modded out by a $C^{*}$ action,
which is the conifold. This is the same type of moduli space found in [11,
10].

Note that the effect of the shift of $\beta^{2}$ and $\bar{\beta}_{2}$ by
$\alpha$ and $\bar{\alpha}$ in (4.10) means that the “origin of $\beta$
space”, which is the tip of the conifold, is a bosonic loop. We can thus view
all the hyperloops that we find here as deformations around some bosonic loop
by some supercharge that it preserves. This is similar to the structure in
[10], but here we have far more general bosonic loops (see Section 6.1) and
choose any of the supercharges that they preserve.

Specific examples of hyperloops of this type are presented in Section 6.2.
Their symmetry algebras are also studied there, as well as a closer inspection
of the connection between them and the hyperloops of [10].

####  4.1.1 The condition $\Pi\neq 0$ from an algebraic point of view

The conditions on $\Pi$ being zero or not can be interpreted from an algebraic
point of view. To do that, let us start by looking at the square of the
supercharge (3.4), which using (A.9) reads

$\displaystyle Q^{2}=-\Pi_{-}J_{-}-\Pi_{0}J_{0}+\Pi_{+}J_{+}-\lambda
L_{\perp}-\frac{1}{2}\lambda_{ab}R^{ab}\,,$ (4.13)

with $\Pi_{\pm}$ and $\Pi_{0}$ the Fourier coefficients of $\Pi$, defined
through

$\Pi\equiv\Pi_{-}e^{-i\varphi}+\Pi_{0}+\Pi_{+}e^{+i\varphi}\,,$ (4.14)

and

$\lambda_{ab}\equiv\epsilon_{\imath\jmath}\bar{\eta}^{\imath}_{a}\eta^{\jmath}_{b}\,,\qquad\lambda\equiv\epsilon^{ab}\lambda_{ab}\,.$
(4.15)

As mentioned in Section 3, $J_{0}$ and $J_{\pm}$ are the generators of the
conformal group along the circle, $R^{ab}$ are
$\mathfrak{s}\mathfrak{u}(2)_{L}$ generators, and $L_{\perp}$ is a combination
of rotation orthogonal to the circle and the unbroken ${\mathfrak{u}}(1)_{R}$
(see Appendix A for further details).

As manifest from (4.13), $Q^{2}$ generically generates
$\mathfrak{sl}(2,\mathbb{R})\oplus\mathfrak{s}\mathfrak{u}(2)_{L}\oplus{\mathfrak{u}}(1)_{L_{\perp}}$,
which is the algebra preserved by the 1/2 BPS Wilson loop. When $\Pi\neq 0$
the conformal generators are part of this preserved algebra (at least in
part). It is now possible to consider subcases of the condition $\Pi\neq 0$ in
which one progressively decouples some of the generators on the right hand
side of (4.13). This imposes conditions on the parameters $\eta$ and
$\bar{\eta}$, which we derive below and which are going to be useful in
Section 6, where we construct specific examples.

We start by considering cases in which the $\mathfrak{s}\mathfrak{u}(2)_{L}$
is “turned off”. In order for the contribution of $R^{ab}$ to $Q^{2}$ to
vanish, one must require that $\lambda_{ab}$ in (4.15) be antisymmetric. This
implies that

$\displaystyle\lambda_{11}=\epsilon_{\imath\jmath}\bar{\eta}^{\imath}_{1}\eta^{\jmath}_{1}=0\,,$
(4.16)

which allows to deduce $\bar{\eta}^{\imath}_{1}\propto\eta^{\imath}_{1}$, and
similarly for $\lambda_{22}$ and $\bar{\eta}^{\imath}_{2},\eta^{\imath}_{2}$.
We may then factorize these parameter in terms of some other quantities
carrying a single index, as follows (bars do not indicate complex conjugation,
as usual)

$\displaystyle\bar{\eta}^{\imath}_{1}$
$\displaystyle=\bar{w}_{1}s^{\imath}\,,$
$\displaystyle\qquad\eta^{\imath}_{1}$ $\displaystyle=w_{1}s^{\imath}\,,$
(4.17) $\displaystyle\bar{\eta}^{\imath}_{2}$
$\displaystyle=\bar{w}_{2}t^{\imath}\,,$
$\displaystyle\qquad\eta^{\imath}_{2}$ $\displaystyle=w_{2}t^{\imath}.$

It remains to impose

$\displaystyle\lambda_{12}+\lambda_{21}=(\epsilon_{\imath\jmath}s^{\imath}t^{\jmath})(\epsilon^{ab}\bar{w}_{a}w_{b})=0\,,$
(4.18)

which can be achieved by setting either $s^{\imath}\propto t^{\imath}$ or
$\bar{w}_{a}\propto w_{a}$. As a consequence, the remaining parameters that
determine $Q^{2}$ are given by

$\lambda=(\epsilon_{\imath\jmath}s^{\imath}t^{\jmath})(\bar{w}_{1}w_{2}+\bar{w}_{2}w_{1})\,,\qquad\Pi=\left(s^{l}e^{+i\varphi/2}+s^{r}e^{-i\varphi/2}\right)^{2}\epsilon^{ab}\bar{w}_{a}w_{b}\,.$
(4.19)

In order to avoid that $\Pi=0$, we must ensure
$\epsilon^{ab}\bar{w}_{a}w_{b}\neq 0$, which implies
$\epsilon_{\imath\jmath}s^{\imath}t^{\jmath}=0$. In particular, the
contribution of $L_{\perp}$ vanishes automatically. In other words,
$Q^{2}\in\mathfrak{so}(2,1)$. More restrictive cases can be easily constructed
by considering special choices of $s^{l}$, $s^{r}$. In particular, setting
$s^{r}=0$ gives $Q^{2}\propto J_{+}$ and similarly $s^{l}=0$ gives
$Q^{2}\propto J_{-}$.

Next, one could maintain the $\mathfrak{s}\mathfrak{u}(2)_{L}$ and set instead
individual Fourier coefficients of $\Pi$ to zero, looking, for example, to the
case
$Q^{2}\in\mathfrak{u}(1)_{J_{0}}\oplus\mathfrak{s}\mathfrak{u}(2)_{L}\oplus\mathfrak{u}(1)_{L_{\perp}}$.
The contributions of $J_{\pm}$ to $Q^{2}$ vanish if and only if

$\displaystyle\epsilon^{ab}\eta^{l}_{a}\bar{\eta}^{l}_{b}=0\,,\qquad\epsilon^{ab}\eta^{r}_{a}\bar{\eta}^{r}_{b}=0\,,$
(4.20)

namely if the $\eta$’s are linearly dependent

$\displaystyle\eta^{l}_{a}$ $\displaystyle=t^{l}w_{a}\,,\qquad$
$\displaystyle\eta^{r}_{a}$ $\displaystyle=t^{r}z_{a}\,,$ (4.21)
$\displaystyle\bar{\eta}^{l}_{a}$ $\displaystyle=\bar{t}^{l}w_{a}\,,\qquad$
$\displaystyle\bar{\eta}^{r}_{a}$ $\displaystyle=\bar{t}^{r}z_{a}\,.$

Without loss of generality, one can take $z,w$ to be normalized, finding the
corresponding parameters

$\displaystyle\Pi$
$\displaystyle=(\epsilon^{ab}z_{a}w_{b})\,(\epsilon_{\imath\jmath}\bar{t}^{\imath}t^{\jmath})\,,$
(4.22) $\displaystyle\lambda_{ab}$
$\displaystyle=(\epsilon_{\imath\jmath}\bar{t}^{\imath}t^{\jmath})z_{(a}w_{b)}+\frac{1}{2}\bar{t}^{\imath}\begin{pmatrix}0&1\\\
1&0\end{pmatrix}_{\imath\jmath}t^{\jmath}(\epsilon^{cd}z_{c}w_{d})\epsilon_{ab}\,.$

One could go on and, for example, turn off $\Pi_{-}$ and $\Pi_{0}$ by imposing

$\displaystyle 0=\epsilon^{ab}\bar{\eta}^{r}_{a}\eta^{r}_{b}\,,\qquad
0=\epsilon^{ab}\bar{\eta}^{r}_{a}\eta^{l}_{b}+\epsilon^{ab}\bar{\eta}^{l}_{a}\eta^{r}_{b}\,,$
(4.23)

which is achieved by taking

$\displaystyle\eta^{r}_{a}=sw_{a},\qquad\bar{\eta}^{r}_{a}=\bar{s}w_{a}\,,$
(4.24)

and yields

$\displaystyle\lambda_{ab}$
$\displaystyle=\bar{s}\left(\eta^{l}_{a}w_{b}-w_{a}\eta^{l}_{b}\right)-tw_{a}w_{b}\,.$
(4.25)

The specific cases considered above do not form an exhaustive classification
of supercharges with $\Pi\neq 0$, but have been selected because they are of
interest in the study of some loops, like the bosonic loops in Section 6.1.
Supercharges whose squares are a linear combination of both
$\mathfrak{s}\mathfrak{u}(2)_{L}$ and conformal generators can nonetheless be
easily constructed.

###  4.2 Deformations with $\Pi=0$

The analysis above gives Wilson loops rather similar to those already studied
in [10] (though far more general). As seen, it requires that the function
$\Pi$ be non-zero. Now we turn to look at the interesting case when

$\Pi=(\bar{\eta}v)_{1}(\eta\bar{v})_{2}-(\bar{\eta}v)_{2}(\eta\bar{v})_{1}=0\,,$
(4.26)

and define

$\xi=\frac{(\eta\bar{v})_{1}}{(\bar{\eta}v)_{1}}=\frac{(\eta\bar{v})_{2}}{(\bar{\eta}v)_{2}}\,,$
(4.27)

thus $\xi(\varphi)\in\mathbb{C}\cup\\{\infty\\}$.

This case is subtle because the supercharge $Q$ in (3.4) annihilates the
rotated scalars (3.12) and, furthermore, the pairs of rotated fields are not
linearly independent

$r^{1}=\xi r^{2}\,,\qquad\bar{r}_{2}=-\xi\bar{r}_{1}\,.$ (4.28)

For convenience we define (assuming $(\bar{\eta}v)_{1}\neq 0$)

$r^{\parallel}=r^{2}\,,\qquad\bar{r}_{\parallel}=-\bar{r}_{1}\,,$ (4.29)

and an orthogonal pair which are not annihilated by $Q$

$r^{\perp}=(\bar{\eta}v)_{2}q^{1}-(\bar{\eta}v)_{1}q^{2}\,,\qquad\bar{r}_{\perp}=(\bar{\eta}v)_{1}\bar{q}_{1}+(\bar{\eta}v)_{2}\bar{q}_{2}\,.$
(4.30)

We then find that

$\displaystyle{Q}r^{\perp}$
$\displaystyle=\Lambda(\xi\psi_{\dot{1}-}+\psi_{\dot{2}+})\,,\qquad{Q}\bar{r}_{\perp}=-\Lambda(\bar{\psi}_{+}^{\dot{1}}-\xi\bar{\psi}_{-}^{\dot{2}})\,,$
(4.31) $\displaystyle{Q}^{2}r^{\perp}$
$\displaystyle=-\Lambda\left((i\partial_{\varphi}\xi-\xi)r^{\parallel}-\frac{2i}{k}\xi(\nu_{I}r^{\parallel}-r^{\parallel}\nu_{I+1})\right),$
$\displaystyle{Q}^{2}\bar{r}_{\perp}$
$\displaystyle=\Lambda\left((i\partial_{\varphi}\xi-\xi)\bar{r}_{\parallel}+\frac{2i}{k}\xi(\nu_{I+1}\bar{r}_{\parallel}-\bar{r}_{\parallel}\nu_{I})\right),$

where

$\displaystyle\Lambda\equiv(\bar{\eta}v)_{1}^{2}+(\bar{\eta}v)_{2}^{2}\,,$
(4.32)

and similarly to (3.16)

$r^{\parallel}\bar{r}_{\perp}+r^{\perp}\bar{r}_{\parallel}=\Lambda\nu_{I}\,,\qquad\bar{r}_{\perp}r^{\parallel}+\bar{r}_{\parallel}r^{\perp}=\Lambda\nu_{I+1}\,.$
(4.33)

We can apply now the same formalism as in the $\Pi\neq 0$ case and take

$\mathcal{L}=\mathcal{L}_{1/2}-i{Q}G+\\{G,H\\}+C\,,\qquad QC=0\,.$ (4.34)

$H$ is the same as above, see (3.19), which in the new notations becomes

$H=\begin{pmatrix}0&\bar{\alpha}r^{\parallel}\\\
\alpha\xi\bar{r}_{\parallel}&0\end{pmatrix}.$ (4.35)

In $G$ we include only $r^{\perp}$ and $\bar{r}_{\perp}$ and $C$ may contain
scalar bilinears as well as the numerical factors discussed before

$G=\begin{pmatrix}0&\bar{\beta}_{\perp}r^{\perp}\\\
\beta^{\perp}\bar{r}_{\perp}&0\end{pmatrix},\qquad
C=\begin{pmatrix}\bar{\beta}_{\parallel}r^{\parallel}\bar{r}_{\parallel}&0\\\
0&\beta^{\parallel}\bar{r}_{\parallel}r^{\parallel}+c\end{pmatrix}.$ (4.36)

$QG$ gives a single linear combination of the fermions
$\xi\psi_{\dot{1}-}+\psi_{\dot{2}+}$. In Appendix C we explore the possibility
of adding another combinations of the fermions, but find that this can only be
done in the case of $\xi=0$, presented in Section 4.2.1 below.

Going back to the deformation (4.26), one can get
$Q{\mathcal{L}}={\mathcal{D}}_{\varphi}^{\mathcal{L}}H$, provided that

$Q^{2}G=[G,H^{2}]+[C,H]\,.$ (4.37)

Unlike the $\Pi\neq 0$ case, here $H$ remains the same regardless of the
deformation.

Besides, one can check that the cubic terms inside $Q^{2}G$ cancel
$[G,H^{2}]+[C,H]$ provided $\beta^{\parallel}=\bar{\beta}_{\parallel}$. The
remaining equations for the terms linear in the scalars are

$\Lambda\bar{\beta}_{\perp}\partial_{\varphi}(e^{i\varphi}\xi)=-ie^{i\varphi}c\bar{\alpha}\,,\qquad\Lambda\beta^{\perp}\partial_{\varphi}(e^{i\varphi}\xi)=-ie^{i\varphi}\xi
c\alpha\,,$ (4.38)

which are simple algebraic relations on $\beta^{\perp}$, $\bar{\beta}_{\perp}$
and $c$.

In the generic case, we can have

$\mathcal{L}=\begin{pmatrix}A_{\varphi,I}+M_{a}{}^{b}r^{a}\bar{r}_{b}-\frac{i}{k}(\tilde{\mu}_{I\;\dot{1}}^{\;\dot{1}}-\tilde{\mu}_{I\;\dot{2}}^{\;\dot{2}})&-i(\bar{\alpha}+\xi\Lambda\bar{\beta}_{\perp})\psi_{\dot{1}-}-i\Lambda\bar{\beta}_{\perp}\psi_{\dot{2}+}\\\
i(\alpha+\Lambda\beta^{\perp})\bar{\psi}_{+}^{\dot{1}}-i\xi\Lambda\beta^{\perp}\bar{\psi}_{-}^{\dot{2}}&A_{\varphi,I+1}+M_{a}{}^{b}\bar{r}_{b}r^{a}-\frac{i}{k}(\tilde{\mu}_{I+1\dot{1}}^{\quad\;\;\;\dot{1}}-\tilde{\mu}_{I+1\dot{2}}^{\quad\;\;\;\dot{2}})+c-\frac{1}{2}\end{pmatrix},$
(4.39)

where

$M_{a}{}^{b}=\begin{pmatrix}0&\frac{i}{k\Lambda}+\xi\alpha\bar{\beta}_{\perp}\\\
\frac{i}{k\Lambda}+\bar{\alpha}\beta^{\perp}&\beta^{\parallel}\end{pmatrix},$
(4.40)

with $a,b=\perp,\parallel$. Plugging in the solutions of (4.38), the resulting
loops generically preserve only one supercharge. However, at some special
points we find supersymmetry enhancement. In fact, we find some very
interesting subclasses of those loops, which are analyzed in detail in Section
6.3.

####  4.2.1 The special cases: $\xi=0$ and $\xi=\infty$

Two further degenerations of the $\Pi=0$ supercharges are when $\xi$ in (4.27)
vanishes or is infinite. Both cases are equivalent under the replacement of
$\eta$ with $\bar{\eta}$ (or $Q^{\dot{2}a+}_{\imath}$ and
$Q^{\dot{1}a-}_{\bar{\imath}}$) and for simplicity we focus on $\xi=0$. This
means that the supercharge $Q$ is comprised of only the four supercharges
$Q^{\dot{1}a-}_{\bar{\imath}}$ and is nilpotent $Q^{2}=0$.

In all cases when $\Pi=0$, we have two scalar fields $r^{\parallel}$ and
$\bar{r}_{\parallel}$ in (4.29) that are annihilated by $Q$. For $\xi=0$, as
can be seen from (4.31), there are also two fermionic field in the
hypermultiplet annihilated by $Q$. Those are $\psi_{\dot{2}+}$ and
$\bar{\psi}^{\dot{1}}_{+}$ and we can therefore insert any distribution of
these fields in the hyperloop while still preserving supersymmetry.

As the bottom left entry in ${\mathcal{L}}_{1/2}$ is comprised of
$\bar{\psi}^{\dot{1}}_{+}$, see (3.1), the matrix $H$ appearing in the
variation of ${\mathcal{L}}_{1/2}$ is upper-triangular, as can indeed be read
off from (4.35). To construct the deformed loops we take $G$ as in (4.36) and
add the extra fermionic fields to $C$. Alternatively, they can also be added
as extra terms into $F$ beyond $QG$

$G=\begin{pmatrix}0&\bar{\beta}_{\perp}r^{\perp}\\\
\beta^{\perp}\bar{r}_{\perp}&0\end{pmatrix},\qquad
C=\begin{pmatrix}\bar{\beta}_{\parallel}r^{\parallel}\bar{r}_{\parallel}&\bar{\delta}\psi_{\dot{2}+}\\\
\delta\bar{\psi}^{\dot{1}}_{+}&\beta^{\parallel}\bar{r}_{\parallel}r^{\parallel}+c\end{pmatrix}.$
(4.41)

Plugging $G$ and $C$ into $Q\mathcal{L}=\mathcal{D}_{\varphi}^{\mathcal{L}}H$,
one gets again the same condition that appeared in (4.37), which can be solved
by

$\delta=c=0\,,\qquad\bar{\beta}_{\parallel}=\beta^{\parallel}\,.$ (4.42)

This gives the superconnection

$\mathcal{L}=\begin{pmatrix}A_{\varphi,I}+M_{a}{}^{b}r^{a}\bar{r}_{b}-\frac{i}{k}(\tilde{\mu}_{I\;\dot{1}}^{\;\dot{1}}-\tilde{\mu}_{I\;\dot{2}}^{\;\dot{2}})&-i\bar{\alpha}\psi_{\dot{1}-}+(\bar{\delta}-i\Lambda\bar{\beta}_{\perp})\psi_{\dot{2}+}\\\
i(\alpha+\Lambda\beta^{\perp})\bar{\psi}_{+}^{\dot{1}}&A_{\varphi,I+1}+M_{a}{}^{b}\bar{r}_{b}r^{a}-\frac{i}{k}(\tilde{\mu}_{I+1\dot{1}}^{\quad\;\;\dot{1}}-\tilde{\mu}_{I+1\dot{2}}^{\quad\;\;\dot{2}})-\frac{1}{2}\end{pmatrix},$
(4.43)

where $M_{a}{}^{b}$ is the same as (4.40) with $\xi=0$. Note that
$\bar{\delta}$ and $\bar{\beta}_{\perp}$ appear only as the combination
$\bar{\delta}-i\Lambda\bar{\beta}_{\perp}$, so we can eliminate any one of
them.

The same answer is found from a different approach in Appendix C, where extra
fermionic fields are added in $F$.

####  4.2.2 The condition $\Pi=0$ from an algebraic point of view

As done for $\Pi\neq 0$ in Section 4.1.1, one can consider the condition
$\Pi=0$ from an algebraic point of view. Here we give a complete
classification of all possible subcases. From the discussion around (4.20),
with
$Q^{2}\in\mathfrak{u}(1)_{J_{0}}\oplus\mathfrak{s}\mathfrak{u}(2)_{L}\oplus\mathfrak{u}(1)_{L_{\perp}}$,
the conditions on $\bar{\eta}^{\imath}_{a},\eta^{\imath}_{a}$ for $\Pi$ to
vanish are easily derived, since one only needs to enforce $\Pi_{0}=0$, so
that
$Q^{2}\in\mathfrak{s}\mathfrak{u}(2)_{L}\oplus\mathfrak{u}(1)_{L_{\perp}}$. By
(4.22), there are two possibilities: either $\epsilon^{ab}z_{a}w_{b}=0$ which
implies $\lambda_{ab}=\lambda_{ba}$ and
$Q^{2}\in\mathfrak{s}\mathfrak{u}(2)_{L}$, or
$\epsilon_{\imath\jmath}\bar{t}^{\imath}t^{\jmath}=0$, which implies
$\lambda_{ab}=-\lambda_{ba}$ and $Q^{2}\in\mathfrak{u}(1)_{L_{\perp}}.$

In the former case, $Q^{2}\in\mathfrak{s}\mathfrak{u}(2)_{L}$, one can let
$z_{a}=w_{a}$ without loss of generality, leading to

$\displaystyle Q^{2}\propto w_{a}w_{b}R^{ab}\,.$ (4.44)

The functions $\xi$ and $\Lambda$ are given by

$\displaystyle\xi=\frac{t^{l}+e^{-i\varphi}t^{r}}{e^{+i\varphi}\bar{t}^{l}+\bar{t}^{r}}\,,\qquad\Lambda=(e^{+i\varphi}\bar{t}^{l}+\bar{t}^{r})^{2}\,.$
(4.45)

In the case $Q^{2}\in\mathfrak{u}(1)_{L_{\perp}}$, one may write instead
$t^{\imath}=t\,s^{\imath},\bar{t}^{\imath}=\bar{t}\,s^{\imath}$, leading to

$\displaystyle Q^{2}\propto\epsilon^{ab}z_{a}w_{b}\,L_{\perp}\,,$ (4.46)

as well as to

$\displaystyle\xi=\frac{te^{-i\varphi}}{\bar{t}}\,,\qquad\Lambda={\bar{t}}^{2}e^{i\varphi}\Big{(}e^{i\varphi}(s^{l})^{2}+e^{-i\varphi}(s^{r})^{2}+2s^{l}s^{r}(\epsilon^{ab}z_{a}w_{b})\Big{)}\,.$
(4.47)

Finally, when both of the conditions above are met the supercharge becomes
nilpotent, $Q^{2}=0$. The parameters are of the form

$\displaystyle\eta^{\imath}_{a}=a\rho^{\imath}w_{a}\,,\qquad\bar{\eta}^{\imath}_{a}=\bar{a}\rho^{\imath}w_{a}\,.$
(4.48)

This factorisation is expected since each term in (4.13) antisymmetrises over
either $\imath,\jmath$ or $a,b$ (or both). The functions $\xi$ and $\Lambda$
take the simple form

$\displaystyle\xi=\frac{ae^{-i\varphi}}{\bar{a}}\,,\qquad\Lambda=\bar{a}^{2}(e^{i\varphi}\rho^{l}+\rho^{r})^{2}\,.$
(4.49)

Note that the function $\xi$ provides a handy way of distinguishing these
cases. Concretely, $\partial_{\varphi}\left(e^{i\varphi}\xi\right)=0$ if and
only if $Q^{2}\in\mathfrak{u}(1)_{L_{\perp}}$. $\xi$ vanishes identically if
and only if $Q$ is composed entirely of barred supercharges.

##  5 Longer quivers and twisted hypers

All the constructions in Section 4 involve only two nodes of the quiver. Here
we turn to hyperloops coupling to more nodes. As a guiding example and
starting point of the deformation, we consider the 1/2 BPS loop on two pairs
of nodes, with undeformed superconnection given by

${\mathcal{L}}_{1/2}=\begin{pmatrix}{\mathcal{A}}_{I}&-i\bar{\alpha}_{I}\psi_{I,\dot{1}-}&0&0\\\
i\alpha_{I}\bar{\psi}^{\dot{1}}_{I,+}&{\mathcal{A}}_{I+1}-\frac{1}{2}&0&0\\\
0&0&{\mathcal{A}}_{I+2}-c&-i\bar{\alpha}_{I+2}\psi_{I+2,\dot{1}-}\\\
0&0&i\alpha_{I+2}\bar{\psi}^{\dot{1}}_{I+2,+}&{\mathcal{A}}_{I+3}-c-\frac{1}{2}\end{pmatrix}.$
(5.1)

We introduce a constant shift $c$ between the two pairs of nodes representing
the effect of a $U(N_{I+1})$ gauge freedom. In this block-diagonal form, there
is no restriction on $c$. The resulting Wilson loop is well defined with
constant $\alpha_{I+2}$ and $\bar{\alpha}_{I+2}$ satisfying
$\alpha_{I+2}\bar{\alpha}_{I+2}=2i/k$. We find (the supertrace sums lines with
signs $+,-,+,-$)

$W=\operatorname{sTr}{\mathcal{P}}\exp
i\oint{\mathcal{L}}\,d\varphi=W_{(I,I+1)}+\exp\left(-i\oint
c\,d\varphi\right)W_{(I+2,I+3)}\,.$ (5.2)

Clearly with this block-diagonal structure, we can take any linear combination
of the two Wilson loops. Adding deformations by the hypermultiplets keeps the
block-diagonal structure, so again it works with any $c$. As already noted in
[10], deformations by twisted hypermultiplets with $\tilde{q}^{\dot{a}}_{I+1}$
are more subtle and fix $c$.

The Wilson loop based on (5.1) still satisfies
$Q{\mathcal{L}}_{1/2}=\mathcal{D}_{\varphi}^{{\mathcal{L}}_{1/2}}H$, this time
with

$H=\begin{pmatrix}0&\bar{\alpha}_{I}r_{I}^{2}&0&0\\\
\alpha_{I}\bar{r}_{I\,2}&0&0&0\\\ 0&0&0&\bar{\alpha}_{I+2}r_{I+2}^{2}\\\
0&0&\alpha_{I+2}\bar{r}_{I+2\,2}&0\end{pmatrix}.$ (5.3)

It is now natural to rotate the fermions from the twisted hypermultiplets

$\tilde{\rho}^{1}_{-}=-(\eta\bar{v})_{a}\tilde{\psi}^{a}_{-}\,,\quad\tilde{\rho}^{2}_{+}=(\bar{\eta}v)_{a}\tilde{\psi}^{a}_{+}\,,\quad\bar{\tilde{\rho}}_{1+}=\epsilon^{ab}(\bar{\eta}v)_{a}\bar{\tilde{\psi}}_{b+}\,,\quad\bar{\tilde{\rho}}_{2-}=\epsilon^{ab}(\eta\bar{v})_{a}\bar{\tilde{\psi}}_{b-}\,,$
(5.4)

such that the supersymmetry transformations are

$Q\tilde{q}_{\dot{1}}=\tilde{\rho}^{2}_{+}\,,\quad
Q\tilde{q}_{\dot{2}}=\tilde{\rho}^{1}_{-}\,,\quad
Q\bar{\tilde{q}}^{\dot{1}}=\bar{\tilde{\rho}}_{2-}\,,\quad
Q\bar{\tilde{q}}^{\dot{2}}=\bar{\tilde{\rho}}_{1+}\,.$ (5.5)

The double variations are then

$\displaystyle Q^{2}\tilde{q}_{\dot{1}}$
$\displaystyle=\Pi\big{(}i\partial_{\varphi}\tilde{q}_{\dot{1}}+{\mathcal{A}}_{I+1}\tilde{q}_{\dot{1}}-\tilde{q}_{\dot{1}}{\mathcal{A}}_{I+2}\big{)}-\frac{2i}{k}(\bar{r}_{2}r^{2}\tilde{q}_{\dot{1}}-\tilde{q}_{\dot{1}}r^{2}\bar{r}_{2})-\frac{1}{2}\epsilon^{ab}(\bar{\eta}v)_{a}(\eta\sigma^{3}\bar{v})_{b}\tilde{q}_{\dot{1}}\,,$
(5.6) $\displaystyle Q^{2}\tilde{q}_{\dot{2}}$
$\displaystyle=\Pi\big{(}i\partial_{\varphi}\tilde{q}_{\dot{2}}+{\mathcal{A}}_{I+1}\tilde{q}_{\dot{2}}-\tilde{q}_{\dot{2}}{\mathcal{A}}_{I+2}\big{)}-\frac{2i}{k}(\bar{r}_{2}r^{2}\tilde{q}_{\dot{2}}-\tilde{q}_{\dot{2}}r^{2}\bar{r}_{2})-\frac{1}{2}\epsilon^{ab}(\bar{\eta}\sigma^{3}v)_{a}(\eta\bar{v})_{b}\tilde{q}_{\dot{2}}\,,$
$\displaystyle Q^{2}\bar{\tilde{q}}^{\dot{1}}$
$\displaystyle=\Pi\big{(}i\partial_{\varphi}\bar{\tilde{q}}^{\dot{1}}+{\mathcal{A}}_{I+2}\bar{\tilde{q}}^{\dot{1}}-\bar{\tilde{q}}^{\dot{1}}{\mathcal{A}}_{I+1}\big{)}-\frac{2i}{k}(r^{2}\bar{r}_{2}\bar{\tilde{q}}^{\dot{1}}-\bar{\tilde{q}}^{\dot{1}}\bar{r}_{2}r^{2})-\frac{1}{2}\epsilon^{ab}(\bar{\eta}\sigma^{3}v)_{a}(\eta\bar{v})_{b}\bar{\tilde{q}}^{\dot{1}}\,,$
$\displaystyle Q^{2}\bar{\tilde{q}}^{\dot{2}}$
$\displaystyle=\Pi\big{(}i\partial_{\varphi}\bar{\tilde{q}}^{\dot{2}}+{\mathcal{A}}_{I+2}\bar{\tilde{q}}^{\dot{2}}-\bar{\tilde{q}}^{\dot{2}}{\mathcal{A}}_{I+1}\big{)}-\frac{2i}{k}(r^{2}\bar{r}_{2}\bar{\tilde{q}}^{\dot{2}}-\bar{\tilde{q}}^{\dot{2}}\bar{r}_{2}r^{2})-\frac{1}{2}\epsilon^{ab}(\bar{\eta}v)_{a}(\eta\sigma^{3}v)_{b}\bar{\tilde{q}}^{\dot{2}}\,.$

Using (3.15), the linear terms above can rewritten as

$\displaystyle\epsilon^{ab}(\bar{\eta}v)_{a}(\eta\sigma^{3}\bar{v})_{b}=-i\partial_{\varphi}\Pi-\lambda\,,\qquad\epsilon^{ab}(\bar{\eta}\sigma^{3}v)_{a}(\eta\bar{v})_{b}=-i\partial_{\varphi}\Pi+\lambda\,,$
(5.7)

such that the double variations become

$\displaystyle Q^{2}\tilde{q}_{\dot{1}}$
$\displaystyle=\Pi\left(i\partial_{\varphi}\tilde{q}_{\dot{1}}-\frac{1}{2}\tilde{q}_{\dot{1}}+\Gamma\tilde{q}_{\dot{1}}+{\mathcal{A}}_{I+1}\tilde{q}_{\dot{1}}-\tilde{q}_{\dot{1}}{\mathcal{A}}_{I+2}\right)-\frac{2i}{k}(\bar{r}_{2}r^{2}\tilde{q}_{\dot{1}}-\tilde{q}_{\dot{1}}r^{2}\bar{r}_{2})\,,$
(5.8) $\displaystyle Q^{2}\tilde{q}_{\dot{2}}$
$\displaystyle=\Pi\left(i\partial_{\varphi}\tilde{q}_{\dot{2}}+\frac{1}{2}\tilde{q}_{\dot{2}}+\bar{\Gamma}\tilde{q}_{\dot{2}}+{\mathcal{A}}_{I+1}\tilde{q}_{\dot{2}}-\tilde{q}_{\dot{2}}{\mathcal{A}}_{I+2}\right)-\frac{2i}{k}(\bar{r}_{2}r^{2}\tilde{q}_{\dot{2}}-\tilde{q}_{\dot{2}}r^{2}\bar{r}_{2})\,,$
$\displaystyle Q^{2}\bar{\tilde{q}}^{\dot{1}}$
$\displaystyle=\Pi\left(i\partial_{\varphi}\bar{\tilde{q}}^{\dot{1}}+\frac{1}{2}\bar{\tilde{q}}^{\dot{1}}+\bar{\Gamma}\bar{\tilde{q}}^{\dot{1}}+{\mathcal{A}}_{I+2}\bar{\tilde{q}}^{\dot{1}}-\bar{\tilde{q}}^{\dot{1}}{\mathcal{A}}_{I+1}\right)-\frac{2i}{k}(r^{2}\bar{r}_{2}\bar{\tilde{q}}^{\dot{1}}-\bar{\tilde{q}}^{\dot{1}}\bar{r}_{2}r^{2})\,,$
$\displaystyle Q^{2}\bar{\tilde{q}}^{\dot{2}}$
$\displaystyle=\Pi\left(i\partial_{\varphi}\bar{\tilde{q}}^{\dot{2}}-\frac{1}{2}\bar{\tilde{q}}^{\dot{2}}+\Gamma\bar{\tilde{q}}^{\dot{2}}+{\mathcal{A}}_{I+2}\bar{\tilde{q}}^{\dot{2}}-\bar{\tilde{q}}^{\dot{2}}{\mathcal{A}}_{I+1}\right)-\frac{2i}{k}(r^{2}\bar{r}_{2}\bar{\tilde{q}}^{\dot{2}}-\bar{\tilde{q}}^{\dot{2}}\bar{r}_{2}r^{2})\,,$

where for latter convenience we introduce

$\Gamma=\frac{1}{2}\left(i\partial_{\varphi}\ln\Pi+\frac{\lambda}{\Pi}+1\right),\qquad\bar{\Gamma}=\frac{1}{2}\left(i\partial_{\varphi}\ln\Pi-\frac{\lambda}{\Pi}-1\right).$
(5.9)

###  5.1 Deformations with $\Pi\neq 0$

We now proceed to deform the loop (5.1) as in (4.1). We take $G$ to be of the
form

$G=\begin{pmatrix}0&\bar{b}_{Ia}r_{I}^{a}&0&0\\\
b_{I}^{a}\bar{r}_{Ia}&0&\bar{d}^{\dot{1}}_{I+1}\tilde{q}_{I+1\,\dot{1}}&0\\\
0&{d}_{I+1\,\dot{1}}\bar{\tilde{q}}^{\dot{1}}_{I+1}&0&\bar{b}_{I+2\,a}r_{I+2}^{a}\\\
0&0&b_{I+2}^{a}\bar{r}_{I+2\,a}&0\end{pmatrix}.$ (5.10)

We allow a coupling to all the scalars in the hypermultiplets, but in the
twisted hypers we restrict to $\tilde{q}_{I+1\,\dot{1}}$ and
$\bar{\tilde{q}}^{\dot{1}}_{I+1}$. The second pair of scalar fields is
examined below.

Using (5.8), the analogue of (4.3) adapted for a longer quiver is

$-iQ^{2}G=\partial_{\varphi}(\Pi G)-i[{\mathcal{L}}_{1/2}^{B},\Pi
G]+i[H^{2},G]-\Pi\hat{{\mathcal{G}}}\,,$ (5.11)

with

$\displaystyle\Pi\hat{{\mathcal{G}}}$
$\displaystyle=\begin{pmatrix}0&\partial_{\varphi}(\Pi\bar{b}_{I\,a})r_{I}^{a}&0&0\\\
\partial_{\varphi}(\Pi
b_{I}^{a})\bar{r}_{I\,a}&0&\partial_{\varphi}(\Pi\bar{d}^{\dot{1}})\tilde{q}_{I+1\,\dot{1}}&0\\\
0&\partial_{\varphi}(\Pi{d}_{\dot{1}})\bar{\tilde{q}}_{I+1}^{\dot{1}}&0&\partial_{\varphi}(\Pi\bar{b}_{I+2\,a})r_{I+2}^{a}\\\
0&0&\partial_{\varphi}(\Pi b_{I+2}^{a})\bar{r}_{I+2\,a}&0\end{pmatrix}$ (5.12)
$\displaystyle\quad+\begin{pmatrix}0&-i\Pi\bar{b}_{I\,1}r^{1}_{I}&0&0\\\ i\Pi
b_{I}^{1}\bar{r}_{I\,1}&0&-i\Pi(c-\Gamma)\bar{d}^{\dot{1}}\tilde{q}_{I+1\,\dot{1}}&0\\\
0&i\Pi(c+\bar{\Gamma})d_{\dot{1}}\bar{\tilde{q}}^{\dot{1}}_{I+1}&0&-i\Pi\bar{b}_{I+2\,1}r_{I+2}^{1}\\\
0&0&i\Pi b^{1}_{I+2}\bar{r}_{I+2\,1}&0\end{pmatrix}.$

Proceeding as before, the analogue of (4.5) sets $B=\\{G,H\\}+\Pi G^{2}$ and
supersymmetry invariance of ${\mathcal{L}}$ now requires solving

$i[C,H+\Pi G]-\Pi\hat{{\mathcal{G}}}=0\,,\qquad
C=\operatorname{diag}(c_{I},c_{I+1},c_{I+2},c_{I+3})\,.$ (5.13)

We recover two copies of the equations (4.9), now for $b_{I}$, $\bar{b}_{I}$,
$b_{I+2}$, and $\bar{b}_{I+2}$. In addition, using
$\Gamma+\bar{\Gamma}=i\partial_{\varphi}\ln\Pi$, we find the two following
equations for $d_{I+1\,\dot{1}}$ and $\bar{d}^{\dot{1}}_{I+1}$

$\displaystyle\partial_{\varphi}(\bar{d}^{\dot{1}}_{I+1})-i(c_{I+1}-c_{I+2}+c+\bar{\Gamma})\bar{d}_{I+1}^{\dot{1}}$
$\displaystyle=0\,,$ (5.14) $\displaystyle\partial_{\varphi}(\Pi
d_{I+1\,\dot{1}})+i(c_{I+1}-c_{I+2}+c+\bar{\Gamma})\Pi{d}_{I+1\,\dot{1}}$
$\displaystyle=0\,.$

Note that these involve not only the numerical factors arising from $C$ but
also the relative shift $c$ that was left arbitrary in (5.1). In particular,
we can make use of this gauge freedom to make the convenient choice
$c_{I+1}=c_{I+2}=0$ and then with $c=-\bar{\Gamma}$, the equations above are
solved by

$\bar{d}^{\dot{1}}_{I+1}=\bar{\delta}^{\dot{1}}_{I+1}\,,\qquad
d_{I+1\,\dot{1}}=\frac{\delta_{I+1\,\dot{1}}}{\Pi}\,,$ (5.15)

with constant $\delta$’s. Other gauges are possible, but they are completely
equivalent to this one.

One can write the explicit expression for ${\mathcal{L}}$ using (4.7). Two
points to note are that in addition to the diagonal bosonic terms and first
off-diagonal fermionic terms, there are also off-off-diagonal bosonic terms
that contain the bilinears $\tilde{q}_{I+1\,\dot{1}}r^{a}_{I+2}$ and
$\bar{\tilde{q}}_{I+1}^{\dot{1}}\bar{r}_{Ia}$. Also, the diagonal terms in the
central nodes now include the modification of the bilinears of the scalars in
the twisted hypermultiplets via

$\widetilde{M}^{\dot{a}}{}_{\dot{b}}\tilde{q}_{I+1\,\dot{a}}\bar{\tilde{q}}_{I+1}^{\dot{b}}\,,\qquad\widetilde{M}=\begin{pmatrix}-i/k+{\bar{\delta}}^{\dot{1}}_{I+1}\delta_{I+1\,\dot{1}}&0\\\
0&i/k\end{pmatrix}.$ (5.16)

Instead of writing the full complicated $4\times 4$ form of the general
${\mathcal{L}}$, we look at some special cases in Section 6.4.

To couple ${\mathcal{L}}$ to $\tilde{q}_{I+1\,\dot{2}}$ and
$\bar{\tilde{q}}^{\dot{2}}_{I+1}$, we take instead

$G=\begin{pmatrix}0&\bar{b}_{Ia}r_{I}^{a}&0&0\\\
b_{I}^{a}\bar{r}_{Ia}&0&\bar{d}^{\dot{2}}_{I+1}\tilde{q}_{I+1\,\dot{2}}&0\\\
0&{d}_{I+1\,\dot{2}}\bar{\tilde{q}}^{\dot{2}}_{I+1}&0&\bar{b}_{I+2\,a}r_{I+2}^{a}\\\
0&0&b_{I+2}^{a}\bar{r}_{I+2\,a}&0\end{pmatrix},$ (5.17)

then (5.11) holds with

$\displaystyle\Pi\hat{{\mathcal{G}}}$
$\displaystyle=\begin{pmatrix}0&\partial_{\varphi}(\Pi\bar{b}_{I\,a})r_{I}^{a}&0&0\\\
\partial_{\varphi}(\Pi
b_{I}^{a})\bar{r}_{I\,a}&0&\partial_{\varphi}(\Pi\bar{d}^{\dot{2}}_{I+1})\tilde{q}_{I+1\,\dot{2}}&0\\\
0&\partial_{\varphi}(\Pi
d_{I+1\,\dot{2}})\bar{\tilde{q}}^{\dot{2}}_{I+1}&0&\partial_{\varphi}(\Pi\bar{b}_{I+2\,a})r_{I+2}^{a}\\\
0&0&\partial_{\varphi}(\Pi b_{I+2}^{a})\bar{r}_{I+2\,a}&0\end{pmatrix}$ (5.18)
$\displaystyle\quad+\begin{pmatrix}0&-i\Pi\bar{b}_{I\,1}r^{1}_{I}&0&0\\\ i\Pi
b_{I}^{1}\bar{r}_{I\,1}&0&-i\Pi(c-\bar{\Gamma}-1)\bar{d}^{\dot{2}}_{I+1}\tilde{q}_{I+1\,\dot{2}}&0\\\
0&i\Pi(c+\Gamma-1)d_{I+1\,\dot{2}}\bar{\tilde{q}}^{\dot{2}}_{I+1}&0&-i\Pi\bar{b}_{I+2\,1}r_{I+2}^{1}\\\
0&0&i\Pi b^{1}_{I+2}\bar{r}_{I+2\,1}&0\end{pmatrix}\,.$

This time, (5.13) gives two equations for $d_{I+1\,\dot{2}}$ and
$\bar{d}^{\dot{2}}_{I+1}$

$\displaystyle\partial_{\varphi}(\bar{d}^{\dot{2}}_{I+1})-i(c_{I+1}-c_{I+2}+c+\Gamma-1)\Pi\bar{d}^{\dot{2}}_{I+1}$
$\displaystyle=0\,,$ (5.19) $\displaystyle\partial_{\varphi}(\Pi
d_{I+1\,\dot{2}})+i(c_{I+1}-c_{I+2}+c+\Gamma-1){d}_{I+1\,\dot{2}}$
$\displaystyle=0\,.$

In this case the convenient gauge is $c_{I+1}=c_{I+2}=0$ where these equations
are solved with $c=-\Gamma+1$ and

$\bar{d}^{\dot{2}}_{I+1}=\bar{\delta}^{\dot{2}}_{I+1}\,,\qquad
d_{I+1\,\dot{2}}=\frac{\delta_{I+1\,\dot{2}}}{\Pi},$ (5.20)

with constant $\delta$’s. Now $\widetilde{M}$ is given by

$\widetilde{M}=\begin{pmatrix}-i/k&0\\\
0&i/k+{\bar{\delta}}^{\dot{2}}_{I+1}\delta_{I+1\,\dot{2}}\end{pmatrix}.$
(5.21)

Notice that we performed the analysis separately for the two pairs of scalars
in the twisted hypermultiplets and the resulting expressions required
different conditions on $c$, namely $c=-\bar{\Gamma}$ and $c=-\Gamma+1$. To
allow ${\mathcal{L}}$ to couple to all scalars of the twisted hypermultiplet
at the same time, these need to be related by a gauge transformation,
requiring

$\hat{c}(\varphi)=-\int_{0}^{\varphi}\left(\bar{\Gamma}-\Gamma+1\right)d\varphi^{\prime}=\int_{0}^{\varphi}\frac{\lambda}{\Pi}\,d\varphi^{\prime}\,,$
(5.22)

to be single valued. Thus, if

$e^{i\hat{c}(2\pi)}=\exp i\oint\frac{\lambda}{\Pi}\,d\varphi=1\,,$ (5.23)

is satisfied, ${\mathcal{L}}$ may couple to all twisted scalars, otherwise it
may couple either to the pair $\tilde{q}_{\dot{1}},\bar{\tilde{q}}^{\dot{1}}$
or to $\tilde{q}_{\dot{2}},\bar{\tilde{q}}^{\dot{2}}$.

To be concrete, if we choose the gauge $c_{I}=c_{I+1}=c_{I+2}=c_{I+3}=0$ and
$c=-\bar{\Gamma}$, a $G$ including all twisted scalars is then composed from
(5.10) and the gauge transformed version of (5.17), giving

$G=\begin{pmatrix}0&\bar{b}_{Ia}r_{I}^{a}&0&0\\\
b_{I}^{a}\bar{r}_{Ia}&0&\bar{d}^{\dot{1}}_{I+1}\tilde{q}_{I+1\,\dot{1}}+e^{i\hat{c}(\varphi)}\bar{d}^{\dot{2}}_{I+1}\tilde{q}_{I+1\,\dot{2}}&0\\\
0&{d}_{I+1\,\dot{1}}\bar{\tilde{q}}^{\dot{1}}_{I+1}+e^{-i\hat{c}(\varphi)}{d}_{I+1\,\dot{2}}\bar{\tilde{q}}^{\dot{2}}_{I+1}&0&\bar{b}_{I+2\,a}r_{I+2}^{a}\\\
0&0&b_{I+2}^{a}\bar{r}_{I+2\,a}&0\end{pmatrix}.$ (5.24)

The construction then follows as before. Differential equations for
$\bar{b}_{Ia}$, $b_{I}^{a}$, $\bar{b}_{I+2\,a}$, $b_{I+2}^{a}$ and for
$d_{I+1\,\dot{1}}$, $\bar{d}_{I+1}^{\dot{1}}$ are as in (4.9) and (5.14) and
are solved by (4.10) and (5.15). As for
$d_{I+1\,\dot{2}},\bar{d}_{I+1}^{\dot{2}}$, we find the equivalent to (5.19)
in the $c=-\bar{\Gamma}$ gauge

$\displaystyle\partial_{\varphi}(\Pi
e^{i\hat{c}(\varphi)}\bar{d}^{\dot{2}}_{I+1})-i\lambda
e^{i\hat{c}(\varphi)}\bar{d}^{\dot{2}}_{I+1}$ $\displaystyle=0\,,$ (5.25)
$\displaystyle\partial_{\varphi}(\Pi
e^{-i\hat{c}(\varphi)}d_{I+1\,\dot{2}})+i\lambda
e^{-i\hat{c}(\varphi)}{d}_{I+1\,\dot{2}}$ $\displaystyle=0\,,$

which is still solved by (5.20).

We found therefore the form of ${\mathcal{L}}$ coupling to both twisted
scalars, under the condition (5.23). Now $\widetilde{M}$ is given by

$\widetilde{M}=\begin{pmatrix}-i/k+{\bar{\delta}}^{\dot{1}}_{I+1}\delta_{I+1\,\dot{1}}&e^{-i\hat{c}(\varphi)}{\bar{\delta}}^{\dot{1}}_{I+1}\delta_{I+1\,\dot{2}}\\\
e^{i\hat{c}(\varphi)}{\bar{\delta}}^{\dot{2}}_{I+1}\delta_{I+1\,\dot{1}}&i/k+{\bar{\delta}}^{\dot{2}}_{I+1}\delta_{I+1\,\dot{2}}\end{pmatrix}.$
(5.26)

A special case of this construction was already carried out in [10]. In the
parameterization of that paper, $\Pi=1$ and $\lambda=\cos\theta$, with
$\theta$ the so-called “latitude” angle. It was then possible to include all
scalar fields in $G$ for $\theta=0$ (see equation (4.9) of [10]). The analog
of the obstruction (5.23) arose there for $\theta\neq 0$ (see the comment
below (5.15) of [10]). The reasoning for that is precisely the fact that
$e^{i\hat{c}(\varphi)}=e^{i\varphi\cos\theta}$ considered there is not single
valued.

###  5.2 Deformations with $\Pi=0$

Generalizing Section 4.2 to allow for twisted hypers, we start again with the
$1/2$ BPS loop with four nodes in (5.1). $H$ is the same as in (5.3), now
written generalizing (4.35) to

$H=\begin{pmatrix}0&\bar{\alpha}_{I}r_{I}^{\parallel}&0&0\\\
-\alpha_{I}\xi\bar{r}_{I\,\parallel}&0&0&0\\\
0&0&0&\bar{\alpha}_{I+2}r^{\parallel}_{I+2}\\\
0&0&-\alpha_{I+2}\xi\bar{r}_{I+2\,\parallel}&0\end{pmatrix}.$ (5.27)

As before, the fact that $\Pi=0$ implies that the variation of the deformed
loop is still a covariant derivative of $H$ regardless of the deformation.
Since $H$ does not include twisted scalars, we do not expect the relative
shift between the two pairs of nodes ($c$ in (5.1)) to be fixed by the
requirement that the deformed loop is supersymmetric. Below we see that this
is indeed the case.

The $\Pi=0$ version of the double transformations (5.6) is

$\displaystyle Q^{2}\tilde{q}_{\dot{1}}$
$\displaystyle=\frac{2i}{k}\xi(\bar{r}_{\parallel}r^{\parallel}\tilde{q}_{\dot{1}}-\tilde{q}_{\dot{1}}r^{\parallel}\bar{r}_{\parallel})+\frac{\lambda}{2}\tilde{q}_{\dot{1}}\,,$
(5.28) $\displaystyle Q^{2}\tilde{q}_{\dot{2}}$
$\displaystyle=\frac{2i}{k}\xi(\bar{r}_{\parallel}r^{\parallel}\tilde{q}_{\dot{2}}-\tilde{q}_{\dot{2}}r^{\parallel}\bar{r}_{\parallel})-\frac{\lambda}{2}\tilde{q}_{\dot{2}}\,,$
$\displaystyle Q^{2}\bar{\tilde{q}}^{\dot{1}}$
$\displaystyle=\frac{2i}{k}\xi(r^{\parallel}\bar{r}_{\parallel}\bar{\tilde{q}}^{\dot{1}}-\bar{\tilde{q}}^{\dot{1}}\bar{r}_{\parallel}r^{\parallel})-\frac{\lambda}{2}\bar{\tilde{q}}^{\dot{1}}\,,$
$\displaystyle Q^{2}\bar{\tilde{q}}^{\dot{2}}$
$\displaystyle=\frac{2i}{k}\xi(r^{\parallel}\bar{r}_{\parallel}\bar{\tilde{q}}^{\dot{2}}-\bar{\tilde{q}}^{\dot{2}}\bar{r}_{\parallel}r^{\parallel})+\frac{\lambda}{2}\bar{\tilde{q}}^{\dot{2}}\,.$

The building blocks are then the $4\times 4$ versions of $G$ and $C$ (we set
$c_{I}=c_{I+2}=0$ for convenience)

$\displaystyle G$
$\displaystyle=\begin{pmatrix}0&\bar{\beta}_{I\,\perp}r^{\perp}_{I}&0&0\\\
\beta_{I}^{\perp}\bar{r}_{I\,\perp}&0&\bar{d}^{\dot{a}}\tilde{q}_{I+1\,\dot{a}}&0\\\
0&d_{\dot{a}}\bar{\tilde{q}}^{\dot{a}}_{I+1}&0&\bar{\beta}_{I+2\,\perp}r^{\perp}_{I+2}\\\
0&0&\beta_{I+2}^{\perp}\bar{r}_{I+2\perp}&0\end{pmatrix},$ (5.29)
$\displaystyle C$
$\displaystyle=\begin{pmatrix}\bar{\beta}_{I\,\parallel}r_{I}^{\parallel}\bar{r}_{I\,\parallel}&0&0&0\\\
0&\beta_{I}^{\parallel}\bar{r}_{I\,\parallel}r^{\parallel}_{I}+c_{I+1}&0&0\\\
0&0&\bar{\beta}_{I+2\,\parallel}r_{I+2}^{\parallel}\bar{r}_{I+2\,\parallel}&0\\\
0&0&0&\beta_{I+2}^{\parallel}\bar{r}_{I+2\,\parallel}r^{\parallel}_{I+2}+c_{I+3}\end{pmatrix}.$

With these in hand, the superconnection
${\mathcal{L}}={\mathcal{L}}_{1/2}-iQG+\\{G,H\\}+C$ is supersymmetric provided
that the same condition as in (4.37) is obeyed.

The analysis for the $\beta$ parameters follows as in the 2-node case. Cubic
terms on the fields cancel for
$\bar{\beta}_{I\,\parallel}=\beta_{I}^{\parallel}$ and
$\bar{\beta}_{I+2\,\parallel}=\beta_{I+2}^{\parallel}$. Linear terms are such
that we find, in addition to (4.38), its $I+2$-node version

$\Lambda\bar{\beta}_{I+2\,\perp}\partial_{\varphi}(e^{i\varphi}\xi)=-ie^{i\varphi}c_{I+3}\bar{\alpha}_{I+2}\,,\qquad\Lambda\beta_{I+2}^{\perp}\partial_{\varphi}(e^{i\varphi}\xi)=ie^{i\varphi}c_{I+3}\alpha_{I+2}\xi\,.$
(5.30)

For the central block containing the $d$ parameters, one realizes that the
cubic term in the double variations (5.28) is exactly equal to $[G,H^{2}]$.
There is no contribution related to $d$ from $[C,H]$, so one is left with the
linear terms arising from $Q^{2}G$

$\begin{pmatrix}\ddots&~{}&~{}&~{}\\\
~{}&0&\dfrac{\lambda}{2}(\bar{d}^{\dot{1}}\tilde{q}_{\dot{1}}-\bar{d}^{\dot{2}}\tilde{q}_{2})&~{}\\\
~{}&-\dfrac{\lambda}{2}(d_{\dot{1}}\bar{\tilde{q}}^{\dot{1}}-d_{\dot{2}}\bar{\tilde{q}}^{\dot{2}})&0&~{}\\\
~{}&~{}&~{}&\ddots\end{pmatrix}=0\,.$ (5.31)

Solutions with nonvanishing $d$ parameters and a non block-diagonal structure
are only possible for supercharges with

$\lambda=0\,.$ (5.32)

In this case there are no constraints on $\bar{d}^{\dot{a}}$ and
$d_{\dot{a}}$, and they can be arbitrary functions. At the level of the
algebra, see Section 4.2.2, this means that loops in this section are
constructed from $Q$’s that square only to $\mathfrak{s}\mathfrak{u}(2)_{L}$
generators $R^{ab}$.

Note that as anticipated the derivation above does not set restrictions on the
relative shift $c$ appearing in (5.1), in contrast to the $\Pi\neq 0$ case. We
examine some special cases of the resulting operators in Section 6.5.

####  5.2.1 The special cases: $\xi=0$ and $\xi=\infty$

The analysis of $\xi=0$ and $\xi=\infty$ follows in analogy with Section
4.2.1. As before, both cases are equivalent under the replacement of $\eta$
and $\bar{\eta}$, so we focus only on the $\xi=0$ case.

Here, since we are considering longer quivers coupling to twisted
hypermultiplets, we need to include in $G$ not only $r^{\perp}$,
$\bar{r}_{\perp}$ but also the twisted scalars that are not annihilated by
$Q$. From (5.5) we see that these are $\tilde{q}_{\dot{1}}$ and
$\bar{\tilde{q}}^{\dot{2}}$, so we have

$G=\begin{pmatrix}0&\bar{\beta}_{I\perp}r_{I}^{\perp}&0&0\\\
\beta_{I}^{\perp}&0&\bar{d}^{\dot{1}}\tilde{q}_{I+1\,\dot{1}}&0\\\
0&d_{\dot{2}}\bar{\tilde{q}}_{I+1}^{\dot{2}}&0&\bar{\beta}_{I+2}r_{I+2}^{\perp}\\\
0&0&\beta_{I+2}^{\perp}\bar{r}_{I+2\perp}&0\end{pmatrix}.$ (5.33)

Conversely, the fields $\tilde{q}_{\dot{2}}$ and $\bar{\tilde{q}}^{\dot{1}}$
are annihilated by $Q$ and are included in the matrix $C$. In addition to
them, we should also include $\tilde{\rho}_{+}^{2}$ and
$\bar{\tilde{\rho}}_{1+}$, which are the linear combination of fermionic
fields from the twisted hypermultiplet that are annihilated by $Q$. Thus, we
have (setting again $c_{I}$ and $c_{I+2}$ to zero for convenience)

$\displaystyle C=$
$\displaystyle\begin{pmatrix}\bar{K}_{I}&\bar{\delta}_{I}\psi_{I\dot{2}+}&\gamma_{1}r_{I}^{\parallel}\tilde{q}_{I+1\,\dot{2}}&0\\\
\delta_{I}\bar{\psi}_{I+}^{\dot{1}}&K_{I}+c_{I+1}&\bar{\delta}_{I+1}\tilde{\rho}_{I+1,+}^{2}&\gamma_{2}\tilde{q}_{I+1\,\dot{2}}r_{I+2}^{\parallel}\\\
\gamma_{3}\bar{\tilde{q}}_{I+1}^{\dot{1}}\bar{r}_{I\parallel}&\delta_{I+1}\bar{\tilde{\rho}}_{I+1,1+}&\bar{K}_{I+2}&\bar{\delta}_{I+2}\psi_{I+2\,\dot{2}+}\\\
0&\gamma_{4}\bar{r}_{I+2\parallel}\bar{\tilde{q}}_{I+1}^{\dot{1}}&\delta_{I+2}\bar{\psi}_{I+2,+}^{\dot{1}}&K_{I+2}+c_{I+3}\end{pmatrix},$
(5.34)

with
$K_{I}\equiv\beta_{I}^{\parallel}\bar{r}_{I\parallel}r_{I}^{\parallel}+\tau_{I+1}\tilde{q}_{I+1\,\dot{2}}\bar{\tilde{q}}^{\dot{1}}_{I+1}$
and
$\bar{K}_{I}\equiv\bar{\beta}_{I\parallel}r_{I}^{\parallel}\bar{r}_{I\parallel}+\tau_{I}\bar{\tilde{q}}_{I-1}^{\dot{1}}\tilde{q}_{I-1\,\dot{2}}$.

As before, the superconnection
${\mathcal{L}}={\mathcal{L}}_{1/2}-iQG+\\{G,H\\}+C$ is supersymmetric provided
that (4.37) is obeyed. This is solved by

$\bar{\beta}_{I\parallel}=\beta_{I}^{\parallel}\,,\qquad\bar{\beta}_{I+2\parallel}=\beta_{I+2}^{\parallel}\,,\qquad\gamma_{1}\bar{\alpha}_{I+2}=\gamma_{2}\bar{\alpha}_{I}\,,$
(5.35)

and by setting the remaining parameters in $C$ to zero, except for
$\bar{\delta}_{I}$, $\bar{\delta}_{I+2}$ and $\delta_{I+1}$, which are left
arbitrary. We write down the resulting operator at the end of Section 6.5.

##  6 Special cases

Having carried out the systematic construction of BPS hyperloops described
above, we turn now to some special examples of the constructions. This
includes making contact with previously described operators and identifying
new ones. Our emphasis is on operators preserving more than one supercharge.

###  6.1 Single node bosonic loops

We start with the simplest possible BPS Wilson loops in three-dimensional
Chern-Simons-matter theories, those involving only a single node and
${\mathcal{L}}$ is a $1\times 1$ block with only the gauge field and bilinears
of the scalars. The first such bosonic loops were constructed by Gaiotto and
Yin in off-shell ${\mathcal{N}}=2$ language in [36]. Analogues of them in
ABJ(M) theory were described in [37, 38, 39] and that description carries over
also to ${\mathcal{N}}=4$ theories. Such loops preserve at most four
supercharges. The other previously identified family of bosonic loops are the
“bosonic latitude” loops of [21, 26, 10], which preserve a pair of
supercharges.

To get such loops in our setting we may decouple the nodes by simply setting
$\beta^{1}=\beta^{2}=\bar{\beta}_{1}=\bar{\beta}_{2}=0$ in the analysis of
Section 4 for the case $\Pi\neq 0$ (we comment below on the case $\Pi=0$).
This eliminates all the fermions in the superconnection ${\mathcal{L}}$, which
becomes block-diagonal with a connection in the $I$-th block taking the form

$\displaystyle{\mathcal{A}}$
$\displaystyle=A_{\varphi}+\frac{i}{k}\Pi^{-1}(r^{1}\bar{r}_{1}-r^{2}\bar{r}_{2})-\frac{i}{k}(\tilde{\mu}\indices{{}^{\dot{1}}_{\dot{1}}}-\tilde{\mu}\indices{{}^{\dot{2}}_{\dot{2}}})\,,$
(6.1)

It is easy to show that these loops preserve at least two supercharges.
Consider in fact the supercharge $Q^{\prime}$ gotten by the replacement
$\bar{\eta}^{\imath}_{a}\to-\bar{\eta}^{\imath}_{a}$ in (3.4)

$Q^{\prime}=\eta^{\imath}_{a}Q^{\dot{2}a+}_{\imath}-\bar{\eta}^{\imath}_{a}(\sigma^{1})_{\imath}^{\
\bar{\imath}}Q^{\dot{1}a-}_{\bar{\imath}}\,.$ (6.2)

Under this change of sign, $\Pi\to-\Pi$, $r^{2}\to-r^{2}$ and
$\bar{r}_{1}\to-\bar{r}_{1}$, such that (6.1) is left invariant. Note that
because $\Pi\neq 0$, $Q$ is the sum of barred and unbarred supercharges and by
the above argument these must be preserved separately.

Alternatively, this can be seen by investigating the bosonic symmetries. In
particular, note that the transverse rotation $T_{\perp}$ keeps the loop fixed
pointwise, and therefore acts trivially on the scalars as well as on the
parallel component of the gauge field, the only fields in the bosonic loop.
Closure of the symmetry algebra then implies that, in addition to $Q$, the
supercharge $[T_{\perp},Q]$ is preserved by the loop. From (A.6) we see that
this generates $Q^{\prime}$, so we come to the same conclusion as above (an
analogous argument can be made using the generator $\bar{R}_{3}$).

A useful way to write the connection (6.1) is in terms of the moment maps
$\mu^{a}{}_{b}$ as

${\mathcal{A}}=A_{\varphi}+\frac{i}{k}\frac{1}{(\chi-\bar{\chi})}\left((\chi+\bar{\chi})(\mu\indices{{}^{1}_{1}}-\mu\indices{{}^{2}_{2}})+2\mu\indices{{}^{2}_{1}}-2\chi\bar{\chi}\,\mu\indices{{}^{1}_{2}}\right)-\frac{i}{k}(\tilde{\mu}\indices{{}^{\dot{1}}_{\dot{1}}}-\tilde{\mu}\indices{{}^{\dot{2}}_{\dot{2}}})\,,$
(6.3)

with

$\chi=\frac{(\eta\bar{v})_{1}}{(\eta\bar{v})_{2}}\,,\qquad\bar{\chi}=\frac{(\bar{\eta}v)_{1}}{(\bar{\eta}v)_{2}}\,,$
(6.4)

which are generally linear fractional transformations of $e^{i\varphi}$ (3.5)
(and as usual, they are not conjugates).

The most degenerate case is when both $\chi$ and $\bar{\chi}$ have no
$\varphi$ dependence. This requires the numerators and denominators to be
proportional to each-other, spanning a two dimensional space of $\eta$’s and
likewise $\bar{\eta}$’s. This implies that the loop preserves 4 supercharges
and having no $\varphi$ dependence, it also preserves the $SO(2,1)$ conformal
group. To recover the Gaiotto-Yin Loop [36] we take
$\chi=1/\bar{\chi}\to\infty$. Other values of $\chi$, $\bar{\chi}$ are related
by the action of the complexification of the broken $SU(2)_{L}$ symmetry.

When $\chi$ is a constant and $\bar{\chi}$ depends on $\varphi$ (or vice
versa), there is only partial degeneracy, and the loops preserve three
supercharges, or are 3/16 BPS. Such loops have not been previously discussed
in the literature.

When both $\chi$ and $\bar{\chi}$ depend on $\varphi$, the loops preserve a
pair of supercharges. A simple example is when they are just monomials, for
example $\chi=-\tan(\theta/2)e^{-i\varphi}$ and
$\bar{\chi}=\cot(\theta/2)e^{-i\varphi}$. The connection takes the form

${\mathcal{A}}=A_{\varphi}-\frac{i}{k}\left(\cos\theta(\mu\indices{{}^{1}_{1}}-\mu\indices{{}^{2}_{2}})+\sin\theta\,e^{-i\varphi}\mu\indices{{}^{1}_{2}}+\sin\theta\,e^{i\varphi}\mu\indices{{}^{2}_{1}}\right)-\frac{i}{k}(\tilde{\mu}\indices{{}^{\dot{1}}_{\dot{1}}}-\tilde{\mu}\indices{{}^{\dot{2}}_{\dot{2}}})\,.$
(6.5)

These are the latitude loops found in [21] and studied in [26, 10]. As the
$\varphi$ dependence breaks conformal invariance, acting with the
(complexified) conformal group $SL_{2}(\mathbb{C})$ on the loop above
generates many other loops, including those where $\chi$ and $\bar{\chi}$ are
proper rational functions and not mere monomials.

There are yet more peculiar bosonic loops that preserve two supercharges, but
are not similar to the latitude loops. Representatives of those have

$\chi=e^{-i\varphi}+\nu\,,\qquad\bar{\chi}=e^{-i\varphi}-\nu\,,$ (6.6)

with an arbitrary parameter $\nu$.

Despite all the machinery in the previous sections, the analysis of the most
general BPS bosonic loop requires yet further techniques, so those will be
explored in a future publication [40]. That exploration will also relax the
condition in this paper that the loops arise from continuous deformations of
the 1/2 BPS loop, which could give rise to further BPS bosonic loops.

###  6.2 Two-node hyperloops with $\Pi\neq 0$

Let us look now at some special examples of the hyperloops with two nodes
constructed in Section 4.1. Examining (4.12), the most symmetric possibility
is that $M$ is proportional to the identity, restoring $SU(2)_{L}$ symmetry.
There are two such solutions. The first with $\beta^{1}=\bar{\beta}_{1}=0$ and
$\beta^{2}\bar{\beta}_{2}=2i/k$, which is just the original 1/2 BPS loop in
(3.1). The second has $\beta^{2}=\bar{\beta}_{2}=0$ and
$\beta^{1}\bar{\beta}_{1}=-2i/k$, which is the second 1/2 BPS loop with the
same symmetries in (3.7) (albeit written in a different gauge).

A less symmetric case is when $M$ is diagonal, but not necessarily
proportional to the identity, so when
$\bar{\beta}_{1}\beta^{2}=\bar{\beta}_{2}\beta^{1}=0$. If
$\beta^{1}=\beta^{2}=0$ or $\bar{\beta}_{1}=\bar{\beta}_{2}=0$, the connection
becomes upper or lower triangular, respectively. As discussed in [11, 9, 10],
the resulting loops are effectively the same as if all the
$\beta^{a}=\bar{\beta}_{a}=0$, since they are all identical as quantum
operators. The interesting case is then when $\bar{\beta}_{1}=\beta^{1}=0$ or
$\bar{\beta}_{2}=\beta^{2}=0$. Taking the former as an example, we find

${\mathcal{L}}=\begin{pmatrix}A_{\varphi,I}+M_{a}^{\
b}r^{a}\bar{r}_{b}-\frac{i}{k}(\tilde{\mu}_{I\ \dot{1}}^{\
\dot{1}}-\tilde{\mu}_{I\ \dot{2}}^{\
\dot{2}})&-i\bar{\beta}_{2}\psi_{I\dot{1}-}\\\
i\beta^{2}\bar{\psi}_{I+}^{\dot{1}}&A_{\varphi,I+1}+M_{\
a}^{b}\bar{r}_{b}r^{a}-\frac{i}{k}(\tilde{\mu}_{I+1\,\dot{1}}{}^{\dot{1}}-\tilde{\mu}_{I+1\,\dot{2}}{}^{\dot{2}})-\frac{1}{2}\end{pmatrix},$
(6.7)

with

$M=\Pi^{-1}\begin{pmatrix}\frac{i}{k}&0\\\
0&\bar{\beta}_{2}\beta^{2}-\frac{i}{k}\end{pmatrix}.$ (6.8)

In addition to the supercharge $Q$, these hyperloops preserve a supercharge
$Q^{\prime}$ arising from same $\eta^{\imath}_{a}$ but with
$\bar{\eta}^{\imath}_{a}\to-\bar{\eta}^{\imath}_{a}$. The argument is
identical to the case of the bosonic loops presented in Section 6.1. In this
case we see that the fermionic terms are unchanged if we keep the same
$\beta$’s and $M\to-M$, so the diagonal entries $M_{1}^{\ 1}r^{1}\bar{r}_{1}$
and $M_{2}^{\ 2}r^{2}\bar{r}_{2}$ are also left invariant. The requirement
that $M$ is diagonal guarantees, therefore, that the loop is also invariant
under $Q^{\prime}$ and is 1/8 BPS.

Thus, for any choice of $Q$ with $\Pi\neq 0$, if we restrict the parameters
such that $\beta^{1}=\bar{\beta}_{1}=0$, we find a family of 1/8 BPS
hyperloops parametrized by $\beta^{2}$ and $\bar{\beta}_{2}$. However, as we
can conjugate ${\mathcal{L}}$ by a constant matrix

${\mathcal{L}}\to\begin{pmatrix}1&0\\\
0&x^{-1}\end{pmatrix}{\mathcal{L}}\begin{pmatrix}1&0\\\ 0&x\end{pmatrix}\,,$
(6.9)

this gauge transformation eliminates one of the parameters, and we end up with
a one (complex) dimensional moduli space.

This is very similar to the discussion in [10], but it is much more general,
as it works with any of the supercharges $Q$ in (3.4) with $\Pi\neq 0$. To
make contact with the constructions in [10] we can look at the moduli space of
1/4 BPS hyperloops studied there, which are all deformations of the usual
bosonic Gaiotto-Yin loops [36]. Those loops preserve a one-dimensional
conformal group, under which the supercharges are charged. Looking at the
algebra (4.13) and requiring only conformal transformations in the square of
the supercharge imposes
$\epsilon_{\imath\jmath}\left(\bar{\eta}_{a}^{\imath}\eta_{b}^{\jmath}+\bar{\eta}_{b}^{\imath}\eta_{a}^{\jmath}\right)=0$.
To realize this, we choose two vectors $\bar{w}_{a}$ and $w_{a}$ (as usual,
bar does not indicate complex conjugation). For an arbitrary vector
$s^{\imath}$, define parameters $\bar{\eta},\eta$ as

$\eta^{\imath}_{a}=w_{a}s^{\imath}\,,\qquad\bar{\eta}^{\imath}_{a}=\bar{w}_{a}s^{\imath}\,.$
(6.10)

The resulting supercharges are all linear combinations of

$w_{a}Q_{\imath}^{\dot{2}a}\,,\qquad\bar{w}_{a}Q_{\bar{\imath}}^{\dot{1}a}\,,\qquad$
(6.11)

whose anticommutators generate the bosonic algebra
$\mathfrak{s}\mathfrak{o}(2,1)\oplus{\mathfrak{u}}(1)$, where the
${\mathfrak{u}}(1)$ summand is generated by
$L_{\perp}+\frac{1}{2}w_{a}\bar{w}_{b}R^{ab}$ (see Section 4.1.1 for details).

In [10] the vector $w_{a}$ was $\delta_{a}^{2}$ and $\bar{w}_{a}$ was
$\delta_{a}^{1}$. Other choices can be achieved by an $SU(2)_{L}$ rotation.
What was more restrictive there is that only a single choice of $Q$ (or
$s^{\imath}$) was used. As long as we turn on only the parameters as in (6.8),
we preserve all the supercharges in (6.11), so any choice (with $\Pi\neq 0$)
is equivalent. When turning on more $\beta$ parameters, we find different
moduli spaces, depending on the exact choice of $Q$. Our analysis here
therefore generalizes also this simple case of deformations of the 1/4 BPS
bosonic loop.

As discussed in Section 6.1, there are several new bosonic loops generated by
our construction that are not related to those in [10]. Clearly their
deformations with $\beta\neq 0$ are also new.

###  6.3 Two-node hyperloops with $\Pi=0$

This case is presented in Section 4.2, where it is shown that the general
deformation is of the form (4.34) with $G$ and $C$ as in (4.36), subject to
the constraints that $\bar{\beta}_{\parallel}=\beta^{\parallel}$ and the
conditions on $\bar{\beta}_{\perp}$, $\beta^{\perp}$ and $c$ in (4.38). The
resulting expression for ${\mathcal{L}}$ is then in (4.39).

A simple way to find loops with enhanced supersymmetry is when the
superconnection is invariant under $\mathfrak{s}\mathfrak{u}(2)_{L}$, which
arises when $M_{a}^{\;b}r^{a}\bar{r}_{b}\propto\nu_{I}$. Looking at the
expression for $\nu$ in (4.33) and $M$ in (4.40), we see that one needs to
impose

$\beta^{\parallel}=0\,,\qquad\xi\alpha\bar{\beta}_{\perp}=\bar{\alpha}\beta^{\perp}\,.$
(6.12)

These equations are consistent with (4.38)555(4.38) is also solved with
$\xi=\xi_{0}e^{-i\varphi}$, with a constant $\xi_{0}\neq 0$, arbitrary
$\beta^{\perp}$, $\bar{\beta}_{\perp}$,
$\beta^{\parallel}=\bar{\beta}_{\parallel}$ and $c=0$., combining all the
parameters to a single periodic function
$\gamma=1-ik\Lambda\bar{\alpha}\beta^{\perp}$ appearing in the superconnection
as

$\mathcal{L}=\begin{pmatrix}A_{\varphi,I}+\frac{i}{k}\gamma\nu_{I}-\frac{i}{k}(\tilde{\mu}_{I\;\dot{1}}^{\;\dot{1}}-\tilde{\mu}_{I\;\dot{2}}^{\;\dot{2}})&-\frac{i\bar{\alpha}}{2}(\gamma+1)\psi_{\dot{1}-}-\frac{i\bar{\alpha}}{2}(\gamma-1)\xi^{-1}\psi_{\dot{2}+}\\\
\frac{i\alpha}{2}(\gamma+1)\bar{\psi}_{+}^{\dot{1}}-\frac{i\alpha}{2}(\gamma-1)\xi\bar{\psi}_{-}^{\dot{2}}&A_{\varphi,I+1}+\frac{i}{k}\gamma\nu_{I+1}-\frac{i}{k}(\tilde{\mu}_{I+1\dot{1}}^{\quad\;\;\dot{1}}-\tilde{\mu}_{I+1\dot{2}}^{\quad\;\;\dot{2}})+c-\frac{1}{2}\end{pmatrix},$
(6.13)

and $c=i\frac{\gamma-1}{2}\partial_{\varphi}\log(\xi e^{i\varphi})$.

The degree of supersymmetry enhancement depends on the choice of supercharge
$Q$. Specifically, following Section 4.2.2, we distinguish three cases.

####  6.3.1 1/8 BPS loops

First, suppose $0\neq Q^{2}\in\mathfrak{s}\mathfrak{u}(2)_{L}$. Putting
together (4.21) and (4.24), one sees that the parameters $\eta$, $\bar{\eta}$
may be cast into the form

$\displaystyle\eta^{\imath}_{a}=t^{\imath}\,w_{a}\,,\qquad\bar{\eta}^{\imath}_{a}=\bar{t}^{\imath}\,w_{a}\,,$
(6.14)

with some vector $w_{a}\neq 0$ and
$\epsilon_{\imath\jmath}t^{\imath}\bar{t}^{\jmath}\neq 0$. Acting on the
resulting supercharge with $\mathfrak{s}\mathfrak{u}(2)_{L}$, we find that,
regardless of the choice of $w_{a}$, the loop preserves the two supercharges
(with a convenient normalization)

$\displaystyle
Q_{1}=\frac{1}{\sqrt{\epsilon_{\imath\jmath}t^{\imath}\bar{t}^{\jmath}}}\left(t^{\imath}\,Q_{\imath}^{{\dot{2}}1}+\bar{t}^{\imath}\,(\sigma_{1})\indices{{}_{\imath}^{\bar{\jmath}}}\,Q^{{\dot{1}}1}_{\bar{\jmath}}\right),\qquad
Q_{2}=\frac{1}{\sqrt{\epsilon_{\imath\jmath}t^{\imath}\bar{t}^{\jmath}}}\left(t^{\imath}\,Q_{\imath}^{{\dot{2}}2}+\bar{t}^{\imath}\,(\sigma_{1})\indices{{}_{\imath}^{\bar{\jmath}}}\,Q^{{\dot{1}}2}_{\bar{\jmath}}\right).$
(6.15)

Using (A.9) it is easy to verify that their anticommutators generate
$\mathfrak{s}\mathfrak{u}(2)_{L}$

$\displaystyle\\{Q_{1},Q_{1}\\}=\frac{1}{2}R_{+}\,,\qquad\\{Q_{1},Q_{2}\\}=-R_{3}\,,\qquad\\{Q_{2},Q_{2}\\}=-\frac{1}{2}R_{-}\,.$
(6.16)

####  6.3.2 1/4 BPS loops and conformal loops

Another case is when the supercharge satisfies $0\neq
Q^{2}\in\mathfrak{u}(1)_{L_{\perp}}$. In this case, as derived in (4.47), we
have $\xi=\xi_{0}e^{-i\varphi}$, which immediately implies $c=0$. As discussed
in Section 4.2.2, the parameters of $Q$ take the form

$\displaystyle\eta^{l}_{a}$ $\displaystyle=ts^{l}w_{a}\,,$
$\displaystyle\qquad\eta^{r}_{a}$ $\displaystyle=ts^{r}z_{a}\,,$ (6.17)
$\displaystyle\bar{\eta}^{l}_{a}$ $\displaystyle=\bar{t}s^{l}w_{a}\,,$
$\displaystyle\qquad\bar{\eta}^{r}_{a}$ $\displaystyle=\bar{t}s^{r}z_{a}\,,$

where both $\epsilon^{ab}w_{a}z_{b}\neq 0$ and $s^{l}s^{r}\neq 0$. Knowing
that the loop is invariant under $\mathfrak{s}\mathfrak{u}(2)_{L}$ we can
project (6.17) to those terms involving either only $w_{a}$ or only $z_{a}$.
Acting then with raising and lowering operators projects further to the two
components $a=1,2$, removing the dependence on $w_{a}$ and $z_{a}$ altogether,
and leaving us with four supercharges

$\displaystyle Q_{1}$
$\displaystyle=tQ^{{\dot{2}}1}_{r}+\bar{t}Q^{{\dot{1}}1}_{\bar{l}}\,,$
$\displaystyle\qquad Q_{2}$
$\displaystyle=tQ^{{\dot{2}}1}_{l}+\bar{t}Q^{{\dot{1}}1}_{\bar{r}}\,,$ (6.18)
$\displaystyle Q_{3}$
$\displaystyle=tQ^{{\dot{2}}2}_{r}+\bar{t}Q^{{\dot{1}}2}_{\bar{l}}\,,$
$\displaystyle\qquad Q_{4}$
$\displaystyle=tQ^{{\dot{2}}2}_{l}+\bar{t}Q^{{\dot{1}}2}_{\bar{r}}\,.$

Examining these, we see that they form doublets of
$\mathfrak{s}\mathfrak{o}(2,1)$ (exchanging $l$ and $r$).

The algebra generated by these supercharges is very simple, with the only non-
vanishing anticommutators

$\displaystyle\\{Q_{1},Q_{4}\\}=-2t\bar{t}L_{\perp}\,,\qquad\\{Q_{2},Q_{3}\\}=2t\bar{t}L_{\perp}\,.$
(6.19)

Note that the bosonic part of this $1/4$ BPS algebra is just
$\mathfrak{u}(1)_{L_{\perp}}$, while $\mathfrak{s}\mathfrak{u}(2)_{L}$ and the
one-dimensional conformal algebra $\mathfrak{s}\mathfrak{o}(2,1)$ act as outer
automorphisms.

We noted that the superconnection (6.13) is invariant under
$\mathfrak{s}\mathfrak{u}(2)_{L}$. It is interesting to check whether it is
also invariant under $\mathfrak{s}\mathfrak{o}(2,1)$. This clearly requires
$\gamma$ to be a constant, as otherwise ${\mathcal{L}}$ is not invariant even
under rotations. Considering then a general conformal generator
$J=a_{+}J_{+}+a_{0}J_{0}+a_{-}J_{-}$ and using (A.4)-(A.5), one finds that the
conformal transformation of ${\mathcal{L}}$ in (6.13) is a total derivative

$\displaystyle J{\cal L}={\cal D}_{\varphi}^{\cal L}(a{\cal L}+H),$ (6.20)

with

$\displaystyle a=a_{+}e^{i\varphi}-ia_{0}+a_{-}e^{-i\varphi},\qquad
H=\left(\begin{array}[]{cc}0&0\\\ 0&a/2\end{array}\right).$ (6.23)

The resulting Wilson loops are then invariant under
$\mathfrak{s}\mathfrak{o}(2,1)\oplus\mathfrak{s}\mathfrak{u}(2)_{L}\oplus\mathfrak{u}(1)_{L_{\perp}}$,
providing a previously unidentified family of conformal 1/4 BPS loops.

Note that the argument here is classical and as the superalgebra (6.19) does
not include the conformal generators, we cannot be sure that it is not spoiled
by quantum corrections.

####  6.3.3 Further 1/8 BPS loops

The last example arising from (6.13) are loops with nilpotent $Q$. Since this
case lies at the intersection of the previous two, we have to impose all the
conditions discussed above. For the parameters, we have

$\displaystyle\eta^{\imath}_{a}=a\rho^{\imath}w_{a}\,,\qquad\bar{\eta}^{\imath}_{a}=\bar{a}\rho^{\imath}w_{a}\,.$
(6.24)

They give a pair of nilpotent supercharges

$\displaystyle Q_{1}$
$\displaystyle=a\rho^{\imath}Q^{\dot{2}1}_{\imath}+\bar{a}\rho^{\imath}(\sigma_{1})\indices{{}_{\imath}^{\bar{\jmath}}}Q^{\dot{2}1}_{\bar{\jmath}}\,,$
(6.25) $\displaystyle Q_{2}$
$\displaystyle=a\rho^{\imath}Q^{\dot{2}2}_{\imath}+\bar{a}\rho^{\imath}(\sigma_{1})\indices{{}_{\imath}^{\bar{\jmath}}}Q^{\dot{2}2}_{\bar{\jmath}}\,,$

whose anticommutator vanishes as well.

Another family of loops with enhanced supersymmetry arises if, instead of
$\mathfrak{s}\mathfrak{u}(2)_{L}$ symmetry (as in (6.13)), we demand conformal
invariance from the beginning. Generalising the discussion in Section 6.3.2,
we impose the equation (6.20) directly on the superconnection (4.39). The off-
diagonal components of this matrix equation are satisfied, as in Section
6.3.2, as long as $\xi=\xi_{0}e^{-i\varphi}$ and $c=0$, which identically
solves the supersymmetry conditions (4.38). Additionally, if we redefine

$\displaystyle\beta^{\perp}=\frac{\alpha}{2\Lambda}(\gamma-1),\qquad\bar{\beta}_{\perp}=\frac{\bar{\alpha}}{2\Lambda\xi}(\bar{\gamma}-1),\qquad\beta^{\parallel}=\frac{i}{k\Lambda}\gamma^{\parallel},$
(6.26)

then we need to impose that $\gamma$, $\bar{\gamma}$ and $\gamma^{\parallel}$
are constants.

The expression for the superconnection (4.39) then becomes

$\mathcal{L}=\begin{pmatrix}A_{\varphi,I}+M_{a}{}^{b}r^{a}\bar{r}_{b}-\frac{i}{k}(\tilde{\mu}_{I\;\dot{1}}^{\;\dot{1}}-\tilde{\mu}_{I\;\dot{2}}^{\;\dot{2}})&-\frac{i\bar{\alpha}}{2}(\bar{\gamma}+1)\psi_{\dot{1}-}-\frac{i\bar{\alpha}}{2}(\bar{\gamma}-1)\xi^{-1}\psi_{\dot{2}+}\\\
\frac{i\alpha}{2}(\gamma+1)\bar{\psi}_{+}^{\dot{1}}-\frac{i\alpha}{2}(\gamma-1)\xi\bar{\psi}_{-}^{\dot{2}}&A_{\varphi,I+1}+M_{a}{}^{b}\bar{r}_{b}r^{a}-\frac{i}{k}(\tilde{\mu}_{I+1\dot{1}}^{\quad\;\;\dot{1}}-\tilde{\mu}_{I+1\dot{2}}^{\quad\;\;\dot{2}})-\frac{1}{2}\end{pmatrix},$
(6.27)

with the couplings to the rotated scalars (4.40) given by

$M_{a}{}^{b}=\frac{i}{k\Lambda}\begin{pmatrix}0&\bar{\gamma}\\\
\gamma&\gamma^{\parallel}\end{pmatrix}.$ (6.28)

The remaining check is whether the diagonal part of equation (6.20) is
satisfied, which imposes that the couplings to the unrotated scalars
$q^{a},\bar{q}_{a}$ are constant. This can be arranged in two ways. Firstly,
by (4.33) we can set $\bar{\gamma}=\gamma,\gamma^{\parallel}=0$ to obtain
scalar terms proportional to $\nu_{I},\nu_{I+1}$ without any explicit
$\varphi$ dependence. These loops are just the conformal $1/4$ BPS loops
described in the previous section.

Alternatively, constant scalar couplings can be obtained for arbitrary
$\gamma,\bar{\gamma},\gamma^{\parallel}$ by demanding instead
$\epsilon^{ab}\bar{\eta}^{l}_{a}\bar{\eta}^{r}_{b}=0$ or, equivalently,
$Q^{2}=0$. In order to derive the symmetries preserved by these loops, we
parametrise the supercharge using (4.48) and act on it with the conformal
generators. This process generates another supercharge, so in total we have

$\displaystyle
Q_{1}=w_{a}\left(aQ^{{\dot{2}}a}_{l}+\bar{a}Q^{{\dot{1}}a}_{\bar{r}}\right),\qquad
Q_{2}=w_{a}\left(aQ^{{\dot{2}}a}_{r}+\bar{a}Q^{{\dot{1}}a}_{\bar{l}}\right).$
(6.29)

Both these supercharges are nilpotent and their anticommutator vanishes. By
construction, $\mathfrak{s}\mathfrak{o}(2,1)$ acts on the algebra as an outer
automorphism.

There is yet another example of supersymmetry enhancement without
$\mathfrak{s}\mathfrak{u}(2)_{L}$ symmetry, but with invariance under
$T_{\perp}$ (but not $L_{\perp}$ in (A.8)). Recalling that $T_{\perp}$ acts
diagonally and separates barred from unbarred supercharges, it is easily seen
that the commutator $Q^{\prime}=[T_{\perp},Q]$ is linearly independent of $Q$,
provided $Q$ comprises both barred and unbarred supercharges (so $\xi\neq
0,\infty$). To see which loops are invariant under $Q^{\prime}$, we note that
keeping $\eta^{\imath}_{a}$ and changing
$\bar{\eta}^{\imath}_{a}\to-\bar{\eta}^{\imath}_{a}$ leaves $\Pi=0$, likewise
$\Lambda$ is unmodified, and $\xi\to-\xi$. Noticing that (4.39) contains terms
proportional to both $\Lambda\beta^{\perp}$ and $\xi\Lambda\beta^{\perp}$, we
have to set $\beta^{\perp}=0$ and similarly for $\bar{\beta}_{\perp}$, which
by (4.38) also fixes $c=0$. The resulting superconnection is

$\mathcal{L}={\mathcal{L}}_{1/2}+\begin{pmatrix}\beta^{\parallel}r^{\parallel}\bar{r}_{\parallel}&0\\\
0&\beta^{\parallel}\bar{r}_{\parallel}r^{\parallel}\end{pmatrix},$ (6.30)

where $\beta^{\parallel}$ can be an arbitrary periodic function of $\varphi$.
One can check that generically the supersymmetry is not enhanced further.

####  6.3.4 The special cases: $\xi=0$ and $\xi=\infty$

When $\xi=0$, the superconnection of loops are the same as (4.39) with
$\xi=c=0$ and $\beta^{\perp}$, $\bar{\beta}_{\perp}$ and $\beta^{\parallel}$
free. If we want to study the $\mathfrak{s}\mathfrak{u}(2)_{L}$ enhanced
points, we should impose $\beta^{\perp}=\beta^{\parallel}=0$ and get the loops

$\mathcal{L}=\mathcal{L}_{1/2}+\begin{pmatrix}0&-i\Lambda\bar{\beta}_{\perp}\psi_{\dot{2}+}\\\
0&0\end{pmatrix}.$ (6.31)

The case $\xi=\infty$ is similar with a term on the lower left corner.

In all of these examples the free parameters $\beta^{\parallel}$,
$\beta^{\perp}$ (and in the last case also $\bar{\beta}_{\perp}$) are any
periodic functions of $\varphi$. The reason is most transparent with regards
to $\beta^{\parallel}$, as $Q$ annihilates $r^{\parallel}\bar{r}_{\parallel}$
and we can insert any density of them along the loop.

In Sections 6.1, 6.2 and 6.3 above, we noted multiple examples of hyperloops
that in addition to $Q$ preserve also $Q^{\prime}$ with
$\bar{\eta}^{\imath}_{a}\to-\bar{\eta}^{\imath}_{a}$. They clearly also
preserve $Q\pm Q^{\prime}$, which are supercharges with $\Pi=0$ and $\xi=0$
and $\xi=\infty$.

###  6.4 Hyperloops with twisted hypers and $\Pi\neq 0$

To couple our hyperloops to the twisted hypermultiplets, the starting point in
Section 5 is a $4\times 4$ superconnection (5.1) which takes a block-diagonal
form and is deformed with parameters $\beta$ and $\delta$. Here we focus on
special examples of these loops. As a first step, we set all the $\beta$’s to
zero. In the absence of the $\delta$ terms, this would give a diagonal
connection with only bosonic fields.

With $\beta=0$ and $\delta\neq 0$, we find instead a block-diagonal form, with
a $2\times 2$ block involving the nodes $I+1$ and $I+2$, and two decoupled
nodes $I$ and $I+3$. We ignore in the following the decoupled nodes and
concentrate only on the remaining $2\times 2$ block. Note that often the
decoupled nodes do not preserve the symmetries of the central block. This can
be remedied in the setting of a circular quiver.

In the case of a deformation with $\delta_{I+1\,\dot{1}}$ and
$\bar{\delta}^{\dot{1}}_{I+1}$, the central block takes the form

${\mathcal{L}}=\begin{pmatrix}A_{\varphi,I+1}+M^{b}_{\
a}\bar{r}_{I\,b}r^{a}_{I}+\widetilde{M}^{\dot{a}}_{\
\dot{b}}\tilde{q}_{I+1\,\dot{a}}\bar{\tilde{q}}_{I+1}^{\dot{b}}-\frac{1}{2}&-i\bar{\delta}^{\dot{1}}_{I+1}\tilde{\rho}_{I+1\,+}^{2}\\\
-i\Pi^{-1}\delta_{I+1\,\dot{1}}\bar{\tilde{\rho}}_{I+1\,2-}&A_{\varphi,I+2}+M_{a}^{\
b}r_{I+2}^{a}\bar{r}_{I+2\,b}+\widetilde{M}^{\
\dot{a}}_{\dot{b}}\bar{\tilde{q}}^{\dot{b}}_{I+1}\tilde{q}_{I+1\,\dot{a}}+\bar{\Gamma}\end{pmatrix}$
(6.32)

with (see (6.1))

$M=\frac{i}{k}\Pi^{-1}\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}\,,\qquad\widetilde{M}=\begin{pmatrix}-i/k+{\bar{\delta}}^{\dot{1}}_{I+1}\delta_{I+1\,\dot{1}}&0\\\
0&i/k\end{pmatrix}.$ (6.33)

This structure is the analog of the two-node quiver with a coupling to a
single pair of scalars in the hypermultiplets as in (6.8). Just as in that
example, these loops have enhanced supersymmetry with the second supercharge
$Q^{\prime}$ given by exchanging
$\bar{\eta}^{\imath}_{a}\to-\bar{\eta}^{\imath}_{a}$. So all these loops are
at least $1/8$ BPS.

Further supersymmetry enhancement arises in the schemes explained in Section
6.1 leading to operators that can preserve either 3 or 4 supercharges. Even
further supersymmetry enhancement arises by setting
$\delta_{I+1\,\dot{1}}\bar{\delta}_{I+1}^{\dot{1}}=2i/k$, as then the loop
enjoys $\mathfrak{s}\mathfrak{u}(2)_{R}$ symmetry. In this case, the analysis
of the previous paragraph is extended to supercharges with
$\dot{1}\leftrightarrow\dot{2}$ and we have a doubling of the amount of
preserved supersymmetry. Note that because of the
$\dot{1}\leftrightarrow\dot{2}$ exchange, these supercharges are not preserved
by the original 1/2 BPS loop, see (3.3). The $1/4$ BPS loop becomes the $1/2$
BPS operator coupling to the single pair of scalars $\tilde{q}_{\dot{1}}$,
$\bar{\tilde{q}}^{\dot{1}}$ from the twisted hypermultiplet. The $1/8$ BPS
operator becomes $1/4$ BPS and for the particular parameterization

$\bar{\eta}_{1}^{r}=\eta_{2}^{l}=\cos\frac{\theta}{2}\,,\qquad\bar{\eta}_{2}^{l}=-\eta_{1}^{r}=\sin\frac{\theta}{2}\,,$
(6.34)

we recover the “fermionic latitude” loops constructed first in ABJM theory in
[21] and generalized to ${\mathcal{N}}=4$ theories in [10], see also [26]. The
$3/16$ BPS operator becomes $3/8$ BPS.

Completely analog constructions arise with
$\delta_{I+1\,\dot{1}}=\bar{\delta}^{\dot{1}}_{I+1}=0$ and nonzero couplings
$\delta_{I+1\,\dot{2}}$ and $\bar{\delta}^{\dot{2}}_{I+1}$. The most symmetric
loop of this class is the second 1/2 BPS loop coupling instead to the pair of
scalars $\tilde{q}_{\dot{2}}$, $\bar{\tilde{q}}^{\dot{2}}$. The cases with all
four $\delta$ parameters non-vanishing is allowed, as long as (5.23) is
satisfied. The analysis follows as before, but
$\mathfrak{s}\mathfrak{u}(2)_{R}$ symmetry is preserved only when we restrict
to a single pair of $\delta$.

###  6.5 Hyperloops with twisted hypers and $\Pi=0$

These operators are considered in Section 5.2, where we find supersymmetric
loops built out of the $G$ and $C$ in (5.29). In particular, the $\beta$
parameters that couple to scalars from the untwisted hypermultiplet satisfy
the same constraints as in the 2-node case, while the couplings to the twisted
scalars, $\bar{d}^{\dot{a}}$ and $d_{\dot{a}}$, are arbitrary periodic
functions as long as $\lambda=0$.

Denoting the superconnection in (4.39) as ${\mathcal{L}}_{\Pi=0}$, the
expression we find for ${\mathcal{L}}$ is

${\mathcal{L}}=\begin{pmatrix}{\mathcal{L}}_{\Pi=0}&\begin{matrix}\bar{\alpha}_{I}\bar{d}^{\dot{a}}r_{I}^{\parallel}\tilde{q}_{I+1\,\dot{a}}&0\\\
\bar{d}^{\dot{1}}\tilde{\rho}_{I+1,+}^{2}+\bar{d}^{\dot{2}}\tilde{\rho}_{I+1,-}^{1}&\bar{d}^{\dot{a}}\bar{\alpha}_{I+2}\tilde{q}_{I+1\,\dot{a}}r_{I+2}^{\parallel}\end{matrix}\\\
\begin{matrix}-d_{\dot{a}}\alpha_{I}\xi\bar{\tilde{q}}^{\dot{a}}_{I+1}\bar{r}_{I\parallel}&d_{\dot{1}}\bar{\tilde{\rho}}_{I+1,2-}+d_{\dot{2}}\bar{\tilde{\rho}}_{I+1,1+}\\\
0&-\alpha_{I+2}d_{\dot{a}}\xi\bar{r}_{I+2\parallel}\bar{\tilde{q}}^{\dot{a}}_{I+1}\end{matrix}&{\mathcal{L}}_{\Pi=0}\end{pmatrix}.$
(6.35)

Note that the coupling to the twisted scalar bilinears is unchanged and the
$\widetilde{M}$ in the central nodes does not receive contributions from the
$d$’s. In general, these loops preserve a single supercharge.

One special case is similar to the $1/4$ BPS hyperloop of Section 6.3, when
$\xi$ (4.27) is of the form $\xi_{0}e^{-i\varphi}$ with constant $\xi_{0}$.
This can arise with either $\xi_{0}=\eta^{r}/\bar{\eta}^{r}$ or
$\xi_{0}=\eta^{l}/\bar{\eta}^{l}$ leading to a two fold degeneracy. This is a
symmetry of the superconnection (6.35) when

$\bar{d}^{\dot{1}}=d_{\dot{2}}=\frac{1}{(\eta\bar{v})_{1}}\,,\quad\bar{d}^{\dot{2}}=d_{\dot{1}}=\frac{1}{(\bar{\eta}v)_{1}}\,,$
(6.36)

and $(\bar{\eta}v)_{1}=(\bar{\eta}v)_{2}$,
$(\eta\bar{v})_{1}=(\eta\bar{v})_{2}$. The resulting hyperloop preserves two
supercharges and, as before, $\mathfrak{s}\mathfrak{o}(2,1)$ acts as an outer
automorphism on the preserved superalgebra. Unlike the 2-nodes case in (6.13),
there is no way to restore $\mathfrak{s}\mathfrak{u}(2)_{L}$ symmetry and find
further supersymmetry enhancement.

####  6.5.1 The special cases: $\xi=0$ and $\xi=\infty$

In Section 5.2.1 the analysis of the case of $\xi=0$ is extended to include
the twisted hypermultiplets. Denoting the superconnection in (4.43) as
${\mathcal{L}}_{\xi=0}$, the extension to include twisted hypermultiplets
gives

${\mathcal{L}}=\begin{pmatrix}{\mathcal{L}}_{\xi=0}&\begin{matrix}r_{I}^{\parallel}(\gamma_{1}\tilde{q}_{I+1\,\dot{2}}+\bar{\alpha}_{I}\bar{d}^{\dot{1}}\tilde{q}_{I+1\,\dot{1}})&0\\\
-i\bar{d}^{\dot{1}}\tilde{\rho}^{2}_{I+1,+}&\frac{\bar{\alpha}_{I+2}}{\bar{\alpha}_{I}}(\gamma_{1}\tilde{q}_{I+1\,\dot{2}}+\bar{\alpha}_{I}\bar{d}^{\dot{1}}\tilde{q}_{I+1\,\dot{1}})r_{I+2}^{\parallel}\end{matrix}\\\
\begin{matrix}0&(\delta_{I+1}-id_{\dot{2}})\bar{\tilde{\rho}}_{I+1,1+}\\\
0&0\end{matrix}&{\mathcal{L}}_{\xi=0}\end{pmatrix}\,.$ (6.37)

Note that, as $\delta_{I+1}$ and $d_{\dot{2}}$ appear only through the linear
combination $\delta_{I+1}-id_{\dot{2}}$, we can eliminate one of them.
Supersymmetry enhancement relying on manifest
$\mathfrak{s}\mathfrak{u}(2)_{L}$ symmetry happens only by setting to zero
off-block-diagonal parameters, in which case we simply recover two decoupled
copies of (6.31).

### Acknowledgements

We would like to acknowledge fruitful discussions with L. Griguolo, E. Pomoni
and Y. Wang. DT is grateful to the Galileo Galilei Institute for hospitality
in the final stages of this work. ND is supported by the Science Technology &
Facilities Council under the grants ST/T000759/1 and ST/P000258/1. The work of
ZK is supported by CSC grant No. 201906340174. MT acknowledges the support of
the Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq). DT
is supported in part by the INFN grant Gauge and String Theory (GAST) and
would like to thank FAPESP’s partial support through the grants 2016/01343-7
and 2019/21281-4.

##  Appendix A Symmetries of the 1/2 BPS Wilson loop

We start by recalling that the symmetries of an ${\mathcal{N}}=4$
superconformal theory on $S^{3}$ form an ${\mathfrak{osp}}(4|4)\cong D(2,2)$
superalgebra, with the bosonic symmetries
$\mathfrak{s}\mathfrak{o}(4,1)\oplus\mathfrak{s}\mathfrak{o}(4).$ These are,
respectively, the three-dimensional conformal algebra and the R-symmetry
algebra. The latter is conveniently thought of as
$\mathfrak{s}\mathfrak{o}(4)\simeq\mathfrak{s}\mathfrak{u}(2)_{L}\oplus\mathfrak{s}\mathfrak{u}(2)_{R}$.
The 16 supercharges transform as conformal spinors under
$\mathfrak{s}\mathfrak{o}(4,1)$ and in the fundamental representations of both
R-symmetry $\mathfrak{s}\mathfrak{u}(2)$’s.

The circular $1/2$ BPS loop breaks part of these symmetries. Specifically, of
the conformal generators, it preserves only the one-dimensional conformal
algebra along the contour of the loop and the rotations in the plane
perpendicular to it

$\displaystyle\mathfrak{s}\mathfrak{o}(2,1)\oplus{\mathfrak{u}}(1)_{\perp}\,.$
(A.1)

$\mathfrak{s}\mathfrak{u}(2)_{L}$ is preserved by the loop, whereas
$\mathfrak{s}\mathfrak{u}(2)_{R}$ is broken to ${\mathfrak{u}}(1)_{R}$.666Of
course, the choice of which of the R-symmetry factors is broken and which one
is preserved is a matter of which 1/2 BPS loop one considers, as explained in
[4].

We denote the conformal generators along the circle by $J_{0}$ and $J_{\pm}$,
with nonvanishing commutators

$[J_{0},J_{\pm}]=\pm J_{\pm}\,,\qquad[J_{+},J_{-}]=2J_{0}\,.$ (A.2)

Parametrising the circle by the angular coordinate $\varphi$, these generators
can be represented by differential operators

$\displaystyle J_{0}=-i\partial_{\varphi},\qquad J_{\pm}=e^{\pm
i\varphi}\partial_{\varphi}\,.$ (A.3)

The action on the fields can be obtained by evaluating the usual conformal
transformations on the circle. Suppressing R-symmetry indices, we find for the
bosonic fields involved in our Wilson loops

$\displaystyle J_{0}A_{\varphi}$
$\displaystyle=-i\partial_{\varphi}A_{\varphi}\,,$ $\displaystyle\qquad
J_{\pm}A_{\varphi}$ $\displaystyle=e^{\pm i\varphi}\left(\partial_{\varphi}\pm
i\right)A_{\varphi}\,,$ (A.4) $\displaystyle J_{0}q$
$\displaystyle=-i\partial_{\varphi}q\,,$ $\displaystyle\qquad J_{\pm}q$
$\displaystyle=e^{\pm i\varphi}(\partial_{\varphi}\pm i/2)q\,,$ $\displaystyle
J_{0}\bar{q}$ $\displaystyle=-i\partial_{\varphi}\bar{q}\,,$
$\displaystyle\qquad J_{\pm}\bar{q}$ $\displaystyle=e^{\pm
i\varphi}(\partial_{\varphi}\pm i/2)\bar{q}\,.$

The second term in the action of $J_{\pm}$ picks up the scaling dimension of
the respective fields. Similarly, for the fermions

$\displaystyle J_{0}\psi$
$\displaystyle=-i\left(\partial_{\varphi}+i\sigma_{3}/2\right)\psi\,,$
$\displaystyle\qquad J_{\pm}\psi$ $\displaystyle=e^{\pm
i\varphi}(\partial_{\varphi}\pm i+i\sigma_{3}/2)\psi\,,$ (A.5) $\displaystyle
J_{0}\bar{\psi}$
$\displaystyle=-i\left(\partial_{\varphi}+i\sigma_{3}/2\right)\bar{\psi}\,,$
$\displaystyle\qquad J_{\pm}\bar{\psi}$ $\displaystyle=e^{\pm
i\varphi}(\partial_{\varphi}\pm i+i\sigma_{3}/2)\bar{\psi}\,.$

We denote by $T_{\perp}$ the generator of rotations
${\mathfrak{u}}(1)_{\perp}$ in the orthogonal plane to the contour, which
commutes with all other preserved conformal generators. The normalization of
$T_{\perp}$ is fixed such that

$\displaystyle T_{\perp}\psi$ $\displaystyle=\frac{i}{2}\sigma_{3}\psi\,,$
$\displaystyle\qquad[T_{\perp},Q^{\dot{a}a}_{\imath}]$
$\displaystyle=\frac{i}{2}Q^{\dot{a}a}_{\imath}\,,$ (A.6) $\displaystyle
T_{\perp}\bar{\psi}$ $\displaystyle=\frac{i}{2}\sigma_{3}\bar{\psi}\,,$
$\displaystyle\qquad[T_{\perp},Q^{\dot{a}a}_{\bar{\imath}}]$
$\displaystyle=-\frac{i}{2}Q^{\dot{a}a}_{\bar{\imath}}\,.$

The generators of $\mathfrak{s}\mathfrak{u}(2)_{L}$ are $R_{\pm},R_{3}$, with
commutation relations

$\displaystyle[R_{3},R_{\pm}]=\pm R_{\pm}\,,\qquad[R_{+},R_{-}]=2R_{3}\,.$
(A.7)

As mentioned above, these symmetries are preserved by the loop. We distinguish
$\mathfrak{s}\mathfrak{u}(2)_{R}$ with bars: $\bar{R}_{\pm},\bar{R}_{3}$. Only
$\bar{R}_{3}$ is preserved by the loop. It is also useful to defined the
twisted generator

$\displaystyle
L_{\perp}\equiv-i\left(T_{\perp}+\frac{i}{2}\bar{R}_{3}\right),$ (A.8)

which mixes the rotations in the perpendicular plane in
${\mathfrak{u}}(1)_{\perp}$ with the $R$-symmetry rotations in
${\mathfrak{u}}(1)_{R}$ [1].

The supercharges preserved by the loop are given in (3.3) and anticommute to

$\displaystyle\\{Q^{\dot{2}a}_{l},Q^{\dot{1}b}_{\bar{l}}\\}$
$\displaystyle=\epsilon^{ab}\left(J_{0}+L_{\perp}\right)+R^{ab}\,,$
$\displaystyle\\{Q^{\dot{2}a}_{l},Q^{\dot{1}b}_{\bar{r}}\\}$
$\displaystyle=\epsilon^{ab}J_{+}\,,$ (A.9)
$\displaystyle\\{Q^{\dot{2}a}_{r},Q^{\dot{1}b}_{\bar{l}}\\}$
$\displaystyle=-\epsilon^{ab}J_{-}\,,$
$\displaystyle\\{Q^{\dot{2}a}_{r},Q^{\dot{1}b}_{\bar{r}}\\}$
$\displaystyle=\epsilon^{ab}\left(J_{0}-L_{\perp}\right)-R^{ab}\,.$

Here, we have contracted the $\mathfrak{s}\mathfrak{u}(2)_{L}$ generators with
the Pauli matrices in the usual fashion and raised one index by
$\epsilon^{ab}$ (with $\epsilon^{12}=1$), such that

$\displaystyle R^{ab}=\begin{pmatrix}R_{+}&-R_{3}\\\
-R_{3}&-R_{-}\end{pmatrix}.$ (A.10)

In order to fully specify the superalgebra, one computes the commutators of
bosonic and fermionic generators using the super-Jacobi identities.
Explicitly, we find that the residual conformal generators act on the
supercharges as follows

$\displaystyle J_{+}\begin{pmatrix}Q_{l}\\\ Q_{r}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0\\\ -Q_{l}\end{pmatrix},$ $\displaystyle\qquad
J_{+}\begin{pmatrix}Q_{\bar{l}}\\\ Q_{\bar{r}}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}-Q_{\bar{r}}\\\ 0\end{pmatrix},$ (A.11)
$\displaystyle J_{-}\begin{pmatrix}Q_{l}\\\ Q_{r}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}-Q_{r}\\\ 0\end{pmatrix},$ $\displaystyle\qquad
J_{-}\begin{pmatrix}Q_{\bar{l}}\\\ Q_{\bar{r}}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}0\\\ -Q_{\bar{l}}\end{pmatrix},$ $\displaystyle
J_{0}\begin{pmatrix}Q_{l}\\\ Q_{r}\end{pmatrix}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}Q_{l}\\\ -Q_{r}\end{pmatrix},$
$\displaystyle\qquad J_{0}\begin{pmatrix}Q_{\bar{l}}\\\
Q_{\bar{r}}\end{pmatrix}$
$\displaystyle=\frac{1}{2}\begin{pmatrix}-Q_{\bar{l}}\\\
Q_{\bar{r}}\end{pmatrix},$ $\displaystyle T_{\perp}\begin{pmatrix}Q_{l}\\\
Q_{r}\end{pmatrix}$ $\displaystyle=\begin{pmatrix}Q_{l}\\\
Q_{r}\end{pmatrix},$ $\displaystyle\qquad
T_{\perp}\begin{pmatrix}Q_{\bar{l}}\\\ Q_{\bar{r}}\end{pmatrix}$
$\displaystyle=-\begin{pmatrix}Q_{\bar{l}}\\\ Q_{\bar{r}}\end{pmatrix}.$

These (anti-)commutators together with the bosonic structure outlined above
define the Lie superalgebra $\mathfrak{sl}(2|2)$. As is easily checked,
$L_{\perp}$ commutes with all supercharges as well as all bosonic generators.
Indeed, $\mathfrak{sl}(2|2)$ is a central extension of the classical Lie
superalgebra $A(1,1)$ by ${\mathfrak{u}}(1)$, so this structure is expected
[41].

##  Appendix B The covariant derivative

Here we explain what it concretely means when a supersymmetry transformation
on a superconnection ${\cal L}$ acts as a total covariant derivative, as in
(3.20)

$Q{\mathcal{L}}={\mathcal{D}}^{\mathcal{L}}_{\varphi}H\,.$ (B.1)

Consider the open Wilson loop (we shall worry about taking the supertrace
later)

$W_{2\pi,0}={\mathcal{P}}\exp i\int_{0}^{2\pi}d\varphi\,{\mathcal{L}}\,,$
(B.2)

and act with $Q$ on the loop. It is crucial that the superconnection
${\mathcal{L}}={\mathcal{L}}^{B}+{\mathcal{L}}^{F}$ is an even supermatrix,
i.e. a matrix whose diagonal entries ${\mathcal{L}}^{B}$ are exclusively
bosonic and whose off-diagonal entries ${\mathcal{L}}^{F}$ are exclusively
fermionic, and likewise for the Wilson loop. Commuting $Q$ through a product
of two such superconnections ${\cal L}_{1}$ and ${\cal L}_{2}$, one gets
$Q({\mathcal{L}}_{2}{\mathcal{L}}_{1})=Q({\mathcal{L}}_{2}){\mathcal{L}}_{1}+\sigma_{3}{\mathcal{L}}_{2}\sigma_{3}Q{\mathcal{L}}_{1}$,
where the Pauli matrix is introduced to flip the sign of the odd part of
${\mathcal{L}}_{1}$.

Acting with $Q$ on $W_{2\pi,0}$, one needs to apply the Leibniz rule, as $Q$
can act on any ${\mathcal{L}}(\varphi)$. Keeping track of the sign changes,
one finds

$QW_{2\pi,0}=i\sigma_{3}\int_{0}^{2\pi}d\varphi\,W_{2\pi,\varphi}\left(\sigma_{3}Q{\mathcal{L}}(\varphi)\right)W_{\varphi,0}\,.$
(B.3)

Now let us assume it exists an $H(\varphi)$, such that
$Q{\mathcal{L}}=\sigma_{3}{\mathcal{D}}_{\varphi}^{\mathcal{L}}(\sigma_{3}H(\varphi))$.
Then, by the standard relations for Wilson loops, one finds

$QW_{2\pi,0}=i\sigma_{3}\int_{0}^{2\pi}d\varphi\,W_{2\pi,\varphi}{\mathcal{D}}_{\varphi}^{\mathcal{L}}(\sigma_{3}H(\varphi))W_{\varphi,0}=iH(2\pi)W_{2\pi,0}-i\sigma_{3}W_{2\pi,0}\sigma_{3}H(0)\,.$
(B.4)

Assuming $H(\varphi)$ to be periodic and taking the supertrace, one gets

$QW=i\operatorname{sTr}(H(0)W_{2\pi,0}-\sigma_{3}W_{2\pi,0}\sigma_{3}H(0))=i\operatorname{Tr}([\sigma_{3}H(0),W_{2\pi,0}])=0\,.$
(B.5)

This implies that the covariant derivative that should appear in the
supersymmetry transformations is

$Q{\mathcal{L}}=\sigma_{3}{\mathcal{D}}_{\varphi}^{\mathcal{L}}(\sigma_{3}H)=\partial_{\varphi}H-i[{\mathcal{L}}_{\textrm{bos}},H]+i\\{{\mathcal{L}}_{\textrm{fer}},H\\}\,.$
(B.6)

In the main text we write this as ${\mathcal{D}}_{\varphi}^{\mathcal{L}}H$,
but we really mean the expression above with the anticommutator of the
fermionic part of the superconnection.

If one prefers working instead with bosonic variations, one can introduce a
Grassmann parameter $\xi$ and write $\delta=\xi Q$. The analogous
supersymmetry condition reads

$\delta{\mathcal{L}}={\mathcal{D}}^{{\mathcal{L}}}_{\varphi}(\xi H)\,.$ (B.7)

##  Appendix C Extra fermionic terms

In this appendix we examine the possibility to add extra fermionic terms to
the $F$ in the superconnection, beyond the term $-iQG$ in (4.34). This term
arises in the case of $\Pi=0$ in Section 4.2, where $G$ includes only two
scalar fields (4.36) and, consequently, $QG$ has only two linear combinations
of the fermions (4.31). To generalize it, we take an extra term related to the
fermions in the original 1/2 BPS connection

$F=-iQG+(D-1)\mathcal{L}_{1/2}^{F}\,.$ (C.1)

Here $D=\operatorname{diag}(\bar{d},d)$.

The result of the analysis below is that such addition is only possible for
$\xi=0$ or $\xi=\infty$, and those cases are already treated in Section 4.2.1.
So this appendix leads to no further hyperloops beyond those described in the
main text.

Taking (C.1) and using the same equations for the variations $QB$ and $QF$ in
(4.6), one gets $\Delta H=(D-1)H$, because there is no derivative term in
$Q^{2}G$. Plugging everything known into (4.6) yields

$\displaystyle-iQ^{2}G$ $\displaystyle=(\partial_{\varphi}D)H-i[B+C,DH]\,,$
(C.2) $\displaystyle QB$ $\displaystyle=\\{QG,DH\\}+i(\det
D-1)\\{\mathcal{L}_{1/2}^{F},H\\}\,.$

Focusing on the second equation for now and using $QH=0$ and
$Q\mathcal{L}_{1/2}^{B}=i\\{\mathcal{L}_{1/2}^{F},H\\}$, one gets

$QB=Q\\{DH,G\\}+(\det D-1)Q\mathcal{L}_{1/2}^{B}\,,$ (C.3)

which is simply solved by

$B=\\{DH,G\\}+(\det D-1)\mathcal{L}_{1/2}^{B}\,.$ (C.4)

Extra terms annihilated by $Q$ are included in $C$.

The case of $\det D=1$ is simply a gauge transformation, changing $\alpha$ and
$\bar{\alpha}$. So we are left with examining the case $\det D\neq 1$. This
results in $B$ having a term proportional to ${\mathcal{L}}_{1/2}^{B}$, which
includes the gauge fields. Since the gauge fields cannot appear in a Wilson
loop with an arbitrary prefactor (they should have prefactor $i$), one needs
to cancel part of this term with factors of the gauge field in $C$. This
amounts to finding a connection annihilated by $Q$, which one can assume to be
purely bosonic:
${\mathcal{L}}^{B\prime}=\operatorname{diag}(\mathcal{A}_{I}^{\prime},\mathcal{A}_{I+1}^{\prime}-1/2)$.
We take

$\mathcal{A}_{I}^{\prime}=A_{\varphi}-\frac{i}{k}(M_{a}{}^{b}r^{a}\bar{r}_{b}+\tilde{\mu}_{\dot{1}}^{\dot{1}}-\tilde{\mu}_{\dot{2}}^{\dot{2}})\,,$
(C.5)

where $a,b\in\\{\parallel,\perp\\}$, and the task is now to find the
coefficient matrix $M_{a}{}^{b}$. Using

$\displaystyle r^{\parallel}(\bar{\psi}_{+}^{\dot{1}}$
$\displaystyle+\xi\bar{\psi}_{-}^{\dot{2}})-(\xi\psi_{\dot{1}-}-\psi_{\dot{2}+})\bar{r}_{\parallel}$
(C.6)
$\displaystyle=\Lambda(M_{\parallel}{}^{\perp}r^{\parallel}+M_{\perp}{}^{\perp}r^{\perp})(\bar{\psi}_{+}^{\dot{1}}-\xi\bar{\psi}_{-}^{\dot{2}})-\Lambda(\xi\psi_{\dot{1}-}+\psi_{\dot{2}+})(M_{\perp}{}^{\parallel}\bar{r}_{\parallel}+M_{\perp}{}^{\perp}\bar{r}_{\perp})\,,$

and imposing $Q\mathcal{A}_{I}^{\prime}=0$ results in

$\displaystyle
Q\left(A_{\varphi}-\frac{i}{k}(-\nu_{I}+\tilde{\mu}_{\dot{1}}^{\dot{1}}-\tilde{\mu}_{\dot{2}}^{\dot{2}})\right)=-\frac{2i}{k}(\xi\psi_{\dot{1}-}\bar{r}_{\parallel}+r^{\parallel}\bar{\psi}_{+}^{\dot{1}})\,,$
(C.7)

which is solved by

$\xi=M_{\perp}{}^{\perp}=0,\quad
M_{\parallel}{}^{\perp}=-M_{\perp}{}^{\parallel}=1/\Lambda\,.$ (C.8)

We see that indeed this works only for $\xi=0$ and therefore it falls under
the cases already analyzed in Section 4.2.1.

To compare with the analysis in Section 4.2.1, we note that for $\xi=0$ there
are many specific features, such as $Q^{2}G=H^{2}=0$. We can also check that
$\mathcal{L}_{1/2}^{B}-\mathcal{L}^{B}$ commutes with $DH$ and the only
remaining supersymmetry conditions is

$\partial_{\varphi}\bar{d}=-ic\bar{d}\,.$ (C.9)

Including the bosonic loop

$\mathcal{A}_{I}^{\prime}=A_{\varphi}-\frac{i}{k\Lambda}(\Lambda
M_{\parallel}{}^{\parallel}r^{\parallel}\bar{r}_{\parallel}+r^{\parallel}\bar{r}_{\perp}-\bar{r}^{\perp}\bar{r}_{\parallel})-\frac{i}{k}(\tilde{\mu}_{\dot{1}}^{\dot{1}}-\tilde{\mu}_{\dot{2}}^{\dot{2}})\,,$
(C.10)

and the analogous expression for ${\mathcal{A}}_{I+1}^{\prime}$ in $C$ with
prefactors $1-\det D$ and combining all the terms, one finally gets the
superconnection

$\mathcal{L}=\begin{pmatrix}A_{\varphi,I}+M_{a}{}^{b}r^{a}\bar{r}_{b}-\frac{i}{k}(\tilde{\mu}_{I\;\dot{1}}^{\;\dot{1}}-\tilde{\mu}_{I\;\dot{2}}^{\;\dot{2}})&-i\bar{\alpha}\bar{d}\psi_{\dot{1}-}-i\Lambda\bar{\beta}_{\perp}\psi_{\dot{2}+}\\\
i(\alpha
d+\Lambda\beta^{\perp})\bar{\psi}_{+}^{\dot{1}}&A_{\varphi,I+1}+M_{a}{}^{b}\bar{r}_{b}r^{a}-\frac{i}{k}(\tilde{\mu}_{I+1\dot{1}}^{\quad\;\;\dot{1}}-\tilde{\mu}_{I+1\dot{2}}^{\quad\;\;\dot{2}})+c-\frac{1}{2}\end{pmatrix},$
(C.11)

with $c=i\partial_{\varphi}\log\bar{d}$ and

$M_{a}{}^{b}=\begin{pmatrix}0&\frac{i}{k\Lambda}\\\
\beta^{\perp}\bar{d}\bar{\alpha}+(2\bar{d}d-1)\frac{i}{k\Lambda}&\beta^{\parallel}\end{pmatrix},$
(C.12)

where $M_{\parallel}{}^{\parallel}$ has been absorbed into
$\beta^{\parallel}$, since both of them are free parameters. One can further
absorb $\bar{d}$ into $\bar{\alpha}$ and $\bar{\beta}_{\perp}$, which sets
$c=0$ and replaces $\alpha\to\alpha\bar{d}$ and
$\beta^{\perp}\to\beta^{\perp}\bar{d}$. Then, with
$\hat{\beta}^{\perp}=(\alpha(d\bar{d}-1)+\Lambda\beta^{\perp}\bar{d})/\Lambda$
the bottom left entry in ${\mathcal{L}}$ becomes
$i(\alpha+\Lambda\hat{\beta}^{\perp})\bar{\psi}_{+}^{\dot{1}}$ and the bottom
left entry in $M_{a}{}^{b}$ becomes
$\bar{\alpha}\hat{\beta}^{\perp}+i/k\Lambda$.

This eliminates the parameters $d$ and $\bar{d}$ from ${\mathcal{L}}$, so they
are completely redundant. Furthermore, we see that these loops are exactly
those found directly in the $\xi=0$ case in (4.43) in Section 4.2.1.

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