††institutetext: Mathematical Institute, University of Oxford, Woodstock Road,
Oxford, OX2 6GG, UK

# Twisted chiral algebras of class $\mathcal{S}$ and mixed Feigin–Frenkel
gluing

Christopher Beem and Sujay Nair

###### Abstract

The correspondence between four-dimensional $\mathcal{N}=2$ superconformal
field theories and vertex operator algebras, when applied to theories of class
$\mathcal{S}$, leads to a rich family of VOAs that have been given the
monicker _chiral algebras of class $\mathcal{S}$_. A remarkably uniform
construction of these vertex operator algebras has been put forward by
Tomoyuki Arakawa in Arakawa:2018egx . The construction of Arakawa:2018egx
takes as input a choice of simple Lie algebra $\mathfrak{g}$, and applies
equally well regardless of whether $\mathfrak{g}$ is simply laced or not. In
the non-simply laced case, however, the resulting VOAs do not correspond in
any clear way to known four-dimensional theories. On the other hand, the
standard realisation of class $\SS$ theories involving non-simply laced
symmetry algebras requires the inclusion of outer automorphism twist lines,
and this requires a further development of the approach of Arakawa:2018egx .
In this paper, we give an account of those further developments and propose
definitions of most chiral algebras of class $\SS$ with outer automorphism
twist lines. We show that our definition passes some consistency checks and
point out some important open problems.

##  1 Introduction

Since their rise to prominence in the late aughts, theories of class $\SS$
Gaiotto:2009we ; Gaiotto:2009hg  have proven to constitute an important and
highly structured family of four-dimensional, ${\cal N}=2$ superconformal
field theories (SCFTs).111Throughout this paper we restrict without further
comment to the ${\cal N}=2$ superconformal case when referencing SCFTs or
“theories” more generally. These theories are best understood as those which
arise in the low energy limit when six-dimensional $(2,0)$ SCFTs are
compactified (in an appropriately supersymmetry-preserving fashion) on a
compact two-manifold (the _UV curve_), possibly in the presence of
codimension-two supersymmetric defects. Many properties of class $\SS$
theories are well-characterised in terms of two-dimensional structures, e.g.,
the marginal (gauge) couplings are identified with complex structure moduli of
the curve. Indeed, a cottage industry of identifying four-dimensional
observables with dual computations taking place in two dimensions on the UV
curve has emerged and achieved a good degree of maturity.

One such observable (to use the term expansively) that is tractable yet
structurally intricate is the associated vertex operator algebra (VOA) or
chiral algebra Beem:2013sza . The VOAs associated to theories of class $\SS$
were first systematically studied in Beem:2014rza  where they went under the
name _chiral algebras of class $\SS$_. In that work, a number of key
properties of these VOAs were identified and some explicit computations
performed in simple cases (with underlying $(2,0)$ theory of type
$\mathfrak{g}=\mathfrak{a}_{1}$ or $\mathfrak{a}_{2}$). A speculative vision
was also put forward wherein general chiral algebras of class $\SS$ might be
uniquely determined by their various duality properties—a _theory space
bootstrap_ —which was an idea that had seen prior success in the case of the
superconformal index Gaiotto:2012xa .

This speculation was largely confirmed several years later in a remarkable
paper by Tomoyuki Arakawa Arakawa:2018egx . As we will review later, Arakawa’s
construction involves a gluing operation, which we refer to as _Feigin–Frenkel
gluing_ , that has no obvious four-dimensional counterpart but which can
heuristically be thought of as gluing UV curves together along interior points
as opposed to marked points. The crucial insight pointing towards the
construction in question is that maximal punctures in class $\SS$ give rise to
critical-level affine Kac-Moody subalgebras of the associated VOA, and the
centres of these subalgebras (their _Feigin-Frenkel centres_) should be
identified amongst the different punctures. This is an affine version of a set
of well-known Higgs branch chiral ring relations for class $\SS$ theories,
which themselves play the key role in the construction of the Moore–Tachikawa
holomorphic symplectic varieties Moore:2011ee  by Ginzburg–Kazhdan
Ginzburg:2018  and Braverman–Finkelberg–Nakajima Nakajima:2015txa ;
Braverman:2016wma ; Braverman:2017ofm ; Braverman:2016pwk .

A curious feature of the construction in Arakawa:2018egx  is that it makes
perfect sense for any choice of underlying simple Lie algebra $\mathfrak{g}$,
whether simply laced or not. This state of affairs is to be contrasted with
the physical reality of class $\SS$, where the basic story applies only for
$\mathfrak{g}$ simply laced (because $\mathfrak{g}$ is the algebra labelling
the parent six-dimensional $(2,0)$ theory), and the incorporation of non-
simply-laced (gauge and global) symmetry algebras normally requires the extra
structure of (discrete) outer automorphism twist lines being introduced on the
UV curve Vafa:1997mh ; Tachikawa:2009 ; Tachikawa:2011 . This raises the
question of how/whether to interpret in four-dimensional terms the VOAs
constructed in Arakawa:2018egx  for the non-simply laced case. Perhaps more
importantly, it leaves open the problem of defining in comparable generality
the _twisted chiral algebras of class $\SS$_ corresponding to theories that
are realised using outer automorphism twist lines.

The purpose of the present work is to address the latter problem. We propose a
definition _à la Arakawa_ for a large family of twisted chiral algebras of
class $\SS$. Our definition allows for the realisation of all trinion chiral
algebras with one untwisted and two twisted punctures. This is still
restricted, however, in that it does not allow for the case of the
$\mathfrak{d}_{4}$-type trinion theory with three twisted punctures, or more
general theories whose realisation require such constituent fixtures. While we
can prove a number of duality properties of these algebras, we encounter an
obstruction in the proof of the most general expected dualities. This
obstruction is analogous to one that appears already in Arakawa:2018egx  in
the case of higher-genus chiral algebras.

The organisation of the rest of this paper is as follows. In Section 2 we
provide an idiosyncratically selective review of class $\SS$ SCFTs and their
generalisation to incorporate outer automorphism twist lines. In Section 3 we
recall the associated vertex operator algebras of class $\SS$ theories and
provide a fairly pedagogical overview of their construction in the simply
laced setting by Arakawa. In Section 4 we propose a definition of the
associated VOA for the simplest twisted trinions and establish generalised
$S$-duality for the twisted chiral algebras of class $\SS$ with respect to a
subset of duality transformations. In Section 5 we make some comments on
possible extensions and future directions. We include a short review of
spectral sequences, which play an important role in our proofs and those of
Arakawa:2018egx  in Appendix A. In Appendices B and C, we prove two important
technical theorems used in our analysis of mixed Feigin–Frenkel gluing. In
Appendix D we give the details of a useful rewriting of class $\SS$ Schur
indices, and in Appendix E we include some calculations regarding the
Drinfel’d–Sokolov reduction from our twisted trinion chiral algebra to a free
symplectic boson VOA.

_Author’s note: while this paper was in the late stages of preparation, the
paperFrenkel:2021bmx  appeared in which their Lemma 9.4 is a finite-
dimensional cousin of our Theorem 4.1._

##  2 Review of (twisted) class $\SS$

To begin, let us review some important features of four-dimensional SCFTs of
class $\SS$, paying special attention to the generalisation to included twist
lines. This is not meant to be a comprehensive overview; much more background
on these theories can be found in, e.g., Gaiotto:2009we ; Gaiotto:2009hg ;
Tachikawa:2011 .

Untwisted theories of class $\SS$ are specified by a triple
$(\mathfrak{g},{\cal C}_{g,s},\\{\Lambda_{i}\\})$, where $\mathfrak{g}$ is the
simply-laced Lie algebra labelling the underlying six-dimensional theory, the
UV curve ${\cal C}_{g,s}$ is a genus $g$ Riemann surface with $s$ marked
points, upon which the $(2,0)$ is compactified, and the $\\{\Lambda_{i}\\}$
for $i=1,\ldots,s$ are a set of labels for the marked points, which amount to
specifications of maximally supersymmetric, codimension-two defect operators
inserted at those points in six dimensions. The only geometric data of ${\cal
C}_{g,s}$ that are relevant are its complex structure moduli, which get
identified with marginal (gauge) couplings of the associated four-dimensional
SCFT.

The six-dimensional $(2,0)$ theories enjoy discrete global symmetries
corresponding to outer automorphisms of the underlying Lie algebra
$\mathfrak{g}$—these can be identified with graph automorphisms of the
corresponding Dynkin diagrams. One may then include twists/monodromies by
these outer automorphism symmetries when compactifying; said differently, one
generalises the compactification procedure to incorporate a local system of
Dynkin diagrams on the UV curve. The data of this local system can be
represented pictorially by including consistent networks of (potentially
oriented) twist lines on the UV curve labelled by elements of the outer
automorphism group. This generalisation of the class $\SS$ construction also
introduces new types of punctures, called _twisted punctures_ , which
correspond to codimension two defects that carry an outer automorphism
monodromy. We will be specialising to the genus zero case in our construction
and so for convenience we introduce the notation, ${\cal C}_{0,m,n}$ to denote
the Riemann sphere with $m$ untiwsted punctures and $2n$ twisted punctures.
One cannot specify, unambiguously, a higher genus surface with twisted
punctures by the genus and number of each type of puncture.

###  2.1 Untwisted and twisted punctures

Marked points in class $\SS$ can be either _regular_ or _irregular_ , and in
this work we will restrict our attention entirely to the regular case.
Regular, untwisted punctures in a theory of type
$\mathfrak{g}\in\\{\mathfrak{a}_{n},\mathfrak{d}_{n},\mathfrak{e}_{n}\\}$ are
labelled by conjugacy classes of homomorphisms
$\Lambda:\mathfrak{sl}(2)\rightarrow\mathfrak{g}$. The defect operator
corresponding to such a puncture enjoys some global symmetry that it then
contributes to the total global flavour symmetry of four dimensional theory.
The factor contributed takes the form
$\mathfrak{f}_{\Lambda}=C_{\mathfrak{g}}\left(\Lambda(\mathfrak{su}(2))\right)$,
i.e., the commutant of the image of $\mathfrak{su}(2)$ in $\mathfrak{g}$. As a
consequence of the Jacobson–Morozov theorem Collingwood:1993 , the marked
points can therefore be equivalently labelled by nilpotent orbits in
$\mathfrak{g}$.

Any simple Lie algebra has at least two especially important nilpotent orbits:
the principal orbit (which is the unique, largest nilpotent orbit) and the
trivial orbit. The trivial orbit corresponds to the trivial homomorphism
$x\mapsto 0,\forall x\in\mathfrak{su}(2)$, while an explicit expression for
the principal orbit can be found in Collingwood:1993 . Their respective
commutants are $\mathfrak{f}=0$ for the principal orbit and
$\mathfrak{f}=\mathfrak{g}$ for the trivial orbit. A puncture labelled by the
trivial embedding is called a maximal puncture, while a puncture labelled by
the principal embedding contributes no flavour symmetry and is equivalent to
having no puncture at all. With an eye towards the twisted case, we will
nevertheless adopt the convention of referring to a hypothetical puncture
labelled by the principal embedding as an empty puncture.

Given a theory where a particular puncture is maximal, the corresponding
theory with that puncture replaced by a sub-maximal puncture can be realised
by partially Higgsing the $\mathfrak{g}$ flavour symmetry associated to that
puncture in the four dimensional theory. A choice of nontrivial $\Lambda$ is
realised by assigning an expectation value to the “moment map” Higgs branch
operator in the conserved current multiplet that lies in the corresponding
nilpotent orbit. Consequently, for many purposes it is sufficient to be able
to construct theories associated to surfaces with maximal punctures, with
other punctures structures being subsequently reached via partial Higgsing.
This will be our approach in this paper.

Lie algebra ($\mathfrak{g}$) |  Outer automorphism twist | Twisted algebra ($\mathfrak{g}_{t}$)
---|---|---
$\mathfrak{a}_{2n}$ | $\mathbb{Z}_{2}$ | $\mathfrak{c}_{n}$
$\mathfrak{a}_{2n-1}$ | $\mathbb{Z}_{2}$ | $\mathfrak{b}_{n}$
$\mathfrak{d}_{n}$ | $\mathbb{Z}_{2}$ | $\mathfrak{c}_{n-1}$
$\mathfrak{d}_{4}$ | $\mathbb{Z}_{3}$ | $\mathfrak{g}_{2}$
$\mathfrak{e}_{6}$ | $\mathbb{Z}_{2}$ | $\mathfrak{f}_{4}$
Table 1: Simply laced Lie algebras $\mathfrak{g}$ and the corresponding
twisted algebras $\mathfrak{g}_{t}$ for different choices of outer
automorphism twist. The Lie algebras $\mathfrak{a}_{2n}$ and
$\mathfrak{d}_{n}$ both give rise to Lie algebras of type $\mathfrak{c}$ after
a twist. The corresponding theories nevertheless have subtle differences—see,
e.g., Chacaltana:2014nya ; Beem:2020pry . The algebra $\mathfrak{d}_{4}$ has
an outer automorphism group of $S_{3}$, the symmetric group on three elements.
Here we will only consider its abelian subgroups $\mathbb{Z}_{2}$ and
$\mathbb{Z}_{3}$.

Twisted punctures are also labelled by $\mathfrak{sl}(2)$
homomorphisms/nilpotent orbits, but now of a different Lie algebra
$\mathfrak{g}_{t}$. Let $\mathfrak{g}_{t}^{\vee}$ be the subalgebra of
$\mathfrak{g}$ invariant under the outer-automorphism monodromy of the
puncture. Then twisted punctures are labelled by nilpotent orbits in the
simple Lie algebra $\mathfrak{g}_{t}$, the Langlands dual to
$\mathfrak{g}_{t}^{\vee}$. The pairs of untwisted algebras and their twisted
counterparts can be found in Table 1. The maximal twisted punctures carries
$\mathfrak{g}_{t}$ flavour symmetry, while submaximal twisted punctures
proceed analogously to the twisted case. Importantly, unlike in the untwisted
case, the empty twisted puncture remains a nontrivial puncture (as it still
carries monodromy on the UV curve; in terms of twist lines there is still a
point where the relevant twist line ends, which distinguishes the point from a
generic point on the UV curve).

Throughout this paper, $\mathfrak{g}$ and $\mathfrak{g}_{u}$ will generally
refer to the type of the parent six-dimensional theory or, equivalently, to
the flavour symmetry algebra contributed by a maximal untwisted puncture,
while we denote by $\mathfrak{g}_{t}$ the flavour algebra of a maximal twisted
puncture as in Table 1. More generally, subscripts $u$ and $t$ will
differentiate between various objects associated to $\mathfrak{g}_{u}$ and
$\mathfrak{g}_{t}$.

###  2.2 Trinions, gluing, and duality

Weak coupling limits of class $\SS$ theories correspond to degeneration limits
of the UV curve, with maximally weakly coupled limits corresponding to pair-
of-pants decompositions. The basic entities are the theories corresponding to
pairs of pants/spheres with three maximal punctures; these are often referred
to as _trinion theories_ or _fixtures_. Even in the untwisted setting, there
are myriad possible (combinations of) marked point labels that can appear on a
single trinion; these have been extensively detailed by (various subsets of)
Chacaltana, Distler, Tachikawa, and Trimm in Chacaltana:2012zy ;
Chacaltana:2010ks ; Chacaltana:2011ze . These (generically strongly coupled)
theories can be combined by exactly marginal gauging to realise general class
$\SS$ theories, but the equivalence of the SCFTs realised in this manner by
inequivalent pair-of-pants decompositions constitutes a highly nontrivial set
of quantum dualities often referred to as generalised $S$-dualities. The class
$\SS$ dictionary identifies the (compactified) moduli space of the UV curve
with the space of complexified gauge couplings of the associated theory. The
various pants decompositions are, therefore, the various weak coupling limits
of the associated theory—all of them related by generalised $S$-duality.

In the untwisted case, all pants decompositions can be reached by iterating
two types of elementary move.

For the first type of move, consider the four puncture sphere ${\cal C}_{0,4}$
with some labelling $1,\dots,4$ of its punctures. We move to the singular
point on the moduli space where the sphere decomposes into two trinions
connected by a long tube, one of which has punctures labelled by $1,2$ and the
other by $3,4$. In Figure 1, this is represented by the duality frame on the
left. By moving to a different singular point, one can decompose ${\cal
C}_{0,4}$ into two connected trinion labelled by $1,3$ and $2,4$—shown on the
right hand side of Figure 1. We call such a move, moving between the various
decompositions of ${\cal C}_{0,4}$, a $4$-move. This can be lifted to a
$4$-move acting on any collection of four punctures on a general ${\cal
C}_{g,s}$. First, one moves to a singular locus on the moduli space where
${\cal C}_{g,s}$ decomposes into a ${\cal C}_{0,4}$ that contains the
punctures of interest and is connected by a long tube to ${\cal C}_{g,s-4}$.
Then one applies a $4$-move to swap between decompositions of ${\cal
C}_{0,4}$, before moving back out of the singular loci.

For surfaces ${\cal C}_{g,s}$ with $g>0$, one must consider another type of
move. For example, consider the one punctured torus ${\cal C}_{1,1}$. The $S$
generator of the modular group acts by swapping the $a$ and $b$ cycles of the
torus. This is a homeomorphism of the torus to itself that is not homotopic to
any iterated $4$-move and so must be a generator, which we call the $ab$-move.
For a surface, ${\cal C}_{g,s}$, there is a natural generalisation of this
move. In the untwisted case, the $ab$-move acts as the $S$-duality
$\tau\mapsto-1/\tau$ on the complex gauge coupling associated to the handle.

12341324 Figure 1: The $4$-move acting on four maximal untwisted punctures.

In the presence of twist lines, trinions can be glued together along either
twisted or untwisted maximal punctures. The possible trinions with twisted
punctures222The case of $\mathfrak{d}_{4}$ trinions with non-abelian twists
has recently been explored in Distler:2021cwz , but we restrict our attention
to the abelian case have also been classified by Chacaltana, Distler,
Tachikawa and Trimm Chacaltana:2012ch ; Chacaltana:2013oka . In the
terminology of Chacaltana:2012ch  we will be restricting our attention to the
case of regular, typical, twisted punctures.

The presence of twist lines introduces new types of duality transformations,
these can once again be split into two classes. On a surface ${\cal
C}_{0,m,n}$, there are three types of $4$-move that one can consider. The
first acts on a collection of four untwisted punctures and swaps between the
various pants decompositions of ${\cal C}_{0,4}$ into untwisted trinions.
Similarly, the second acts on two pairs of twisted punctures and swaps between
the pants decomposistions of ${\cal C}_{0,0,2}$. The third move is different,
it acts on a pair of untwisted punctures and a pair of twisted punctures. the
surface ${\cal C}_{0,1,1}$ has two pants decompositions, described in Figure
2. The third type of $4$-move swaps between these two frames. Unlike the
previous moves, the type of gluing in each frame is different. On the left
hand side, one gauges the diagonal $\mathfrak{g}_{u}$ action whereas one
gauges a diagonal $\mathfrak{g}_{t}$ action on the right. Iterating these
$4$-moves allows on to traverse a large strand of the duality web of ${\cal
C}_{0,m,n}$. For the non-abelian case of $\mathfrak{d}_{4}$ there are more
intricate dualities, explored in Distler:2021cwz , which we will not discuss
here.

12341324 Figure 2: The two degeneration limits of ${\cal C}_{0,2,1}$. Note
that the gluing on the left is untwisted but twisted on the right. We mark
untwisted punctures by unfilled circles and twisted punctures by filled
circles. We connect the twisted punctures by dashed twist-lines for clarity.

For twisted surfaces at higher genus, the $ab$-move has another variation.
Take, for instance, the $\mathfrak{d}_{n}$ theory on a maximal three punctured
sphere. We can construct a gauge theory by self-gluing two punctures on the
sphere together, which corresponds to gauging the diagonal action of
$Spin(2n)$ on the two theories. We may also gauge with respect to a diagonal
action that has been twisted by an outer automorphism, i.e., $Spin(2n)$ acts
as $g\otimes\sigma(g)$ for (a lift of) an outer automorphism $\sigma$.
Pictorially, we represent this by a cylinder, with a twist line around it,
connecting the punctures. This results in a genus one surface with a twist
line running around the $a$-cycle of the torus.

Now, consider, a $\mathfrak{d}_{n}$ theory with one maximal puncture and two
maximal, $\mathbb{Z}_{2}$, twisted punctures. Again, we can construct a gauge
theory, by self-gluing the two twisted punctures together. This time, we gauge
with respect to the diagonal action. This results in a genus one surface with
a twist line running along the $b$-cycle. These two theories are known to be
$S$-dual, and the associated UV curves are related by the action of the
$ab$-move. The two degeneration limits are shown in Figure 3. Unlike the
untwisted case, the $ab$-move changes the gauge group as well as moving to a
weakly coupled point.

12341324 Figure 3: The $ab$-move swapping between two decompositions of the
once punctured torus with a twist line.

We are primarily interested in observables that do not depend on exactly
marginal deformations, such as gauge couplings. Such observables are blind to
continuous paths in the moduli space and only see the various degeneration
points. In the untwisted case, the $ab$-move is trivial for such observables
and the action of the $4$-moves can be phrased in terms of the action of
permutations on punctures. The $4$-moves, in the untwisted case, act by
permuting the flavour symmetries associated to the punctures, which shows up
as permutations on the flavour fugacities (see the next section) of the index
or permuting the moment-maps of the Higgs branch and the associated vertex
algebra. It is, therefore, useful to phrase the action of $S$-duality on these
observables in terms of the action of a permutation group. For a surface
${\cal C}_{g,n}$ the $4$-moves generate an action of $S_{n}$.

The twisted case, however, is not so simple. The first and second types of
$4$-move, acting on four punctures of the same type, can be rephrased as the
action of $S_{4}$ on each collection of untwisted or twisted punctures. The
third type of four move, i.e. the move of Figure 2, cannot be rephrased in
this way. The two duality frames are not simply related by the action of a
permutation on their moment maps or flavour fugacities. Instead one must check
the non-trivial agreement between both types of BRST gauging on each side.
Matters are further complicated by the $ab$-move of Figure 3. Like the
$4$-move of Figure 2, both sides of the duality feature a different type of
BRST gauging, and again one must check the agreement between both gluings.

In Section 4.7, after we have constructed the twisted vertex algebras, we will
construct the vertex algebraic version of these moves. We are unable to
establish invariance across the full duality web, but we do prove invariance
under the various types of $4$-moves we have defined. In the genus zero case,
to which we are mostly specialising, this does recover the full duality web
(under some reasonable assumptions about the properties of gauging).

###  2.3 The superconformal index

The superconformal index of a four dimensional ${\cal N}=2$ SCFT is defined as
a graded trace over the radially quantised Hilbert space of the theory
Kinney:2005ej ; Rastelli:2014jja ,

$\mathcal{I}(p,q,t,\mathbf{x})=\mathop{\mathrm{Tr}}\nolimits\,(-1)^{F}e^{-\beta(E-2j_{2}-2R+r)}p^{\frac{E+2j_{1}-2R-r}{2}}q^{\frac{E-2j_{1}-2R-r}{2}}t^{R+r}\prod_{i=1}^{\mathrm{rk}\mathfrak{g}_{F}}x_{i}^{T_{i}}~{},$
(1)

where $p,q,t$ are superconformal fugacities and
$\mathbf{x}=(x_{1},x_{2},\dots,x_{\mathrm{rk}\,\mathfrak{g}_{F}})$ are further
fugacities conjugate to generators, $T_{i}$, of a Cartan subalgebra of the
flavour symmetry $\mathfrak{g}_{F}$. A standard argument implies that the
index can only receive a nonvanishing contribution from states obeying

$E-2j_{2}-2R+r=0~{},$ (2)

and so is actually independent of $\beta$. As written, the index counts
minimally supersymmetric states. There are a number of limits with enhanced
supersymmetry, and we will be specifically interested in the Schur limit
Gadde:2011uv ,

$q\rightarrow t~{},\quad p\text{ arbitrary}~{}.$ (3)

In this limit, superconformal representation theory ensures that the index is
in fact independent of $p$ and takes the schematic form

$\mathcal{I}(q)=\mathop{\mathrm{Tr}}\nolimits(-1)^{F}q^{E-R}~{},$ (4)

with the regularising $\beta$-dependent term suppressed. The Schur index only
counts states whose quantum numbers satisfy the further linear relations

$E-j_{1}-j_{2}=2R~{},\qquad r=j_{1}-j_{2}~{}.$ (5)

This index is a key ingredient in the SCFT/VOA correspondence Beem:2013sza .
As a graded vector space, the associated VOA can be identified with the space
of Schur operators, _i.e._ , the local operators satisfying the above
constraints, with the holomorphic dimension of a given state being given by

$h=\frac{E+(j_{1}+j_{2})}{2}=E-R~{},$ (6)

so the Schur index is precisely the vacuum (super)character of the associated
vertex algebra Beem:2013sza .

For theories of class $\mathcal{S}$, the form of the full superconformal index
has been shown to follow from duality properties Gaiotto:2012xa , though for
our purposes here we will restrict attention to the Schur limit. The index
takes the form of a sum over highest weights in the set of integrable dominant
weight representations $P^{+}$ (i.e., a sum over finite-dimensional
$\mathfrak{g}$ representations), weighted by some “structure constants”
$C_{\lambda\lambda\lambda}$, which are functions of $q$. For a UV curve ${\cal
C}_{g,s}$, which can be realised by gluing $2g-2+s$ trinions, the index takes
the form Aganagic:2005 ; Gadde:20102d ; Gadde:2011uv ; Alday:2013kda

${\cal
I}(q;\mathbf{x}_{1},\mathbf{x}_{2},\dots,\mathbf{x}_{s})=\sum_{\lambda\in
P^{+}}(C_{\lambda\lambda\lambda}(q))^{2g-2+s}\prod_{i=1}^{s}{\cal
K}(q;\mathbf{x}_{i})\chi^{\lambda}(\mathbf{x}_{i})~{},$ (7)

where $\mathbf{x}_{i}$ are the flavour fugacities arising from the flavour
symmetries of the $i\,$th puncture. The $\chi^{\lambda}$ are Schur
polynomials, i.e., characters of the finite-dimensional $\mathfrak{g}$
representation with highest weight $\lambda$. The ${\cal K}$ factors are
defined by

${\cal K}(\mathbf{x})=\sqrt{{\cal I}^{V}(\mathbf{q;x})}~{},$ (8)

where ${\cal I}^{V}$ is the Schur index for a free vector multiplet in the
adjoint representation of $\mathfrak{g}$.

To introduce non-maximal punctures at the level of the index amounts to a
fugacity replacement (with a subtraction of certain divergences) that is
detailed in Gadde:2011uv ; Rastelli:2014jja ; Beem:2014rza . Since reducing to
an empty puncture amounts to removing the puncture all together, one has the
relation

$C_{\lambda\lambda\lambda}(q)=\frac{1}{{\cal
K}(\times)\chi^{\lambda}(\times)}~{},$ (9)

where $\times$ represents the regularised fugacity replacement for the empty
puncture, which is purely a function of $q$.

We can recast the above in more representation theoretic terms by using
Appendix A of Lemos:2014lua , which contains a useful dictionary. Let ${\cal
R}$ denote the set of roots of $\mathfrak{g}$, then

${\cal
K}(\mathbf{x})=\frac{1}{(q;q)_{\infty}^{\mathrm{rk}~{}\mathfrak{g}}}\prod_{\alpha\in{\cal
R}}\frac{1}{(qe^{\alpha}(\mathbf{x});q)_{\infty}}~{},$ (10)

where $e^{\alpha}:T\rightarrow\mathbb{C}^{\times}$ are the roots of the simply
connected Lie group, $G$, with $Lie(G)=\mathfrak{g}$ and maximal torus $T$.
For the empty puncture, we have

${\cal
K}(\times)=\prod_{i=1}^{\mathrm{rk}\,\mathfrak{g}}\frac{1}{(q^{d_{i}},q)_{\infty}}~{},$
(11)

which one can identify with the character of the Feigin–Frenkel centre,
$\mathfrak{z}(\mathfrak{g})$. Similarly, the summand, ${\cal
K}(q;\mathbf{x})\chi^{\lambda}(\mathbf{x})$ can be identified with the
character of the critical level Weyl module of heighest weight $\lambda$,
$\mathbb{V}_{\lambda}$ as

${\cal
K}(q;\mathbf{x})\chi^{\lambda}(\mathbf{x})=\mathop{\mathrm{Tr}}\nolimits_{\mathbb{V}_{\lambda}}(q^{D}\mathbf{x}^{T})~{},$
(12)

where $D$ is the quasiconformal weight. For clarity’s sake we shall often
suppress the flavour fugacities and use the notation ${\rm
ch}~{}V=\mathop{\mathrm{Tr}}\nolimits_{V}(q^{D})$. We will provide more
details on the Feigin–Frenkel centre and Weyl modules in Section 3, but for
now we will continue with our rewriting. At the critical level, the Weyl
modules are not irreducible—instead they have a unique simple quotient
$\mathbb{L}_{\lambda}$. The characters of the Weyl module and their simple
quotients are related by

${\rm ch}~{}\mathbb{L}_{\lambda}=\frac{{\rm ch}~{}\mathbb{V}_{\lambda}}{{\rm
ch}~{}\mathfrak{z}_{\lambda}}=\frac{{\cal
K}(q;\mathbf{x})\chi^{\lambda}(\mathbf{x})}{{\cal
K}(\times)\chi^{\lambda}(\times)}~{},$ (13)

where $\mathfrak{z}_{\lambda}$ is the Drinfel’d Sokolov reduction of
$\mathbb{V}_{\lambda}$ and will also be introduced in Section 3. The index of
${\cal C}_{g,s}$ can then be completely rewritten in terms of characters of
Weyl modules as

${\cal
I}(q;\mathbf{x_{1}}\mathbf{x_{2}},\dots,\mathbf{x}_{s})=\sum_{\lambda\in
P^{+}}({\rm ch}~{}\mathfrak{z}_{\lambda})^{2-2g}\prod_{i=1}^{s}{\rm
ch}~{}\mathbb{L}_{\lambda}=\sum_{\lambda\in P^{+}}({\rm
ch}~{}\mathfrak{z}_{\lambda})^{2}\prod_{i=1}^{s}\frac{{\rm
ch}~{}\mathbb{V}_{\lambda}}{{\rm ch}~{}\mathfrak{z}_{\lambda}}~{}.$ (14)

The Macdonald limit (and so Schur limit by further specialisation) of the
superconformal index in the twisted setting was studied for type
$\mathfrak{d}_{n}$ theories in Lemos:2012ph . According to the analysis there,
the presence of an single twisted puncture restricts the sum over $P^{+}$ to
representations that are invariant under the action of the outer automorphism
twist, which is equivalent to summing over the set of highest weight
representations of the twisted algebra $\mathfrak{g}_{t}$ (which we denote by
$P^{+}_{t}$). The overall structure constants are also modified, though they
are expressed in terms of the same building blocks. For a surface of genus $g$
with $s$ untwisted punctures and $2s_{t}$ $\mathbb{Z}_{2}$-twisted punctures
such that no twist lines wrap any cycles, the index then takes the form

$\mathcal{I}(q;\mathbf{x}_{1},\dots,\mathbf{x}_{s},\mathbf{y}_{1},\dots,\mathbf{y}_{2s_{t}})=\sum_{\lambda^{\prime}\in
P^{+}_{t}}\frac{\prod_{i=1}^{s}\mathcal{K}_{\mathfrak{g}_{u}}(q;\mathbf{x}_{i})\chi^{\lambda=\lambda^{\prime}}_{\mathfrak{g}_{u}}(\mathbf{x}_{i})\prod_{j=1}^{2s_{t}}\mathcal{K}_{\mathfrak{g}_{t}}(q;\mathbf{y}_{j})\chi^{\lambda^{\prime}}_{\mathfrak{g}_{t}}(\mathbf{y}_{j})}{{(\mathcal{K}_{\mathfrak{g}_{u}}(\times)\chi^{\lambda=\lambda^{\prime}}_{\mathfrak{g}_{u}}(\times))}^{2g-2+s_{u}+2s_{t}}}~{}.$
(15)

Here $\mathbf{x}_{i}$ are fugacities for untwisted punctures and
$\mathbf{y}_{j}$ are fugacities for the twisted ones. We have adopted the
notation of Lemos:2012ph , where $\lambda=\lambda^{\prime}$ means that the
highest weight $\lambda$ of $\mathfrak{g}_{u}$ is the one that restricts to
$\lambda^{\prime}$ when passing to outer automorphism invariants. Later on we
shall introduce our own notation for these weights and elaborate on this
restriction. For the non-abelian twist of $\mathfrak{d}_{4}$, a TQFT form of
the index was proposed in Distler:2021cwz . This agrees with the heuristics we
have so far observed—namely the sum is restricted to the integral dominant
weights of $\mathfrak{d}_{4}$ that are invariant under the action of the
twists that are present.

One can also rewrite the twisted index in terms of Weyl modules as

$\mathcal{I}(q;\mathbf{x}_{1},\dots,\mathbf{x}_{s},\mathbf{y}_{1},\dots,\mathbf{y}_{2s_{t}})=\sum_{\lambda^{\prime}\in
P_{t}^{+}}({\rm
ch}~{}\mathfrak{z}_{\lambda}^{u})^{2-2g}\bigg{(}\prod_{i=1}^{s}\mathbb{L}_{\lambda=\lambda^{\prime}}^{u}\bigg{)}\bigg{(}\prod_{j=1}^{2s_{t}}\mathbb{L}_{\lambda^{\prime}}^{t}\bigg{)}~{},$
(16)

where we use the superscripts $u,t$ to distinguish between the (simple
quotients of) Weyl modules over each algebra.

It may be worth remarking that all structure constants are of “untwisted type”
in spite of the ${\cal K}$ factors and Schur functions of twisted type. This
means that when closing a twisted puncture, the specialised puncture factor in
the numerator won’t cancel against a corresponding factor in the denominator.
This is the index-level incarnation of the fact that minimal twisted punctures
are nontrivial and cannot be completely erased from the UV curve.

###  2.4 Residual gauge symmetry and derived structures

An interesting trichotomy in four-dimensional $\mathcal{N}=2$ theories is
between those theories with “pure” Higgs branches (there is no unbroken
continuous gauge symmetry at generic points), those with “enhanced” Higgs
branches (generic Higgs vacua still support massless Abelian vector
multiplets), and those with interacting Higgs branches (where generic Higgs
vacua support a nontrivial interacting SCFT). To the best of our knowledge, in
the presence of regular punctures only pure and enhanced Higgs branches can
arise, whereas interacting Higgs branches are common in Argyres–Douglas type
theories that can be engineered with irregular punctures.

In the untwisted class $\SS$ setting, there is a simple characterisation of
under what circumstances a theory will have an enhanced rather than a pure
Higgs branch: theories of type $\mathfrak{a_{n}}$ ,$\mathfrak{d_{n}}$, or
$\mathfrak{e_{n}}$ with genus $g$ have generic residual gauge symmetry with
rank equal $n\times g$. This admits a straightforward heuristic explanation:
maximal Higgsing will involve first closing all punctures and then moving to
the maximal Higgs branch of the unpunctured theory. For the unpunctured
theory, one can adopt a six-dimensional perspective and first consider moving
to a point on the tensor branch of the $(2,0)$ where the effective theory is
that of $n$ free tensor multiplets, and subsequently compactifying.

For twisted theories, the situation is a bit more complicated. To illustrate,
we consider the $\mathfrak{d_{2}}$ theory. Due to the accidental isomorphism
$\mathfrak{d}_{2}\cong\mathfrak{a}_{1}\times\mathfrak{a}_{1}$, we can recast
theories of type $\mathfrak{d}_{2}$ into $\mathfrak{a}_{1}$ theories. The
twisted subalgebra is just $\mathfrak{a}_{1}$, and a full twisted puncture
becomes a conventional (untwisted) puncture of the $\mathfrak{a}_{1}$ theory.
In particular, the $Spin(4)$ gauge theory with $N_{f}=4$ flavours can be
engineered via compactification of the $\mathfrak{d}_{2}$ theory on a sphere
two maximal and two minimal twisted punctures. Equivalently, it can be
described as the $\mathfrak{a}_{1}$ theory compactified on a genus-one surface
with two punctures. Thus, the $\mathfrak{a}_{1}$ surface is a double cover of
the $\mathfrak{d}_{2}$ surface, treating the twist lines as branch cuts. The
$\mathfrak{d}_{2}$ theory has residual gauge symmetry at a generic point of
the Higgs branch—despite being superficially of genus zero. We will see that
this phenomenon is characteristic of the twisted theories.

Somewhat more generally, for $SO(2n)$ superconformal QCD—realised in type
$\mathfrak{d}_{n}$ using a sphere with four twisted punctures (two minimal and
two maximal)—a generic point of the Higgs branch has precisely a residual
$U(1)$ gauge symmetry Argyres:1996hc . Though for $n\geqslant 3$ there is not
an immediate relation to a higher genus class $\mathcal{S}$ theory like the
$\mathfrak{d}_{2}$ case, we can nevertheless observe that the presence of
outer automorphism twist lines means that there is a natural covering space
for the UV curve (thought of as the space where the corresponding local system
of Dynkin diagrams has no monodromy), and in this example the covering space
is genus one. We believe that in the general case of twisted class
$\mathcal{S}$ theories, it is precisely the value of the genus (zero or
nonzero) of this covering space that controls whether a theory in question has
a pure or an enhanced Higgs branch. More generally, we take away the lesson
that in the presence of twisted punctures, genus zero theories should perhaps
nevertheless be thought of as more analgous to higher-genus untwisted theories
than genus-zero untwisted theories.

From an algebraic perspective, we can think of the presence of an enhanced
Higgs branch as a shadow of an underlying _derived structure_. In particular,
the Hall–Littlewood chiral ring for higher genus (untwisted) class
$\mathcal{S}$ theories is known to include additional fermionic sectors beyond
the Higgs branch chiral ring, which can be understood as a consequence of the
Higgs branches of these theories being more naturally understood as derived
symplectic varieties CJBDBS:HL . We propose that for twisted class
$\mathcal{S}$ theories as well there will be a correspondence between the
presence of an enhanced Higgs branch and higher-fermion-number (derived)
elements of the Hall–Littlewood chiral ring.

For our purposes in this paper, this nontrivial derived structure is most
relevant as an indicator that the BRST cohomology that encodes the associated
vertex operator algebra of a gauge theory (see the review in the next section)
is not concentrated in degree zero. Consequently, we expect a construction in
the twisted setting to be able to mimic the corresponding untwisted genus-zero
story only so long as there are not too many twisted punctures. This
expectation will be borne out in the detailed analysis of Section 4.6.

##  3 Chiral algebras of class $\SS$ after Arakawa

Four-dimensional ${\cal N}=2$ SCFTs give rise to their associated vertex
operator algebras by way of the cohomological construction of Beem:2013sza .
In the case of SCFTs of class $\SS$, the associated vertex algebras are called
chiral algebras of class $\SS$ and were studied in Beem:2014rza . A systematic
and rigorous definition and analysis of these vertex algebras in the untwisted
case was developed in Arakawa:2018egx . In this section we first present
several imporant constructions that are employed in definition of chiral
algebras of class $\SS$ in Arakawa:2018egx , before ultimately recalling the
definition from that work.

###  3.1 Critical level current algebras and the Kazhdan–Lusztig category

If an SCFT has a (continuous) global flavour symmetry with Lie algebra
$\mathfrak{f}$, then the superconformal primaries in the corresponding
conserved current multiplets are moment map operators in the Higgs branch
chiral ring, which are Schur operators. Their images in the associated vertex
algebra are affine currents that generate an $\hat{\mathfrak{f}}$ affine Kac-
Moody (AKM) vertex algebra at level

$k_{2d}=-\frac{1}{2}k_{4d}~{},$ (17)

where $k_{4d}$ is the (positive for unitary theories) four dimensional flavour
central charge. Theories of class $\SS$ have flavour symmetries arising from
marked points, and for a maximal puncture the four-dimensional level of the
$\mathfrak{g}_{u}$ or $\mathfrak{g}_{t}$ flavour symmetry is
$k_{4d}=2h^{\vee}_{\mathfrak{g}_{u,t}}$. The corresponding
$\hat{\mathfrak{g}}_{u,t}$ current subalgebras in the associated VOA are
therefore at the _critical level_
$k_{2d}=\kappa_{c}=-h^{\vee}_{\mathfrak{g}_{u,t}}$. This is an important value
of the level where there is a large null ideal.333In fact, the null ideal is a
commutative vertex operator subalgebra (centre) known as the _Feigin–Frenkel
centre_ , to which we will return momentarily.

Null states in these current subalgebras need not be null with respect to the
full VOA. To describe the situation, it is useful to introduce the universal
affine vertex algebra $V^{k_{2d}}(\mathfrak{g})$, where _none_ of the null
states are set to zero,

$V^{k}(\mathfrak{g})\coloneqq\mathrm{Ind}_{\mathfrak{g}[\\![t]\\!]\oplus\mathbb{C}\mathbbm{1}}^{\hat{\mathfrak{g}}_{k}}\mathbb{C}=U(\hat{\mathfrak{g}}_{k})\otimes_{U(\mathfrak{g}[\\![t]\\!]\oplus\mathbb{C}K)}\mathbb{C}~{}.$
(18)

Here $\mathfrak{g}[\\![t]\\!]$ acts trivially on $\mathbb{C}$ and $K$ is the
central element, which acts as the constant level $k\in\mathbb{C}$. This can
be thought of as the specialisation of the generic-level current algebra to
any particular level $k$ without taking any further quotient if this turns out
to be reducible. (By contrast, taking the maximal quotient gives the simple
affine vertex algebra $V_{k_{2d}}(\mathfrak{g})$, whose underlying graded
vector space can jump as a function of $k$.) $V^{k}(\mathfrak{g})$ is
generically conformal with stress tensor given by the Sugawara construction.
At the critical level, this construction fails and the vertex algebra is no
longer conformal. Rather it is _quasiconformal_ and still satisfies many nice
properties one expects from a conformal vertex algebra Frenkel:2004 .

Chiral algebras of class $\SS$ for theories with at least one maximal marked
point will have the structure of modules over (potentially several copies of)
$V^{\kappa_{c}}(\mathfrak{g})$, and this turns out to be a useful organising
structure. The Kazhdan–Lusztig category (at the critical level),
$\mathrm{KL}$, is the full subcategory of modules of the universal affine
vertex algebra $V^{\kappa_{c}}(\mathfrak{g})$ whose objects are
$\mathbb{Z}$-graded $V^{\kappa_{c}}(\mathfrak{g})$ modules that have a locally
finite action of $\mathfrak{g}$,444A module $M$ over $V^{k}(\mathfrak{g})$ is
a smooth $\hat{\mathfrak{g}}_{k}$ modules, i.e., for each $x\in\mathfrak{g}$
and $m\in M$, $(xt^{n})m=0$ for sufficiently large $n\in\mathbb{N}$.

$\forall m\in M~{},\quad{\rm dim}\left(U(\mathfrak{g})m\right)<\infty~{}.$
(19)

This definition allows a full $\mathbb{Z}$-grading, so includes modules with
conformal weight unbounded below. For a VOA associated to a physical four-
dimensional theory thought of as an element of $\mathrm{KL}$, this would be
pathological. We take $\mathrm{KL}^{ord}$ to denote the full subcategory of
$\mathrm{KL}$ admitting a $\mathbb{Z}_{\geqslant 0}$-grading such that each
homogeneous subspace is finite dimensional—these can be thought of as positive
energy representations.

An important example of an object in $\mathrm{KL}^{ord}$ are the Weyl modules
$\mathbb{V}_{\lambda}$, defined by

$\mathbb{V}_{\lambda}\coloneqq\mathrm{Ind}_{\mathfrak{g}[\\![t]\\!]\oplus\mathbb{C}K}^{\hat{\mathfrak{g}}_{\kappa_{c}}}V_{\lambda}~{},$
(20)

where $V_{\lambda}$ is the finite-dimensional irreducible representation of
$\mathfrak{g}$ with highest (integral, dominant) weight $\lambda$. The
universal affine vertex algebra is therefore the vacuum module,
$\mathbb{V}_{0}$. The Weyl modules have a unique simple quotient,
$\mathbb{L}_{\lambda}$.

A vertex algebra object in $\mathrm{KL}$ is a vertex algebra $V$ with a vertex
algebra morphism $\mu_{V}:V^{\kappa_{c}}(\mathfrak{g})\rightarrow V$, such
that $V$ is a direct limit of objects in $\mathrm{KL}^{ord}$. The chiral
algebras of class $\SS$ will be vertex algebra objects in $\mathrm{KL}$. Given
two vertex algebra objects, $V_{1},V_{2}\in\mathrm{KL}$, the product
$V_{1}\otimes V_{2}$ is a vertex algebra object in
$\mathrm{KL}_{-\kappa_{g}}$, the Kazhdan–Lusztig category at level
$-\kappa_{g}=-2h^{\vee}$.

###  3.2 The Feigin–Frenkel centre

The critical-level universal affine vertex algebra,
$V^{\kappa_{c}}(\mathfrak{g})$ has a large centre $\mathfrak{z}(\mathfrak{g})$
known as the Feigin–Frenkel centre. This centre plays a starring role in
Arakawa’s construction of chiral algebras of class $\SS$, though its
properties may not be common knowledge to all four-dimensional physicists.

The Feigin–Frenkel centre is a commutative vertex algebra with strong
generators $\\{P_{i}(z)\\}$ of (quasi)conformal weight $\Delta_{i}=d_{i}$,
where $d_{i}$ are the degrees of the fundamental invariants of $\mathfrak{g}$
Frenkel:2007  (see Table 2). For many purposes, it is useful to work in terms
of the universal enveloping algebra of the Fourier modes of the generators of
$\mathfrak{z}(\mathfrak{g})$, which is denoted by ${\cal Z}$. This is a
(commutative) polynomial ring on infinitely many variables,

${\cal
Z}\coloneqq\mathbb{C}[P_{i(n)}~{}|~{}i=1,\dots,\mathrm{rk}\,\mathfrak{g},\,n\in\mathbb{Z}]~{},$
(21)

where we are adopting the mathematician’s conventions for the grading of
Fourier modes,

$P_{i}(z)=\sum_{n\in\mathbb{Z}}P_{i,(n)}z^{-n-1}~{}.$ (22)

By contrast, the physicist’s gradings are shifted by the (quasi)conformal
weights minus one,

$P_{i}(z)=\sum_{n\in\mathbb{Z}}P_{i,n}z^{-n-\Delta_{i}}~{}.$ (23)

Two important subalgebras of ${\cal Z}$ are as follows,

${\cal Z}_{<0}=\\{P_{i,n}\in{\cal Z}~{}|~{}n<0\\}~{},\quad{\cal
Z}_{(<0)}=\\{P_{i,(n)}\in{\cal Z}~{}|~{}n<0\\}~{}.$ (24)

The second of these can be identified with $\mathfrak{z}(\mathfrak{g})$ itself
with respect to the operation of normally ordered product (which is
commutative and associative since $\mathfrak{z}(\mathfrak{g})$ is a
commutative vertex algebra), while the first includes a subset of modes that
encode the singular terms subleading to the zero mode in the OPE action of the
$P_{i}(z)$.

A positive-energy representation of $\mathfrak{z}(\mathfrak{g})$ is a ${\cal
Z}$ module $M$ with grading $M=\bigoplus_{d\in p+\mathbb{N}}M_{d}$ for some
$p\in\mathbb{C}$, and any object in $\mathrm{KL}^{ord}$ is in particular a
positive-energy representation of $\mathfrak{z}(\mathfrak{g})$. Henceforth,
${\cal Z}$-Mod will denote the category of ${\cal Z}$ modules that are
positive-energy representations of $\mathfrak{z}(\mathfrak{g})$.

Important examples of such objects are the Weyl modules
$\mathbb{V}_{\lambda}$, which is an object in ${\cal Z}$-Mod, but is simply
annihilated by all $P_{i,n}$ with $n>0$. Let $\chi_{\lambda}:{\cal
Z}\rightarrow\mathbb{C}$ be the character defined by
$P_{i}V_{\lambda}=\chi_{\lambda}(P_{i,0})V_{\lambda}$, where $P_{i}$ is the
$i$th fundamental invariant of $\mathfrak{g}$ and $\chi_{\lambda}(P_{i,n})=0$
otherwise. The zero modes, $P_{i,0}$, act on $\mathbb{V}_{\lambda}$ by the
character $\chi_{\lambda}$. The potential for interesting structure then lies
with its behaviour as a module over ${\cal Z}_{<0}$. Defining the quotient

$\mathfrak{z}_{\lambda}\coloneqq{\cal Z}_{<0}/\mathrm{Ann_{{\cal
Z}_{<0}}}(\mathbb{V}_{\lambda})~{},$ (25)

the action of ${\cal Z}_{<0}$ on $\mathbb{V}_{\lambda}$ factors through
$\mathfrak{z}_{\lambda}$, which manifestly acts freely.

One can “glue together” Weyl modules using the Feigin–Frenkel centre by taking
tensor products over $\mathfrak{z}_{\lambda}$. For example, one can define the
bimodule
$\mathbb{V}_{\lambda}\otimes_{\mathfrak{z}_{\lambda}}\mathbb{V}_{\lambda^{*}}$,
where $\lambda^{*}$ is the dual weight to $\lambda$ and where the action of
$\mathfrak{z}_{\lambda}$ on $\mathbb{V}_{\lambda^{*}}$ is twisted by the
Cartan involution $\tau:\mathfrak{z}\rightarrow\mathfrak{z}$ frenkel2004
.555This involution corresponding to $\alpha\mapsto-w_{0}(\alpha)$, where
$w_{0}$ is the longest element of the Weyl group. The morphism $\tau$ sends a
finite dimensional, highest-weight representation $V_{\lambda}$ of
$\mathfrak{g}$ to its dual
$V_{\lambda}^{*}=Hom_{\mathfrak{g}}(V_{\lambda},\mathbb{C})$. For simple Lie
algebras, the Weyl group elements $w_{0}$ can be found in Bourbaki:2002 . For
non-simply laced Lie algebras and the algebras $\mathfrak{e}_{7}$ and
$\mathfrak{e}_{8}$, $w_{0}=-1$ and the involution is trivial. For
$\mathfrak{a}_{n}$ and $\mathfrak{d}_{2n+1}$ the involution, $\tau$, is
precisely the $\mathbb{Z}_{2}$ outer automorphism twist. For
$\mathfrak{d}_{2n}$, the involution is again trivial. This module then has two
critical-level actions of $\hat{\mathfrak{g}}$, one on each Weyl module. But
the action of the Feigin–Frenkel centres are identified, so this is a module
over
$\mathbb{V}_{2,0}\colonequals(V^{\kappa_{c}}(\mathfrak{g}))^{\otimes_{\mathfrak{z}(\mathfrak{g})}2}$.

This generalises to the case of $s$ copies of the critical level current
algebra,

$\mathbb{V}_{\lambda,s}=\mathbb{V}_{\lambda}\otimes_{\mathfrak{z}_{\lambda}}\mathbb{V}_{\lambda^{*}}\otimes_{\mathfrak{z}_{\lambda}}\dots\otimes_{\mathfrak{z}_{\lambda}}\mathbb{V}_{\lambda^{(*)}}~{},$
(26)

where the final module is $\mathbb{V}_{\lambda}$ if $s$ is odd, or
$\mathbb{V}_{\lambda^{*}}$ if $s$ is even. This is a module over
$\mathbb{V}_{s,0}\colonequals(V^{\kappa_{c}}(\mathfrak{g}))^{\otimes_{\mathfrak{z}(\mathfrak{g})}s}$.
We denote the contragredient dual module by $D(\mathbb{V}_{\lambda,s})$.

A key property of Weyl modules is that they are projective over
$t^{-1}\mathfrak{g}[t^{-1}]$ and their contragredient duals are injective over
$t\mathfrak{g}[t]$. Similarly $\mathbb{V}_{\lambda,s}$ and its contragredient
dual are projective/injective over each $t^{-1}\mathfrak{g}[t^{-1}]$ and
$t\mathfrak{g}[t]$ action, respectively. These modules play an important role
in the analysis of Arakawa:2018egx , and this will continue to be the case for
twisted chiral algebras of class $\SS$.

###  3.3 BRST and semi-infinite cohomology

Exactly marginal gauging of flavour symmetries in four dimensions is realised
at the level of the associated VOA by a BRST construction Beem:2013sza . In
particular, the associated VOA of an $\mathfrak{f}$-valued vector multiplet is
the subalgebra of an adjoint-valued $(b,c)$ ghost system that is annihilated
by the zero mode of the $b$ ghosts—these are sometimes called symplectic
fermions, or the “small algebra” of the ghost system. The vertex algebra of a
gauge theory is realised in terms of the vertex algebra ${\cal V}$ of the
original ungauged theory and these symplectic fermions. One constructs a BRST
current,

$J_{\rm
BRST}(z)=(J_{i}c^{i})(z)+\frac{1}{2}f_{ij}\,^{k}(c^{i}c^{j}b_{k})(z)~{},$ (27)

where the $J_{i}$ generate an $\hat{\mathfrak{f}}$ affine current subalgebra
of ${\cal V}$ and $f_{ij}\,^{k}$ are the structure constants of
$\mathfrak{f}$. Precisely when that current subalgebra has level
$\kappa_{\mathfrak{f}}=-2h^{\vee}_{\mathfrak{f}}$, the zero mode squares to
zero so defines a differential on $C^{\bullet}={\cal V}\otimes\langle
b,c\rangle$. The grading on $C^{\bullet}$ is by ghost number, which in many
cases can be identified with $U(1)_{r}$ charge.

Restricting to the kernel of $b_{0}$ (i.e., dropping the zero-modes of the $c$
ghosts) and taking $\mathfrak{f}$ invariants defines the relative subcomplex
$C_{rel}^{\bullet}$,

$C_{rel}^{\bullet}\coloneqq\\{v\in{\cal V}\otimes\langle
b,c\rangle~{}|~{}b^{i}_{(0)}v=0\text{ for}\,i=1,\dots,{\rm
dim}\,\mathfrak{f}\\}^{\mathfrak{f}}~{}.$ (28)

The BRST differential descends to the relative subcomplex and its cohomology
defines the relative BRST cohomology. This gives the vertex algebra of the
gauged four dimensional theory. Importantly, one cannot neglect the higher
cohomology groups in this construction.666Indeed, in examples such as ${\cal
N}=4$ super Yang-Mills theory and the genus-two theory of type
$\mathfrak{a}_{1}$ where explicit computations can be performed, there is
cohomology outside of degree zero. More generally we expect cohomology outside
degree zero at least for theories with enhanced Higgs branches.

The above construction can be realised as an example of the semi-infinite
cohomology functor of Voronov Voronov:1993aw ; this more abstract formalism is
somewhat useful in the analysis of chiral algebras of class $\SS$. The state
space of the $\mathfrak{f}$-valued ghost system is the same thing as the space
of _semi-infinite forms_ on $\mathfrak{f}$ and denoted
$\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}\mathfrak{f}$. For an
$\hat{\mathfrak{f}}$-module $M$ at level $\kappa_{\mathfrak{f}}$, the _Feigin
standard complex_ is

$C^{\frac{\infty}{2}+\bullet}\left(\hat{\mathfrak{f}},M\right)=\left(\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}\hat{\mathfrak{f}}\otimes
M,d\right)~{},$ (29)

where $d=J_{{\rm BRST},(0)}$. The semi-infinite cohomology of
$\hat{\mathfrak{f}}$ with coefficients in $M$ is just the cohomology of this
complex,

$\mathrm{H}^{\frac{\infty}{2}+\bullet}\left(\hat{\mathfrak{f}},M\right)=H^{\bullet}\left(C^{\frac{\infty}{2}}\left(\hat{\mathfrak{f}},M\right),d\right)~{}.$
(30)

Voronov identifies this cohomology as a two-sided derived functor of the
functor of semi-variants, and proves an important vanishing theorem that
characterises situations where semi-infinite cohomology is concentrated in
degree zero.

######  Theorem 3.1.

(Theorem 2.1 of Voronov:1993aw ) Let $M\in\mathcal{O}$ be a semijective
module, i.e., it is injective as a $\mathfrak{f}_{+}$ module and projective as
a $\mathfrak{f}_{-}$ module. Then

$\mathrm{H}^{\frac{\infty}{2}+p}(\mathfrak{f},M)=\begin{cases}\makebox[19.91684pt][l]{$M^{\mathfrak{f}_{+}}_{\mathfrak{f}_{-}}$}\quad
p=0~{},\\\ \makebox[19.91684pt][l]{$0$}\quad p\neq 0~{},\end{cases}$ (31)

where $M^{\mathfrak{f}_{+}}_{\mathfrak{f}_{-}}$ is the projection of
$\mathfrak{f}_{+}$-invariants onto the $\mathfrak{f}_{-}$-coinvariants. More
concretely,

$M^{\mathfrak{f}_{+}}_{\mathfrak{f}_{-}}=\\{m\in
M\,|\,\mathfrak{f}_{+}m=0\\}{\bigg{/}}\\{m\in
M\,|\,\mathfrak{f}_{+}m=0\quad\text{and}\quad m=xm^{\prime}\,\,\,\text{for
some }x\in\mathfrak{f}_{-}\\}~{}.$ (32)

The relative BRST cohomology described above is also understood as relative
semi-infinite cohomology with respect to the
$\mathfrak{f}\subset\hat{\mathfrak{f}}_{\kappa}$. As in Voronov:1993aw ;
Arakawa:2018egx , we denote this by

$\mathrm{H}^{\frac{\infty}{2}+\bullet}(\hat{\mathfrak{f}}_{\kappa},\mathfrak{f},M)~{},$
(33)

so the prescription for gauging a flavour symmetry $\mathfrak{f}$ in the
associated VOA amounts to taking the relative semi-infinite cohomology. The
procedure for gluing together two chiral algebras of class $\SS$ $V_{1}$ and
$V_{2}$ of type $\mathfrak{g}$ then amounts to taking the relative semi-
infinite cohomology,

$V_{1}\circ
V_{2}\colonequals\mathrm{H}^{\frac{\infty}{2}+\bullet}(\hat{\mathfrak{g}}_{-\kappa_{g}},\mathfrak{g},V_{1}\otimes
V_{2})~{}.$ (34)

An application of Theorem 3.1 in the relative context gives the following
vanishing theorem, which is important in establishing generalized $S$-duality
at the level of vertex algebras.

######  Theorem 3.2.

(Section 3.11.2 of Voronov:1993aw  and Proposition 3.4 of Arakawa:2018egx )
Suppose $V\in\mathrm{KL}_{-\kappa_{g}}$ and $V$ is projective as a
$U(t^{-1}\mathfrak{g}[t^{-1}])$ module and injective as a
$U(t\mathfrak{g}[t])$ module, i.e., $V$ is a semijective object in
$\mathrm{KL}$. Then,

$\mathrm{H}^{\frac{\infty}{2}+p}(\hat{\mathfrak{g}}_{-\kappa_{g}},\mathfrak{g},V)=0\quad\text{for
}p\neq 0~{}.$ (35)

###  3.4 Drinfel’d–Sokolov reduction

Starting with chiral algebras of class $\SS$ for theories with only maximal
punctures, one realises the VOAs with non-maximal punctures by a chiral
analogue of the Higgsing procedure described in Section 2. Namely, this type
of Higgsing is accomplished via quantum Drinfel’d–Sokolov (DS) reduction,
which we review briefly below.777This identification of Higgsing with DS
reduction was presented as a _conjecture_ in Beem:2014rza . It remains an open
problem to prove this from first principles—there is by now extensive evidence
for the claim.

A nontrivial homomorphism
$\Lambda:\mathfrak{su}(2)\hookrightarrow\mathfrak{g}$ induces a grading on
$\mathfrak{g}$ by the image of the Cartan generator,888If the grading is
genuinely half-integral and not integral, then there is a subtlety and one
must introduce an alternate Cartan generator, whose grading is integral, and
impose modified constraints in what follows. See Feher:1992ed ; deBoer:1992sy
; deBoer:1993iz  for more details.

$\mathfrak{g}=\bigoplus_{m\in\frac{1}{2}\mathbb{Z}}\mathfrak{g}_{m}\,,\quad\mathfrak{g}_{m}\coloneqq\\{t\in\mathfrak{g}\,|\,\mathrm{ad}_{\Lambda(h)}\,(t)=mt\\}~{}.$
(36)

Let $\\{t_{\alpha}\\}$ denote a basis for the negatively graded part of
$\mathfrak{g}$, with corresponding affine currents $J_{\alpha}$ in
$V^{k_{c}}(\mathfrak{g})$. Drinfel’d–Sokolov reduction can be thought of as
imposing the constraint

$J_{\alpha}=\delta_{\alpha\alpha_{-}}~{},$ (37)

where $t_{\alpha_{-}}$ is the image of the negative weight generator
$f\in\mathfrak{su}(2)$. This constraint is imposed via BRST by introducing a
ghost system $(c^{\alpha},b_{\alpha})$ valued in the adjoint of
$\bigoplus_{m<0}\mathfrak{g}_{m}$ and forming the BRST current

$J_{DS}(z)=\sum_{\alpha}\left((J_{\alpha}-\delta_{\alpha\alpha_{-}})c^{\alpha}\right)(z)-\frac{1}{2}\sum_{\alpha\beta}^{\gamma}f_{\alpha\beta}^{\phantom{\alpha\beta}\gamma}(b_{\gamma}c^{\alpha}c^{\beta})(z)~{},$
(38)

with the BRST differential $d_{DS}=J_{{\rm DS},(0)}$. We define the DS
reduction of a vertex algebra ${\cal V}$, which has the structure of a
$V^{k}(\mathfrak{g})$ module, as the cohomology

$H^{\bullet}_{f,DS}(V)=H_{\rm BRST}^{\bullet}(V\otimes\langle
c^{\alpha},b_{\beta}\rangle,d_{DS})~{}.$ (39)

Taking the coefficients to be exactly $V^{k}(\mathfrak{g})$ gives the ${\cal
W}$-algebra ${\cal
W}(f,\hat{\mathfrak{g}}_{\kappa})=H^{0}_{f,DS}(V^{\kappa}(\mathfrak{g}))$.

Like the BRST operation associated to gauging, DS reduction can be phrased in
terms of semi-infinite cohomology (see, e.g., Arakawa2017introduction ). We
will make extensive use of the result Arakawa:2007rep .

######  Theorem 3.3 (Main Theorem 1 of Arakawa:2007rep ).

For any $M\in\mathrm{KL}$, $H^{i}_{f,DS}(M)=0$ for $i\neq 0$.

The case of principal DS reduction at the critical level is especially
interesting because ${\cal
W}(f_{prin},\hat{\mathfrak{g}}_{\kappa_{c}})\cong\mathfrak{z}(\mathfrak{g})$.
For non vacuum modules, there is an isomorphism Frenkel:2010

$H^{0}_{f_{prin},DS}(\mathbb{V}_{\lambda})\cong\mathfrak{z}_{\lambda}~{}.$
(40)

In general we get an exact functor
$H^{0}_{f_{prin},DS}:\mathrm{KL}\rightarrow{\cal Z}$-Mod. We will mostly be
interested in principal DS reduction in what follows, and will suppress the
explicit $f_{prin}$ argument from now on.

Pictorially, principal DS reduction corresponds to removing a puncture. The
key idea in the construction of chiral algebras of class $\mathcal{S}$ is to
develop a method to _invert_ this operation and add punctures back onto the UV
curve. We discuss this in the following sections.

###  3.5 Feigin–Frenkel gluing

For chiral algebras of class $\SS$ with maximal punctures, the current
subalgebras are all at the critical level, so there could theoretically be
several copies of the Feigin–Frenkel centre present. This turns out to not be
the case; in fact the current subalgebras all _share a common Feigin–Frenkel
centre_ , so the chiral algebras of class $\SS$ with $s$ punctures are modules
over $\mathbb{V}_{0,s}$.

This phenomenon of shared FF centres is a chiral analogue of certain well-
known Higgs branch chiral ring relations for theories of class $\SS$. We
recall that for theory of type $\mathfrak{a}_{n}$ associated to a UV curve
with $s$ punctures, there are $s$ moment map operators $\mu_{s}$ subject to
the relations

$\mathop{\mathrm{Tr}}\nolimits\mu_{1}^{k}=\mathop{\mathrm{Tr}}\nolimits\mu_{2}^{k}=\cdots=\mathop{\mathrm{Tr}}\nolimits\mu_{s}^{k}~{},\qquad
k=2,\ldots,n+1~{}.$ (41)

More generally, for the $\mathfrak{d}_{n}$ and $\mathfrak{e}_{n}$ theories
there are analogous relations corresponding to the respective fundamental
invariants of those algebras. These play a central role in the construction of
Higgs branches of class $\SS$ by Ginzburg–Kazhdan Ginzburg:2018 .

It is not immediate that these Higgs chiral ring relations lead to the
identification of Feigin–Frenkel centres. This is because the Feigin–Frenkel
generators are related to, but not equal to, the corresponding Higgs branch
operators.999To be precise, the Feigin–Frenkel operators are identified with
the corresponding Higgs branch operators upon passing to the associated graded
of the $R$ filtration Beem:2017ooy . To illustrate, consider the case $k=2$ of
(41). The Higgs branch operators associated to the quadratic fundamental
invariant are related to the Segal–Sugawara operators, $P_{1,i}$ but also
receive a nonzero contribution from the VOA stress-energy tensor $T$
Beem:2018duj ,

$P_{1,i}=\mathop{\mathrm{Tr}}\nolimits\mu^{2}_{i}+\alpha T~{},$ (42)

where $\alpha$ is a fixed (nonzero) constant computed in Beem:2018duj  and
$P_{1,i}$ is the quadratic Feigin–Frenkel generator associated to the $i$’th
puncture. The Higgs chiral ring relations set the
$\mathop{\mathrm{Tr}}\nolimits\mu^{2}_{i}$ equal and, importantly, there is a
_unique_ $\hat{{\cal C}}_{0(0,0)}$ multiplet (the four-dimensional stress
tensor multiplet) so the operator $T$ is the same for each $i$. As a result,
the quadratic generators of the Feigin–Frenkel centre are identified across
different punctures, i.e., $P_{1,1}=P_{1,2}=\dots P_{1,s}$. For higher order
invariants, the same argument doesn’t go through without additional
information about the structure of the VOA; in the case of the cubic invariant
of $\mathfrak{a}_{n}$ there will be mixing between Higgs branch operators
($\widehat{\cal B}_{3}$ multiplets) and ${\cal C}_{1(0,0)}$ multiplets, but
the uniqueness of the latter is not apparent. Nevertheless, precisely for the
class $\mathcal{S}$ theories, the known expression for the Schur index
indicates that the identification of the higher Feigin–Frenkel generators
should indeed hold.

As in the definition of $\mathbb{V}_{0,2}$, given two modules $M,N\in{\cal
Z}$-Mod, we can equip $M\otimes N$ with a ${\cal Z}$-Mod structure via a
twisted diagonal action,

$P\otimes 1-1\otimes\tau(P)\quad\forall P\in{\cal Z}~{}.$ (43)

The twist by $\tau$ is a matter of convention, but will prove convenient in
later constructions when Feigin–Frenkel gluing and gauging appear
together—roughly speaking, one can anticipate this issue by observing that
when gauging together two VOAs, the gauge invariant states arise from pairing
representations with highest weights $\lambda$ and $\lambda^{*}$ respectively,
so if one wishes to be able to identify a common action of ${\cal Z}$ on these
two representations then there should be a twist by $\tau$.

As an abelian Lie algebra, ${\cal Z}$ has a semi-infinite structure and so can
be used to define semi-infinite cohomology with coefficients in some object of
${\cal Z}$-Mod. Specifically, we will have coefficients of the form $M\otimes
N$ for $M,N\in{\cal Z}$-Mod. The Feigin standard complex is then given by

$C^{\bullet}({\cal Z},M\otimes N)\coloneqq M\otimes
N\otimes\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}{\cal Z}~{},$ (44)

with BRST current

$Q(z)=\sum_{i=1}^{\mathrm{rk}\,\mathfrak{g}}(P_{i}c^{i})(z)~{},$ (45)

whose zero mode $Q_{(0)}$ acts as a differential for the cochain complex.
Suppose we have two objects $V_{1}$ and $V_{2}$ in ${\cal Z}$-Mod, we define
Feigin–Frenkel (FF) gluing as

$V_{1}\ast V_{2}\coloneqq\mathrm{H}^{\frac{\infty}{2}+0}({\cal Z},V_{1}\otimes
V_{2})~{}.$ (46)

In the case when $V_{1}$ and $V_{2}$ are vertex algebras, $V_{1}\ast V_{2}$ is
also a vertex algebra. The vector space $\mathbf{V}_{1}\otimes V_{2}$ also has
the structure of a ${\cal Z}$-module via the left (or right) action of ${\cal
Z}$ on the tensor factors, which have become identified through the gluing.

As was the case with the gauge theory BRST problem, we have a vanishing
theorem here.

######  Theorem 3.4 (Theorem 9.10 of Arakawa:2018egx ).

Let $M\in{\cal Z}$-Mod be free as a ${\cal Z}_{(<0)}$ module, then

$\mathrm{H}^{\frac{\infty}{2}+i}({\cal Z},M)=0\quad\textrm{for}~{}~{}i<0~{}.$

This is a weaker conclusion than in the vanishing theorem 3.2, as the
cohomology is not necessarily concentrated in degree zero. Nevertheless for
many purposes it is sufficient.101010From a derived perspective, truncating
the cohomology at degree zero is somewhat unnatural. We will return to this
point in a later section.

We see that the BRST procedure enforces that the action of the Feigin–Frenkel
centre on $V_{1}$ and $V_{2}$ are identified. This is a chiral version of the
Hamiltonian reduction procedure described in Ginzburg:2018 .

###  3.6 Vertex algebras at genus zero

The pieces are now all in place to recount the construction of (genus zero)
untwisted chiral algebras of class $\SS$ from Arakawa:2018egx .

#### Chiral differential operators and the cylinder

The starting point of the construction of Arakawa:2018egx  is the cylinder
VOA; from here one can define the cap chiral algebra by Drinfel’d–Sokolov
reduction and, it turns out, construct all genus zero VOAs by FF gluing. The
form of the cylinder algebra for $\mathfrak{a}_{n}$ theories was identified
concretely in Beem:2014rza , but it was subsequently recognised in
Arakawa:2015ro  that this reproduced a more general, and purely algebraic,
construction that makes no explicit reference to four dimensional physics. The
construction is universal and depends only on the choice of an algebraic group
$G$, which here we take to be semisimple.

Starting from such a $G$, the arc space $J_{\infty}G$ is a scheme over
$\mathbb{C}$ defined as

$J_{\infty}G\coloneqq Hom_{Sch/\mathbb{C}}(D,G)~{},$ (47)

where $D$ is the formal disc $D=\mathrm{Spec}\,\mathbb{C}[\\![t]\\!]$. See
Arakawa2017introduction  for more concrete discussion of arc spaces. The Lie
algebra $\mathfrak{g}=Lie(G)$ acts on ${\cal O}(G)$ via derivations, and this
action lifts to one of $J_{\infty}(\mathfrak{g})\cong\mathfrak{g}[\\![t]\\!]$
on ${\cal O}(J_{\infty}G)$. Therefore, ${\cal O}(J_{\infty}G)$ has the
structure of a $\mathfrak{g}[\\![t]\\!]\oplus\mathbb{C}K$ module where $K$
acts as the level $k\in\mathbb{C}$. One can further produce a
$\hat{\mathfrak{g}}_{\kappa_{c}}$ module via induction, which defines the
_chiral differential operators_ on $G$ (at the critical level) ag  (see also
Gorbounov:1992 ; Gorbounov:2001 ; Gorbounov:2004 ; Malikov:1999 ),

$\mathcal{D}^{ch}_{G}=U(\hat{\mathfrak{g}}_{\kappa_{c}})\otimes_{U(\mathfrak{g}[\\![t]\\!]\oplus\mathbb{C}K)}{\cal
O}(J_{\infty}G)~{}.$ (48)

This has the structure of a conformal vertex algebra Gorbounov:2001  with
central charge

$c_{2d}=2\,{\rm dim}\,G~{},$ (49)

which matches with the central charge for a twice punctured sphere (see, e.g.,
Chacaltana:2012zy ).

By construction, $\mathcal{D}^{ch}_{G}$ is a vertex algebra object in
$\mathrm{KL}$ and there is an embedding of the universal affine vertex algebra
$\pi_{L}:V^{\kappa_{c}}(\mathfrak{g})\hookrightarrow\mathcal{D}^{ch}_{G}$.
This vertex algebra homomorphism is induced by the embedding of $\mathfrak{g}$
as left invariant vector fields of $G$. The Lie algebra $\mathfrak{g}$ is also
isomorphic to the right invariant vector fields of $G$ and this embedding is
also lifted to a vertex algebra homomorphism
$\pi_{R}:V^{\kappa_{c}}(\mathfrak{g})\hookrightarrow\mathcal{D}^{ch}_{G}$,
such that the images of $\pi_{L}$ and $\pi_{R}$ commute Gorbounov:2001 . The
left and right embeddings of $V^{\kappa_{c}}(\mathfrak{g})$ restrict to
embeddings of the Feigin–Frenkel centre $\mathfrak{z}(\mathfrak{g})$, and the
two embeddings of the Feigin–Frenkel centre coincide frenkel2004 ,

$\pi_{L}(\mathfrak{z}(\mathfrak{g}))\cong\pi_{R}(\mathfrak{z}(\mathfrak{g}))\cong(\mathcal{D}^{ch}_{G})^{\mathfrak{g}[t]\times\mathfrak{g}[t]}~{}.$
(50)

$\mathcal{D}^{ch}_{G}$ is free as a module over
$U(t^{-1}\mathfrak{g}[t^{-}1])$ and cofree over $U(t\mathfrak{g}[t])$
Arakawa:2018egx ; thus the conditions of Theorem 3.2 are met and for any
$M\in\mathrm{KL}$, the cohomology $\mathcal{D}^{ch}_{G}\circ M$ is
concentrated in degree zero, and furthermore Arakawa:2018egx ,

$\mathrm{H}^{\frac{\infty}{2}+i}(\hat{\mathfrak{g}}_{-\kappa_{g}},\mathfrak{g},\mathcal{D}^{ch}_{G}\otimes
M)\cong\mathrm{H}^{\frac{\infty}{2}+i}(\hat{\mathfrak{g}}_{-\kappa_{g}},\mathfrak{g},M\otimes\mathcal{D}^{ch}_{G})\cong\delta_{i,0}M~{}.$
(51)

It is easy to see that $\mathcal{D}^{ch}_{G}$ is the unique vertex algebra
object in $\mathrm{KL}$, which satisfies this property. Physically, the chiral
algebra of the cylinder must act as the identity object under $\circ$, when
restricted to the chiral algebras of class $\SS$. This is a weaker condition
than what is satisfied by $\mathcal{D}^{ch}_{G}$. One could a imagine
situation in which the chiral algebra of the cylinder acts as the identity for
the chiral algebras of class $\SS$ but not on all of $\mathrm{KL}$. In this
case, the chiral algebra of the cylinder may not necessarily be isomorphic to
$\mathcal{D}^{ch}_{G}$. However, this pathological possibility seems to not be
realised.

#### Equivariant affine ${\cal W}$-algebras and the cap

Starting from the cylinder VOA, the cap algebra is recovered by completely
reducing one maximal puncture. Arakawa has named the resulting VOA the
(principal) _equivariant affine $W$-algebra_ $\mathbf{W}_{G}$ (it is an affine
analogue of the equivariant $W$-algebra of Losev:2007Qua ),

$\mathbf{W}_{G}\coloneqq H^{0}_{DS}(\mathcal{D}^{ch}_{G})~{}.$ (52)

In the usual way, this vertex algebra inherits a conformal structure from the
cylinder, with central charge

$c_{\mathbf{W}_{G}}={\rm
dim}\,\mathfrak{g}+\mathrm{rk}\,\mathfrak{g}+24\kappa_{g}(\rho,\rho^{\vee})~{},$
(53)

where $\rho$ is half the sum of all positive roots and $\rho^{\vee}$ is the
half sum of all positive coroots.

By Propositions $6.4$ and $6.5$ of Arakawa:2018egx , $\mathbf{W}_{G}$ is free
over $U(t^{-1}\mathfrak{g}[t^{-1}])$ and cofree over $U(t\mathfrak{g}[t])$,
and so is the $\hat{\mathfrak{g}}_{-\kappa_{g}}$ module $\mathbf{W}_{G}\otimes
M$. The cohomology when gauging is therefore concentrated in degree zero.
Theorem $6.8$ of Arakawa:2018egx  then establishes that for any
$M\in\mathrm{KL}$, and in particular for vertex algebra objects in
$\mathrm{KL}$,

$\mathrm{H}^{\frac{\infty}{2}+\bullet}(\hat{\mathfrak{g}}_{-\kappa_{g}},\mathfrak{g},\mathbf{W}_{G}\otimes
M)=H^{0}_{DS}(M)~{}.$ (54)

When $M$ is a chiral algebra of class $\SS$, this corresponds, pictorially, to
the fact that gluing a cap and a surface together along a maximal puncture has
the effect of closing the puncture.

Feigin–Frenkel gluing a cap onto another vertex algebra provides a sort of
inverse to the principal DS reduction functor; by FF gluing a cap onto a
vertex algebra $V\in{\cal Z}$-Mod we provide it with a
$V^{\kappa_{c}}(\mathfrak{g})$ action, and it becomes a vertex algebra object
in $\mathrm{KL}$. The cap is free over ${\cal Z}_{(<0)}$ Arakawa:2018egx  and
so $\mathbf{W}_{G}\ast-:{\cal Z}\text{-Mod}\rightarrow\mathrm{KL}$ is a left-
exact functor that acts, almost, as an inverse to $H^{0}_{DS}$. For any vertex
algebra object $V$ in $\mathrm{KL}$, Theorem $9.11$ of Arakawa:2018egx
guarantees that

$V\cong\mathrm{H}^{\frac{\infty}{2}+0}({\cal Z},\mathbf{W}_{G}\otimes
H^{0}_{DS}(V))~{},$ (55)

and for the subcategory of $V$-module objects in $\mathrm{KL}$,

$\mathrm{H}^{\frac{\infty}{2}+0}({\cal Z},\mathbf{W}_{G}\otimes
H^{0}_{DS}(-))~{},$ (56)

is the identity functor. Indeed, this means that the cylinder VOA can be
recovered from the equivariant affine $W$-algebra by FF gluing two caps
together Arakawa:2018egx ,

$\mathrm{H}^{\frac{\infty}{2}+0}({\cal
Z},\mathbf{W}_{G}\otimes\mathbf{W}_{G})\cong\mathcal{D}^{ch}_{G}~{}.$ (57)

This is only almost an inverse functor because $\mathrm{KL}$ and ${\cal
Z}$-Mod are not equivalent. If we restrict to the subcategory
$\mathrm{KL}_{0}$ of ${\cal Z}$-Mod, which is the image of $H^{0}_{DS}$, then
$H^{0}_{DS}(\mathrm{H}^{\frac{\infty}{2}+0}({\cal Z},\mathbf{W}_{G}\otimes
V))\cong V$ for any $V\in\mathrm{KL}_{0}$.

#### General genus zero VOAs

This situation leads to a conceptually straightforward construction of all
genus zero chiral algebras of class $\SS$: one starts with
$\mathcal{D}^{ch}_{G}$ and repeatedly applies $\mathbf{W}\ast-$ to add
punctures. If one takes for granted the general expectations of chiral
algebras of class $\SS$ as outlined in Beem:2014rza , then this must give the
correct answer. However, the formalism of Arakawa:2018egx  allows Arakawa to
go further and demonstrate, _a priori_ , the expected behaviour of these VOAs
_vis à vis_ their gluing and duality properties.

To this end, the construction given in Arakawa:2018egx  is superficially
different, amounting to gluing together $s$ cap VOAs _at once_ rather than
iteratively adding punctures. Take the chain complex

$C^{\bullet}\left(\bigoplus_{i=1}^{s-1}{\cal
Z}^{i,i+1},\mathbf{W}^{s}\right)\coloneqq\mathbf{W}^{s}\otimes\bigg{(}\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}(\mathfrak{z}(\mathfrak{g}))\bigg{)}^{s-1}~{},$
(58)

for $s>1$, with differential

$\begin{split}Q(z)&=\sum_{i=1}^{s-1}Q^{i,i+1}(z)~{},\\\
Q^{i,i+1}(z)&=\sum_{j=1}^{\mathrm{rk}\,\mathfrak{g}}(\rho_{i}(P_{j})-\rho_{i+1}(\tau(P_{j})))\rho_{gh,i}(c^{j})(z)~{},\end{split}$
(59)

where $\rho_{i}$ represents the action of $\mathfrak{z}(\mathfrak{g})$ on the
$i$-th factor of $\mathbf{W}$ and $\rho_{gh,i}(c^{j})$ acts on the $i$-th
factor of the ghost system
$\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}(\mathfrak{z}(\mathfrak{g}))$.
The vertex algebra of a sphere of type $\mathfrak{g}$ with $s$ maximal
punctures is then defined to be

$\begin{split}\mathbf{V}_{G,1}&=\mathbf{W}~{},\\\
\mathbf{V}_{G,s}&=\mathrm{H}^{\frac{\infty}{2}+0}(\bigoplus_{i=1}^{s-1}{\cal
Z}^{i,i+1},\mathbf{W}^{s})~{}.\end{split}$ (60)

It is a lemma of Arakawa:2018egx , that this agrees with the recursive
construction of the genus zero vertex algebras.

As a consequence of their construction in terms of the cylinder VOA, the
vertex algebras $\mathbf{V}_{G,s}$ are endowed with an increasing filtration
Arakawa:2018egx , ${\cal F}^{\bullet}\mathbf{V}_{G,s}$ such that ${\cal
F}^{0}\mathbf{V}_{G,s}=0$ and

$\frac{{\cal F}^{i}\mathbf{V}_{G,s}}{{\cal
F}^{i-1}\mathbf{V}_{G,s}}\cong\mathbb{V}_{s,\lambda}~{},$ (61)

for some integral, dominant weight $\lambda$. They further possess a
decreasing filtration ${\cal G}^{\bullet}\mathbf{V}_{G,s}$, with ${\cal
G}^{0}\mathbf{V}_{G,s}=\mathbf{V}_{G,s}$ and successive quotients isomorphic
to $D(\mathbb{V}_{s,\lambda})$ for some integral dominant $\lambda$. This
means that the genus zero chiral algebras are limits of projective and
injective objects in $\mathrm{KL}$, and so, importantly, satisfy the
requirements of Theorem 3.2. This means that when gluing these VOAs together
via gauge theory BRST, the cohomology
$\mathrm{H}^{\frac{\infty}{2}+\bullet}(\hat{\mathfrak{g}}_{-\kappa_{g}},\mathfrak{g},\mathbf{V}_{G,s}\otimes\mathbf{V}_{G,s^{\prime}})$
is concentrated in degree zero, and via a spectral sequence argument this
establishes the associativity of gauging for the genus zero chiral
algebras.111111Furthermore, all cohomologies being concentrated in degree zero
is compatible with the expectation that in genus zero there is no residual
gauge symmetry on the Higgs branch, and so no Hall–Littlewood chiral ring
beyond the Higgs chiral ring.

An interesting feature of this construction is that it trivialises
$S$-duality. The caps involved in the construction are identical and the
gluing is done simultaneously. The $\mathbf{V}_{G,s}$ are manifestly invariant
under permutations. Since the construction makes no reference to a choice of
pants decomposition, there is a disconnect to the physical chiral algebras of
class $\SS$—built out of the BRST gauging of trinion algebras. Arakawa has
shown in Arakawa:2018egx  that the operations of FF-gluing and BRST gauging
are compatible, in the sense that the cap gluing construction agrees with the
pants decomposition.

##  4 Twisted trinions from mixed Feigin–Frenkel gluing

Arakawa’s construction is universal and is well defined for any semisimple
$\mathfrak{g}$—including non simply laced Lie algebras.121212The same is true
for the putative construction of Moore–Tachikawa symplectic varieties in
Ginzburg:2018 . At the same time, there is no construction for vertex algebras
that have both $\mathfrak{g}_{u}$ and $\mathfrak{g}_{t}$ symmetries (both
twisted and untwisted punctures). A calculation comparing the character of
Arakawa:2018egx  and the index of Lemos:2012ph  shows that the two vertex
algebras cannot be isomorphic; we show this in Appendix D. There are some
notable exceptions; namely, the index of the twisted cylinder and the
character of $\mathcal{D}^{ch}_{G_{t}}$ agree. Consequently, so must the index
of the twisted cap and the character of $\mathbf{W}_{G_{t}}$.131313Strictly
speaking, the characters of the cylinder and cap VOAs don’t exist due to
infinite-dimensional weight spaces. However, one can proceed formally by
working term-by-term in the sum over integral dominant weights; this can be
understood from a vertex algebra perspective as considering the decomposition
into blocks belonging to $\mathrm{KL}^{\lambda}$ for each $\lambda\in P^{+}$.
This suggests that $\mathcal{D}^{ch}_{G_{t}}$ and $\mathbf{W}_{G_{t}}$ will be
key building blocks for our construction; indeed, the algebras associated to
the sphere with two maximal twisted punctures and one maximal/one minimal
twisted puncture should be identified with $\mathcal{D}^{ch}_{G_{t}}$ and
$\mathbf{W}_{G_{t}}$, respectively.

In this section, we posit a construction for more general genus zero mixed
vertex algebras. Our proposal takes advantage of Feigin–Frenkel gluing to
define the mixed trinion with one untwisted and two twisted punctures. From
this, in principle, all other genus zero vertex algebras can be constructed.
Recalling our discussion of residual gauge symmetries in Section 2.4, we have
some expectations regarding the behaviour of these mixed vertex algebras under
gauging. In particular, unlike the untwisted case, we expect that the BRST
cohomology of the _untwisted_ gauging of two such vertex algebras will not be
concentrated in degree zero, but will rather enjoy some derived enhancement.

To establish such properties regarding untwisted and twisted gauging, we need
to examine the decomposition of the mixed vertex algebras into semijective
objects in both twisted and untwisted $\mathrm{KL}$ categories. This will be
depend in a crucial way on the structure of critical-level Weyl modules over
$\hat{\mathfrak{g}}_{t,\kappa_{c}}$ as modules over the untwisted
Feigin–Frenkel centre $\mathfrak{z}(\mathfrak{g}_{u})$. This will lead us to a
technical interlude involving opers.

In this section we assume that we are dealing with commuting twists, either
$\mathbb{Z}_{2}$ or $\mathbb{Z}_{3}$. We will remark on some peculiarities of
the $\mathbb{Z}_{3}$ case near the end of the section.

###  4.1 The (un)twisted Feigin–Frenkel centre and the mixed trinion

The mixed vertex algebras should simultaneously be vertex algebra objects in
$\mathrm{KL}_{u}$ and in $\mathrm{KL}_{t}$, so they will admit actions of both
Feigin–Frenkel centres. The construction of $\mathbf{V}_{G,s}$ suggests that
the action of these Feigin–Frenkel centres should be identified, but of course
the twisted and untwisted centres are not isomorphic. It will be useful,
therefore, to first examine how the actions of these two Feigin–Frenkel
centres interact with each other.

Let $\sigma\in\mathrm{Out}(\mathfrak{g}_{u})$ be an outer automorphism,
$\mathfrak{g}_{u}^{\sigma}$ be the $\sigma$-invariant subalgebra of
$\mathfrak{g}_{u}$, and
$\mathfrak{g}_{t}=\left(\mathfrak{g}_{u}^{\sigma}\right)^{\vee}$. There exists
a projection Fuchs1996 ,

$\pi_{\sigma}:\mathfrak{h}_{u}\twoheadrightarrow\mathfrak{h}_{t}~{},$ (62)

from the Cartan subalgebra of $\mathfrak{g}_{u}$ to that of $\mathfrak{g}_{t}$
that projects to elements that are invariant under $\sigma$. The outer
automorphism lifts to an automorphism of $U(\mathfrak{g}_{u})$ and we have a
surjection

$Z(U(\mathfrak{g}_{u}))\twoheadrightarrow Z(U(\mathfrak{g}_{t}))~{},$ (63)

which is just the projection of the centre of $U(\mathfrak{g}_{u})$ to its
$\langle\sigma\rangle$-coinvariants, i.e., we set the $\sigma$-non-invariant
generators of $Z(\mathfrak{g}_{u})$ to zero. The action of $\sigma$ can be
lifted to $\hat{\mathfrak{g}}_{u,\kappa_{c}}$ according to
$\sigma(xt^{n})=\sigma(x)t^{n}$. This gives a projection

${\cal Z}_{u}\twoheadrightarrow{\cal Z}_{t}\cong({\cal Z}_{u})_{\sigma}~{},$
(64)

where $({\cal Z}_{u})_{\sigma}$ is the space of
$\langle\sigma\rangle$-coinvariants of the untwisted Feigin–Frenkel centre.

The projection $\pi_{\sigma}$ also induces an embedding of weight spaces

$\iota:P^{+}_{t}\hookrightarrow P^{+}_{u}~{},$ (65)

with image $\iota(P_{t}^{+})$ equal to the subset of elements in $P^{+}_{u}$
that are invariant under the action of $\sigma$. For example if
$\mathfrak{g}_{u}=D_{n}$ and $\mathfrak{g}_{t}=C_{n-1}$ (so $\sigma$ is the
generator of the $\mathbb{Z}_{2}$ outer automorphism), we have

$\iota(\lambda_{1},\lambda_{2},\dots,\lambda_{n-1})=(\lambda_{1},\lambda_{2},\dots,\lambda_{n-1},\lambda_{n-1})~{}.$
(66)

In Lemos:2012ph , this was indicated with the notation
$\lambda^{\prime}=\lambda$. We will abuse notation and use $\lambda$ for both
the weight in $P_{t}^{+}$ and its image under $\iota:P_{t}^{+}\hookrightarrow
P_{u}^{+}$. So $V_{\lambda}^{t}$ denotes the finite dimensional irreducible
representation of $\mathfrak{g}_{t}$ with highest weight $\lambda$, and
$V^{u}_{\lambda}$ is the finite dimensional irreducible representation of
$\mathfrak{g}_{u}$ with highest weight $\iota(\lambda)$.

Given an object $M\in{\cal Z}_{t}$-Mod, we can lift it to a module in ${\cal
Z}_{u}$-Mod via the restriction of scalars associated to ${\cal
Z}_{u}\twoheadrightarrow{\cal Z}_{t}$, giving a functor ${\cal
Z}_{t}\text{-Mod}\rightarrow{\cal Z}_{u}\text{-Mod}$. The mixed vertex
algebras must simultaneously be objects in the $\mathrm{KL}$ categories of
$\mathfrak{g}_{u}$ and $\mathfrak{g}_{t}$. In the case where there are no
untwisted punctures, the algebra must still have a ${\cal Z}_{u}$-Mod
structure. This is because any such algebra can be obtained via DS reduction
of a vertex algebra object in $\mathrm{KL}_{u}$, and the DS reduction functor
lands in ${\cal Z}_{u}$-Mod.

The first mixed object we would like to construct is the vertex algebra
corresponding to ${\cal C}_{0,1,1}$, which has two $\mathrm{KL}_{t}$ actions
and one $\mathrm{KL}_{u}$ action. The most natural guess for a construction of
this algebra would be to FF glue an untiwsted cap ($\mathbf{W}_{u}$) to the
twisted cylinder ($\mathcal{D}^{ch}_{t}$) via the untwisted Feigin–Frenkel
centre. The preceeding discussion assures us that $\mathcal{D}^{ch}_{t}$ can
be lifted from ${\cal Z}_{t}$-Mod to ${\cal Z}_{u}$-Mod, and as such we
propose that the mixed trinion associated to ${\cal C}_{0,1,1}$ is

$V_{1,1}\coloneqq\mathrm{H}^{\frac{\infty}{2}+0}({\cal
Z}_{u},\mathbf{W}_{u}\otimes\mathcal{D}^{ch}_{t})~{}.$ (67)

This mixed trinion is the most important vertex algebra of our construction.
Indeed, it is the basic building block of the TQFT along with the untwisted
trinion $\mathbf{V}_{G_{u},3}$. Before proceeding to more general surfaces, it
will be useful to establish the key properties of $\mathbf{V}_{1,1}$.

The genus zero vertex algebras, $\mathbf{V}_{G,s}$ have filtrations whose
succesive quotients are $\mathbb{V}_{\lambda,s}$ or
$D(\mathbb{V}_{\lambda,s})$, for some integral dominant $\lambda$. This is key
in establishing that $\mathbf{V}_{G,s}$ are semijective in $\mathrm{KL}_{u}$,
which in turn implies that gauging is concentrated in degree zero due to
Theorem 3.2. Such a filtration would be extremely useful in establishing the
properties of $\mathbf{V}_{1,1}$ under the various types of gluing.

Ideally, in the mixed case the successive quotients would simultaneously be
objects in $\mathrm{KL}_{u}$ and $\mathrm{KL}_{t}$, and the natural proposal
for such an object would be of the form

$\mathbb{V}_{\lambda^{\prime}}^{u}\otimes_{\mathfrak{z}^{u}_{\lambda^{\prime}}}\mathbb{V}_{\lambda}^{t}$
(68)

where $\lambda\in P_{t}^{+}$ and $\lambda^{\prime}\in P_{u}^{+}$. While it is
obvious that ${\cal Z}_{t}$ modules can be lifted to ${\cal Z}_{u}$ modules by
way of the projection ${\cal Z}_{u}\twoheadrightarrow{\cal Z}_{t}$, it is not
so obvious that this should hold for the quotient modules
$\mathfrak{z}^{u}_{\lambda^{\prime}}$, which are more complicated.
Furthermore, there is a question of how $\lambda$ and $\lambda^{\prime}$ are
related. Altogether, there is room for doubt over whether the suggested tensor
product over $\mathfrak{z}^{u}_{\lambda^{\prime}}$ is even well-defined. To
show that such a lift does indeed exist—and that the tensor product is well
defined—for special values of $\lambda^{\prime}$, we will work in the language
of opers.

$\mathfrak{g}_{u}$ | $\mathfrak{z}(\mathfrak{g}_{u})$ | $\mathfrak{g}_{t}$ | $\mathfrak{z}(\mathfrak{g}_{t})$
---|---|---|---
$\mathfrak{a}_{2n-1}$ | $P_{2},P_{3},\dots,P_{2n}$ | ${\mathfrak{b}}_{n}$ | $P_{2},P_{4},\dots,P_{2n}$
$\mathfrak{a}_{2n}$ | $P_{2},P_{3},\dots,P_{2n+1}$ | ${\mathfrak{c}}_{n}$ | $P_{2},P_{4},\dots,P_{2n}$
$\mathfrak{d}_{n}$ | $\tilde{P}_{n};P_{2},P_{4},\dots,P_{n},\dots,P_{2n-2}$ | ${\mathfrak{c}}_{n-1}$ | $P_{2},P_{4},\dots,P_{2n-2}$
$\mathfrak{e}_{6}$ | $P_{2},P_{5},P_{6},P_{8},P_{9},P_{12}$ | $\mathfrak{f}_{4}$ | $P_{2},P_{6},P_{8},P_{12}$
$\mathfrak{d}_{4}$ | $\tilde{P}_{4},P_{2},P_{4},P_{6}$ | $\mathfrak{g}_{2}$ | $P_{2},P_{6}$
Table 2: The monomial generators of the Feigin–Frenkel centres of the
untwisted algebra $\mathfrak{g}_{u}$ and its associated twisted algebra
$\mathfrak{g}_{t}$. Note that algebras of type $\mathfrak{d}_{n}$ have two
generators of degree $n$, only one of which is invariant under the outer
automorphism. For the $\mathfrak{d}_{4}$ case, neither generator of degree
four is invariant under the $\mathbb{Z}_{3}$ outer automorphism.

###  4.2 Opers and the Feigin–Frenkel centre

This subsection is largely self-contained and may be skipped over by the
reader who is less interested in these technical details. At the end of this
subsection, we formulate a theorem in the language of opers that implies a
lifting of $\mathfrak{z}_{t,\lambda}$ modules to
$\mathfrak{z}_{u,\lambda^{\prime}}$ modules. In the following subsection we
will move on to investigate the properties of $\mathbf{V}_{1,1}$.

We will not require the full machinery of opers (as can be defined on general
algebraic curves), but instead will proceed with only a selective review of
opers on the (punctured) disk with a choice of local coordinate. For more
details, we direct the reader to Frenkel:2004 ; Frenkel:2007 .

Let $D=\mathrm{Spec}\,\mathbb{C}[\\![t]\\!]$ be the formal disc and
$D^{\times}=\mathrm{Spec}\,\mathbb{C}(\\!(t)\\!)$ be the formal punctured
disc. Let $G$ be a semisimple algebraic Lie group with $Lie(G)=\mathfrak{g}$,
and $B\subset G$ a Borel subgroup of $G$ with $Lie(B)=\mathfrak{b}$. A
$G$-oper on $D$ is the triple,

$(\mathscr{F},\nabla,\mathscr{F}_{B})~{},$ (69)

where $\mathscr{F}$ is a principal $G$-bundle on $D$, $\nabla$ is a
$G$-connection on $\mathscr{F}$, and $\mathscr{F}_{B}$ is a reduction of
$\mathscr{F}$ to the Borel subgroup $B$ such that $\mathscr{F}_{B}$ is
transversal to $\nabla$. A $G$-oper on $D^{\times}$ is defined analogously.

The space of opers on the disc can be made more concrete as follows. Let
$\Lambda_{p}(h)$ be the image of the Cartan generator of $\mathfrak{sl}_{2}$
under the principal embedding,
$\Lambda_{p}:\mathfrak{sl}_{2}\rightarrow\mathfrak{g}$. This defines gradings

$\mathfrak{g}=\bigoplus_{i}\mathfrak{g}_{i}~{},\quad\mathfrak{b}=\bigoplus_{i}\mathfrak{b}_{i}~{},$
(70)

on $\mathfrak{g}$ and $\mathfrak{b}$ by eigenvalues of
$\mathrm{ad}\,\Lambda_{p}(h)$. Let $p_{-1}=\Lambda_{p}(f)$ be the image of the
$\mathfrak{sl}_{2}$ lowering operator and let $p_{1}=\Lambda_{p}(e)$ be the
image of the raising operator. We define $V^{can}$ to be the subspace of
$\mathfrak{n}=\bigoplus_{i>0}\mathfrak{b}_{i}$ that is invariant under
$p_{1}$, i.e., it is the subspace spanned by highest weight vectors with
respect to the embedded $\mathfrak{sl}_{2}$. The subspace has the
decomposition

$V^{can}=\bigoplus_{d_{i}}V_{d_{i}}^{can}~{},$ (71)

where $d_{i}$ are the exponents of $\mathfrak{g}$. Let $\\{p_{i}\\}$ be a
basis of $V^{can}$ such that $p_{i}\in V^{can}_{d_{i}}$.141414In the case of
$\mathfrak{d}_{2n}$, the subspace $V^{can}_{2n}$ is two dimensional and we
have two linearly independent vectors $p_{2n}$ and $\tilde{p}_{2n}$. There is
a transparent similarity to the generators of $\mathfrak{z}(\mathfrak{g})$.

We denote the space of all such $G$-opers on the $D$ and $D^{\times}$ by
$\mathrm{Op}_{G}\,D$ and $\mathrm{Op}_{G}\,D^{\times}$, respectively. Let $t$
be a formal coordinate on $D$. By an appropriate gauge transformation, any
$G$-oper can be put into the canonical form Frenkel:2007 ,

$\nabla=d+p_{-1}+\sum_{i=1}^{\mathrm{rk}\,\mathfrak{g}}v_{i}(t)p_{i}~{},$ (72)

where $v_{i}(t)\in\mathbb{C}[\\![t]\\!]$ or $v_{i}(t)\in\mathbb{C}(\\!(t)\\!)$
for $D$ or $D^{\times}$, respectively. A celebrated theorem of Feigin and
Frenkel states that Feigin:1984 ,

$\mathfrak{z}(\mathfrak{g})\cong\mathrm{Fun}(\mathrm{Op}_{{}^{L}G}\,D)~{},$
(73)

and

${\cal Z}\cong\mathrm{Fun}(\mathrm{Op}_{{}^{L}G}\,D^{\times})~{},$ (74)

where ${}^{L}G$ is the Langlands dual group of $G$.

The Feigin–Frenkel centre acts on the Weyl modules $\mathbb{V}_{\lambda}$, and
this action factors through a certain quotient of the algebra of functions on
opers Frenkel:2010 . Namely,

${\cal
Z}\xrightarrow{\sim}\mathrm{Fun}\,\mathrm{Op}_{{}^{L}G}(D^{\times})\twoheadrightarrow\mathrm{Fun}\,\mathrm{Op}_{{}^{L}G}^{\lambda}\cong\mathrm{End}_{\hat{\mathfrak{g}}_{\kappa_{c}}}(\mathbb{V}_{\lambda})~{}.$
(75)

The quotient arises as follows. The subscheme
$\mathrm{Op}_{{}^{L}G}(D)^{\lambda,reg}\subset\mathrm{Op}_{{}^{L}G}(D)$ is
that of opers on the disc with _regular singularity, with residue $\lambda$,
and with no monodromy_ (see Frenkel:2007  for more details). The subscheme
$\mathrm{Op}_{{}^{L}G}^{\lambda}\subset\mathrm{Op}_{{}^{L}G}(D^{\times})$ is
the image of $\mathrm{Op}_{{}^{L}G}(D)^{\lambda,reg}$ under the natural
inclusion

$\mathrm{Op}_{{}^{L}G}(D)\hookrightarrow\mathrm{Op}_{{}^{L}G}(D^{\times})~{}.$
(76)

The kernel of the projection (75) onto the opers without monodromy is the
annihilator ideal

$I_{\lambda}=\mathrm{Ann}_{\mathfrak{z}}\mathbb{V}_{\lambda}~{}.$ (77)

Thus, we have

$\mathfrak{z}_{\lambda}\coloneqq{\cal
Z}_{<0}/I_{\lambda}\cong\mathrm{Fun}\,\mathrm{Op}_{{}^{L}G}^{\lambda}~{}~{},$
(78)

where $\mathfrak{z}_{\lambda}$ is the quotient we introduced in (25).

The particulars of our case lead to some simplifications. We take
${}^{L}G_{u}$ to be the Langlands dual of the simply connected Lie group,
$G_{u}$, with Lie algebra $\mathfrak{g}_{u}$. Since $\mathfrak{g}_{u}$ is
simply laced, $\mathrm{Lie}(^{L}G_{u})\cong\mathfrak{g}_{u}$.

From our discussion in Section 4.1, we have the following projection

$\mathrm{Fun\,Op}_{{}^{L}G_{u}}(D^{\times})\twoheadrightarrow(\mathrm{Fun\,Op}_{{}^{L}G_{u}}(D^{\times}))_{\sigma}\cong\mathrm{Fun\,Op}_{{}^{L}G_{t}}(D^{\times})~{},$
(79)

which gives a closed embedding of the schemes

$\mathrm{Op}_{{}^{L}G_{t}}(D^{\times})\cong(\mathrm{Op}_{{}^{L}G_{u}}(D^{\times}))^{\sigma}\hookrightarrow\mathrm{Op}_{{}^{L}G_{u}}(D^{\times})~{},$
(80)

where $(\mathrm{Op}_{{}^{L}G_{u}}(D^{\times}))^{\sigma}$ is the spectrum of
$(\mathrm{Fun\,Op}_{{}^{L}G_{u}}(D^{\times}))_{\sigma}$. This is nothing more
than the surjection ${\cal Z}_{u}\twoheadrightarrow{\cal Z}_{t}$ of Section
4.1, rephrased in the language of opers.

To restrict this surjection to the no-monodromy opers, it is necessary that
$\lambda^{\prime}$ be outer automorphism invariant, specifically
$\lambda^{\prime}=\iota(\lambda)$. We will continue to abuse notation and
denote $\iota(\lambda)$ by $\lambda$. We then have the following.

######  Theorem 4.1.

The restriction of the closed immersion

$\mathrm{Op}_{{}^{L}G_{t}}(D^{\times})\hookrightarrow\mathrm{Op}_{{}^{L}G_{u}}(D^{\times})~{}.$

to the subscheme $\mathrm{Op}_{{}^{L}G_{t}}^{\lambda}$ factors as

$\mathrm{Op}_{{}^{L}G_{t}}^{\lambda}\hookrightarrow\mathrm{Op}_{{}^{L}G_{u}}^{\lambda}\hookrightarrow\mathrm{Op}_{{}^{L}G_{u}}(D^{\times})~{},$

with each map a closed immersion. Equivalently, the natural surjection

$\mathrm{Fun\,Op}_{{}^{L}G_{u}}(D^{\times})\twoheadrightarrow(\mathrm{Fun\,Op}_{{}^{L}G_{u}}(D^{\times}))_{\sigma}~{},$

restricts to a surjection

$\mathrm{Fun\,Op}^{\lambda}_{{}^{L}G_{u}}\twoheadrightarrow(\mathrm{Fun\,Op}^{\lambda}_{{}^{L}G_{u}})_{\sigma}~{},$

on the quotient algebras.

We present a proof of this theorem that makes use of Miura opers and Cartan
connections in Appendix B. The theorem amounts precisely to the statement that
there is a surjection
$\mathfrak{z}^{u}_{\lambda}\twoheadrightarrow\mathfrak{z}^{t}_{\lambda}$, and
this will be crucial in establishing that our construction of mixed vertex
algebras of class $\SS$ makes sense and that the vertex algebras so defined
satsify certain desirable properties.

###  4.3 Properties of the mixed trinion

In what follows we will make use of the following technical proposition.

######  Proposition 4.2.

Let $N$ be a $\mathfrak{z}^{t}_{\lambda}$-module. Then $N$ is an object of
$KL_{u,0}$, i.e., there exists some $M\in\mathrm{KL}_{u}$ such that

$N=H^{0}_{DS}(u,M)~{}.$

###### Proof.

We prove this by explicitly constructing an object in $\mathrm{KL}_{u}$ whose
DS reduction is isomorphic to $N$ as an object of
$\mathfrak{z}^{t}_{\lambda}$-mod. From Theorem 4.1, the
$\mathfrak{z}^{t}_{\lambda}$ action can be lifted to an action of
$\mathfrak{z}^{u}_{\lambda}$.

Let $\mathbb{V}_{\lambda}^{u}$ be the Weyl module of
$\hat{\mathfrak{g}}_{u,\kappa_{c}}$ with highest weight $\iota(\lambda)$. The
tensor product,

$\mathbb{V}_{\lambda}^{u}\otimes_{\mathfrak{z}_{\lambda}^{u}}N~{},$

is well-defined, where the $\mathfrak{z}_{\lambda}^{u}$ action on $N$ is from
the lift. By construction, this is an object in $\mathrm{KL}_{u}$ with respect
to the $\hat{\mathfrak{g}}_{u,\kappa_{c}}$ action on the untwisted Weyl
module. Let us consider its DS reduction,

$H^{0}_{DS}(u,\mathbb{V}_{\lambda}^{u}\otimes_{\mathfrak{z}_{\lambda}^{u}}N)~{}.$

By way of the Künneth theorem, noting that
$H^{0}_{DS}(\mathbb{V}_{\lambda})\cong\mathfrak{z}_{\lambda}^{u}$ Frenkel:2010
is manifestly free over itself, we have that

$H^{0}_{DS}(u,\mathbb{V}_{\lambda}^{u}\otimes_{\mathfrak{z}_{\lambda}^{u}}N)\cong\mathfrak{z}^{u}_{\lambda}\otimes_{\mathfrak{z}_{\lambda}^{u}}N\cong
N~{},$

as desired. ∎

With this in hand we can establish an important theorem concerning the DS
reduction of our mixed trinion vertex algebra.

######  Theorem 4.3.

We have the following isomorphism:

$H^{0}_{DS}(u,\mathbf{V}_{1,1})\cong\mathcal{D}^{ch}_{t}~{},$

so $\mathcal{D}^{ch}_{t}\in\mathrm{KL}_{u,0}$.

This is the statement that the mixed vertex algebra we have constructed can
indeed be identified with the UV curve ${\cal C}_{0,1,1}$ insofar as closing
the maximal untwisted puncture results in the cylinder of type
$\mathfrak{g}_{t}$. The proof of the above theorem is not entirely
straightforward because $H^{0}_{DS}(u,\mathrm{H}^{\frac{\infty}{2}+0}({\cal
Z}_{u},\mathbf{W}_{u}\otimes-))$ is not necessarily the identity on a generic
object in ${\cal Z}_{u}$-Mod. The full proof of the the theorem is relegated
to Appendix C; here we provide a sketch.

The proof proceeds by first establishing that at the level of formal
characters,

${\rm ch}~{}H^{0}_{DS}(u,\mathbf{V}_{1,1})_{\lambda}\leqslant{\rm
ch}~{}\mathcal{D}^{ch}_{t,[\lambda]}~{}.$ (81)

In words, each weight space (with fixed generalised eigenvalue under the
action of the Feigin–Frenkel zero modes) of $H^{0}_{DS}(u,\mathbf{V}_{1,1})$
is of dimension less than or equal to that of the corresponding weight space
of $\mathcal{D}^{ch}_{t}$. This is argued by leveraging the fact that
$\mathcal{D}^{ch}_{t}$ has an increasing filtration with subquotients
$\mathbb{V}^{t}_{\lambda,2}$, which are in $\mathrm{KL}_{u,0}$ by Proposition
4.2. We show that passing (in a careful sense) to the associated graded of
this filtration can only increase the dimensions of the weight spaces, and on
the associated graded the composition of FF gluing and DS reduction acts as
the identity; this leads to (81). Since $\mathcal{D}^{ch}_{t}$ is simple, we
need only construct a non-zero homomorphism $\mathcal{D}^{ch}_{t}\rightarrow
H^{0}_{DS}(u,\mathbf{V}_{1,1})$ to establish the isomorphism. The construction
of such a homomorphism follows an adaptation of the proof of Theorem $9.9$ of
Arakawa:2018egx  to this twisted setting.

Theorem 4.3 will serve as the foundation which lets us build up a number of
other important properties of the genus zero mixed vertex algebras.

######  Proposition 4.4.

The mixed trinion $\mathbf{V}_{1,1}$ is semijective in $\mathrm{KL}_{t}$.

###### Proof.

The cylinder $\mathcal{D}^{ch}_{t}$ has an increasing filtration frenkel2004 ,

$0=N_{0}\subset N_{1}\subset N_{2}\subset\dots~{},\quad N=\bigcup
N_{i}\cong\mathcal{D}^{ch}_{t}~{},$

whose successive quotients take the form

$N_{i}/N_{i-1}\cong\mathbb{V}_{\lambda}^{t}\otimes_{\mathfrak{z}^{t}_{\lambda}}\mathbb{V}_{\lambda^{*}}^{t}~{},$

for some $\lambda\in P_{t}^{+}$ and $\lambda^{*}$ the dual representation.
Each $N_{i}/N_{i-1}$ is an object in $\mathfrak{z}_{\lambda}$, so Proposition
4.2 applies and $N_{i}/N_{i-1}\in\mathrm{KL}_{u,0}$. Applying our Theorem 4.3
and Theorem $9.14$ of Arakawa:2018egx , the mixed vertex algebra
$\mathbf{V}_{1,1}\cong\mathbf{W}_{u}\ast_{u}\mathcal{D}^{ch}_{t}$ has an
increasing filtration

$0=M_{0}\subset M_{1}\subset M_{2}\subset\dots~{},\quad M=\bigcup
M_{i}\cong\mathbf{V}_{1,1}~{},$

with successive quotients $M_{i}/M_{i-1}$ isomorphic to
$\mathrm{H}^{\frac{\infty}{2}+0}({\cal Z}_{u},\mathbf{W}_{u}\otimes
N_{i}/N_{i+1})$. But from Proposition 4.2, $N_{i}/N_{i-1}$ is in
$\mathrm{KL}_{u,0}$ with $N_{i}/N_{i-1}\cong
H^{0}_{DS}(u,\mathbb{V}^{u}_{\lambda}\otimes_{\mathfrak{z}^{u}_{\lambda}}N_{i}/N_{i-1})$.
Thus

$M_{i}/M_{i-1}\cong\mathbb{V}_{\lambda}^{u}\otimes_{\mathfrak{z}^{u}_{\lambda}}\left(\mathbb{V}^{t}_{\lambda}\otimes_{\mathfrak{z}^{t}_{\lambda}}\mathbb{V}_{\lambda^{*}}^{t}\right)~{}.$

The action of $U(t^{-1}\mathfrak{g}_{t}[t^{-1}])$ on either of the Weyl
modules $\mathbb{V}^{t}_{\lambda,\lambda^{*}}$ is projective, so
$\mathbf{V}_{1,1}$ is projective over $U(t^{-1}\mathfrak{g}_{t}[t^{-1}])$. To
establish that $\mathbf{V}_{1,1}$ is injective over $U(t\mathfrak{g}_{t}[t])$,
we can repeat the same argument after taking $(\mathcal{D}^{ch}_{t})^{op}$ and
using the identification

$(\mathcal{D}^{ch}_{t})^{op}\cong\mathcal{D}^{ch}_{t}~{}.$

∎

We observe that $\mathbf{V}_{1,1}$ is not semijective in $\mathrm{KL}_{u}$.
Inituitively, this is because the extra generators of the Feigin–Frenkel
centre must be set to zero when glued to the twisted cylinder, and these
relations spoil projectivity. More precisely, we consider the vacuum vector
$|{0}\rangle$. Any state element $P_{i,-n}|{0}\rangle$ can be written as
$\phi|{0}\rangle$ for some $\phi\in U(t^{-1}\mathfrak{g}_{u}[t^{-1}])$ a
regular element. However, the modes $P_{i,-n}$, which are not invariant under
$\sigma$, must act as zero.As there are regular elements in
$U(t^{-1}\mathfrak{g}_{u}[t^{-1}])$ which act as zero, $\mathbf{V}_{1,1}$
cannot be torsion free—so cannot be projective—over
$U(t^{-1}\mathfrak{g}_{u}[t^{-1}])$.

The semijectivity of $\mathbf{V}_{1,1}$ in $\mathrm{KL}_{t}$ is in accordance
with our expectations regarding enhanced Higgs branches/residual gauge
symmetries. The twisted class $\SS$ theories for surfaces ${\cal C}_{0,m,1}$
formed by gluing ${\cal C}_{0,1,1}$ along twisted punctures have no residual
gauge symmetry, so the gauge theory gluing $\mathbf{V}_{1,1}\circ_{t}-$ should
be concentrated in cohomological degree zero. We have just established this
for our $\mathbf{V}_{1,1}$ algebra by showing semijiectivity in
$\mathrm{KL}_{t}$. On the other hand, gluing along the untwisted puncture may
lead to a higher-genus local system covering space (cf. 2.4), which falls in
line with our observation that $\mathbf{V}_{1,1}$ is not semijective in
$\mathrm{KL}_{u}$.

Having shown that $\mathbf{V}_{1,1}$ has the expected properties under
$\circ_{t}$ and $\circ_{u}$, we move on to some more intrinsic properties of
$\mathbf{V}_{1,1}$. Though our construction ensures that $\mathbf{V}_{1,1}$ is
a vertex algebra, it is not at all clear that it has the properties expected
from four-dimensional unitarity. Namely, $\mathbf{V}_{1,1}$ must be a conical,
conformal vertex algebra with negative central charge. Let us first address
the issue of the character.

######  Proposition 4.5.

The character of the vertex algebra $\mathbf{V}_{1,1}$ is given by

$\mathrm{Tr}_{\mathbf{V}_{1,1}}(q^{L_{0}}\mathbf{a}\,\mathbf{b}_{1}\mathbf{b}_{2})=\sum_{\lambda\in
P_{t}^{+}}\frac{{\cal K}_{u}(\mathbf{a})\chi^{\lambda}_{u}(\mathbf{a}){\cal
K}_{t}(\mathbf{b}_{1})\chi_{t}^{\lambda}(\mathbf{b}_{1}){\cal
K}_{t}(\mathbf{b}_{2})\chi_{t}^{\lambda}(\mathbf{b}_{2})}{{\cal
K}_{u}(\times)\chi_{u}^{\lambda}(\times)}~{},$

where $\mathbf{a}$ is a $G_{u}$ fugacity and the $\mathbf{b}_{i}$ are $G_{t}$
fugacities. Furthermore, $\mathbf{V}_{1,1}$ is conical.

###### Proof.

By Theorem 4.3, we have that

$H^{0}_{DS}(u,\mathbf{V}_{1,1})\cong\mathcal{D}^{ch}_{t}~{}.$

As a graded vector space $\mathbf{V}_{1,1}\cong\bigoplus_{\lambda\in
P_{u}^{+}}\mathbf{V}_{1,1,\lambda}$, and by Proposition $8.4$ of
Arakawa:2018egx .

$\mathrm{ch}\mathbf{V}_{1,1,\lambda}=q^{\lambda(\rho^{\vee})}\mathrm{ch}\,\mathbb{L}_{\lambda}\mathrm{ch}H^{0}_{DS}(u,\mathbf{V}_{1,1,\lambda})~{}.$

From the structure of the cylinder, we know that
$H^{0}_{DS}(u,\mathbf{V}_{1,1,\lambda})$ is zero unless it is in the image of
$\iota:P_{t}^{+}\hookrightarrow P_{u}^{+}$. Therefore, we have

$\mathrm{ch}\,\mathbf{V}_{1,1,\lambda}=q^{\lambda(\rho^{\vee})}\mathrm{ch}\,\mathbb{L}^{u}_{\lambda}\mathrm{ch}\mathbb{V}^{t}_{\lambda}\otimes_{\mathfrak{z}^{t}_{\lambda}}\mathbb{V}_{\lambda^{*}}^{t}~{}.$

Recalling Section 2.3 and the appendix of Lemos:2014lua , we can rewrite this
in the notation of ${\cal K}$-factors, giving the desired result.

To show that $\mathbf{V}_{1,1}$ is conical, note that the cylinder,
$\mathcal{D}^{ch}_{t}$, is non-negatively graded and
$\lambda(\rho^{\vee})\geqslant 0$ since it is integral dominant, with equality
only for $\lambda=0$. This establishes that $\mathbf{V}_{1,1}$ is non-
negatively graded by weight. The character ${\rm
ch}~{}\mathbf{V}_{1,1,0}=1+\dots$ since $\mathbb{L}_{\lambda=0}^{u}$ and
$\mathcal{D}^{ch}_{t,\lambda=0}$ are both conical. Thus the mixed trinion is
conical. ∎

######  Proposition 4.6.

The vertex algebra $\mathbf{V}_{1,1}$ is conformal with central charge

$c_{\mathbf{V}_{1,1}}=2{\rm dim}\,\mathfrak{g}_{t}+{\rm
dim}\mathfrak{g}_{u}-\mathrm{rk}\mathfrak{g}_{u}-24\rho_{u}\cdot\rho_{u}^{\vee}~{}.$

###### Proof.

This proof relies on ideas from the proof of Proposition $10.7$ of
Arakawa:2018egx , but requires modifications. The vertex algebras
$\mathbf{W}_{u}$, $\mathcal{D}^{ch}_{t}$, and the ghost system
$\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}(\mathfrak{z}(\mathfrak{g}_{u}))$
are all conformal, and we denote their respective conformal vectors by
$\omega_{\mathbf{W}}$, $\omega_{\mathcal{D}^{ch}}$ and $\omega_{gh}$. Clearly,
$\omega=\omega_{\mathbf{W}}+\omega_{\mathcal{D}^{ch}}+\omega_{gh}$ is a
conformal vector for the complex,
$\mathbf{W}_{u}\otimes\mathcal{D}^{ch}_{t}\otimes\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}(\mathfrak{z}(\mathfrak{g}_{u}))$.
We write

$\omega(z)=\sum_{m\in\mathbb{Z}}L_{m}z^{-m-1}~{},$

for the associated field.

By Lemma $9.4$ of Arakawa:2018egx , the Feigin–Frenkel centre of
$\mathbf{W}_{u}$ is preserved by the action of $L_{m}$ for $m\geqslant-1$. For
a generator $P_{i}\in\mathfrak{z}(\mathfrak{g}_{u})$,

$\omega(z)P_{i}(w)\sim\frac{\partial
P_{i}}{z-w}+\frac{(d_{i}+1)P_{i}}{(z-w)^{2}}+\sum_{j=2}^{d_{i}+2}\frac{(-1)^{j}j!}{(z-w)^{j+1}}q_{j}^{(i)}(w)~{},$

where $q_{j}^{(i)}$ is some homogeneous state in
$\mathfrak{z}(\mathfrak{g}_{u})$ with weight $d_{i}-j+2$. Let us denote by
$\widetilde{P_{i}}$ the image of $P_{i}$ under the projection
$\mathfrak{z}(\mathfrak{g}_{u})\twoheadrightarrow\mathfrak{z}(\mathfrak{g}_{t})$.
One then has

$\omega(z)\widetilde{P_{i}}(w)\sim\frac{\partial\widetilde{P_{i}}}{z-w}+\frac{(d_{i}+1)\widetilde{P_{i}}}{(z-w)^{2}}+\sum_{j=2}^{d_{i}+2}\frac{(-1)^{j}j!}{(z-w)^{j+1}}\widetilde{q_{j}}^{(i)}(w)~{},$

where we think of $\widetilde{P_{i}}$ as a state in
$\mathfrak{z}(\mathfrak{g}_{t})\subset\mathcal{D}^{ch}_{t}$. Let $Q(z)$ be the
BRST differential for Feigin–Frenkel gluing. We have that

$Q_{(0)}(z)\omega(w)=\sum_{i=1}^{\mathrm{rk}\,\mathfrak{g}}\sum_{j=2}^{d_{i}+1}\partial^{j}\big{(}\rho_{\mathbf{W}}(q_{j}^{(i)})-\rho_{\mathcal{D}^{ch}_{t}}(\tau(q_{j}^{(i)}))\big{)}c_{i}~{},$

where $\rho_{W}:\mathfrak{z}(\mathfrak{g}_{u})\hookrightarrow\mathbf{W}_{u}$
and
$\rho_{\mathcal{D}^{ch}_{t}}:\mathfrak{z}(\mathfrak{g}_{u})\hookrightarrow\mathcal{D}^{ch}_{t}$
denote the action of the untwisted Feigin-Frenkel centres on $\mathbf{W}_{u}$
and on $\mathcal{D}^{ch}_{t}$ via the projection to
$\mathfrak{z}(\mathfrak{g}_{t})$. Unfortunately, $\omega$ does not descend
directly to cohomology, so correction terms must be introduced to construct a
putative conformal vector in cohomology.

If the right hand side of the above equation equals $Q_{(0)}\chi$ for some
state $\chi$, then $\widetilde{\omega}=\omega+\chi$ is $Q$-closed and defines
a vector in $\mathbf{V}_{1,1}$. to show that such a $\chi$ exists, it is
sufficient to show that $\widetilde{q_{j}}^{(i)}=\pi(q_{j}^{(i)})$.

The action of $L_{m}$ for $m\geqslant-1$ on $\mathfrak{z}(\mathfrak{g}_{u})$
is given by the action of $\mathrm{Der}({\cal O}_{D})$ on
$\mathrm{Op}_{{}^{L}G_{u}}(D)$, which correspond to infinitesmal coordinate
changes on the formal disc Frenkel:2007 . We postpone the full argument to
Appendix B for brevity, but by (173) the action of $\mathrm{Der}({\cal
O}_{D})$ intertwines and $\widetilde{q_{j}}^{(i)}=\pi(q_{j}^{(i)})$. Thus
$\widetilde{\omega}\in\mathbf{V}_{1,1}$.

Now, we wish to show that $\widetilde{\omega}$ is a conformal vector. The
vector $\chi$ can be written as

$\chi=\sum_{i=1}^{\mathrm{rk}\,\mathfrak{g}_{u}}\sum_{j=2}^{d_{i}+2}\partial^{j}(\rho_{\mathbf{W}}\otimes\rho_{\mathcal{D}^{ch}_{t}}\otimes\rho_{gh})(z_{ij})~{},$

for some
$z_{ij}\in\mathfrak{z}(\mathfrak{g}_{u})\otimes\mathfrak{z}(\mathfrak{g}_{u})\otimes\mathrm{\bigwedge}\,^{\frac{\infty}{2}+0}(\mathfrak{z}(\mathfrak{g}_{u}))$.
Therefore, $\widetilde{\omega}_{(i)}=\omega_{(i)}$ for $i=0,1$, so the OPEs
agree up to the quadratic pole. Since $\mathbf{V}_{1,1}$ is non-negatively
graded by Proposition 4.5, Lemma 3.1.2 of Frenkel:2007  says that all we need
to check is that $\widetilde{\omega}_{(3)}\widetilde{\omega}=c/2|{0}\rangle$
for some $c\in\mathbb{C}$., i.e., the quartic pole in the $Vir\times Vir$ OPE
is a multiple of the identity. However, as $\mathbf{V}_{1,1}$ is conical, the
only operator of dimension zero that can appear in the OPE is the identity.
Thus, $\widetilde{\omega}$ is a conformal vector of $\mathbf{V}_{1,1}$.

Finally, we wish to show that $\widetilde{\omega}$ and $\omega$ have the same
central charge in cohomology. Note that
$\mathbf{V}_{1,1}=\sum_{\Delta\in\mathbb{N}}\mathbf{V}_{1,1}^{\Delta}$ with
${\rm dim}\,\mathbf{V}_{1,1}^{\Delta}<\infty$ and is conical—so
$\mathbf{V}_{1,1}$ is of CFT type. As a result, Lemma 4.1 of Moriwaki_2020
applies. Namely, for any $x\in\mathbf{V}_{1,1}$, if $\Delta(x)\geqslant 2$ and
$x_{(0)}\widetilde{\omega}=0$ then
$x\in\mathrm{im}_{\widetilde{\omega}_{(-1)}}(\mathbf{V}_{1,1})$.

Now suppose $\omega^{\prime}$ was some other conformal vector in
$\mathbf{V}_{1,1}$, such that $\omega^{\prime}_{(1)}$ agrees with
$\widetilde{\omega}_{(1)}$. Then, we have that
$(\omega^{\prime}-\widetilde{\omega})_{(0)}\widetilde{\omega}=0$, so
$\omega^{\prime}-\widetilde{\omega}=\partial x$ for some
$x\in\mathbf{V}_{1,1}$ with $\Delta=1$. Since
$\widetilde{\omega}_{(i)}=\omega^{\prime}_{(i)}$ for $i=0,1$, we must have
that $\partial x_{(i=0,1)}=0$ so $x$ is central in
$\mathbf{V}_{1,1}^{\Delta=1}$. Repeating the argument with
$(\omega-\widetilde{\omega})_{(1)}$ and using the Borcherds identities, leads
us to conclude that $\partial x=0$, so $\widetilde{\omega}$ is unique.

Under DS reduction, $H^{0}_{DS}(u,\mathbf{V}_{1,1})\cong\mathcal{D}^{ch}_{t}$
and the image of $\widetilde{\omega}$ gives rise to a conformal vector in
$\mathcal{D}^{ch}_{t}$. By a similar argument as the preceeding, one can show
that this is the unique conformal vector which agrees with the grading on
$\mathcal{D}^{ch}_{t}$ (see the proof of Proposition $10.7$ in Arakawa:2018egx
). The central charge of $\mathcal{D}^{ch}_{t}$ is
$c_{\mathcal{D}^{ch}_{t}}=2{\rm dim}\,\mathfrak{g}_{t}$ and the central charge
of the image of $\widetilde{\omega}$ is related to $c_{\mathbf{V}_{1,1}}$ by

$c_{\mathcal{D}^{ch}_{t}}=c_{\mathbf{V}_{1,1}}+\mathrm{rk}\,\mathfrak{g}_{u}-{\rm
dim}\mathfrak{g}_{u}+24\rho_{u}\cdot\rho^{\vee}_{u}~{},$

under DS reduction. ∎

With these propositions established, we know that $\mathbf{V}_{1,1}$ obeys
many of the desirable properties one would expect from the chiral algebra
associated to ${\cal C}_{0,1,1}$. However, we have not explicitly tied this
object to the construction of Beem:2013sza . One might reasonably wonder
whether the object we have constructed is necessarily the mixed trinion of
class $\SS$. We have the following uniqueness result.

######  Proposition 4.7.

The mixed trinion $\mathbf{V}_{1,1}$ is the unique vertex algebra object in
$\mathrm{KL}_{u}$ such that

$H^{0}_{DS}(u,\mathbf{V}_{1,1})\cong\mathcal{D}^{ch}_{t}~{}.$

###### Proof.

This follows easily from the fact that $\mathcal{D}^{ch}_{t}$ is an object in
$\mathrm{KL}_{u,0}$. Suppose $V\in\mathrm{KL}_{u}$ is a vertex algebra object
such that $H^{0}_{DS}(u,V)\cong\mathcal{D}^{ch}_{t}$. Then it must be the case
that $V\cong\mathbf{W}_{u}\ast_{u}\mathcal{D}^{ch}_{t}$, since
$\mathbf{W}_{u}\ast_{u}(H^{0}_{DS}(u,-))$ is the identity in
$\mathrm{KL}_{u}$. However,
$\mathbf{V}_{1,1}\coloneqq\mathbf{W}_{u}\ast_{u}\mathcal{D}^{ch}_{t}$, so
indeed $V\cong\mathbf{V}_{1,1}$. ∎

Suppose now that $\widetilde{\mathbf{V}}_{1,1}$ is the vertex algebra
canonically associated to ${\cal C}_{0,1,1}$ via the four-dimensional
construction of Beem:2013sza . This must be a vertex algebra object in
$\mathrm{KL}_{u}$, since it has an action of
$V^{\kappa_{c}}(\mathfrak{g}_{u})$ coming from the untwisted puncture and
inherits a suitable grading from the physical grading of superconformal
quantum numbers. Performing untwisted DS reduction on
$\tilde{\mathbf{V}}_{1,1}$ must produce the cylinder ${\cal C}_{0,0,1}$, which
has corresponding vertex algebra $\mathcal{D}^{ch}_{t}$.151515To see that one
must recover the cylinder upon DS reduction, even though this doesn’t
correspond to a physical four-dimensional theory, one may proceed as follows.
The physical vertex algebra $\widetilde{\mathbf{V}}_{1,1}$ must satisfy
$\widetilde{\mathbf{V}}_{1,1}\circ_{t}\widetilde{\mathbf{V}}_{1,1}$.
Performing DS reduction, we must have
$H^{0}_{DS}(u,\widetilde{\mathbf{V}}_{1,1}\circ_{t}\widetilde{\mathbf{V}}_{1,1})\cong\widetilde{\mathbf{V}}_{1,1}$.
By means of a spectral sequence argument (we delay this until the next
subsection) one can rearrange the order of cohomologies to show
$H^{0}_{DS}(u,\widetilde{\mathbf{V}}_{1,1})\circ_{t}\widetilde{\mathbf{V}}_{1,1}\cong\widetilde{\mathbf{V}}_{1,1}$,
which implies that
$H^{0}_{DS}(u,\widetilde{\mathbf{V}}_{1,1})=\mathcal{D}^{ch}_{t}$. Proposition
4.7 then applies, so we have

$\widetilde{\mathbf{V}}_{1,1}\cong\mathbf{W}_{u}\ast_{u}\mathcal{D}^{ch}_{t}\cong\mathbf{V}_{1,1}~{}.$
(82)

Thus far, this has been fairly abstract. Let us provide some concrete
observations and predictions. We have argued that the mixed trinion
$\mathbf{V}_{1,1}$, as we have constructed it, is the unique vertex algebra
that could be associated to ${\cal C}_{0,1,1}$. Let us consider the vertex
algebra associated to ${\cal C}_{f}$, which is a genus zero surface with one
maximal untwisted puncture, one maximal twisted puncture and an empty twisted
puncture. The corresponding vertex algebra is
$\mathbf{V}_{f}=H^{0}_{DS}(t,\mathbf{V}_{1,1})$.

The surface ${\cal C}_{f}$ does correspond to a physical SCFT, and in
particular, when $\mathfrak{g}_{u}=\mathfrak{d}_{n}$ and
$\mathfrak{g}_{t}=\mathfrak{c}_{n-1}$ the corresponding SCFT is a free
hypermultiplet theory (hence the subscript $f$). In this case,
$\mathbf{V}_{f}$ should be a symplectic boson vertex algebra with a commuting
$\hat{\mathfrak{g}}_{u,\kappa_{c}}\times\hat{\mathfrak{g}}_{t,\kappa_{c}}$
subalgebra. Unfortunately, this does not hold for the other choices of
$\mathfrak{g}_{u}$ (at generic rank). There is, however, another example that
has appeared in recent literature. The even rank $\mathfrak{a}$-type Lie
algebras $\mathfrak{g}_{u}=\mathfrak{a}_{2n}$ have, as their twisted algebras,
$\mathfrak{g}_{t}=\mathfrak{c}_{n}$—unlike the $\mathfrak{d}_{n}$ theories,
these SCFTs have global Witten anomalies. For
$\mathfrak{g}_{u}=\mathfrak{a}_{2}$ and
$\mathfrak{g}_{t}=\mathfrak{c}_{1}=\mathfrak{a}_{1}$, the SCFT is the ${\cal
T}_{X}$ theory of Buican:2017fiq . This was identified as the rank-two $H_{2}$
$F$-theory SCFT Beem:2019snk , and the class $\SS$ realisation was given in
Beem:2020pry . In the $\mathfrak{e}_{6}$ and $\mathfrak{a}_{2n+1}$ cases,
these vertex algebras remain unstudied to the best of our knowledge.

It is perhaps of technical interest to note that $\mathbf{V}_{f}$ should be
conical, despite the fact that it is obtained from the Feigin–Frenkel gluing
of just two caps. In the cases where the two caps are of the same type, one
obtains the cylinder $\mathcal{D}^{ch}_{u}$ or $\mathcal{D}^{ch}_{t}$—neither
of which are conical.

The identification of $\mathbf{V}_{f}$ with a symplectic boson system for
$\mathfrak{g}_{u}=\mathfrak{d}_{n}$ leads to a curious observation. Take the
symplectic boson system, $\SS{\cal B}$ associated to the bifundamental
representation of $\mathfrak{g}_{u}\times\mathfrak{g}_{t}$, its DS reduction
must be isomorphic to the DS reduction of $\mathbf{V}_{f}$. This implies that
for $\mathfrak{g}_{u}=\mathfrak{d}_{n}$,

$H^{0}_{DS}(u,\SS{\cal B})\cong
H^{0}_{DS}(u,\mathbf{V}_{f})\cong\mathbf{W}_{t}~{}.$ (83)

Indeed, this presents an alternate construction for the equivariant affine
${\cal W}$-algebra, $\mathbf{W}_{{\it cf.}_{n-1}}$. While we have not
established this isomorphism rigorously, we present some supporting evidence
for the lowest rank cases in Appendix E.

###  4.4 Rearrangment lemmas

Having established many key properties of the mixed trinion, which is the
building block of the twisted chiral algebras of class $\SS$, we would like to
extend our results to other genus zero, mixed vertex algebras. As we increase
the number of twisted and untwisted punctures, we are naturally required to
consider how the various types of gluing interact with each other. It will be
useful in this endeavour to have a collection of rearrangement lemmas that
establish the extent to which the various gluings associate.

In this subsection we establish a series of technical rearrangement lemmas
concerning the interplay between the various cohomological operations we have
defined thus far. The proofs of these lemmas, which are modifications of
proofs of Arakawa:2018egx , rely heavily on the machinery of spectral
sequences. The reader who is uninterested in highly technical details may wish
to skip this section and pick up in the following, where we extend our
construction to the vertex algebras associated to ${\cal C}_{0,m,n}$.

Since we are interested in the interplay of gluings, we will need to consider
objects that have multiple, commuting $\hat{\mathfrak{g}}_{\kappa_{c}}$
actions. To be completely precise, one should decorate each $\circ$ in this
section with subscripts to indicate the diagonal action that is being gauged.
This would be somewhat cumbersome, so we will overload the $\circ$ notation
and rely on context to make the relevant actions clear. Our lemmas only ever
concern two such actions, for the sake of argument we call them the left and
right actions. Suppose $U,V,W$ are in $\mathrm{KL}$, such that $V$ has two
actions of $\hat{g}_{\kappa_{c}}$ and $V$ is in $\mathrm{KL}$ with respect to
both actions. One should then interpret the symbol $U\circ(V\circ W)$ as the
semi-infinite cohomology
$\mathrm{H}^{\frac{\infty}{2}+\bullet}(\hat{\mathfrak{g}}_{-\kappa_{g}},\mathfrak{g},U\otimes\mathrm{H}^{\frac{\infty}{2}+\bullet}(\hat{\mathfrak{g}}_{-\kappa_{g}},\mathfrak{g},V\otimes
W))$, where the diagonal action of $\hat{\mathfrak{g}}_{-\kappa_{g}}$ is with
respect to the right $\hat{\mathfrak{g}}_{\kappa_{c}}$ action on $V$ and the
sole $\hat{\mathfrak{g}}_{\kappa_{c}}$ action on $W$. Similarly, the diagonal
action on
$U\otimes\mathrm{H}^{\frac{\infty}{2}+\bullet}(\hat{\mathfrak{g}}_{-\kappa_{g}},\mathfrak{g},V\otimes
W)$ is with respect to the sole $\hat{\mathfrak{g}}_{\kappa_{c}}$ action on
$U$ and the $\hat{\mathfrak{g}}_{\kappa_{c}}$ action on $V\circ W$ induced by
the left action on $V$.

Similarly, for objects $U,V,W\in{\cal Z}$-Mod, we will denote the semi-
infinite cohomology $\mathrm{H}^{\frac{\infty}{2}+0}({\cal
Z},U\otimes\mathrm{H}^{\frac{\infty}{2}+0}({\cal Z},V\otimes W))$ by
$U\ast(V\ast W)$. Here one should take the action on $V\otimes W$ for the
first cohomology and on $U\otimes V$ for the second—recall that the ${\cal
Z}$-action on $V$ descends to the cohomology $V\ast W$.

Hereafter, it should be understood that when there are many $\mathrm{KL}_{u}$
or $\mathrm{KL}_{t}$ actions present, we choose two such actions for the
purposes of the rearrangement lemmas. For the chiral algebras of class $\SS$
(both untwisted and twisted), all the moment maps (from the same algebra) are
related by discrete automorphisms, and one can make such a choice without loss
of generality.

First, we recast some of the results of Arakawa:2018egx  as rearrangement
lemmas.

######  Lemma 4.8.

Let $U,V,W$ be vertex algebra objects in $\mathrm{KL}$ such that $U$ is
semijective in $\mathrm{KL}$ and $V$ has two $\mathrm{KL}$ actions, then

$U\circ(V\circ W)\cong(U\circ V)\circ W~{}.$

###### Proof.

This proof is similar to the proof of Theorem 10.11 of Arakawa:2018egx .
Consider the bicomplex

$C^{\bullet\bullet}=U\otimes V\otimes
W\otimes\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}(\mathfrak{g})\otimes\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}(\mathfrak{g})~{},$

with the differential $d_{1}$ acting on $U\otimes V$ and the first
$\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}(\mathfrak{g})$ and $d_{2}$
acting on $V\otimes W$ and the second
$\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}(\mathfrak{g})$. It is easy to
see that $d_{1}d_{2}+d_{2}d_{1}=0$, so the total complex
$C_{tot}^{p}=\bigoplus_{m+n=p}C^{m,n}$ is a cochain complex with the
differential $d=d_{1}+(-1)^{m}d_{2}$ Weibel:1994 . There are two spectral
sequences converging to the total cohomology of $(C_{tot},d)$

$\begin{split}{}_{I}E^{p,q}_{2}=\mathrm{H}^{\frac{\infty}{2}+p}(\hat{\mathfrak{g}}_{-\kappa_{g}},\mathfrak{g},U\otimes\mathrm{H}^{\frac{\infty}{2}+q}(\hat{\mathfrak{g}}_{-\kappa_{g}},\mathfrak{g},V\otimes
W))~{},\\\
{}_{II}E^{p,q}_{2}=\mathrm{H}^{\frac{\infty}{2}+p}(\hat{\mathfrak{g}}_{-\kappa_{g}},\mathfrak{g},W\otimes\mathrm{H}^{\frac{\infty}{2}+q}(\hat{\mathfrak{g}}_{-\kappa_{g}},\mathfrak{g},U\otimes
V))~{}.\end{split}$

By Theorem 3.2, the cohomologies
$\mathrm{H}^{\frac{\infty}{2}+p}(\hat{\mathfrak{g}}_{-\kappa_{g}},\mathfrak{g},U\otimes-)$
are concentrated in degree zero, so both spectral sequences collapse at the
second page. The only nonzero entries are ${}_{I}E_{2}^{0,q}$ and
${}_{II}E_{2}^{p,0}$. Thus, we have

${}_{I}E^{0,p}_{2}\cong\ _{II}E^{p,0}_{2}\cong H^{p}_{tot}(C_{tot},d)~{},$

which gives the desired isomorphism. ∎

The vertex algebra objects $\mathbf{V}_{G,s}$ are semijective in $\mathrm{KL}$
so the composition of vertex algebras is associative—in the untwisted setting.
Furthermore, gauging at genus zero is always concentrated in degree zero. In
the twisted setting, we will have to work harder.

We have a similar result for Feigin–Frenkel gluing.

######  Lemma 4.9.

Suppose $M_{1},M_{2},M_{3}\in{\cal Z}$-Mod and suppose that $M_{1}$ and
$M_{3}$ are free over ${\cal Z}_{(<0)}$. Then

$M_{1}\ast(M_{2}\ast M_{3})\cong(M_{1}\ast M_{2})\ast M_{3}~{}.$

###### Proof.

This proof is similar to the proof of Proposition 10.2 of Arakawa:2018egx .
Consider the bicomplex

$C^{\bullet\bullet}=M_{1}\otimes M_{2}\otimes
M_{3}\otimes\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}(\mathfrak{z}(\mathfrak{g}))\otimes\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}(\mathfrak{z}(g))~{},$

with differentials $d_{12}$ acting on $M_{1}\otimes
M_{2}\otimes\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}(\mathfrak{z}(\mathfrak{g}))$
and $d_{23}$ acting on $M_{2}\otimes
M_{3}\otimes\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}(\mathfrak{z}(\mathfrak{g}))$.
The two differentials anticommute, so we can form the total complex
$C_{tot}^{n}=\bigoplus_{p+q=n}C^{p,q}$ with total differential
$d_{tot}=d_{12}+(-1)^{q}d_{23}$. There are two spectral sequences converging
to the total cohomology $H_{tot}$ of $C_{tot}$, whose second pages are given
by

$\begin{split}{}_{I}E_{2}^{p,q}\coloneqq\mathrm{H}^{\frac{\infty}{2}+p}({\cal
Z},M_{1}\otimes\mathrm{H}^{\frac{\infty}{2}+q}({\cal Z},M_{2}\otimes
M_{3}))~{},\\\
{}_{II}E_{2}^{p,q}\coloneqq\mathrm{H}^{\frac{\infty}{2}+p}({\cal
Z},\mathrm{H}^{\frac{\infty}{2}+q}({\cal Z},M_{1}\otimes M_{2})\otimes
M_{3})~{}.\end{split}$

By Theorem 3.4, the entries $E^{p,q}_{2}$ in either spectral sequence vanish
if $p<0$ or $q<0$. Thus, we have $E_{2}^{00}=E_{\infty}^{00}$, which gives the
isomorphism

$H^{0}_{tot}(C_{tot},d_{tot})\cong\mathrm{H}^{\frac{\infty}{2}+0}({\cal
Z},M_{1}\otimes\mathrm{H}^{\frac{\infty}{2}+0}({\cal Z},M_{2}\otimes
M_{3}))\cong\mathrm{H}^{\frac{\infty}{2}+0}({\cal
Z},\mathrm{H}^{\frac{\infty}{2}+0}({\cal Z},M_{1}\otimes M_{2})\otimes
M_{3})~{},$

as desired. ∎

Since the algebras $\mathbf{V}_{G,s}$ are free over ${\cal Z}_{<0}$
(Proposition $10.2$ of Arakawa:2018egx ), the above lemma applies, and
Feigin–Frenkel gluing $*_{u}$ is associative. We hold off on analysing
associativity of $\ast$ for the twisted algebras, since it is a challenge to
understand the twisted FF gluing between two mixed vertex algebras.

We next consider the combination of the two gluing operations, $\ast$ and
$\circ$.

######  Lemma 4.10.

Suppose $U\in{\cal Z}$-Mod, $V\in\mathrm{KL}$ and $\mathbf{W}\in\mathrm{KL}$.
Additionally, suppose $U$ is free over ${\cal Z}_{(<0)}$ and $W$ is
semijective in $\mathrm{KL}$. Then we have the isomorphism

$U\ast(V\circ W)\cong(U\ast V)\circ W~{}.$

###### Proof.

Consider the bicomplex

$C^{\bullet\bullet}=U\otimes V\otimes
W\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}(\mathfrak{z}(\mathfrak{g}))\otimes\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}(\mathfrak{g})~{},$

with differentials $d_{\mathfrak{g}}$ acting on $V\otimes
W\otimes\otimes\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}(\mathfrak{g})$
and $d_{\mathfrak{z}}$ acting on $U\otimes
V\otimes\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}(\mathfrak{z}(\mathfrak{g}))$.
The differentials anticommute, so we can form the total complex
$C^{p}_{tot}=\bigoplus_{m+n=p}C^{p,q}$ with differential
$d_{tot}=d_{\mathfrak{g}}+(-1)^{q}d_{\mathfrak{z}}$. There are two spectral
sequences converging to the total cohomology:

$\begin{split}{}_{I}E_{2}^{p,q}=\mathrm{H}^{\frac{\infty}{2}+p}({\cal
Z},U\otimes\mathrm{H}^{\frac{\infty}{2}+q}(\hat{\mathfrak{g}}_{-\kappa_{g}},\mathfrak{g},V\otimes
W))~{},\\\
{}_{II}E_{2}^{p,q}=\mathrm{H}^{\frac{\infty}{2}+p}(\hat{\mathfrak{g}}_{-\kappa_{g}},\mathfrak{g},\mathrm{H}^{\frac{\infty}{2}+q}({\cal
Z},U\otimes V)\otimes W)~{}.\end{split}$ (84)

The cohmology $\mathrm{H}^{\frac{\infty}{2}+p}({\cal Z},U\otimes-)$ vanishes
for $p<0$ and the cohomology
$\mathrm{H}^{\frac{\infty}{2}+p}(\hat{\mathfrak{g}}_{-\kappa_{g}},\mathfrak{g},-\otimes
W)$ is concentrated in degree zero. Thus both spectral sequences will collapse
at the second page and we have

${}_{I}E_{2}^{0,0}\cong\ _{II}E_{2}^{0,0}\cong
H^{0}_{tot}(C_{tot},d_{tot})~{},$ (85)

as desired. ∎

Now we come to the rearrangement of twisted and untwisted gluing. The results
here are more limited; the obvious generalisations of the above spectral
sequence arguments often fail in the twisted setting, since the mixed vertex
algebras are not semijective in $\mathrm{KL}_{u}$. Nevertheless, we will
manage to demonstrate that some properties of the gluing of twisted algebras
are as we expect. To start with, we have a result for the interchange of
twisted and untwisted gauging.

######  Lemma 4.11.

Suppose $V_{1}$ is in $\mathrm{KL}_{u}$, $V_{2}$ is in $\mathrm{KL}_{u}$ and
in $\mathrm{KL}_{t}$, and $V_{3}$ is in $\mathrm{KL}_{t}$. Furthermore,
suppose that $V_{3}$ is semijective for $\mathfrak{g}_{t}$. Then, we have the
isomorphism

$V_{1}\circ_{u}(V_{2}\circ_{t}V_{3})\cong(V_{1}\circ_{u}V_{2})\circ_{t}V_{3}~{}.$

###### Proof.

As usual, the proof is via a spectral sequence, but with the new feature that
cohomology is not necessarily concentrated in degree zero. We define the
bicomplex

$C^{p,q}=V_{1}\otimes V_{2}\otimes
V_{3}\otimes\mathrm{\bigwedge}\,^{\frac{\infty}{2}+p}(\hat{\mathfrak{g}}_{u})\otimes\mathrm{\bigwedge}\,^{\frac{\infty}{2}+q}(\hat{\mathfrak{g}}_{t})~{},$

with the differential $d_{u}$ and $d_{t}$ acting on $V_{1}\otimes
V_{2}\otimes\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}(\hat{\mathfrak{g}}_{u})$
and $V_{2}\otimes
V_{2}\otimes\mathrm{\bigwedge}\,^{\frac{\infty}{2}+\bullet}(\hat{\mathfrak{g}}_{t})$
respectively. The differentials anticommute, so we can form the total complex
$C_{tot}^{m}=\bigoplus_{p+q=m}C^{p,q}$ with the differential
$d_{tot}=d_{u}+(-1)^{p}d_{t}$. There are two spectral sequences converging to
the total cohomology, given by

$\begin{split}{}_{I}E_{2}^{p,q}=\mathrm{H}^{\frac{\infty}{2}+p}(\hat{\mathfrak{g}}_{u,-\kappa_{g}},\mathfrak{g}_{u},V_{1}\otimes\mathrm{H}^{\frac{\infty}{2}+p}(\hat{\mathfrak{g}}_{t,-\kappa_{g}},\mathfrak{g}_{t},V_{2}\otimes
V_{3}))~{},\\\
{}_{II}E^{p,q}_{2}=\mathrm{H}^{\frac{\infty}{2}+p}(\hat{\mathfrak{g}}_{t,-\kappa_{g}},\mathfrak{g}_{t},\mathrm{H}^{\frac{\infty}{2}+q}(\hat{\mathfrak{g}}_{u,-\kappa_{g}},\mathfrak{g}_{u},V_{1}\otimes
V_{2})\otimes V_{3})~{}.\end{split}$

Since $V_{3}$ satisfies the conditions of Theorem 3.2, the spectral sequences
will collapse on the second page and ${}_{I}E^{p,q}_{2}=0$ for $q\neq 0$ and
${}_{II}E^{p,q}_{2}=0$ for $p\neq 0$. Thus we have the isomorphism

$H^{m}_{tot}=\,_{I}E^{m,0}_{2}=\,_{II}E^{0,m}_{2}~{}.$

∎

We can also show that the twisted gauging and untwisted Feigin–Frenkel gluing
are nicely compatible,

######  Lemma 4.12.

Let $M$ be an object of ${\cal Z}_{u}$-Mod such that it is free over ${\cal
Z}_{u,<0}$ and let $W$ be semijective in $\mathrm{KL}_{t}$. Suppose, $V$ is a
vertex algebra object in $\mathrm{KL}_{t}$, then we have the follwoing
isomorphism.