# Monopole Operators and Bulk-Boundary Relation in Holomorphic Topological
Theories

Keyou Zeng

###### Abstract

We study the holomorphic twist of $3d$ $\mathcal{N}=2$ supersymmetric field
theories, discuss the perturbative bulk local operators in general, and
explicitly construct non perturbative bulk local operators for abelian gauge
theories. Our construction is verified by matching the character of the
algebra with the superconformal index. We test a conjectural relation between
the self-Ext of boundary algebras and bulk algebras in various cases,
including Landau-Ginzburg models with an arbitrary superpotential and some
abelian gauge theories. In the latter cases, monopole operators appear in the
self-Ext of a perturbative boundary algebra. We briefly discuss the higher
structures in both boundary and bulk algebras.

###### Contents

  1. 1 Introduction
    1. 1.1 Future directions
  2. 2 Review of holomorphic twisted $3d$ $\mathcal{N}=2$ theory
    1. 2.1 Twisted theory in BV formalism
    2. 2.2 AKSZ formalism
  3. 3 Bulk algebras and their characters
    1. 3.1 Chiral multiplets
    2. 3.2 Theories with gauge fields
      1. 3.2.1 Perturbative algebra
      2. 3.2.2 Non-perturbative algebra
    3. 3.3 Superconformal index and operator counting
      1. 3.3.1 Chern-Simons line bundle twist
      2. 3.3.2 Berezinian and Half-form twist
  4. 4 Examples
    1. 4.1 XYZ model
    2. 4.2 $U(1)$ pure gauge theory
    3. 4.3 $N_{f}=\tilde{N}_{f}=1$ SQED
    4. 4.4 $U(1)_{-\frac{1}{2}}$ \+ a chiral
    5. 4.5 Bulk algebras and dualities
  5. 5 Review of boundary conditions and boundary algebras
    1. 5.1 Boundary algebras for chiral multiplets
    2. 5.2 Boundary algebras for vector multiplets
    3. 5.3 Boundary algebra with vector and chiral multiplets
  6. 6 Self-Ext and bulk local operator algebra
    1. 6.1 Chiral multiplet
      1. 6.1.1 Single chiral multiplet with D b.c.
      2. 6.1.2 XYZ model with NDD b.c.
      3. 6.1.3 XYZ model with NNN b.c.
      4. 6.1.4 XYZ model with DDD b.c.
      5. 6.1.5 Chiral multiplets with arbitrary superpotential
    2. 6.2 Koszul duality in boundary chiral algebra
    3. 6.3 Including gauge fields
      1. 6.3.1 Perturbative analysis for vector multiplet
      2. 6.3.2 $U(1)$ gauge theory at level 1
      3. 6.3.3 A failure case
      4. 6.3.4 $U(1)_{-\frac{1}{2}}+$ chiral with $(\mathcal{N},N)$ b.c.
      5. 6.3.5 SQED with $(\mathcal{N},N,N)$ b.c.
    4. 6.4 Algebraic structure from Ext
      1. 6.4.1 Hochschild cup product
      2. 6.4.2 A generalization of Deligne’s conjecture
  7. A Classical BV-formalism in finite dimension
  8. B Berezinian and its properties
  9. C Computing some complexes

##  1 Introduction

Twisting supersymmetric field theories [1, 2] has been a very powerful and
successful tool for extracting mathematical structures from physical quantum
field theories. Twisted theories, as they were originally discovered, involve
fully topological twist, which renders the original field theories into
topological field theories. Classical examples include A and B model
topological string [2] in $2d$, Rozansky-Witten theory [3] in $3d$, Donaldson-
Witten theory[1], Vafa-Witten theory[4] and Kapustin-Witten theory[5] in $4d$.

Fully topological twist depends on the choice of a supercharge $Q$ that
transforms as a scalar under a ”twisted” Lorentz group action. The existence
of such a scalars require the existence of a relatively large amount of
supersymmetry. Actually, the twisting procedure can be modified to adapt to a
generic nilpotent supercharge. Early examples include [6, 7, 8] and this
technique has been systematically developed in e.g. [9, 10] recently. After
such a twist, the resulting theories become holomorphic in most space time
directions. In even spacetime dimension $d=2n$, the choice of nilpotent
supercharge specifies a complex structure on $\mathbb{R}^{2n}=\mathbb{C}^{n}$.
All correlation functions of the twisted theory will depend holomorphically on
$\mathbb{C}^{n}$. In odd spacetime dimension $d=2n+1$, the choice of nilpotent
supercharge specifies a splitting
$\mathbb{R}^{2n+1}=\mathbb{C}^{n}\times\mathbb{R}$. Correlation functions of
the twisted theory will depend holomorphically on $\mathbb{C}^{n}$ and be
independent of $\mathbb{R}$.

In this paper we study twisted $3d$ $\mathcal{N}=2$ theories following [11,
12]. The purpose of this paper is twofold. First, we try to give a more
detailed description of the bulk algebras of the twisted theories. The study
of monopole operators will play an important role here. A monopole operator is
defined by prescribed local singular behavior of fields. The strategy we
employ for studying them relies on the state-operator correspondence, which
has also been used in many previous works [13, 14, 15] that successfully
identified their spectrum and various quantum numbers. In this paper, we
combine state-operator correspondence with the method of geometric
quantization to give a description of the full local operator algebras for
abelian gauge theories. The algebra is decomposed into different sectors
labeled by monopole (topological) charges and each sector represents a family
of gauge invariant dressed monopole operators. As a justification of our
construction, we compute the characters of the local operator algebras and our
results agree nicely with existing literature on superconformal index of $3d$
$\mathcal{N}=2$ theory [16, 17, 18, 19].

Then, we exam a conjectural relation between the bulk and boundary local
operator algebras [12]. This relation is actually a general feature of
topological quantum field theories[20], which links boundary information with
bulk information. The $2d$ TQFT version of this conjecture is the isomorphism
between the bulk algebra and the Hochschild cohomology of boundary algebra. In
$3d$, this conjecture becomes the isomorphism between the bulk algebra and the
derived center of boundary chiral algebra, which is computed by self-Ext of
the boundary vacuum module. This technique has also been used in [21, 22] to
study Higgs and Coulomb branch operators of $3d$ $\mathcal{N}=4$ theories.

This paper is organized as follows. In Section 2, we review the twisted $3d$
$\mathcal{N}=2$ theories in BV formalism and in AKSZ formalism. In section 3,
we analyze the bulk local operators of the theories. A description of the
perturbative algebras will be provided for the general situations. We describe
non perturbative algebras for theories with abelian gauge group, by utilizing
state operator correspondence. We also discuss character of the local operator
algebra and its relation with the $3d$ superconformal index. In section 4 we
study local operator algebras and their characters in some explicit examples.
The BRST cohomologies of the local operator algebras will be computed in the
first few spin sectors. The implications of $3d$ mirror symmetry on the local
operators will be discussed. In section 5 we briefly review boundary
conditions and the corresponding boundary chiral algebras. In Section 6 we
compute the self-Ext of various boundary chiral algebras under different
boundary conditions and prove that they agree with bulk algebra in some non-
trivial cases. A failure case of this technique will also be provided.
Finally, we touch on the algebraic structures on the self-Ext and its relation
with the algebraic structure of bulk local operators.

###  1.1 Future directions

We outline several other motivations for this work and a partial list of
related open issues.

##### Three manifold invariants

String Theory/M-Theory predicts the existence of the $6d$ superconformal field
theories living on M5 branes. A twisted compactification of a $6d$ theory
labeled by a simply-laced Lie algebra $\mathfrak{g}$ on a manifold $M$ of
dimension $d$ defines a $(6-d)$ dimensional supersymmetric field theory
$T[M,\mathfrak{g}]$. In the IR, the theories $T[M,\mathfrak{g}]$ only depend
on part of the geometry of $M$. Specifically, they all have a nilpotent
supercharge whose cohomology is invariant under diffeomorphism of $M$.

For $d=3$, theories $T[M^{3},\mathfrak{g}]$ are a family of $3d$
$\mathcal{N}=2$ supersymmetric field theories. For
$\mathfrak{g}=\mathfrak{su}(2)$, theories $T[M^{3},\mathfrak{su}(2)]$ are
explicitly constructed in [23] based on a triangulation of the $3$ manifold
$M^{3}$ into tetrahedra, and a gluing of the “building block” theory
$T_{\Delta}$ through the triangulation. This constructions leads to the $3d$
$\mathcal{N}=2$ abelian Chern-Simons matter theories whose twisted versions
are studied in this paper. Superconformal index of $T[M^{3}]$ are studied in
[19] and gives us an invariant of the three manifold $M^{3}$. The bulk algebra
$\mathrm{Obs}_{T[M^{3}]}$ then provides a ”categorification” of the $3d$ index
in [19] (In the sense that we replace a number by a graded vector space).
Techniques in this paper can be applied to describe monopole operators in
$T[M^{3}]$, and to give an explicit construction of the three manifold
invariant $\mathrm{Obs}_{T[M^{3}]}$.

##### Line operators

Twisted $3d$ $\mathcal{N}=2$ theories admit a large collection of line
operators extending in the real direction. They come in two basic varieties:
Wilson lines and vortex lines. Together they generate a category $\mathcal{C}$
that is expected to be a chiral category [24] and hold a wealth of algebraic
information of the theory. For example in this guise the bulk boundary
relation studied in this paper can be adapted to the statement that local
operators are the same as (or can be defined as) self-interfaces of the
trivial line defect. More explicitly we have

$\mathrm{Obs}_{\text{loc}}=\mathrm{Hom}_{\mathcal{C}}(\mathbb{1}_{\mathcal{C}},\mathbb{1}_{\mathcal{C}})\,.$
(1.1)

In topological twist of $3d$ $\mathcal{N}=4$ theories, line operators was
systematically studied in [25]. It will be interesting to identify the
category $\mathcal{C}$ in various $\mathcal{N}=2$ setups.

Mirror symmetry always makes nontrivial predictions for mathematical objects
extracted from mirror quantum field theories. For the category of line
operators, mirror duality of two $3d$ $\mathcal{N}=2$ theories $\mathcal{T}$
and $\mathcal{T}^{\prime}$ then suggest a equivalence of categories
$\mathcal{C}_{\mathcal{T}}\cong\mathcal{C}_{\mathcal{T}^{\prime}}$. For $3d$
$\mathcal{N}=4$ theories, such examples are explored in [25] and recently in
[26].

##### Integrable system

We can consider the twisted $3d$ $\mathcal{N}=2$ theories defined on a
spacetime of the form $\Sigma\times\mathbb{R}$, where $\Sigma$ is a Riemann
surface. We can also consider line defect passing through $\Sigma$ and this
will lead to a punctured surface with some labels on the punctured points. The
twisted theory will assign a Hilbert space $\mathcal{H}(\Sigma)$ to the
surface after quantization [27]. Bringing a bulk operator to the surface
$\Sigma$ defines an action of the operator on the Hilbert space
$\mathcal{H}(\Sigma)$.

$H^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\mathrm{Obs}_{\text{loc}},Q)\to\mathrm{End}(\mathcal{H}(\Sigma))\,.$
(1.2)

If we place bulk local operators at different points separated from the lines
defects that we inserted, then as a consequence of the twisted $3d$ theory
being topological along $\mathbb{R}$, the action of the operators on
$\mathcal{H}(\Sigma)$ all commute. This is a general pattern of quantum
integrable system, where bulk local operators play the roles of commuting
Hamiltonian. For example, for theories with only vector multiplet at critical
Chern Simons level, this construction will lead to Hitchin integrable system
[28, 29] and specifically the Gaudin integrable system [30, 31] for $\Sigma$ a
punctured sphere. It will be interesting to figure out the integrable systems
associated with more general $3d$ $\mathcal{N}=2$ theories possibly with both
vector and chiral multiplets.

##### Higher algebra and Deligne’s conjecture

Structures of topological quantum field theories, or quantum field theories in
general, impose structures on the space of observables [32]. For the case of
$d$ dimensional TQFT, the algebraic structures of the local operators at the
chain level are captured by the notion of $E_{d}$ algebra. At the level of
cohomology, an $E_{d}$ algebra becomes a shifted version of Poisson algebra,
known as $P_{d}$ algebra. The bulk boundary relations then predict that the
derived center of the boundary will carry these structures. For example in
$2d$, we have the well know Gerstenhaber structure on the Hochschild
cohomology $HH^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(A)$ [33]. If we look at
the chain level this becomes Deligne’s conjecture (now a theorem) on the
existence of $E_{2}$ structure on the Hochschild complex [34, 35]. In $d\geq
2$ dimension, ”higher” Deligne’s conjecture state that $E_{(d-1)}$ Hochschild
cohomology is furnished with an $E_{d}$ algebra structure [36, 20].

We wish to find analog statements for the holomorphic topological theories in
$3d$. The general structure of the local operators should be captured by a
higher analog of chiral algebra that also includes the OPE structures along
$\mathbb{R}$. Though such a structure is still mysterious to us, if we take
cohomology, this structure becomes the more familiar (shifted) Poisson vertex
algebra [37, 38]. Therefore we hope that the self-Ext
$\mathrm{Ext}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}_{V-\text{mod}}(V,V)$ of a
vertex algebra $V$ can be endowed with a shifted Poisson vertex algebra
structure. If we look at the chain level, $\mathrm{RHom}_{V-\text{mod}}(V,V)$
should have the higher analog of vertex algebra structure. Moreover, when $V$
comes equipped with a stress energy tensor, namely $V$ is a conformal vertex
algebra, the bulk theory of it becomes topological (see [12] for more detail).
In this case, the self-RHom should be endowed with the structure of an $E_{3}$
algebra. We explain this story in more detail in Section 6.4.

##  2 Review of holomorphic twisted $3d$ $\mathcal{N}=2$ theory

In this paper we mainly work in the flat Euclidean spacetime
$\mathbb{R}_{t}\times\mathbb{C}_{z,\bar{z}}$. The $3d$ $\mathcal{N}=2$ SUSY
algebra has generators $Q_{\pm},\bar{Q}_{\pm}$ and commutation relations

$\begin{split}&\\{Q_{+},\bar{Q}_{+}\\}=-2i\partial_{\bar{z}},\quad\quad\\{Q_{-},\bar{Q}_{-}\\}=2i\partial_{z}\,,\\\
&\\{Q_{+},\bar{Q}_{-}\\}=\\{Q_{-},\bar{Q}_{+}\\}=i\partial_{t}\,.\end{split}$
(2.1)

In [11, 12] a class of $3d$ holomorphic topological theory were studied. Such
a theory arises from a $3d$ $\mathcal{N}=2$ supersymmetric field theory after
performing a holomorphic twist. This amounts to changing the cohomological
degree of fields using the $R$ symmetry and adding the supercharge
$\bar{Q}_{+}$ into the BRST differential. By adding $\bar{Q}_{+}$ into the
BRST differential, many $\bar{Q}_{+}$-exact terms can be removed and we get a
new (and much simpler) quantum field theory. The fact that $\partial_{t}$ and
$\partial_{\bar{z}}$ are $\bar{Q}_{+}$-exact tells us that all correlation
functions of the new theory will be independent of $t$ and $\bar{z}$. We call
such a theory holomorphic topological, meaning that it depends topologically
in the $t$ direction and holomorphic in the $z$ direction. This explains the
name holomorphic twist or, to be more precise, holomorphic topological twist.

In this section we review the $3d$ $\mathcal{N}=2$ theory in the twisted
formalism introduced in [11, 12]. The advantage of directly working in the
twisted formalism is that the field content, the action functional and the
equations of motion are greatly simplified, while many structures that are
contained in the $\bar{Q}_{+}$ cohomology of the original physical theory are
still preserved in the twisted formalism. We then recast the theory into AKSZ
formalism. This encodes classical information of the theory into a geometric
structure, which paves the way for performing geometric quantization later.

###  2.1 Twisted theory in BV formalism

A $3d$ $\mathcal{N}=2$ SUSY theory containing both vector multiplets and
chiral multiplets is specified by the following data.

  1. 1.

A compact gauge group $G$. (In twisted formalism, we always work with its
complexification. So we also use $G$ to denote the complexification of the
gauge group in the following).

  2. 2.

A unitary representation $V$ of $G$. (After complexification, $V$ becomes a
complex linear representation.)

  3. 3.

A $G$ invariant polynomial $W:V\to\mathbb{C}$, called superpotential.

  4. 4.

An integer (or half-integer) $k$ called Chern-Simons level.

To perform the holomorphic twist we require additional data of a $U(1)_{R}$
symmetry, under which $V$ decompose into subspaces with different $R$ charges:
$V=\oplus_{r}V^{(r)}$. Under the $R$ charge assignment, the superpotential $W$
must be a quasi-homogeneous function of $R$-charge $2$.

In this paper we directly work with the twisted theories. These theories are
generally defined on a $3$-manifold $M$ with a transverse holomorphic
foliation(THF). A $3$-manifold is equiped with a THF if it has local
coordinate patches $(t,z)\in\mathbb{R}\times\mathbb{C}$ and the transition
functions take the form

$(f(t,z,\bar{z}),h(z),\bar{h}(\bar{z}))\,,$ (2.2)

where $h(z)$ is a holomorphic function. The above form of coordinate
transformation implies that the (anti)holomorphic 1-forms
$\Omega^{1,0}$($\Omega^{0,1}$) are globally well defined. And we have well
defined $\partial,\bar{\partial}$ operators. The $dt$ direction, on the other
hand, is not globally defined. However, we can define the quotient spaces
$\Omega^{1}/\Omega^{1,0}$ and $\Omega^{1}/\Omega^{0,1}$. The projection map
$p:\Omega^{1}\to\Omega^{1}/\Omega^{1,0}$ can be written in local coordinates
as

$p:\;\;f_{t}dt+f_{z}dz+f_{\bar{z}}d\bar{z}\to f_{t}dt+f_{\bar{z}}d\bar{z}\,.$
(2.3)

Using this projection, we can define a modified differential operator as

$\hat{d}=p\cdot d:\;f\to\partial_{t}fdt+\partial_{\bar{z}}fd\bar{z}\,.$ (2.4)

We also introduce a dg algebra

$\mathcal{A}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}=\Omega^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}/(\Omega^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}\wedge\Omega^{1,0})\,.$
(2.5)

In a local coordinate it takes the form $C^{\infty}(U)[dt,d\bar{z}]$. The
differential $\hat{d}$ naturally extend to
$\mathcal{A}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}$ and we still denote it by
$\hat{d}$. There are natural variant of
$\mathcal{A}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}$:

$\mathcal{A}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},k}=\mathcal{A}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}\otimes\Omega^{k,0}\,.$
(2.6)

In local coordinate it takes the form $C^{\infty}(U)[dt,d\bar{z}]dz^{k}$. The
operator $\hat{d}$ is still well defined on $\mathcal{A}^{*,k}$ and takes the
form $\hat{d}=dt\partial_{t}+d\bar{z}\partial{\bar{z}}$ in a local coordinate.
The wedge product gives us a natural pairing

$\mathcal{A}^{i,k}\otimes\mathcal{A}^{j,l}\to\mathcal{A}^{i+j,k+l}\,.$ (2.7)

Integration on $M$ gives us a map

$\int:\mathcal{A}^{2,1}=\Omega^{3}(M,\mathbb{C})\to\mathbb{C}\,.$ (2.8)

In this paper we study the local properties of these theories, hence it
suffices to work in a local coordinate. The cohomology of
$(\mathcal{A}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},k},\hat{d})$ in a local
coordinate consist of those function $\mathrm{Hol}(D)$ which are independent
of $t$ and holomorphic in $z$.

A convenient way to analyze gauge theories is through the BV formalism. By
adding ghost, anti-field, anti-ghost, a gauge theory can be described by a
differential graded (dg) Lie algebra equipped with an invariant pairing of
degree $-3$. For the holomorphic twisted $3d$ $\mathcal{N}=2$ theory, the
field content in BV formalism can be organized into the following
”superfields”

  * •

A gauge field:
$\mathbf{A}=c+A+B^{*}\in\mathcal{A}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}\otimes\mathfrak{g}[1]$.

  * •

An coadjoint valued field:
$\mathbf{B}=B+A^{*}+c^{*}\in\mathcal{A}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},1}\otimes\mathfrak{g}^{*}$.

  * •

A matter field in representation $V$:
$\mathbf{\Phi}=\phi+\eta^{*}+\psi^{*}\in\oplus_{k}\mathcal{A}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},k}\otimes
V^{(k)}$.

  * •

A matter field in the dual representation $V^{*}$:
$\mathbf{\Psi}=\psi+\eta+\phi^{*}\in\oplus_{k}\mathcal{A}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},1-k}\otimes(V^{(k)})^{*}[1]$.

where the symbol $[1]$ indicates a shift of cohomological degree by 1, so that
the gauge field $A$ has degree 0. Here, the fields $\mathbf{A},\mathbf{B}$ are
obtained form twisting the vector multiplets of the physical $3d$
$\mathcal{N}=2$ theory and $\mathbf{\Phi},\mathbf{\Psi}$ are obtained from
twisting the chiral multiplets. Thus, we will also use the names ”vector
multiplets” and ”chiral multiplets” to call them in the twisted formalism.

We can write down the full BV action functional in terms of the above data

$S_{BV}=\int\langle\mathbf{B},\hat{d}\mathbf{A}+\frac{1}{2}[\mathbf{A},\mathbf{A}]\rangle+\langle\mathbf{\Psi},(\hat{d}+\mathbf{A})\mathbf{\Phi}\rangle+W(\mathbf{\Phi})+\frac{k}{8\pi}\langle\mathbf{A},\partial\mathbf{A}\rangle\,.$
(2.9)

Here the inner product $\langle-,-\rangle$ is induced from the dual pairing
between $\mathfrak{g},\mathfrak{g}^{*}$ and $V,V^{*}$ and wedge product of
differential forms. The integration map $\int$ is defined by 2.8 that picks up
the top degree forms.

We mention that in the presence of a Chern-Simons term, the gauge theory part
is actually equivalent to a physical Chern-Simons theory. This fact can be
obtained via a field redefinition [12]. We provide another explanation in the
next section.

Another important structure in BV formalism of QFT is the degree $1$ bracket
that pairs fields/ghost with anti-fields/anti-ghost, called BV bracket. In our
theory, the BV bracket is given by

$\\{\mathbf{\Phi}(x),\mathbf{\Psi}(y)\\}_{\text{BV}}=\\{\mathbf{A}(x),\mathbf{B}(y)\\}_{\text{BV}}=\delta(x-y)\mathrm{dVol}\,.$
(2.10)

The BRST operator is easy to calculate from the BV bracket using the formula
$Q=\\{S,-\\}_{\text{BV}}$. We find

$\begin{split}Q\mathbf{A}&=\hat{d}\mathbf{A}+[\mathbf{A},\mathbf{A}]\,,\\\
Q\mathbf{B}&=\hat{d}_{\mathbf{A}}\mathbf{B}+\mu(\mathbf{\Phi},\mathbf{\Psi})-\frac{k}{4\pi}\partial\mathbf{A}\,,\\\
Q\mathbf{\Phi}&=\hat{d}_{\mathbf{A}}\mathbf{\Phi}\,,\\\
Q\mathbf{\Psi}&=\hat{d}_{\mathbf{A}}\mathbf{\Psi}+\frac{\delta
W}{\delta\mathbf{\Phi}}\,,\\\ \end{split}\ $ (2.11)

where we used (shifted) moment map $\mu:V\otimes V^{*}[1]\to\mathfrak{g}^{*}$
associated with the action of $G$ on $V$.

###  2.2 AKSZ formalism

Roughly speaking, the geometry of the (classical) BV formalism is given by a
dg manifold equipped with a $-1$ shifted symplectic form. A BV action
functional is an even function $S$ such that $Q=\\{S,-\\}$. In some cases, the
BV field theory can be formulated as an AKSZ sigma model [39]. In the AKSZ
construction, we have a dg manifold $N$ equipped with a volume form of degree
$l$ and a dg manifold $M$ equipped with a symplectic form of degree $-1-l$.
Then we consider the mapping space $\mathrm{Map}(N,M)$, which can be shown to
have the structure of a classical BV field theory. The natural $(-1)$-shifted
sympletic structure $\omega$ is described as follows. Given $f:N\to M$, the
tangent space of $\mathrm{Map}(N,M)$ at $f$ is
$T_{f}\mathrm{Map}(N,M)=\Gamma(N,f^{*}TM)$. Then the symplectic form on the
mapping space $\mathrm{Map}(N,M)$ can be written as

$\omega(\alpha,\beta)=\int_{N}\mathrm{dVol}_{N}(\alpha,\beta)_{M},\;\;\alpha,\beta\in\Gamma(N,f^{*}TM)\,,$
(2.12)

where $(\;,\;)_{M}$ denote the pairing on $T_{M}$ that comes from the
symplectic structure on $M$.

In this section, we show that the holomorphic twisted $3d$ $\mathcal{N}=2$
theory described above can also be formulated as an AKSZ sigma model.

We first consider the vector multiplet with gauge group $G$ and Chern Simons
level $0$, the field content is

$(\mathbf{A},\mathbf{B})\in\mathcal{A}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}\otimes\mathfrak{g}[1]\oplus\mathcal{A}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},1}\otimes\mathfrak{g}^{*}\,.$
(2.13)

As a graded manifold, these fields describe maps
$\mathbb{R}_{dR}\times\mathbb{C}_{\bar{\partial}}\to\mathfrak{g}[1]\oplus\mathfrak{g}^{*}$.
We can observe that the BRST differential is inherited from the Chevalley-
Eilenberg differential of
$C^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\mathfrak{g};\mathrm{Sym}\mathfrak{g}^{*})=\mathbb{C}[T^{*}[1]B\mathfrak{g}]$.
Besides, the BV bracket comes from the natural symplectic form on the shifted
cotangent bundle $T^{*}[1]B\mathfrak{g}$. Therefore, in the AKSZ formulation
the twisted vector multiplet (without a CS term) can be formulated as the
mapping space:
$\mathrm{Map}(\mathbb{R}_{dR}\times\Sigma_{\bar{\partial}},T^{*}[1]B\mathfrak{g})$.
We should be careful here. By writing the target as $T^{*}[1]B\mathfrak{g}$,
we only defined a perturbation theory. The ”global” version of this theory is

$\mathrm{Map}(\mathbb{R}_{dR}\times\Sigma_{\bar{\partial}},T^{*}[1]BG)\,.$
(2.14)

We can also understand theory with a Chern-Simons term in a similar fashion.
First we rewrite the mapping space as follows

$\mathrm{Map}(\mathbb{R}_{dR}\times\Sigma_{\bar{\partial}},T^{*}[1]BG)=\mathrm{Map}(\mathbb{R}_{dR}\times
T[1]\Sigma_{\bar{\partial}},BG)\,.$ (2.15)

Adding a Chern-Simons term has the effect of turning
$T[1]\Sigma_{\bar{\partial}}$ into $\Sigma_{dR}$. Thus we obtained the
standard description of Chern-Simons theory [39]

$\mathrm{Map}((\mathbb{R}\times\Sigma)_{dR},BG)\,.$ (2.16)

We can also make the dependence on complex structure more explicit, by writing

$\mathrm{Map}((\mathbb{R}\times\Sigma)_{dR},BG)=\mathrm{Map}(\mathbb{R}_{dR},T^{*}_{\mathbf{k}}\mathrm{Map}(\Sigma_{\bar{\partial}},BG)=\mathrm{Map}(\mathbb{R}_{dR},T^{*}_{\mathbf{k}}\mathrm{Bun}_{G}(\Sigma))\,.$
(2.17)

In this way we understand Chern-Simons theory as a topological quantum
mechanics with target $T^{*}_{\mathbf{k}}\mathrm{Bun}_{G}(\Sigma)$ as a
twisted cotangent bundle of $\mathrm{Bun}_{G}(\Sigma)$. Here we actually
understand the Chern-Simons level $\mathbf{k}$ as a Cartan-Killing form, which
can be further identified as a symplectic structure on $BG$ and integrated to
a closed two form on $\mathrm{Bun}_{G}(\Sigma)$. From this perspective we can
regard the theory without Chern-Simons term as a topological quantum mechanics
with target $T^{*}\mathrm{Bun}_{G}(\Sigma)=\mathrm{Higgs}_{G}(\Sigma)$, the
Hitchin moduli space of $G$ bundle on $\Sigma$. This suggests a close relation
between this theory and the Hitchin system.

For the chiral multiplet, the field content is

$(\mathbf{\Phi},\mathbf{\Psi})\in\bigoplus_{k}\mathcal{A}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},k}\otimes
V^{(k)}\oplus\mathcal{A}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},1-k}\otimes(V^{(k)})^{*}[1]\,.$
(2.18)

They can be described by the mapping space
$\mathrm{Map}(\mathbb{R}_{dR}\times\Sigma_{\bar{\partial}},T^{*}[1]V)$.
Turning on a superpotential $W$ deforms the differential $Q$ by a term
$\frac{\delta W}{\delta\mathbf{\Phi}}\frac{\partial}{\partial\mathbf{\Psi}}$.
As is reviewed in Appendix A, this amounts to turning $T^{*}[1]V$ into
$\mathrm{dCrit}(W)$, the derived critical locus of $W$. Thus we conclude that
the AKSZ formulation for chiral multiplet with superpotential $W$ is given by
the mapping space:

$\mathrm{Map}(\mathbb{R}_{dR}\times\Sigma_{\bar{\partial}},\mathrm{dCrit}(W))\,.$
(2.19)

More generally, for theory with both vector and chiral multiplets, the AKSZ
formulation is given by the mapping space

$\mathrm{Map}(\mathbb{R}_{dR}\times\Sigma_{\bar{\partial}},\text{dCrit}(W:V/G\to\mathbb{C}))\,.$
(2.20)

Or equivalently

$\mathrm{Map}(\mathbb{R}_{dR}\times\Sigma_{\bar{\partial}},\text{dCrit}(W)\mathbin{/\mkern-6.0mu/}G)\,.$
(2.21)

We can check that this is compatible with previous special cases. For $G$
trivial, the target automatically becomes $\text{dCrit}(W)$. And for $V=0$,
$V/G=BG$. The derived critical locus for trivial superpotential is simply the
$1$ shifted cotangent bundle, which gives us $T^{*}[1]BG$.

##  3 Bulk algebras and their characters

In this section, we study the bulk local operator algebra for the holomorphic
twisted $3d$ $\mathcal{N}=2$ theory. We study both perturbative and non
perturbative algebra. This includes the consideration of monopole operators,
which cannot be expressed as polynomial functions of the fields. After
appropriately identifying the operators and their quantum numbers, we study
the character of the algebra, which reproduces the $3d$ superconformal index
of the corresponding SUSY theory.

At the perturbative level, the local operator algebra
$\mathrm{Obs}^{\mathrm{per}}$ is easy to describe. Perturbative local
operators are polynomial functions in the fields
$(\mathbf{\Phi},\mathbf{\Psi}),(\mathbf{B},\mathbf{A})$, subject to the BRST
differential 2.11. As we have mentioned, locally, cohomology of
$(\mathcal{A}^{\mathbin{\vbox{\hbox{{$\bullet$}}}},k},\hat{d})$ consists of
those functions $\mathrm{Hol}(D)$ that are independent in $t$ and holomorphic
in $z$. Therefore, by only taking the bottom component of the superfields and
restricting to fields that are holomorphic in $z$ we find a smaller but quasi-
isomorphic algebra. This algebra consist of functions in the fields
$\\{\phi(z),\psi(z),b(z),c(z)\\}$. The differential can be schematically
written as

$\begin{split}Qc&=\frac{1}{2}[c,c]\,,\\\
Qb&=[c,b]+\mu(\phi,\psi)-\frac{k}{4\pi}\partial c\,,\\\ Q\phi&=[c,\phi]\,,\\\
Q\psi&=[c,\psi]+\frac{\delta W(\phi)}{\delta\phi}\,.\\\ \end{split}$ (3.1)

We can also explain this result in a geometric manner. We consider a cylinder
$C_{\epsilon}=[-\epsilon,\epsilon]\times D_{\epsilon}$ where $D_{\epsilon}$ is
a disk of radius $\epsilon$ in $\mathbb{C}$. The perturbative local operators
are functions on the (derived) space of solution to the equation of motion on
$C_{\epsilon}$ as $\epsilon\to 0$.

$\mathrm{Obs}^{\mathrm{per}}=\lim_{\epsilon\to
0}\mathbb{C}[\mathrm{EOM}(C_{\epsilon})]\,.$ (3.2)

The AKSZ formalism of the theory immediately tells us that perturbatively,
$\mathrm{EOM}(C_{\epsilon})$ is

$\mathrm{Map}([-\epsilon,\epsilon]_{dR}\times(D_{\epsilon})_{\bar{\partial}},\mathrm{dCrit}(W:V/\mathfrak{g}\to\mathbb{C}))\,.$
(3.3)

This space describes maps constant along the real direction and holomorphic on
$D_{\epsilon}$. Equivalently we can ignore $[-\epsilon,\epsilon]_{dR}$ and we
replace the disk by the formal disk
$\mathbb{D}=\mathrm{Spec}(\mathbb{C}[[z]])$ as $\epsilon\to 0$. Then the space
of solutions to the equations of motion can be described by algebraic maps
$\\{\mathbb{D}\to\mathrm{dCrit}(W:V/\mathfrak{g}\to\mathbb{C})\\}=J_{\infty}(\mathrm{dCrit}(W:V/\mathfrak{g}\to\mathbb{C}))$.
Then we find the perturbative local operator algebra to be the space of
functions:
$\mathbb{C}[J_{\infty}(\mathrm{dCrit}(W:V/\mathfrak{g}\to\mathbb{C}))]$, which
is exactly the same as the algebra of functions of the fields
$\\{\phi(z),\psi(z),b(z),c(z)\\}$ with differential 3.1.

However, even at the perturbative level, this answer is not correct, as we
need to be more careful about the ghost. In a quantum field theory with gauge
symmetry, we can think of introducing ghosts as a homological method to
compute gauge invariant local operators. However, for a compact gauge group,
taking $G$ invariant is already an exact functor and there is no need to do
this homologically, and it suffices to only introduce higher order ghost
modes. Similar problems appears in [12] in the discussion of the boundary
algebra, where more details about (derived) invariants of Lie algebra and Lie
group is provided. The upshot is that, instead of introducing constant ghost
mode in the local operator algebra, we impose $G$ invariant by hand.
Therefore, the perturbative operator algebra should be something that looks
like $\mathbb{C}[\partial c,\partial^{2}c,\dots]^{G}$ instead of
$\mathbb{C}[c,\partial c,\dots]$.

From a physical perspective, the first order derivative of the $c$ ghost is
cohomologous to the gaugino in the physical SUSY theory before twisting [12].
In the computation of the superconformal index in physics literature[18, 16],
only gaugino and its derivatives contribute to the index and it always
involves an integration over the gauge $G$ fugacities. This essentially means
that only $G$ invariant polynomials that only involve higher modes of the
ghost $c$ contribute to the bulk algebra.

Accordingly, when $G$ is a compact group, we write the perturbative local
operator algebra as

$\mathrm{Obs}^{\mathrm{per}}=\mathbb{C}[J_{\infty}(\mathrm{dCrit}(W)\mathbin{/\mkern-6.0mu/}G)]\,.$
(3.4)

The symplectic quotient here is understood in a derived sense. The derived
quotient by $J_{\infty}G=G[[z]]$ is divided into two parts using the
decomposition $G[[z]]=G\ltimes zG[[z]]$. Taking the derived quotient by
$zG[[z]]$ amounts to adding ghost valued in the Lie algebra and taking
quotient by $G$ amounts to taking $G$ invariant by hand.

A full description of the local operators should also include monopole
operators, which are defined by specifying some singular behavior of fields
around a point. A general strategy for dealing with these operators is to use
state-operator correspondence. For an $n$ dimensional TQFT, state-operator
correspondence tells us that the space of local operators can be identified
with the Hilbert space of state $Z(S^{n-1})$ on the $n-1$ sphere surrounding
the point. More generally, local operators of a $n$ dimensional CFT are in
one-to-one correspondence with states in the radially quantized Hilbert space
of the theory. This method is used in [14, 13, 15] to construct monopole
operators in ”physical” 3d theories, and is also closely related to the BFN
construction of the Coulomb branch operators [40, 41]. The theory we consider
is holomorphic topological. Therefore, instead of the sphere $S^{2}$, it will
be more convenient to consider the following punctured cylinder:

$C^{\times}=D_{\epsilon}\times[-\epsilon,\epsilon]\backslash\\{(0,0)\\}\,.$
(3.5)

We take the limit $\epsilon\to 0$ in the end. Our goal is to construct the
Hilbert space associated to this punctured cylinder. The standard procedure
for doing this consist of two steps, we first construct the phase space on
this cylinder as a symplectic manifold and them perform geometric
quantization.

$\Rightarrow$$\times$$\times$$=$$\mathbb{B}$

First we analyze the phase space $\mathrm{EOM}(C^{\times})$ as the derived
space of solutions to the equations of motion on $C^{\times}$. It is important
to note that this space describes maps that are independent of $t$. Therefore
we can make the following simplification. We take two sets of solution on
$C_{\epsilon}$, which is determined only by the data on $D_{\epsilon}$. Then
we glue the two sets of solutions together via an isomorphism over the
punctured disk $D^{\times}_{\epsilon}$. We expect to have a equivalence
between the phase space on $C^{\times}$ and the space constructed via the
gluing procedure. Again, taking $\epsilon\to 0$ we replace disk by formal disk
$\mathbb{D}$. The above analysis suggest us that the phase space can be
constructed formally as solutions on the ”ravioli”, or ”formal bubble”
$\mathbb{B}=\mathbb{D}\bigsqcup_{\mathbb{D}^{\times}}\mathbb{D}$ defined by
gluing two formal disk through a punctured disk. More detailed analysis of the
phase space is performed in the following sections.

###  3.1 Chiral multiplets

For theory with only chiral multiplets, we do not expect to have any non-
perturbative local operators. If we try to solve the equations of motion, any
solution will be constant in $t$, and there couldn’t exist non-perturbative
objects that can localize in the $t$ direction. In this case, the smallest
dimensional non-perturbative objects are line defects, instead of local
operators. Therefore, the perturbative description of the local operators
should suffice, and we get from 3.4 functions on the infinite jet of the
derived critical locus of $W$.

$\mathrm{Obs}=\mathbb{C}[J_{\infty}\mathrm{dCrit}(W)]\,.$ (3.6)

Although monopole operators are absent, the state-operator correspondence is
still valid to describe the local operators. This provides us with an
alternative way to compute the space of local operators, which should give us
the same answer as the perturbative analysis. We will do this in the following
because this process exhibits us the simplest example of computation on the
formal bubble, which we will generalize later.

For chiral multiplets without superpotential the phase space we need to
consider is $\mathrm{Map}(\mathbb{B}_{\bar{\partial}},T^{*}[1]V)$. The
”derived” structure of this phase space is essential for our analysis
111Physicist usually only care about algebra at the degree zero part of the
cohomology. However in our setup the whole complex becomes important because
elements of non zero cohomological degree might come from physical fields
before twisting.. For example, ring of functions on
$\mathbb{B}_{\bar{\partial}}$ can be modeled on the Čech cohomology, which
give us the following complex

$0\to\mathbb{C}[[z]]\oplus\mathbb{C}[[z]]\overset{d}{\to}\mathbb{C}((z))\to
0\,,$ (3.7)

where the differential $d$ is given by $d(f(z),g(z))=f(z)-g(z)$ for
$(f(z),g(z))\in\mathbb{C}[[z]]\oplus\mathbb{C}[[z]]$. Similarly, the phase
space of the free chiral multiplet is modeled on the complex

$0\to(T^{*}[1]V)[[z]]^{\oplus 2}\overset{d}{\to}(T^{*}[1]V)((z))\to 0\,.$
(3.8)

More precisely, we have direct sum of complexes
$0\to(V^{(r)}[[z]]dz^{r})^{\oplus 2}\to V^{(r)}((z))dz^{r}\to 0$ and
$0\to(V^{(r)*}[1][[z]]dz^{1-r})^{\oplus 2}\to V^{(r)*}[1]((z))dz^{1-r}\to 0$.
Here we can absorb the factor $dz^{r}$ into $V$ and $dz^{1-r}$ into
$V^{*}[1]$. By this notation $V$ now have twisted spin grading and so does
$V^{*}$. To correctly incorporate the spin grading of them the shifted
cotangent bundle $T^{*}[1]V$ need to be corrected to $T^{*}[1](1)V$, where by
$(1)$ we shift the twisted spin grading. To simplify the notation we still
denote it by $T^{*}[1]V$. Remembering the shifting of spin grading is
important in the computation of the characters of the algebra.

The above complex has cohomology

$\begin{split}\mathcal{P}&=T^{*}[1]V[[z]]\oplus(T^{*}[1]V((z))/T^{*}[1]V[[z]])[-1]\\\
&=V[[z]]\oplus
V^{*}[1][[z]]\oplus(V^{*}[1][[z]])^{*}\oplus(V[[z]])^{*}\,.\end{split}$ (3.9)

The symplectic form is given by the natural pairing between $V[[z]]$ and
$(V[[z]])^{*}$ and the pairing between $V^{*}[1][[z]]$ and
$(V^{*}[1][[z]])^{*}$.

To perform the geometric quantization, we need to choose a polarization. The
choice that reproduce our perturbative result is to consider the Lagrangian
fibration

$\pi:\mathcal{P}\to T^{*}[1]V\otimes\check{H}^{0}(\mathbb{B})=V[[z]]\oplus
V^{*}[1][[z]]\,.$ (3.10)

By this choice of polarization we consider functions on $\mathcal{P}$ that is
constant along the fiber of $\pi$, which is equivalent to functions on
$(T^{*}[1]V)[[z]]=J_{\infty}T^{*}[1]V$.

We could, of course, chose other polarization. For example, we could take the
polarization to be the Lagrangian fibration over

$\mathrm{Map}(\mathbb{B}_{\bar{\partial}},V)=V[[z]]\otimes(V^{*}[1][[z]])^{*}\,.$
(3.11)

This looks like the polarization much often used in other related setups.
However, global functions on $V((z))/V[[z]][-1]$ are not well defined. If we
naively take functions on $V[z^{-1}]z^{-1}[-1]$ we meet the problem of
operators of arbitrary negative spin. In this case, we expect the Hilbert
space to be functions on $V\otimes\check{H}^{0}(\mathbb{B})$ tensoring with
”distributions” on $V\otimes\check{H}^{1}(\mathbb{B})$ (or global sections of
the dualizing sheaf). Similar phenomenal appears in [12] in studying the
boundary chiral algebra, where Dolbeault homology instead of cohomology was
used to construct boundary operators, and in [41] where equivariant Borel-
Moore homology was used to define Coulomb branch operators. In any case, the
space of local operators we get should be the only reasonable answer by
appropriate construction, which are functions on $J_{\infty}T^{*}[1]V$:

$\mathrm{Obs}=\mathbb{C}[J_{\infty}T^{*}[1]V]\,.$ (3.12)

To turn on a superpotential, we simply replace $T^{*}[1]V$ by
$\mathrm{dCrit}(W)$ in the above analysis. This gives us
$\mathrm{Obs}=\mathbb{C}[J_{\infty}\mathrm{dCrit}(W)]$, which is the same as
the perturbative analysis.

Its useful to write down an explicit expression for the local operators. Here
we take the usual physical notation. We denote $\partial^{n}\phi_{i}$ the
$n$-th $z$ derivatives of the bottom component of the field $\Phi_{i}$ at
$z=0$, and $\partial^{n}\psi_{i}$ the $n$-th $z$ derivatives of the bottom
component of the field $\Psi_{i}$ at $z=0$. They serve as the coordinates on
the infinite jet of $V$ and $V^{*}[1]$ respectively. Then the space of local
operators can be written as
$\mathbb{C}[\\{\partial^{n}\phi_{i},\partial^{n}\psi_{i}\\}_{n\geq 0,i}]$. The
superpotential $W$ gives us a non-zero differential

$Q\partial^{n}\psi=\partial^{n}(\frac{\delta W(\phi)}{\delta\phi})\,.$ (3.13)

This expression will be useful later when we compare the Ext computation with
the direct bulk computation.

###  3.2 Theories with gauge fields

In this section, we take vector multiplets into consideration. The existence
of monopole operators makes these theories much more complicated than theories
with only chiral multiplets. We will not attempt to give a systematic study of
vector multiplets. Instead, we focus on theories with only abelian gauge
group. Their moduli spaces of equations of motion have simpler structure,
which allows us to avoid many technical details but is still able to give us
some nontrivial results.

####  3.2.1 Perturbative algebra

Let’s first consider pure gauge theories. According to our previous discussion
3, perturbative local operators consist of $G$ invariant functions of fields
$b(z),c(z)$ with constant ghost modes removed. This is exactly the cochain
complex of relative Lie algebra cohomology. We can also obtain this result
through our ”definition” 3.4. By taking $V=0$, we have

$\mathrm{Obs}^{\text{per}}=\mathbb{C}[J_{\infty}(\mathfrak{g}^{*}/G)]=\mathbb{C}[\mathfrak{g}^{*}[[z]]/G[[z]]]\,.$
(3.14)

As we have explained, we take $zG[[z]]$ invariant through introducing ghost
valued in the Lie algebra of $zG[[z]]$ and $G$ invariant is taken by hand.
Then we get relative Lie algebra cohomology

$\mathrm{Obs}^{\text{per}}=C^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\mathfrak{g}[[z]],\mathfrak{g},\mathrm{Sym}(\mathfrak{g}^{*}[[z]])^{*})\,.$
(3.15)

When $\mathfrak{g}$ is semisimple, we can identify $\mathfrak{g}^{*}$ with
$\mathfrak{g}$ by the Cartan-Killing form. Then the local operator algebra can
be equivalently written as

$\mathrm{Obs}^{\text{per}}=C^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\mathfrak{g}[[z,\varepsilon]],\mathfrak{g})\,,$
(3.16)

where $\varepsilon$ is a odd parameter satisfying $\varepsilon^{2}=0$. This
relative Lie algebra cohomology is computed in [42]. Here we follow the
notation of [43]. Denote
$C_{\mathfrak{g}^{*}}:=\mathfrak{g}^{*}/G=\mathrm{Spec}(\mathbb{C}[\mathfrak{g}^{*}]^{G})$.
We choose generators $P_{i},i=1,\dots l$ of $\mathbb{C}[\mathfrak{g}^{*}]^{G}$
of degree $d_{i}+1$. Then
$\mathbb{C}[C_{\mathfrak{g}^{*}}]=\mathbb{C}[P_{1},\dots,P_{l}]$. Define the
local Hitchin space as

$C_{\mathfrak{g}^{*},K}=\Gamma(\mathbb{D},K\times_{\mathbb{C}^{\times}}C_{\mathfrak{g}^{*}})\,,$
(3.17)

where $K$ is the canonical line bundle on the disk. We can identify the ring
of functions $\mathbb{C}[C_{\mathfrak{g}^{*},K}]$ on the local Hitchin space
as the polynomial algebra $\mathbb{C}[\\{P_{i,n}\\}_{i=1,\dots,l;n\geq 0}]$.
It was proved in [42] that, there is an isomorphism of graded algebras

$H^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\mathfrak{g}[[z,\varepsilon]],\mathfrak{g})=\Omega^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(C_{\mathfrak{g}^{*},K})\,.$
(3.18)

The ghost number zero part
$H^{0}(\mathfrak{g}[[z,\varepsilon]],\mathfrak{g})=\mathcal{O}(C_{\mathfrak{g}^{*},K})$
of this algebra have been extensively studied in the context of vertex
algebra, hitchin system and geometric Langlands. We define the center
$\mathfrak{z}(\hat{\mathfrak{g}})=\mathrm{End}_{\hat{\mathfrak{g}}_{\kappa_{c}}}V_{\kappa_{c}}$
of the vacume verma module $V_{\kappa_{c}}$ at critical level.
$\mathfrak{z}(\hat{\mathfrak{g}})$ has a filtration induced from the
filtration of $V_{\kappa_{c}}$. Then we have an isomorphism of the the
associated graded

$\text{gr
}\mathfrak{z}(\hat{\mathfrak{g}})\cong\mathcal{O}(C_{\mathfrak{g}^{*},K})\,.$
(3.19)

Moreover, there is an isomorphism of filtered algebras [44, 45]

$\mathfrak{z}(\hat{\mathfrak{g}})\cong\mathcal{O}(\mathrm{Op}_{{{}^{L}}\mathfrak{g}}(\mathbb{D}))\,,$
(3.20)

where $\mathrm{Op}_{{{}^{L}}\mathfrak{g}}(\mathbb{D})$ is the space of
${{}^{L}}\mathfrak{g}$-opers on $\mathbb{D}$. The center
$\mathfrak{z}(\hat{\mathfrak{g}})$ can be further identified with the
classical $\mathcal{W}$ algebra $\mathcal{W}_{\infty}({{}^{L}}\mathfrak{g})$
[44]. The appearance of the Langlands dual Lie algebra ${{}^{L}}\mathfrak{g}$
here is very interesting. The proof of 3.20 in [44, 45] and the appearance of
the Langlands dual Lie algebra is essentially a consequence of T-duality in
$2d$. We hope that there could also be an S-duality argument for it.

The algebras $\mathcal{O}(C_{\mathfrak{g}^{*},K})$ and
$\mathfrak{z}(\hat{\mathfrak{g}})$ also play important roles in the Hitchin
integrable system and its quantization. There is a classical Hitchin
homomorpism [29, 28]

$h^{\text{cl}}:\mathcal{O}(C_{\mathfrak{g}^{*},K})\to\Gamma(\mathrm{Bun}_{G},p^{*}\mathcal{O}_{T^{*}\mathrm{Bun}_{G}})\,,$
(3.21)

and its quantization [29]

$h:\mathfrak{z}(\hat{\mathfrak{g}})\to\Gamma(\mathrm{Bun}_{G},\mathcal{D}^{\prime}_{T^{*}\mathrm{Bun}_{G}})\,.$
(3.22)

This provides an instance of the connection between the twisted $3d$
$\mathcal{N}=2$ theory and integrable system that we outlined in the
introduction 1.2.

When there is a bare Chern-Simons term, we turn on a differential
$\varepsilon\partial_{z}$ on $\mathfrak{g}[[z,\varepsilon]]$ and make it into
a dg Lie algebra. We have a quasi-isomorphism

$\mathfrak{g}[[z,\varepsilon]]_{\text{deformed}}:=(\mathfrak{g}[[z,\varepsilon]],\varepsilon\partial_{z})\to\mathfrak{g}\,.$
(3.23)

This map induces a quasi-isomorphism on Lie algebra cochain

$C^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\mathfrak{g},\mathfrak{g})\to
C^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\mathfrak{g}[[z,\varepsilon]]_{\text{deformed}},\mathfrak{g})\,.$
(3.24)

Therefore the perturbative algebra for vector multiplet with a Chern-Simons
term becomes trivial in cohomology. We can also see this by noting that, turn
on the Chern-Simons term simply turn on the De Rham differential on
$\Omega^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(C_{\mathfrak{g}^{*},K})$ [43].

In the presence of chiral multiplets, perturbative operator algebra is given
by 3.4. However, we don’t have too much knowledge about its cohomology at the
moment.

####  3.2.2 Non-perturbative algebra

Now we turn to discuss the full non-perturbative algebra. In this paper, we
only consider the special case of $U(1)$ gauge theory since the structure of
the phase space is more accessible. The complexified gauge group is
$G=\mathbb{C}^{\times}$, and the phase space for pure gauge theory is
$\mathrm{Map}(\mathbb{B}_{\bar{\partial}},T^{*}[1]BG)$. Geometrically, the
space $\mathrm{Map}(\mathbb{B}_{\bar{\partial}},T^{*}[1]BG)$ describe a
$\mathbb{C}^{\times}$ bundle on $\mathbb{B}_{\bar{\partial}}$ together with a
coadjoint valued section. The space of holomorphic $\mathbb{C}^{\times}$
bundles on $\mathbb{B}$ as a set is given by $\mathbb{Z}$. For each integer
$m$, we get a $\mathbb{C}^{\times}$ bundle $P_{m}$ by defining its transition
function on the overlap $\mathbb{D}^{\times}$ to be $z^{m}$. Therefore our
phase space has a decomposition:

$\mathrm{EOM}(\mathbb{B})=\prod_{m\in\mathbb{Z}}\mathrm{EOM}^{m}(\mathbb{B})\,.$
(3.25)

Each component $\mathrm{EOM}^{n}(\mathbb{B})$ describe the space of coadjoint
valued sections of $P_{m}$ module gauge transformation. It contains rich
structure as a ”derived space”. It has a tangent space (complex)
$\mathrm{R}\Gamma(\mathbb{B},P_{m}\times_{\mathbb{C}^{\times}}(\mathfrak{gl}_{1}^{*}\oplus\mathfrak{gl}_{1}[1]))$.
Computation of its cohomology is similar as the chiral multiplet case, and we
find

$\mathfrak{gl}_{1}^{*}[[z]]\oplus\mathfrak{gl}_{1}[1][[z]]\oplus(\mathfrak{gl}_{1}^{*}((z))/\mathfrak{gl}_{1}^{*}[[z]])[-1]\oplus(\mathfrak{gl}_{1}((z))/\mathfrak{gl}_{1}[[z]])$
(3.26)

We used the fact that the $\mathbb{C}^{\times}$ action on $\mathfrak{gl}_{1}$
is trivial. The symplectic structure is given by the natural pairing between
$\mathfrak{gl}_{1}^{*}[[z]]$ and
$(\mathfrak{gl}_{1}((z))/\mathfrak{gl}_{1}[[z]])$, and the pairing between
$\mathfrak{gl}_{1}[1][[z]]$ and
$(\mathfrak{gl}_{1}^{*}((z))/\mathfrak{gl}_{1}^{*}[[z]])[-1]$.

Next we construct the polarization. To avoid operators of arbitrary negative
spin we don’t use the Lagrangian fibration
$T^{*}\mathrm{Map}(\mathbb{B}_{\bar{\partial}},BG)\to\mathrm{Map}(\mathbb{B}_{\bar{\partial}},BG)$.
Instead, we take polarized sections to be constant along
$(\mathfrak{gl}_{1}^{*}((z))/\mathfrak{gl}_{1}^{*}[[z]])[-1]\oplus(\mathfrak{gl}_{1}((z))/\mathfrak{gl}_{1}[[z]])$
directions as in Section 3.1. After this polarization we find the Hilbert
space to be the space of functions on

$\prod_{m\in\mathbb{Z}}J_{\infty}\mathfrak{gl}_{1}^{*}/J_{\infty}\mathbb{C}^{\times}\,,$
(3.27)

where, as before, the quotient is understand in the derived sense. Its easy to
see that for $m=0$ we reproduce the perturbative algebra.

For theories with chiral multiplet, the analysis is similar. To simplify the
notation we consider the case when the superpotential is zero. The phase
space, as before, has $\mathbb{Z}$ disconnected components labeled by
$\mathbb{C}^{\times}$ bundle $P_{m}$. For each component labeled by $P_{m}$,
the tangent complex is given by

$\mathrm{R}\Gamma(\mathbb{B},P_{m}\times_{\mathbb{C}^{\times}}(T^{*}[1]V\oplus\mathfrak{gl}_{1}^{*}\oplus\mathfrak{gl}_{1}[1])\,.$
(3.28)

As an illustration, we first compute
$\mathrm{R}\Gamma(\mathbb{B},P_{m}\times_{\mathbb{C}^{\times}}V)$. The
$\mathbb{C}^{\times}$ action on $V$ is no longer trivial. We decompose $V$
into wight space $V=\oplus_{w\in\mathbb{Z}}V_{w}$. Then
$\mathrm{R}\Gamma(\mathbb{B},P_{m}\times_{\mathbb{C}^{\times}}V)$ decompose
into

$\oplus_{w}\mathrm{R}\Gamma(\mathbb{B},P_{m}\times_{\mathbb{C}^{\times}}V_{w})\,.$
(3.29)

Each summand is computed by the following Čech complex

$0\to V_{w}[[z]]\oplus V_{w}[[z]]\overset{d_{w}}{\longrightarrow}V_{w}((z))\to
0\,,$ (3.30)

where the differential $d$ is given by $d_{w}(f(z),g(z))=f(z)-z^{wm}g(z)$ for
any $(f(z),g(z))\in V_{w}[[z]]\oplus V_{w}[[z]]$. We find that

$H^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\mathbb{B},P_{n}\times_{\mathbb{C}^{\times}}V_{w})=\begin{cases}V_{w}[[z]]z^{wm}\oplus(V_{w}((z))/V_{w}[[z]])[-1],&wm\geq
0\\\ V_{w}[[z]]\oplus(V_{w}((z))/V_{w}[[z]]z^{wm})[-1],&wm<0\end{cases}\,.$
(3.31)

Similarly,

$H^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\mathbb{B},P_{m}\times_{\mathbb{C}^{\times}}V_{w}^{*}[1])=\begin{cases}V_{w}^{*}[[z]][1]\oplus(V_{w}^{*}((z))/V_{w}^{*}[[z]]z^{-wm}),&wm\geq
0\\\
V_{w}^{*}[[z]][1]z^{-wm}\oplus(V_{w}^{*}((z))/V_{w}^{*}[[z]]),&wm<0\end{cases}\,.$
(3.32)

The symplectic pairing is given by the pairing between
$H^{0}(\mathbb{B},P_{m}\times_{\mathbb{C}^{\times}}V_{w})$ and
$H^{0}(\mathbb{B},P_{m}\times_{\mathbb{C}^{\times}}V_{w}^{*}[1])$ and the
pairing between $H^{1}(\mathbb{B},P_{m}\times_{\mathbb{C}^{\times}}V_{w})$ and
$H^{-1}(\mathbb{B},P_{m}\times_{\mathbb{C}^{\times}}V_{w}^{*}[1])$. Our
previous experience suggest that we should chose the polarization to be the
Lagrangian fibration over

$\Gamma(\mathbb{B},P_{m}\times_{\mathbb{C}^{\times}}T^{*}[1]V_{w})=\begin{cases}V_{w}[[z]]z^{wm}\oplus
V_{w}^{*}[[z]][1],&wm\geq 0\\\ V_{w}[[z]]\oplus
V_{w}^{*}[[z]]z^{-wm}[1],&wm<0\end{cases}\,.$ (3.33)

Summing over weight, and combining with the gauge fields, we find that the
local operator algebra is given by functions on the space

$\prod_{m\in\mathbb{Z}}\Gamma(\mathbb{B},P_{m}\times_{\mathbb{C}^{\times}}T^{*}[1]V\oplus\mathfrak{gl}_{1}^{*})/J_{\infty}\mathbb{C}^{\times}\,.$
(3.34)

This construction easily generalize to the case of an arbitrary abelian gauge
theory. For example, when the gauge group is $G=U(1)^{r}$, and we have chiral
multiplet in a representation $V$ of $G$, then local operators can be
constructed from the following space

$\prod_{m\in\mathbb{Z}^{r}}\Gamma(\mathbb{B},P_{m}\times_{\mathbb{C}^{\times}}T^{*}[1]V\oplus(\mathfrak{gl}_{1}^{*})^{r})/(J_{\infty}\mathbb{C}^{\times})^{r}\,.$
(3.35)

For non abelian gauge group, we expect that, for pure gauge theory, the bulk
algebra can be given by derived $J_{\infty}G$ invariants of the WZW vacuum
module $\mathrm{WZW}_{k}[\mathfrak{g}]$. However, it is not clear what the
full algebra will look like for the most general cases.

###  3.3 Superconformal index and operator counting

An important tool in studying physical $3d$ $\mathcal{N}=2$ theories is the
superconformal index. It computes the following partition function for a
theory defined on $S^{2}\times S^{1}$ [46, 17]

$I(t_{a};x)=\mathrm{Tr}((-1)^{F}e^{-\beta(E-R-
j_{3})}x^{E+j_{3}}\prod_{a}t_{a}^{F_{a}})\,.$ (3.36)

Here $E$ is the energy, $R$ is the R-charge, $j_{3}$ is the third component of
the angular momentum, and $F_{a}$ run over the global flavor symmetry.
Standard physical arguments tells us that only states with $E=R+j_{3}$
contributes to the index. Therefore we can simplify the index as

$I(t_{a};x)=\mathrm{Tr}((-1)^{F}q^{\frac{R}{2}+j_{3}}\prod_{a}t_{a}^{F_{a}})\,.$
(3.37)

where $q=x^{2}$. This index is usually computed by localization technique [16,
18].

In this paper we study the holomorphic twisted version of physical $3d$
$\mathcal{N}=2$ theories. To any such a theory we consider its local operator
algebra $\mathrm{Obs}$. This algebra comes equipped with twisted spin grading
$J$, fermionic grading $F$, and other gradings $F_{a}$ associated with some
global symmetry. Then we can define the character

$\chi(\mathrm{Obs})=\mathrm{Tr}_{\mathrm{Obs}}((-1)^{F}q^{J}\prod_{a}t_{a}^{F_{a}})\,.$
(3.38)

The twisting procedure identify the twisted spin with $J=\frac{R}{2}+j_{3}$
and keep other flavor symmetry unchanged. It turns out that this character
defined for the holomorphic twisted theory coincide with the superconformal
index of the physical theory. On the one hand, this provide us an alternative
method to compute the superconformal index by counting local operators. On the
other hand, we can think of the local operators as providing a
”categorification” of the index.

For example, for a free chiral multiplet, the space of local operator is
$\mathbb{C}[\\{\partial^{n}\phi,\partial^{n}\psi\\}_{n\geq 0}]$. This theory
has a flavor symmetry that rotate the field $\phi$ and $\psi$. Suppose the
spin and flavor charge of the fields is given as follows

| $\phi$ | $\psi$
---|---|---
$T$ | 1 | -1
$J$ | 0 | 1

Then the operator $\partial^{n}\phi$ contribute $q^{n}x$ to the index and
$\partial^{n}\psi$ contribute $q^{n+1}x^{-1}$ to the index. We can easily
write down the index of a free chiral

$\chi(\mathrm{Obs}_{\text{single
chiral}})=\frac{(qx^{-1},q)_{\infty}}{(x,q)_{\infty}}\,,$ (3.39)

where we used the $q$-Pochhammer symbols

$(x;q)_{\infty}=\prod_{n\geq 0}(1-q^{n}x)\,.$ (3.40)

####  3.3.1 Chern-Simons line bundle twist

In this section, we consider theory with a Chern-Simons term. In the presence
of a Chern-Simons term, the Hilbert space is no longer functions on the
polarized phase space, but sections of a line bundle $\mathcal{L}^{\otimes
k}$, with $k$ being the Chern-Simons coefficient. $\mathcal{L}$ is also called
determinant line bundle. Details of the construction of this line bundle for
an arbitrary gauge group are given in [47, 48], here we present its
construction for the abelian group $\mathbb{C}^{\times}$.

A $G$ bundle on the formal bubble $\mathbb{B}$ can be defined by a transition
function on $\mathbb{D}$ module gauge transformations on the two copies of
$\mathbb{D}$. Therefore we can regard $\mathrm{Map}(\mathbb{B},BG)$ as a
double quotient

$G[[z]]\backslash G((z))/G[[z]]=G[[z]]\backslash\mathrm{Gr}_{G}\,.$ (3.41)

The affine Grassmannian $\mathrm{Gr}_{G}$ describe $G$ bundle on the disk
$\mathbb{D}$ trivialized on the punctured disk $\mathbb{D}^{\times}$.

Let $R=\mathbb{C}_{\text{fund}}$ be the fundamental representation of
$G=\mathbb{C}^{\times}$. For a point in $\mathrm{Map}(\mathbb{B},BG)$
represented by a $P\in\mathrm{Gr}_{\mathbb{C}^{\times}}$, we get a associated
bundle $R_{P}$ on $\mathbb{D}$. For $P$ equivalent to the $P_{m}$ that we
introduced in 3.2.2, we can describe the space of sections of $R_{P}$ as

$R[[z]]z^{m},\;\;\text{for }m\in\mathbb{Z}\,.$ (3.42)

We define a map $F:\Gamma(\mathbb{D},R_{P})\to R[[z]]$ by the following
composition of maps

$\Gamma(\mathbb{D},R_{P})\hookrightarrow\Gamma(\mathbb{D}^{\times},R_{P})=R((z))\to
R[[z]]\,.$ (3.43)

Then we define the fiber of the Chern-Simons line bundle at $P$ to be the
following

$\mathcal{L}_{P}=\det(\ker F)\otimes\det(\text{coker}F)^{-1}\,.$ (3.44)

We are interested in the character of this space under the action of gauge
group $\mathbb{C}^{\times}$ and spin rotation $\mathbb{C}^{\times}_{q}$.
Suppose $P=P_{m}$ and $m<0$. We have $\Gamma(\mathbb{D},R_{P})=R[[z]]z^{-m}$,
the map $F$ is injective and has cokernel

$\mathrm{coker}F=\bigoplus_{0\leq i\leq-m}Rz^{i}\,.$ (3.45)

We can compute the character

$\chi(\mathrm{coker}F)=\prod_{i=0}^{m-1}\chi(\det
Rz^{i})=\prod_{i=0}^{-m-1}sq^{-i}=s^{m}q^{-\frac{(m+1)m}{2}}\,.$ (3.46)

For $m\geq 0$, the map $F:R[[z]]z^{-m}\to R[[z]]$ is surjective and has kernel

$\ker F=\bigoplus_{-m\leq i\leq-1}Rz^{i}\,.$ (3.47)

It has character

$\chi(\ker F)=\prod_{i=-1}^{-m}sq^{-i}=s^{-m}q^{\frac{(m+1)m}{2}}\,.$ (3.48)

We find that the character of $\mathcal{L}_{P}$ is

$s^{-m}q^{\frac{(m+1)m}{2}}\,\;\;\text{ for }m\in\mathbb{Z}\,.$ (3.49)

####  3.3.2 Berezinian and Half-form twist

The final step in geometric quantization is the so called half-form
correction. Here we briefly explain the idea of it. Suppose that in the
quantization problem, the polarization is given by a Lagrangian fibration over
$X$. If we consider the space of polarized sections as the Hilbert space
$\mathcal{H}$ of quantum states, we meet the problem of defining the inner
product on $\mathcal{H}$. The inner product should be defined via integration,
but integration of polarized sections over the whole phase space diverge, and
there is no natural choice of a measure on $X$. So we are unable to define a
inner product on the space of polarized sections. To remedy this situation, we
consider the following half-form bundle

$(\det\Omega^{1}X)^{\frac{1}{2}}\,.$ (3.50)

Suppose $X$ has dimension $n$. Then $\det\Omega^{1}X$ has the familiar form
$\wedge^{n}\Omega^{1}X=\Omega^{n}X$ as top degree differential form. We tensor
the quantum line bundle with the half-form bundle. Then two sections can be
paired and integrated over $X$. This gives us the notion of inner product on
the Hilbert space.

In our situation, we get a supermanifold $\mathcal{X}$ after polarization.
Differential form of top degree no longer make sense on $\mathcal{X}$, as the
one forms in the Grassmannian odd directions are Grassmannian even variables.
The right definition of integration over a supermanifold $\mathcal{X}$ is
provided by the Berezinian integration, where the integration map is defined
on the space of Berezinian, which replace the determinant, of one form

$\int:\;\operatorname{Ber}\Omega^{1}\mathcal{X}\to\mathbb{C}\,.$ (3.51)

We provide the definition and properties of the Berezinian in Appendix B.

The half-form bundle in our case becomes

$(\operatorname{Ber}\Omega^{1}\mathcal{X})^{\frac{1}{2}}\,.$ (3.52)

In the case of free chiral multiples, we get $J_{\infty}T^{*}[1]V$ after
polarization. The above discussion suggest that we consider the following line
bundle

$\sqrt{\operatorname{Ber}\Omega^{1}(J_{\infty}T^{*}[1]V)}=\sqrt{\operatorname{Ber}(T^{*}[1]V[[z]])^{*}}\otimes\mathcal{O}_{J_{\infty}T^{*}[1]V}\,.$
(3.53)

The space $(J_{\infty}T^{*}[1]V)^{*}$ is infinite dimensional and its
Berezinian is not well defined. Its character is not well defined as well. The
power of gauge fugacity $s$ diverges if we try to compute the character. The
superconformal index of a free chiral in physical computation also coincides
with 3.39 without any correction, this suggests that we should ignore the half
form correction in this case.

It becomes more interesting when the chiral multiplet is coupled with a $U(1)$
vector multiplet. In this case we have
$\Gamma(\mathbb{B},P_{m}\times_{\mathbb{C}^{\times}}T^{*}[1]V)$ as the base of
fibration for the monopole charge $m$ sector. Although it is again an infinite
dimensional space, we get a finite and well defined object by dividing the
infinite part we found before. Here, we consider the following ”regularized”
half form correction

$L_{h}(m)=\left(\frac{\operatorname{Ber}\Gamma(\mathbb{B},P_{m}\times_{\mathbb{C}^{\times}}T^{*}[1]V_{w})^{*}}{\operatorname{Ber}(T^{*}[1]V[[z]])^{*}}\right)^{\frac{1}{2}}=\left(\frac{\operatorname{Ber}\Gamma(\mathbb{B},P_{m}\times_{\mathbb{C}^{\times}}T^{*}[1]V_{w})}{\operatorname{Ber}(T^{*}[1]V[[z]])}\right)^{-\frac{1}{2}}\,.$
(3.54)

Using the properties of Berezinian that

$\operatorname{Ber}(V_{1}\oplus
V_{2})=\operatorname{Ber}(V_{1})\otimes\operatorname{Ber}(V_{2})\,,$ (3.55)

we can rewrite 3.54 as the Berezinian of a finite dimensional space. We
substitute 3.33 into the expression and found that each wight space $V_{w}$
contribute the half form bundle by the following

$L_{h,w}(m)=\begin{cases}(\operatorname{Ber}(\oplus_{i=0}^{wm-1}V_{w}z^{i}))^{\frac{1}{2}},&wm\geq
0\\\
(\operatorname{Ber}(\oplus_{i=0}^{-wm-1}V^{*}_{w}[1]z^{i}))^{\frac{1}{2}},&wm<0\end{cases}\,.$
(3.56)

We have

$L_{h}(m)=\bigotimes_{w}L_{h,w}(m)\,.$ (3.57)

For $wm\geq 0$, the space $V_{w}z^{i}$ has degree $0$, so the Berezinian
becomes the usual determinant. We can compute the character of the determinant
by taking product of the characters of a basis of it. We have

$\chi(\operatorname{Ber}(V_{w}z^{i}))=s^{w\dim V_{w}}q^{-i\dim V_{w}}\,,$
(3.58)

where $s$ is the gauge $\mathbb{C}^{\times}$ fugacity. For the character of
the Berezinian of $V^{*}_{w}[1]z^{i}$, we consider the
$\mathbb{C}^{\times}\times\mathbb{C}^{\times}_{q}$ action on
$V^{*}_{w}[1]z^{i}$ which induce an action on the Berezinian. The action on
$V^{*}_{w}[1]z^{i}$ is given by a diagonal matrix
$\mathrm{diag}(s^{-w}q^{-i-1},s^{-w}q^{-i-1},\dots,s^{-w}q^{-i-1})$ that sit
completely in degree $1$. The Berezinian of this diagonal matrix is
$(s^{-w}q^{-i-1})^{-\dim V_{w}}$, we get

$\chi(\operatorname{Ber}(V^{*}_{w}[1]z^{i}))=(-1)^{\dim V_{w}}s^{w\dim
V_{w}}q^{(i+1)\dim V_{w}}\,,$ (3.59)

where we also include a $(-1)$ factor coming from the $\mathbb{Z}_{2}$ grading
of the Berezinian B. We temporarily ignore the issue of possibly other flavor
symmetry acting on $V$. We find that, for $wm\geq 0$

$\begin{split}\chi(L_{h,w}(m))=&\prod_{i=0}^{wm-1}\chi(\operatorname{Ber}(V_{w}z^{i})^{\frac{1}{2}})\\\
=&s^{\frac{1}{2}mw^{2}\dim V_{w}}q^{\frac{-wm(wm-1)}{4}\dim
V_{w}}\,,\end{split}$ (3.60)

and for $wm<0$

$\begin{split}\chi(L_{h,w}(m))=&\prod_{i=0}^{-wm-1}\chi(\operatorname{Ber}(V^{*}_{w}[1]z^{i}))\\\
=&(-1)^{mw\dim V_{w}}s^{-\frac{1}{2}mw^{2}\dim V_{w}}q^{\frac{wm(wm-1)}{4}\dim
V_{w}}\,.\end{split}$ (3.61)

We can get the character of the half form bundle by

$\chi(L_{h}(m))=\prod_{w}\chi(L_{h,w}(m))\,.$ (3.62)

In summary, for theory with Chern-SImons level $k$, the space of local
operators can be identified with sections of the line bundle
$\mathcal{L}_{h}\otimes\mathcal{L}_{CS}^{\otimes k}$. In fact, this line
bundle contains information about ”one loop correction” in the original
physical theory. For example, we can consider the gauge $\mathbb{C}^{\times}$
character of this bundle restricted to the monopole charge $m=\pm 1$ sectors

$\chi_{\mathbb{C}^{\times}}(\mathcal{L}_{h}\otimes\mathcal{L}_{CS}^{\otimes
k}(\pm 1))=s^{\mp k+\sum_{w}\frac{1}{2}w|w|\dim V_{w}}\,.$ (3.63)

This correspond to the $\mathbb{C}^{\times}$ gauge charge of the two ”bare”
monopole operators, and physically comes from the one loop effective Chern-
Simons level. This is compatible with results in physics literature [49].
Similarly, the quantum correction of spin of the monopole operator is also
contained in this construction.

Using the decomposition 3.33, we see that the local operator algebra can be
decomposed into different monopole sectors

$\mathrm{Obs}=\bigoplus_{m\in\mathbb{Z}}\mathrm{Obs}^{m}\,.$ (3.64)

Each sector $\mathrm{Obs}^{m}$ consists of a family of dressed monopole
operators computed by the cohomology of some complex. The ”dressing” is
provided by some polynomial of the fields, carrying a certain
$\mathbb{C}^{\times}$ gauge charge that cancels the gauge charge of the line
bundle 3.63 on that monopole component. We will provide some explicit examples
in the next section.

##  4 Examples

In this section, we apply the general discussion above to some specific
examples. We explicitly write down the chain complexes of the operator
algebras. Unfortunately, we are unable to compute the whole cohomologies of
these complexes. We only compute the cohomologies of operators of small spins
as an illustration. On the other hand, the character can be easily computed by
counting operators directly from the cochain complex.

###  4.1 XYZ model

We first consider the XYZ model, which is one of the most important examples
in this paper. This theory has three chiral multiplets denoted
$(\mathbf{X},\mathbf{\Psi_{X}})$, $(\mathbf{Y},\mathbf{\Psi_{Y}})$,
$(\mathbf{Z},\mathbf{\Psi_{Z}})$, and is equipped with a cubic superpotential

$W=\mathbf{X}\mathbf{Y}\mathbf{Z}\,.$ (4.1)

Using the notation in Section 3.1, the operator algebra is generated by
bosonic operators $\partial^{n}X,\partial^{n}Y,\partial^{n}Z$ for $n\geq 0$
and fermionic operators
$\partial^{n}\psi_{X},\partial^{n}\psi_{Y},\partial^{n}\psi_{Z}$ for $n\geq
0$. We write down the complex of local operator algebra

$\mathrm{Obs}_{XYZ}=\left(\mathbb{C}[\\{\partial^{n}X,\partial^{n}Y,\partial^{n}Z,\partial^{n}\psi_{X},\partial^{n}\psi_{Y},\partial^{n}\psi_{Z}\\}_{n\geq
0}],Q\right)\,,$ (4.2)

with the differential $Q$ given as follows 222More precisely we have
$Q\partial^{n}\psi_{X}=\sum_{0\leq l\leq
n}\binom{n}{l}\partial^{n-l}Y\partial^{l}Z$. However we can always get rid of
this coefficient by a field redefinition
$\partial^{n}\psi_{X}\to\frac{1}{n!}\partial^{n}\psi_{X}$ etc.

$\begin{split}Q\partial^{n}\psi_{X}&=\sum_{0\leq l\leq
n}\partial^{n-l}Y\partial^{l}Z,\\\ Q\partial^{n}\psi_{Y}&=\sum_{0\leq l\leq
n}\partial^{n-l}Z\partial^{l}X,\\\ Q\partial^{n}\psi_{Z}&=\sum_{0\leq l\leq
n}\partial^{n-l}X\partial^{l}Y.\\\ \end{split}$ (4.3)

The XYZ model has two flavor symmetry, denoted $A$ and $T$. The spin and
flavor symmetry charges of various fields is given as follows

| $X$ | $Y$ | $Z$ | $\psi_{X}$ | $\psi_{Y}$ | $\psi_{Z}$
---|---|---|---|---|---|---
$A$ | $2$ | $-1$ | $-1$ | $-2$ | $1$ | $1$
$T$ | $0$ | $1$ | $-1$ | $0$ | $-1$ | $1$
$J$ | $0$ | $\frac{1}{2}$ | $\frac{1}{2}$ | $1$ | $\frac{1}{2}$ | $\frac{1}{2}$

It’s very hard to compute the cohomology of the whole complex. However, the
complex is graded by spin $J$ and flavor symmetry $T$ and so does the
cohomology. For each spin $J$ and $T$ charge, the cohomology can be computed
by hand. Here, we list part of the cohomology of small spins.

| $J=0$ |  $J=1$ |  $J=2$ | …
---|---|---|---|---
$T=0$ | $\\{X^{n}\\}_{n\geq 0}$ |  $\\{X^{n}\partial X,X^{n}(X\psi_{X}-Y\psi_{Y})\\}_{n\geq 0}$, $Y\psi_{Y}-Z\psi_{Z}$ |  $\\{X^{n}\partial^{2}X,X^{n}(\partial X)^{2}$, $X^{n}\partial(X\psi_{X}-Y\psi_{Y})$, $X^{n}(\partial X\psi_{X}-Y\partial\psi_{Y}+\partial Z\psi_{Z})\\}_{n\geq 0}$, $\partial YZ-Y\partial Z$, $\partial(Y\psi_{Y}-Z\psi_{Z})$, $X\partial(Y\psi_{Y}-Z\psi_{Z})$ |
$T=2$ |  |  $Y^{2}$ |  $Y^{2}(Z\psi_{Z}-Y\psi_{Y})$, $Y\partial Y$ |
… |  |  |  |

| $J=\frac{1}{2}$ |  $J=\frac{3}{2}$ |  $J=\frac{5}{2}$ | …
---|---|---|---|---
$T=1$ | $Y$ |  $\partial XY$, $\partial Y$, $Y(Y\psi_{Y}-Z\psi_{Z})$ |  $(\partial X)^{2}Y$, $\partial^{2}XY$, $Y\partial(Y\psi_{Y}-Z\psi_{Z})$, $Y\partial(X\psi_{X}-Y\psi_{Y})$ , $\partial^{2}Y$ |
$T=-1$ | $Z$ |  $\partial XZ$, $\partial Z$, $Z(Y\psi_{Y}-Z\psi_{Z})$ |  $(\partial X)^{2}Z$, $\partial^{2}XZ$, $Z\partial(Y\psi_{Y}-Z\psi_{Z})$, $Z\partial(X\psi_{X}-Z\psi_{Z})$ , $\partial^{2}Z$ |
… |  |  |  |

Although the whole cohomology is hard to compute, its ghost number zero part
is easy to find. It can be identified with functions on the infinite jet of
the ”non derived” critical locus

$\mathbb{C}[\\{\partial^{n}X,\partial^{n}Y,\partial^{n}Z\\}_{n\geq
0}]/\langle\\{\partial^{n}(XY),\partial^{n}(YZ),\partial^{n}(ZX)\\}_{n\geq
0}\rangle\,.$ (4.4)

Similarly for chiral mutiplets with an arbitrary superpotential, the ghost
number zero part of the cohomology is

$\mathbb{C}[J_{\infty}\mathrm{Crit}(W)]=\mathbb{C}[\\{\partial^{n}\phi_{i}\\}_{n\geq
0,i}]/\langle\\{\partial^{n}\left(\frac{\delta
W}{\delta\phi_{i}}\right)\\}_{n\geq 0,i}\rangle\,.$ (4.5)

The character of the XYZ model can be computed as if there is no
superpotential, and is simply the product of the characters of the three
chiral multiplets. We have

$\chi(\mathrm{Obs}_{XYZ})=\frac{(a^{-2}q,q)_{\infty}}{(a^{2},q)_{\infty}}\frac{(ax^{-1}q^{\frac{1}{2}},q)_{\infty}}{(a^{-1}xq^{\frac{1}{2}},q)_{\infty}}\frac{(axq^{\frac{1}{2}},q)_{\infty}}{(a^{-1}x^{-1}q^{\frac{1}{2}},q)_{\infty}}\,.$
(4.6)

###  4.2 $U(1)$ pure gauge theory

As our first example of gauge theory, we consider pure abelian
$G=\mathbb{C}^{\times}$ vector multiplet without matter. The perturbative
algebra is very simple. We have

$\mathrm{Obs}_{U(1)}^{\text{per}}=\mathbb{C}[J_{\infty}\mathfrak{gl}_{1}/J_{\infty}\mathbb{C}^{\times}]\,.$
(4.7)

As the adjoint action of $\mathbb{C}^{\times}$ on $\mathfrak{gl}_{1}$ is
trivial, all operators are $\mathbb{C}^{\times}$ invariant. So the operator
algebra consist of the field $b$ and all its derivative and derivatives of the
field $c$ . Explicitly this gives us

$\mathrm{Obs}_{U(1)}^{\text{per}}=\mathbb{C}[\\{\partial^{n+1}c,\partial^{n}b\\}_{n\geq
0}]\,.$ (4.8)

Non perturbative algebra is analyzed in Section 3.2.2. The phase space has
connected component labeled by $\pi_{1}(\mathbb{C}^{*})=\mathbb{Z}$. For each
connected component, we see from 3.33 that the operator algebra is a copy of
$\mathrm{Obs}^{\text{per}}$. Therefore we get

$\mathrm{Obs}_{U(1)}=\bigoplus_{m\in\mathbb{Z}}\mathbb{C}[\\{\partial^{n+1}c,\partial^{n}b\\}_{n\geq
0}]=\mathbb{C}[v,v^{-1},\\{\partial^{n+1}c,\partial^{n}b\\}_{n\geq 0}]\,.$
(4.9)

We can also write it as
$\mathrm{Obs}_{U(1)}=\mathbb{C}[J_{\infty}T^{*}[1]\mathbb{C}^{\times}]$, which
is the same as the bulk algebra of a free chiral valued in
$\mathbb{C}^{\times}$.

The character of this theory is trivial because $\partial^{n+1}c$ and
$\partial^{n}b$ contribute $-q^{n+1}$ and $q^{n+1}$ to the character
respectively, and they cancel with each other.

###  4.3 $N_{f}=\tilde{N}_{f}=1$ SQED

The $\mathcal{N}=2$ SQED has one $G=\mathbb{C}^{\times}$ vector multiplet
$(\mathbf{B},\mathbf{A})$ and two chiral multiplets
$(\mathbf{\Phi}_{+},\mathbf{\Psi}_{+})$,
$(\mathbf{\Phi}_{-},\mathbf{\Psi}_{-})$ with gauge charge $+1,-1$,
respectively. There is no superpotential and Chern Simons term. The global
symmetry group of SQED comprises a $U(1)_{A}$ axial symmetry, $U(1)_{T}$
topological symmetry, and the usual $U(1)_{R}$ symmetry. The charges of
various fields under gauge symmetry, flavor symmetries and spin are

| $\phi_{+}$ | $\phi_{-}$ | $\psi_{+}$ | $\psi_{-}$ | $c$ | $b$
---|---|---|---|---|---|---
Gauge | $1$ | $-1$ | $-1$ | $1$ | $0$ | $0$
$A$ | $1$ | $1$ | $-1$ | $-1$ | $0$ | $0$
$T$ | $0$ | $0$ | $0$ | $0$ | $0$ | $0$
$J$ | $0$ | $0$ | $1$ | $1$ | $0$ | $0$

It’s very easy to write down the local operator algebra in the perturbative
sector. We have

$\mathrm{Obs}_{\mathrm{SQED}}^{\text{per}}=(\mathbb{C}[\\{\partial^{n+1}c,\partial^{n}b,\partial^{n}\phi_{\pm},\partial^{n}\psi_{\pm}\\}_{n\geq
0}]^{\mathbb{C}^{\times}},Q)\,,$ (4.10)

where the differential $Q$ is given by

$\displaystyle Q\partial^{n}b$ $\displaystyle=\sum_{0\leq l\leq
n}\partial^{l}\phi_{+}\partial^{n-l}\psi_{+}-\partial^{l}\phi_{-}\partial^{n-l}\psi_{-}\,,$
(4.11) $\displaystyle Q\partial^{n}\phi_{\pm}$ $\displaystyle=\pm\sum_{1\leq
l\leq n}\partial^{l}c\partial^{n-l}\phi_{\pm}\,,$ $\displaystyle
Q\partial^{n}\psi_{\pm}$ $\displaystyle=\mp\sum_{1\leq l\leq
n}\partial^{l}c\partial^{n-l}\psi_{\pm}\,.$

As before, we decompose the cohomology by spin charges and do the computation
of cohomology for small spins

| $J=0$ |  $J=1$ |  $J=2$ | …
---|---|---|---|---
$T=0$ | $\\{(\phi_{+}\phi_{-})^{n}\\}_{n\geq 0}$ |  $\\{(\phi_{+}\phi_{-})^{n}\partial(\phi_{+}\phi_{-})$, $(\phi_{+}\phi_{-})^{n}(\phi_{+}\psi_{+}+\phi_{-}\psi_{-})\\}_{n\geq 0}$, $\partial c$ |  $\\{(\phi_{+}\phi_{-})^{n}\partial^{2}(\phi_{+}\phi_{-})$, $(\phi_{+}\phi_{-})^{n}(\partial(\phi_{+}\phi_{-}))^{2}$, $(\phi_{+}\phi_{-})^{n}(\partial(\phi_{+}\psi_{+}+\phi_{-}\psi_{-}))$, $(\phi_{+}\phi_{-})^{n}(b\partial c+\partial\phi_{+}\psi_{+}-\phi_{-}\partial\psi_{-})\\}_{n\geq 0}$, $\psi_{+}\psi_{-}$, $\partial^{2}c$, $\phi_{+}\phi_{-}\partial^{2}c$ |

As we have discussed in Section 3.2.2, for operators at a non zero monopole
sector $T=m$, we need to consider the following space

$\Gamma(\mathbb{B},P_{m}\times_{\mathbb{C}^{\times}}(T^{*}[1]V))\,.$ (4.12)

Here $V=\mathbb{C}^{2}$ and can be decompose into
$\mathbb{C}_{+}\oplus\mathbb{C}_{-}$ on which the $\mathbb{C}^{\times}$
weights are $+1,-1$ respectively. Note that operators are functions on the
space of field. So operators from functions on $\mathbb{C}_{+}[[z]]$ has $-1$
gauge charge and operators from functions on $\mathbb{C}_{-}[[z]]$ has $+1$
gauge charge. The computation of 4.12 is already explained in 3.2.2 and we
find that

$\Gamma(\mathbb{B},P_{m}\times_{\mathbb{C}^{\times}}(T^{*}[1]V))=\begin{cases}z^{m}\mathbb{C}[[z]]\oplus\mathbb{C}[[z]]\oplus\mathbb{C}[[z]][1]\oplus
z^{m}\mathbb{C}[[z]][1],&m\geq 0\\\ \mathbb{C}[[z]]\oplus
z^{-m}\mathbb{C}[[z]]\oplus
z^{-m}\mathbb{C}[[z]][1]\oplus\mathbb{C}[[z]][1],&m<0\\\ \end{cases}\,.$
(4.13)

In summary, the local operator algebra at monopole charge $m$ sector is given
by333More precisely, the local operator algebra at monopole charge $m$ sector
should be written as $\left(L(m)[\\{\partial^{n+1}c,\dots\\}_{n\geq
0}]\right)^{\mathbb{C}^{\times}}$, where $L(m)\cong\mathbb{C}$ correspond to
the line bundle of 3.3.1,3.3.2 and carry both $\mathbb{C}^{\times}$ gauge
charge and spin charge. We will keep using the notation 4.14 in this paper,
but keep in mind the various quantum number of $L(m)$

$\begin{split}&\mathrm{Obs}^{m}_{\text{SQED}}\\\
&=\begin{cases}(\mathbb{C}[\\{\partial^{n+1}c,\partial^{n}b,\partial^{n}\phi_{+},\partial^{m+n}\phi_{-},\partial^{-m+n}\psi_{+},\partial^{n}\psi_{-}\\}_{n\geq
0}]^{\mathbb{C}^{\times}},Q),&\text{ for }m\geq 0\\\
(\mathbb{C}[\\{\partial^{n+1}c,\partial^{n}b,\partial^{-m+n}\phi_{+},\partial^{n}\phi_{-},\partial^{n}\psi_{+},\partial^{-m+n}\psi_{-}\\}_{n\geq
0}]^{\mathbb{C}\times},Q),&\text{ for }m<0\end{cases}\,.\end{split}$ (4.14)

The differential $Q$ is obtained from the differential 4.11 by discarding all
terms not in the complex. The full local operator algebra is

$\mathrm{Obs}_{\text{SQED}}=\bigoplus_{m\in\mathbb{Z}}\mathrm{Obs}^{m}_{\text{SQED}}\,.$
(4.15)

We give some examples of operators of non-zero monopole charge.

| $J=\frac{1}{2}$ |  $J=\frac{3}{2}$ |  $J=\frac{5}{2}$ | …
---|---|---|---|---
$T=1$ | $v_{+}$ |  $v_{+}\phi_{+}\partial\phi_{-}$, $v_{+}\partial c$, $v_{+}b$ |  $v_{+}(\phi_{+}\partial\phi_{-})^{2}$, $v_{+}\partial(\phi_{+}\partial\phi_{-})$, $v_{+}(\phi_{+}\partial\psi_{+}-\partial\phi_{-}\psi_{-})$, $v_{+}b^{2}$, $v_{+}\partial^{2}c$ |
$T=-1$ | $v_{-}$ |  $v_{-}\partial\phi_{+}\phi_{-}$, $v_{-}\partial c$, $v_{-}b$ |  $v_{-}(\partial\phi_{+}\phi_{-})^{2}$, $v_{-}\partial(\partial\phi_{+}\phi_{-})$, $v_{-}(\partial\phi_{+}\psi_{+}-\phi_{-}\partial\psi_{-})$, $v_{+}b^{2}$, $v_{+}\partial^{2}c$ |

To compute the character for each monopole sector, we need to take the half
form correction into account. The gauge and spin fugacities are already
considered in 3.3.2, so we only need to considered the flavor
$\mathbb{C}^{\times}_{A}$ fugacity here. When $m\geq 0$ we have

$L_{h}(m)=(\operatorname{Ber}(\oplus_{i=0}^{m-1}\mathbb{C}_{+}z^{i}))^{\frac{1}{2}}\otimes(\operatorname{Ber}(\oplus_{i=0}^{m-1}\mathbb{C}^{*}_{-}[1]z^{i}))^{\frac{1}{2}}\,.$
(4.16)

$\mathbb{C}_{+}$ has $A$ charge $-1$ and $\mathbb{C}_{-}^{*}$ has $A$ charge
$+1$. Combining with spin and gauge fugacity 3.3.2 gives us the character

$\begin{split}\chi(L_{h}(m))&=(-1)^{m}a^{-\frac{1}{2}m}s^{\frac{1}{2}m}q^{\frac{-m(m-1)}{4}}a^{-\frac{1}{2}m}s^{-\frac{1}{2}m}q^{\frac{m(m+1)}{4}}\\\
&=(-1)^{m}a^{-m}q^{\frac{1}{2}m}\,.\end{split}$ (4.17)

For $m<0$ we have

$L_{h}(m)=(\operatorname{Ber}(\oplus_{i=0}^{-m-1}\mathbb{C}_{-}z^{i}))^{\frac{1}{2}}\otimes(\operatorname{Ber}(\oplus_{i=0}^{-m-1}\mathbb{C}^{*}_{+}[1]z^{i}))^{\frac{1}{2}}\,.$
(4.18)

$\mathbb{C}_{-}$ has $A$ charge $-1$ and $\mathbb{C}_{+}^{*}$ has $A$ charge
$+1$. The character is

$\begin{split}\chi(L_{h}(m))&=(-1)^{m}a^{\frac{1}{2}m}s^{-\frac{1}{2}m}q^{\frac{m(m-1)}{4}}a^{\frac{1}{2}m}s^{\frac{1}{2}m}q^{\frac{-m(m+1)}{4}}\\\
&=(-1)^{m}a^{m}q^{-\frac{1}{2}m}\,.\end{split}$ (4.19)

Together we have

$\chi(L_{h}(m))=(-1)^{m}a^{-|m|}q^{\frac{|m|}{2}}\,.$ (4.20)

We can now compute the full character of the algebra. Contributions from
$\partial^{n+1}c$ and $\partial^{n}b$ cancel. $\partial^{n}\phi_{+}$
contribute $saq^{n}$, $\partial^{n}\phi_{-}$ contribute $s^{-1}aq^{n}$,
$\partial^{n}\psi_{+}$ contribute $-s^{-1}a^{-1}q^{n+1}$ and
$\partial^{n}\psi_{-}$ contribute $-sa^{-1}q^{n+1}$. Therefore, for $m\geq 0$
we have

$\chi(\mathrm{Obs}_{\text{SQED}}^{m})=(-1)^{m}a^{-|m|}q^{\frac{|m|}{2}}\int\frac{ds}{2\pi
is}\frac{(sa^{-1}q;q)_{\infty}(s^{-1}a^{-1}q^{1+m};q)_{\infty}}{(sa;q)_{\infty}(s^{-1}aq^{m};q)_{\infty}}\,.$
(4.21)

Similarly for $m<0$ we have

$\chi(\mathrm{Obs}_{\text{SQED}}^{m})=(-1)^{m}a^{-|m|}q^{\frac{|m|}{2}}\int\frac{ds}{2\pi
is}\frac{(sa^{-1}q^{1-m};q)_{\infty}(s^{-1}a^{-1}q;q)_{\infty}}{(saq^{-m};q)_{\infty}(s^{-1}a;q)_{\infty}}\,.$
(4.22)

We can simplify our expression by making a change of the integration variable
$s\to sq^{\frac{m}{2}}$. This gives us

$\chi(\mathrm{Obs}_{\text{SQED}}^{m})=(-1)^{m}a^{-|m|}q^{\frac{|m|}{2}}\int\frac{ds}{2\pi
is}\frac{(sa^{-1}q^{1+\frac{|m|}{2}};q)_{\infty}(s^{-1}a^{-1}q^{1+\frac{|m|}{2}};q)_{\infty}}{(saq^{\frac{|m|}{2}};q)_{\infty}(s^{-1}aq^{\frac{|m|}{2}};q)_{\infty}}\,.$
(4.23)

The index of the full local operator algebra is the sum

$\chi(\mathrm{Obs}_{\text{SQED}})=\sum_{m\in\mathbb{Z}}x^{m}\chi(\mathrm{Obs}_{\text{SQED}}^{m})=\sum_{m\in\mathbb{Z}}x^{m}a^{-|m|}q^{\frac{|m|}{2}}\int\frac{ds}{2\pi
is}\frac{(s^{\pm}a^{-1}q^{1+\frac{|m|}{2}};q)_{\infty}}{(s^{\pm}aq^{\frac{|m|}{2}};q)_{\infty}}\;.$
(4.24)

Here $x$ is the fugacity of topological symmetry and we absorbed the phase
factor $(-1)^{m}$ into $x^{m}$.

###  4.4 $U(1)_{-\frac{1}{2}}$ \+ a chiral

In this section we consider a $G=\mathbb{C}^{\times}$ vector multiplet
$(\mathbf{A},\mathbf{B})$ at Chern Simons level $-\frac{1}{2}$ coupled to a
chiral multiplet $(\mathbf{\Phi},\mathbf{\Xi})$ of charge $1$ under the gauge
group.

For the operator algebra at the monopole zero sector, we get the dg algebra

$(\mathbb{C}[\\{\partial^{n}\varphi,\partial^{n}\xi,\partial^{n}b,\partial^{n+1}c\\}_{n\geq
0}]^{\mathbb{C}^{\times}},Q)\,.$ (4.25)

with differential $Q$ given by

$\begin{split}Q\partial^{n}\varphi&=\sum_{1\leq l\leq
n}\partial^{m}c\partial^{n-l}\varphi\,,\\\ Q\partial^{n}\xi&=-\sum_{1\leq
l\leq n}\partial^{l}c\partial^{n-l}\,,\\\ Q\partial^{n}b&=\sum_{0\leq l\leq
n}\partial^{l}\varphi\partial^{n-l}\xi\,.\end{split}$ (4.26)

We list the cohomology for small spins

| $J=1$ | $J=2$ |  $J=3$ | …
---|---|---|---|---
$T=0$ | $\partial c$ |  $\partial^{2}c$, $\xi\partial\varphi+b\partial c$ |  $\partial^{3}c,\,\partial c\partial^{2}c$, $b\xi\partial\varphi+\frac{1}{2}b^{2}\partial c$, $\xi\partial^{2}\varphi-\varphi\partial^{2}\xi+2b\partial^{2}c+\partial c\partial b$ |

For monopole charge $m>0$, the operator algebras can be computed by the
following complexes

$\mathrm{Obs}^{m}=(\mathbb{C}[\\{\partial^{n}\varphi,\partial^{m+n}\xi,\partial^{n}b,\partial^{n+1}c\\}_{n\geq
0}]^{\mathbb{C}^{\times}},Q),\quad\text{ for }m>0.$

The differential is the truncation of 4.26. As before, we give some examples
of operators here

| $J=0$ | $J=1$ | $J=2$ | …
---|---|---|---|---
$T=1$ | $v_{+}$ |  $v_{+}\partial c$, $v_{+}b$ |  $v_{+}b\partial c$, $v_{+}b^{2}$, $v_{+}\partial^{2}c$ |
$T=2$ | $v_{+}^{2}$ |  $v_{+}^{2}\partial c$, $v_{+}^{2}b$ |  $v_{+}^{2}b\partial c$, $v_{+}^{2}b^{2}$, $v_{+}^{2}\partial^{2}c$, $v_{+}^{2}\partial b$ |

According to 3.3.1 3.3.2, the half form correction and the Chern-Simons term
contribute a factor

$s^{-\frac{1}{2}m}q^{\frac{m(m+1)}{4}}(s^{-m}q^{\frac{(m+1)m}{2}})^{-\frac{1}{2}}=1$
(4.27)

Then we have the character

$\chi(\mathrm{Obs}^{m})=\int\frac{ds}{2\pi
is}\frac{(z^{-1}q;q)_{\infty}}{(sq^{m};q)_{\infty}},\quad m\geq 0$ (4.28)

By making change of integration variable, $s\to sq^{\frac{m}{2}}$, we get

$\chi(\mathrm{Obs}^{m})=\int\frac{ds}{2\pi
is}\frac{(s^{-1}q^{1+\frac{m}{2}};q)_{\infty}}{(sq^{\frac{m}{2}};q)_{\infty}},\quad
m\geq 0\,.$ (4.29)

For the monopole charge $m$ sector with $m<0$, the half form and line bundle
twist contribute a factor

$(-1)^{m}s^{\frac{1}{2}m}q^{\frac{-m(m+1)}{4}}(s^{-m}q^{-\frac{(m+1)m}{2}})^{-\frac{1}{2}}=(-s)^{m}q^{\frac{-m(m+1)}{2}}\,.$
(4.30)

After the change of variable $s\to sq^{\frac{m}{2}}$ as we have done before,
we find that the character for $m<0$ is given by

$\chi(\mathrm{Obs}^{m})=\int\frac{ds}{2\pi
is}(-s)^{m}q^{\frac{-m}{2}}\frac{(s^{-1}q^{1-\frac{m}{2}};q)_{\infty}}{(sq^{-\frac{m}{2}};q)_{\infty}},\quad
m<0\,.$ (4.31)

We can summarize the expression for $m\geq 0$ and $m<0$ into one expression

$\chi(\mathrm{Obs})=\sum_{m\in\mathbb{Z}}x^{m}\int\frac{ds}{2\pi
is}(-q^{\frac{1}{2}})^{-\frac{1}{2}(m-|m|)}s^{\frac{1}{2}(m-|m|)}\frac{(s^{-1}q^{1+\frac{|m|}{2}};q)_{\infty}}{(sq^{\frac{|m|}{2}};q)_{\infty}}\,.$
(4.32)

It is interesting to note that the character here contain a relative phase
factor $(-1)^{-\frac{1}{2}(m-|m|)}$ that only contribute when $m<0$. This
phase factor cannot be derived from the usual localization computation of the
index. However, it is predicted in [19] to incorporate various mirror
dualities for the index. In our setup, this phase factor naturally comes from
the fermionic grading of the Berezinian.

A closely related theory is the so called $U(1)_{\frac{1}{2}}+$ chiral theory,
where we have a $G=\mathbb{C}^{\times}$ vector multiplet at Chern Simons level
$\frac{1}{2}$ coupled to a chiral multiplet of charge $1$ under the gauge
group. This theory, together with the $U(1)_{-\frac{1}{2}}+$ chiral theory and
the free chiral theory, consist of the simplest example of $3d$
$\mathcal{N}=2$ mirror “triality” [23].

$\text{free chiral}\leftrightarrow U(1)_{\frac{1}{2}}+\text{ chiral
}\leftrightarrow U(1)_{-\frac{1}{2}}+\text{ chiral }\,.$ (4.33)

###  4.5 Bulk algebras and dualities

In this section, we consider local operator algebras of different theories
related by infrared dualities. It was argued in [12] that the $Q$ cohomology
of bulk local operator algebra is constant along the RG flow. Therefore for
two IR dual theories $\mathcal{T},\mathcal{T^{\prime}}$ (two theories that
flow to the same IR, also known as mirror dual theories [50, 51]), the
corresponding local operator algebras
$\mathrm{Obs}_{\mathcal{T}},\mathrm{Obs}_{\mathcal{T^{\prime}}}$ should have
the same $Q$ cohomology.

$H^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\mathrm{Obs}_{\mathcal{T}},Q_{\mathcal{T}})\approx
H^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\mathrm{Obs}_{\mathcal{T}^{\prime}},Q_{\mathcal{T}^{\prime}})\,.$
(4.34)

The mirror map always mixes the monopole operators and perturbative operators
on the two sides. We should be able to observe this in cohomology. A even
stronger statement is that the two complexes
$(\mathrm{Obs}_{\mathcal{T}},Q_{\mathcal{T}}),(\mathrm{Obs}_{\mathcal{T^{\prime}}},Q_{\mathcal{T}^{\prime}})$
are quasi-isomorphic. If true, this will provide us with a very strong and
non-trivial test of $3d$ mirror symmetry. Although we are unable to verify
this statement at the moment, we provide some evidence by comparing the
cohomology of operators of small spins for mirror dual theories.

We have computed some operators for the $U(1)_{-\frac{1}{2}}+\text{ chiral }$
theory. From what we have found in Section 4.4, we can observe an exact match
of cohomologies of local operators for the two theories. Some examples are
listed as below

$T=0$ | $J=1$ | $J=2$ |  $J=3$ …
---|---|---|---
free chiral | $\phi\psi$ | $\partial(\phi\psi)$ | $\phi\partial\psi-\partial\phi\psi$ | $\partial^{2}(\phi\psi)$ | $\phi^{2}\psi\partial\psi$ | …
$U(1)_{-\frac{1}{2}}+$ chiral | $\partial c$ | $\partial^{2}c$ | $\xi\partial\varphi+b\partial c$ | $\partial^{3}c$ | $\partial c\partial^{2}c$

$T=0$ | $J=3$ | …
---|---|---
free chiral | $\partial\phi\partial\psi$ | $\partial^{2}\phi\psi-\phi\partial^{2}\psi$ | …
$U(1)_{-\frac{1}{2}}+$ chiral | $b\xi\partial\varphi+\frac{1}{2}b^{2}\partial c$ | $\xi\partial^{2}\varphi-\varphi\partial^{2}\xi+2b\partial^{2}c+\partial c\partial b$

$T=1$ | $J=0$ | $J=1$ | $J=2$ | …
---|---|---|---|---
free chiral | $\phi$ | $\phi^{2}\psi$ | $\partial\phi$ | $\phi\partial\phi\psi$ | $\partial^{2}\phi$ | $\phi\partial(\phi\psi)$ | …
$U(1)_{-\frac{1}{2}}+$ chiral | $v_{+}$ | $v_{+}\partial c$ | $v_{+}b$ | $v_{+}b\partial c$ |  $v_{+}b^{2}$, | $v_{+}\partial^{2}c$

The mirror duality indeed maps monopole operators on one side to perturbative
operators on the other side. The identification $\phi\leftrightarrow v_{+}$ is
compatible with study of corresponding physical theories [23].

Another classical example is given by the duality between SQED and XYZ
model[51]. By comparing the cohomology of local operators with different
quantum numbers, we can find an identification of operators under the duality.
For $T$ charge $0$ sector we have

$T=0$ | $J=1$ | $J=2$ | …
---|---|---|---
XYZ | $X^{n}$ | $X^{n}\partial X$ | $X^{n}(X\psi_{X}-Y\psi_{Y})$ | $Y\psi_{Y}-Z\psi_{Z}$ | …
SQED | $(\phi_{+}\phi_{-})^{n}$ | $(\phi_{+}\phi_{-})^{n}\partial(\phi_{+}\phi_{-})$ | $(\phi_{+}\phi_{-})^{n}(\phi_{+}\psi_{+}+\phi_{-}\psi_{-})$ | $\partial c$

$T=0$ | $J=3$
---|---
XYZ | SQED
$A=-2$ | $\partial YZ-Y\partial Z$ | $\psi_{+}\psi_{-}$
$A=0$ | $\partial(Y\psi_{Y}-Z\psi_{Z})$ | $\partial^{2}c$
$A=2$ | $X\partial(Y\psi_{Y}-Z\psi_{Z})$ | $(\phi_{+}\phi_{-})\partial^{2}c$
$A=2n$ | $X^{n}\partial(X\psi_{X}-Y\psi_{Y})$ | $(\phi_{+}\phi_{-})^{n}\partial(\phi_{+}\psi_{+}-\phi_{-}\psi_{-})$
$X^{n}(\partial X\psi_{X}-Y\partial\psi_{Y}+\partial Z\psi_{Z})$ | $(\phi_{+}\phi_{-})^{n}(b\partial c+\partial\phi_{+}\psi_{+}-\partial\phi_{-}\psi_{-})$
$A=2n+2$ | $X^{n}\partial^{2}X$ | $(\phi_{+}\phi_{-})^{n}\partial^{2}(\phi_{+}\phi_{-})$
$A=2n+4$ | $X^{n}(\partial X)^{2}$ | $(\phi_{+}\phi_{-})^{n}(\partial(\phi_{+}\phi_{-}))^{2}$

For nonzero $T$ charge sector, we also find the following identification

$T=1$ | $J=\frac{1}{2}$ | $J=\frac{3}{2}$ | $J=\frac{5}{2}$ | …
---|---|---|---|---
XYZ | $Y$ | $\partial Y$ | $\partial XY$ | $Y(Y\psi_{Y}-Z\psi_{Z})$ | $\partial^{2}Y$ | $(\partial X)^{2}Y$ | …
SQED | $v_{+}$ | $v_{+}b$ | $v_{+}\phi_{+}\partial\phi_{-}$ | $v_{+}\partial c$ | $v_{+}b^{2}$ | $v_{+}(\phi_{+}\partial\phi_{-})^{2}$

$T=1$ | $J=\frac{5}{2}$ | …
---|---|---
XYZ | $\partial^{2}XY$ | $Y\partial(X\psi_{X}-Y\psi_{Y})$ | $Y\partial(Y\psi_{Y}-Z\psi_{Z})$ | …
SQED | $v_{+}\partial(\phi_{+}\partial\phi_{-})$ | $v_{+}(\phi_{+}\partial\psi_{+}-\partial\phi_{-}\psi_{-})$ | $v_{+}\partial^{2}c$

With the above identification of operators for the two theories, we can study
how the Poisson bracket behave under the mirror map. For example, in the XYZ
model we have the following bracket

$\\{X,Y\partial Z-Z\partial Y\\}=0\,.$ (4.35)

The corresponding bracket in the SQED is

$\\{\phi_{+}\phi_{-},\psi_{+}\psi_{-}\\}=\phi_{-}\psi_{-}-\phi_{+}\psi_{+}=-Qb\,.$
(4.36)

This observation suggests that we expect the Poisson bracket to be preserved
under the isomorphism only at the level of cohomology. If we find a quasi
isomorphism between the two complexes $(\mathrm{Obs}_{\text{XYZ}},Q)$ and
$(\mathrm{Obs}_{\text{SQED}},Q)$, it will not preserve the Poisson bracket (or
$\lambda$-bracket) at the cochain level. This inconsistency can be resolved by
realizing that the bulk algebra actually has a higher analog of chiral algebra
structure at the chain level. We will discuss more this structure in Section
6.4. The upshot is that the quasi isomorphism between the two complexes should
be accompanied by an infinite series of higher maps between them so that the
full algebraic structure of the bulk algebras is preserved. Finding all the
”higher” morphisms between the algebras will be interesting.

##  5 Review of boundary conditions and boundary algebras

Half-BPS boundary conditions of physical $3d$ $\mathcal{N}=2$ theory
preserving $2d$ $\mathcal{N}=(0,2)$ supersymmetry are automatically compatible
with the holomorphic twist, therefore lead to boundary conditions of twisted
theory. These half BPS boundary conditions are systematically studied in [52],
and are reinterpreted in the twisted formalism in [12]. Here we briefly review
some of the results here.

There are four basic classes of boundary conditions for the vector and chiral
multiplet.

  * •

Dirichlet boundary condition for the chiral multiplet, abbreviated D, by
imposing $\mathbf{\Phi}=0$ on the boundary.

  * •

Neumann boundary condition for the chiral multiplet, abbreviated N, by
imposing $\mathbf{\Psi}=0$ on the boundary.

  * •

Dirichlet boundary condition for the vector multiplet, abbreviated
$\mathcal{D}$, by imposing $\mathbf{A}=0$ on the boundary.

  * •

Neumann boundary condition for the vector multiplet, abbreviated
$\mathcal{N}$, by imposing $\mathbf{B}=0$ on the boundary.

There could be more complicated boundary conditions, but in this paper, we
only study combinations of the above four.

###  5.1 Boundary algebras for chiral multiplets

We consider $n$ chiral multiplet with a superpotential $W$. Suppose the
boundary condition is chosen such that the first $k$ chiral multiplets are
given Neumann boundary condition, and the last $n-k$ chiral multiplets are
given Dirichlet boundary condition. Namely, we set

$\mathbf{\Psi}_{i}=0,\;i=1,\dots,k,\quad\quad\mathbf{\Phi}_{i}=0,\;i=k+1,\dots,n$

at the boundary.

By passing to cohomology, the boundary algebra is generated by bosonic
operators $\\{\phi_{i}(z),i=1,\dots,k\\}$ and fermionic operatos
$\\{\psi_{j}(z),j=k+1\dots,n\\}$. There are two situations to be discussed
depending on the superpotential. First, if the superpotential vanish at the
boundary

$W|_{\partial}=0\,,$ (5.1)

then the boundary chiral algebra is specified by OPE

$\psi_{j}\psi_{l}=\frac{\hslash}{z}\partial_{j}\partial_{l}W|_{\partial}\,,$
(5.2)

and BRST differential

$Q\psi_{j}=\partial_{j}W|_{\partial}\,.$ (5.3)

Another possibility is that the superpotential does not vanish at the
boundary. In this case, this boundary condition is anomalous due to the
superpotential. We can easily see this by considering the classical master
equation:

$\\{S_{BV},S_{BV}\\}=2\\{\int\langle\mathbf{\Psi}^{i},\hat{d}\mathbf{\Phi}_{i}\rangle,\int
W(\mathbf{\Phi})\\}=2\int\hat{d}W(\mathbf{\Phi})=-2\int_{\Sigma}W(\mathbf{\Phi}^{\partial})\,.$
(5.4)

This term measure the failure of the classical master equation to hold in the
presence of boundary, and we see that it is proportional to $W|_{\partial}$.
Here, we can add boundary fermion to cancel this anomaly. We introduce $2d$
boundary fermion in the BV formalism, and we use superfield notation
compatible with the $3d$ theories. We have fields

$\mathbf{\Gamma}_{\alpha}\in\Omega^{k_{\alpha},\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\Sigma),\;\;\mathbf{\Gamma}^{\alpha}\in\Omega^{1-k_{\alpha},\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\Sigma)\,.$
(5.5)

They are equipped with the following BV bracket

$\\{\mathbf{\Gamma}_{\alpha},\mathbf{\Gamma}^{\beta}\\}=\delta_{\alpha\beta}\delta(x-y)d\mathrm{Vol}\,.$
(5.6)

The BV action functional is given by

$S=\sum_{\alpha}\int_{\Sigma}\mathbf{\Gamma}_{\alpha}\wedge\bar{\partial}\mathbf{\Gamma}^{\alpha}\,.$
(5.7)

This BV action gives $\bar{\partial}$ as the BRST operator. Now suppose that
we have polynomials
$E_{\alpha}(\mathbf{\Phi}^{\partial}),J^{\alpha}(\mathbf{\Phi}^{\partial})$,
which are of charge $k_{\alpha},1-k_{\alpha}$ respectively under the
$U(1)_{R}$ symmetry and satisfy:

$E_{\alpha}(\mathbf{\Phi}^{\partial})J^{\alpha}(\mathbf{\Phi}^{\partial})=W(\mathbf{\Phi}^{\partial})\,.$
(5.8)

Note that we can always find such a factorization of superpotential by adding
enough boundary fermion. We can add the following boundary term to the
Lagrangian:

$S^{\partial}=\sum_{\alpha}\int_{\Sigma}\mathbf{\Gamma}_{\alpha}\wedge\bar{\partial}\mathbf{\Gamma}^{\alpha}+\mathbf{\Gamma}^{\alpha}E_{\alpha}(\mathbf{\Phi}^{\partial})+\mathbf{\Gamma}_{\alpha}J^{\alpha}(\mathbf{\Phi}^{\partial})\,.$
(5.9)

This boundary term contains the BV action of the $2d$ fermion and the coupling
terms on the boundary. Then we can show that the combined system
$S^{\text{bulk}}+S^{\partial}$ satisfies the classical master equation.

For the boundary chiral algebra, we now have the new fields
$(\mathbf{\Gamma}_{\alpha},\mathbf{\Gamma}^{\alpha})$. Since the cohomology of
the complex $(\Omega^{k,\mathbin{\vbox{\hbox{{$\bullet$}}}}},\bar{\partial})$
consist of holomorphic $k$-form, we only need to consider the bottom component
of the fields just like what we have done in the $3d$ case. Therefore the
boundary algebras have new operators generated by
$\Gamma_{\alpha}(z),\Gamma^{\alpha}(z)$. The original OPE 5.2 and BRST
differential 5.3 do not change, but we have the following new OPE

$\Gamma_{\alpha}(z)\widetilde{\Gamma}^{\alpha}(0)=\frac{\delta_{\alpha\beta}}{z}\,,$
(5.10)

and new BRST differential

$\begin{split}Q\Gamma_{\alpha}&=E_{\alpha}(\Phi_{i})\,,\\\
Q\Gamma^{\alpha}&=J^{\alpha}(\Phi_{i})\,.\end{split}$ (5.11)

We can flip the boundary condition of a bulk chiral from Neumann to Dirichlet
by introducing a boundary fermi multiplet. We start with the boundary
condition with the first $m$ chiral multiplets given Neumann boundary
condition, and the last $n-m$ chiral multiplets given Dirichlet boundary
condition. Suppose the superpotential vanish at the boundary. We couple this
system with a boundary fermi multiplet
$(\mathbf{\Gamma}(z),\widetilde{\mathbf{\Gamma}}(z))$ by a $E$ term $E=X$ and
no $J$ term as in 5.9. The vertex algebra of the coupled system is argued [12]
to be equivalent to that of another boundary condition where the first $m-1$
chiral multiplets are given Neumann boundary condition, and the last $n-m+1$
chiral multiplets are given Dirichlet boundary condition. This isomorphism is
only expected to be true at the level of cohomology.

We need to mention that if we only care about the vertex algebra structure at
the level of cohomology, then the dg vertex algebra model proposed in this
section is enough. However, this model does not fully characterize the
algebraic structure of boundary algebra with general superpotential. This is
related to the fact that the superpotential might have non zero higher
derivatives (of order $3$ or higher). That information should be contained in
the structure of boundary algebra but is lost in our model. In this case, the
boundary chiral algebra should form an $A_{\infty}$ analog of chiral algebra,
where we have higher operations coming from the higher derivatives of the
superpotential. It should produce the familiar $A_{\infty}$ algebra upon
reduction on a circle. Theses higher operations might sound unfamiliar but
will be important for the conjectural bulk-boundary relation to work as we
will see later.

###  5.2 Boundary algebras for vector multiplets

First, we consider the Neumann boundary condition for the gauge fields, which
set $\mathbf{B}=0$ at the boundary. The boundary algebra is generated by
$\mathbf{A}\in\mathcal{A}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}\otimes\mathfrak{g}[1]$
and by passing to cohomology the boundary algebra is generated by the ghost
fields $c^{a}$. The OPE structure is trivial but we have a nontrivial BRST
operator.

$Qc^{a}=f^{a}_{bc}c^{b}c^{c}\,.$ (5.12)

As we have discussed in Section 3.2, When the gauge group is compact, we
should only introduce ghosts for non-constant gauge transformations and impose
$G$ invariance by hand by only considering $G$ invariant operators. In
summary, the boundary algebra is:

$C^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(z\mathfrak{g}[z])^{G}\,.$ (5.13)

Explicitly, The boundary algebra is generated by $G$ invariant combinations of
$\partial_{z}c^{a},\partial_{z}^{2}c^{a},\dots$ having trivial OPE and
equipped with the BRST operator

$Q\partial_{z}^{n}c^{a}=\sum_{\begin{subarray}{c}l+k=n\\\
l,k>0\end{subarray}}f^{a}_{bc}\partial_{z}^{l}c^{b}\partial_{z}^{k}c^{c}\,.$
(5.14)

Now we consider the Dirichlet boundary conditions for a gauge field. This set
$\mathbf{A}=0$ at the boundary. Naively, the boundary algebra is generated by
fields $B_{a}$, the bottom component of
$\mathbf{B}\in\mathcal{A}^{1,\mathbin{\vbox{\hbox{{$\bullet$}}}}}\otimes\mathfrak{g}^{*}$.
The BRST differential is trivial and the OPE is given by

$B_{a}(z)B_{b}(0)\sim\frac{f^{c}_{ab}}{z}B_{c}+\frac{k}{z^{2}}\delta_{ab}-\frac{h^{\vee}}{z^{2}}\delta_{ab}\,.$
(5.15)

where $h^{\vee}$ is the dual coxter number. We denote this vertex algebra by
$V_{k-h^{\vee}}(\mathfrak{g})$. This is exactly the vertex algebra associated
to vacuum representation of level $k-h^{\vee}$ of the affine Kac-Moody algebra
$\widehat{\mathfrak{g}}$.

There are further non-perturbative corrections to the boundary algebra with
Dirichlet boundary conditions. To describe the correct space of local
operators on the boundary, we should again use the operator-state
correspondence by constructing the Hilbert space associated with the hemi-
sphere ending on the boundary. A systematic analysis is performed in [12],
which shows that the boundary algebra is computed as Dolbeault homology of
affine Grassmannian

$H_{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\text{Gr}_{G},\mathcal{L}^{\otimes(h-k)})=H^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\text{Gr},\mathcal{L}^{\otimes(k-h)})^{*}\,.$
(5.16)

This space is also known [53, 54] to be isomorphic to $L_{k-h}$, the
integrable highest weight representation of $\widehat{\mathfrak{g}}_{k-h}$. It
has been proved in [55] that $L_{k,0}$ possess the structure of a vertex
operator algebra. This is exactly the vertex operator algebra associated with
WZW model.

###  5.3 Boundary algebra with vector and chiral multiplets

Now suppose we have both vector and chiral multiplets. The boundary conditions
could be any combinations of Neumann and Dirichlet conditions. In this paper,
we will focus on the case when the vector multiples take Neumann boundary
conditions, which, as we have seen, has much simpler structure. The boundary
conditions for the chiral multiplets can be arbitrary. As in Section 5.1, we
give Neumann boundary conditions to the first $k$ chiral multiplets and
Dirichlet boundary conditions to last $n-k$ chiral multiplets. We also add
boundary fermions
$(\mathbf{\Gamma}^{\alpha},\widetilde{\mathbf{\Gamma}}_{\alpha})$ with
$E_{\alpha}$ and $J^{\alpha}$ terms to cancel the boundary anomaly. Including
the gauge fields, the boundary algebra is generated by

$\phi_{i}(z),\psi_{j}(z),\Gamma^{\alpha}(z),\widetilde{\Gamma}_{\alpha}(z),\;i=1,\dots,k,j=k+1,\dots,n\,.$
(5.17)

and their $z$ derivatives, as well as the $z$ derivatives of ghost $c_{a}(z)$.
The nontrivial OPE is the same as before:

$\psi_{j}\psi_{l}=\frac{\hslash}{z}\partial_{j}\partial_{l}W|_{\partial},\quad\Gamma_{\alpha}(z)\widetilde{\Gamma}^{\alpha}(0)=\frac{\delta_{\alpha\beta}}{z}\,.$
(5.18)

The BRST transformation is more complicated. Schematically, they can be
written as

$\begin{split}Q\phi=c\cdot\phi,\;Q\psi=dW|_{\partial}+c\cdot\psi\,,\\\
Q\Gamma=c\cdot\Gamma+E,\;Q\widetilde{\Gamma}=c\cdot\widetilde{\Gamma}+J\,.\end{split}$
(5.19)

Due to the non-trivial OPE, the algebra is no longer a polynomial algebra.
Hence extra attention need to be paid when working with the BRST operator in
this case. When computing normal ordered product of operators, some unexpected
terms may appear in the BRST transformation. We will see this in one of our
examples.

In the presence of gauge fields, we should also be careful about the boundary
gauge anomalies. For the boundary theory to be consistent at the quantum
level, all gauge anomalies must be canceled. Details of boundary anomalies and
their cancellation are provided in [52].

##  6 Self-Ext and bulk local operator algebra

It is conjectured in [12] that, the algebra of bulk local operators can be
computed by the self-Ext of the vacuum module of the boundary VOA. We explain,
following [21, 12], why this might be true. First, by bringing a bulk local
operator $\mathcal{O}\in\mathrm{Obs}$ to the boundary, we get an action of
bulk operators $\mathrm{Obs}$ on boundary vertex algebra
$\mathcal{V}_{\partial}[\mathcal{B}]$. This can be encoded as homomorphism

$\mathrm{Obs}\to\mathrm{End}(\mathcal{V}_{\partial}[\mathcal{B}])\,.$ (6.1)

On the other hand, the algebra of boundary charges
$\oint\mathcal{V}_{\partial}[\mathcal{B}]$ acts on the space of local boundary
operators via boundary OPE, and give us an homomorphism

$\oint\mathcal{V}_{\partial}[\mathcal{B}]\to\mathrm{End}(\mathcal{V}_{\partial}[\mathcal{B}])\,.$
(6.2)

Since the actions of bulk operator and boundary charges on
$\mathcal{V}_{\partial}[\mathcal{B}]$ are defined by operator product in
$\mathbb{R}$ and $\mathbb{C}$ direction respectively, they must commute with
each other. Therefore the image of $\mathrm{Obs}$ in
$\mathrm{End}(\mathcal{V}_{\partial}[\mathcal{B}])$ lie in
$\mathrm{End}_{\oint\mathcal{V}_{\partial}[\mathcal{B}]}(\mathcal{V}_{\partial}[\mathcal{B}])$,
and we get a ”bulk boundary” map

$\beta:\;\mathrm{Obs}\to\mathrm{End}_{\oint\mathcal{V}_{\partial}[\mathcal{B}]}(\mathcal{V}_{\partial}[\mathcal{B}])\,.$
(6.3)

The algebra
$\mathrm{End}_{\oint\mathcal{V}_{\partial}[\mathcal{B}]}(\mathcal{V}_{\partial}[\mathcal{B}])$
is also known as the center $Z(\mathcal{V}_{\partial}[\mathcal{B}])$ of the
VOA $\mathcal{V}_{\partial}[\mathcal{B}]$, so we also write
$\beta:\mathrm{Obs}\to Z(\mathcal{V}_{\partial}[\mathcal{B}])$. It was argued
in [12] that this center is neither injective nor surjective in general.
However, we would expect this ”bulk boundary” map to contain more information
if we consider its derived generalization. So to say, we conjecture that there
is an isomorphism

$\beta_{\text{der}}:\mathrm{Obs}\to
Z_{\text{der}}(\mathcal{V}_{\partial}[\mathcal{B}])\,,$ (6.4)

for boundary condition $\mathcal{B}$ that is large enough (namely, when
$\mathcal{B}$ is a generator of the category of boundary conditions). Here,
the center is replaced by the derived center, and is computed by
$\mathrm{Ext}^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}_{\oint\mathcal{V}_{\partial}[\mathcal{B}]}(\mathcal{V}_{\partial}[\mathcal{B}],\mathcal{V}_{\partial}[\mathcal{B}])$.

To explain this conjecture, we recall the $2d$ TQFT version of this statement.
For a $2d$ TQFT, the boundary conditions form a category $\mathcal{C}$
(usually a dg category). For any boundary conditions $\mathcal{B}$,
$\mathrm{Hom}_{\mathcal{C}}(\mathcal{B},\mathcal{B})=\mathrm{End}_{\mathcal{C}}(\mathcal{B})$
has the structure of a (dg) algebra. We call that the boundary condition
$\mathcal{B}$ is large enough (or a generator of the category) if the category
of boundary conditions is equivalent to the category of (dg) module of
$\mathrm{End}_{\mathcal{C}}(\mathcal{B})$. On the one hand, in the axiomatic
approach of TQFT [20], we take it as a definition that the bulk operator
algebra is given by the derived center of the category $\mathcal{C}$, which is
computed by the Hochschild cohomology of
$A_{\partial}[\mathcal{B}]=\mathrm{End}_{\mathcal{C}}(\mathcal{B})$ for large
enough $\mathcal{B}$. On the other hand, in the physical approach of TQFT, the
bulk operators can be identified as closed string states and can be computed
directly. The isomorphism between the bulk operator algebra and Hochschild
cohomology of boundary algebra $A_{\partial}[\mathcal{B}]$ is considered in
various cases [56, 57, 58].

The analogous statement should hold in $3d$ TQFT. For A and B twist of $3d$
$N=4$ theory, this was used in [21, 22] to compute Higgs and Coulomb branches
operators. In the following sections, we perform the calculation for various
$3d$ holomorphic twist theories. The expected relation is verified for
”Landau-Ginzburg” model with arbitrary superpotential. For theories with
vector multiplets, it is much harder to compute the Ext, due to the rather
complicated structure of the boundary chiral algebra. At the moment, we are
only able to check some examples with abelian gauge group and in the Neumann
boundary condition. Even for these simple examples, it is fascinating to see
that bulk monopole operators show up in the self-Ext of boundary algebra that
is purely perturbative.

###  6.1 Chiral multiplet

In this section, we illustrate the above ideas by looking at theories with
only chiral multiplets. Although much of the analysis can be done in an
abstract way by properties of $R\mathrm{Hom}$, we still perform the
calculation in a very concrete manner by writing down explicitly the Koszul
resolution.

Imposing a boundary condition usually ”freezes” half of the bulk degrees of
freedom. It’s very interesting to see how the ”frozen” half of degrees of
freedom naturally arises from the Koszul resolution and eventually gives us
the right bulk algebra. It’s also very interesting to see how the various
terms in the differential of bulk algebra are obtained from the Ext
calculation.

####  6.1.1 Single chiral multiplet with D b.c.

We start by considering the simplest example, a single chiral multiplet
$(\mathbf{\Phi},\mathbf{\Psi})$ without any superpotential. We consider the
Dirichlet boundary condition. According to our previous discussion, the
boundary algebra is generated by a field $\psi(z)$. There is no BRST operator
and non-trivial OPE. The algebra of charges is generated by

$\psi_{n}=\oint z^{n-1}\psi(z),\;n\in\mathbb{Z}\,.$ (6.5)

All these modes commute because there is no OPE. We have

$\oint\mathcal{V}_{\partial}[D]=\mathbb{C}[\\{\psi_{n}\\}_{n\in\mathbb{Z}}]\,.$
(6.6)

The vacuum module $\mathcal{M}_{vac}$ is generated by a vacuum vector
$|0\rangle$ annihilated by charges $\psi_{n}$ for $n\geq 0$. Equivalently we
can write

$\mathcal{M}_{vac}=\mathbb{C}[\\{\psi_{n}\\}_{n<0}]\,.$ (6.7)

This vacuum module has a free resolution by adjoining
$\oint\mathcal{V}_{\partial}[D]$ infinite many bosonic variables
$\lambda_{n},n\geq 0$, this is also know as the Koszul resolution [59]. We
have

$\widetilde{\mathcal{M}}_{vac}=\oint\mathcal{V}_{\partial}[D]\otimes\mathbb{C}[\\{\lambda_{n}\\}_{n\geq
0}]\,.$ (6.8)

The differential $d$ send $\lambda_{n}\to\psi_{n}$, for $n\geq 0$. The self-
Ext is computed as maps of $\oint\mathcal{V}_{\partial}[D]$ module from
$\widetilde{\mathcal{M}}_{vac}$ to $\mathcal{M}_{vac}$. We have444We use
restricted dual throughout this section

$\begin{split}\mathrm{Hom}_{\oint\mathcal{V}_{\partial}[D]}(\widetilde{\mathcal{M}}_{vac},\mathcal{M}_{vac})&=\mathrm{Hom}_{\mathbb{C}}(\mathbb{C}[\\{\lambda_{n}\\}_{n\geq
0}],\mathcal{M}_{vac})\\\ &=\mathcal{M}_{vac}[\\{\lambda_{n}^{*}\\}_{n\geq
0}]\,.\end{split}$ (6.9)

This complex has no differential, hence the cohomology is the complex itself.
If we change our notation

$\psi_{-n-1}\to\partial^{n}\psi,\;\;\lambda_{n}^{*}\to\partial^{n}\phi\,,$
(6.10)

the self-Ext can be written as

$\mathbb{C}[\\{\partial^{n}\psi,\partial^{n}\phi\\}_{n\geq 0}]\,.$ (6.11)

This is exactly the bulk algebra for a single free chiral.

We can consider a complex mass deformation of the single free chiral by adding
a superpotential:

$W(\mathbf{\Phi})=\frac{1}{2}M\mathbf{\Phi}^{2}\,.$ (6.12)

With this deformation, the boundary chiral algebra in D b.c. have the
following OPE

$\psi(z)\psi(0)\sim\frac{M}{z}\,.$ (6.13)

The algebra of charges now have the following anti commutation 555In this
paper we write both commuting and anti commuting bracket by $[-,-]$. With this
notation the bracket is defined by $[f,g]=fg-(-1)^{|f||g|}gf$, where $|a|$ is
the cohomological degree of $a$ relation

$[\psi_{n},\psi_{m}]=M\delta_{n+m,-1}\,.$ (6.14)

Then the associated algebra of charges is no longer the polynomial algebra
6.6, but a Clifford algebra generated by relations 6.14. However, we can still
identify the vacume module as 6.7 and the resolution 6.8 is still valid as a
resolution of the vacume module. But we need to be careful about the
differential here. The $\psi_{n}$ in the differential $d\lambda_{n}=\psi_{n}$
is now understand as a multiplication by $\psi_{n}$. The self-Ext is computed
by the complex 6.9, but now with a non vanishing differential. Given a
$f\in\mathrm{Hom}_{\mathbb{C}}(\mathbb{C}[\\{\lambda_{n}\\}_{n\geq
0}],\mathcal{M}_{vac})$. By definition, its differential is computed by

$\begin{split}df(\lambda_{n_{1}}\dots\lambda_{n_{k}})&=\sum
f(\lambda_{n_{1}}\dots d\lambda_{n_{j}}\dots\lambda_{n_{k}})\\\
&=\sum\psi_{n_{j}}f(\lambda_{n_{1}}\dots\hat{\lambda}_{n_{j}}\dots\lambda_{n_{k}}).\quad\quad\text{
for }n_{j}\geq 0\,.\end{split}$ (6.15)

$\psi_{n_{j}}$ acts on $\mathcal{M}_{vac}$ via the relation 6.14. We found
that it is identified with $\partial_{\psi_{-1-n_{j}}}$. We further identify
$\mathrm{Hom}_{\mathbb{C}}(\mathbb{C}[\\{\lambda_{n}\\}_{n\geq
0}],\mathcal{M}_{vac})=\mathcal{M}_{vac}[\\{\lambda_{n}^{*}\\}_{n\geq 0}]$.
Then the differential on $\mathcal{M}_{vac}[\\{\lambda_{n}^{*}\\}_{n\geq 0}]$
is given by

$d=M\sum_{n\geq 0}\lambda_{n}^{*}\partial_{\psi_{-1-n}}\,.$ (6.16)

By the change of variable 6.10, the self-Ext is then computed by the complex

$(\mathbb{C}[\\{\partial^{n}\psi,\partial^{n}\phi\\}_{n\geq 0}],Q)\,,$ (6.17)

with differential $Q\partial^{n}\psi=M\partial^{n}\phi$. This is exactly the
cochain complex computing the bulk algebra of a massive chiral. It has
cohomology
$H^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\mathbb{C}[\\{\partial^{n}\psi,\partial^{n}\phi\\}_{n\geq
0}],Q)=\mathbb{C}$.

####  6.1.2 XYZ model with NDD b.c.

Let’s consider the XYZ model with NDD boundary condition. In this case the
boundary algebra is generated by $X(z),\psi_{Y}(z),\psi_{Z}(z)$. According to
our discussion in Section 5.1, this algebra has no BRST differential, but has
a nontrivial boundary OPE

$\psi_{Y}(z)\psi_{Z}(0)\sim\frac{1}{z}X(0)\,.$ (6.18)

The algebra of charges is generated by

$X_{n}=\oint z^{n}X(z)dz,\;\;\psi_{Y,n}=\oint
z^{n}\psi_{Y}(z)dz,\;\;\psi_{Z,n}=\oint z^{n}\psi_{Z}(z)dz\,.$ (6.19)

with (anti)commutation relations inherited from the OPE

$[\psi_{Y,n},\psi_{Z,m}]=X_{n+m}\,.$ (6.20)

If we define a super Lie algebra
$\mathfrak{g}_{NDD}=\mathrm{span}_{\mathbb{C}}\\{X,\psi_{Y},\psi_{Z}\\}$ with
one bosonic basis $X$ and two fermionic basis $\psi_{Y},\psi_{Z}$ and only one
nontrivial commutator $[\psi_{Y},\psi_{Z}]=X$. Then we observe that the
algebra of boundary charges is nothing but the universal enveloping algebra

$\oint\mathcal{V}_{\partial}=U(\mathfrak{g}_{NDD}((z)))\,.$ (6.21)

The vacuum module $\mathcal{M}_{vac}$ is generated by a vacuum vector
$|0\rangle$ which is annihilated by charges $X_{n},\psi_{Y,n},\psi_{Z,n}$ for
$n\geq 0$. In other words

$\mathcal{M}_{vac}=U(\mathfrak{g}_{NDD}((z)))\otimes_{U(\mathfrak{g}_{NDD}[z])}\mathbb{C}\,.$
(6.22)

Here $\mathbb{C}$ is the trivial one dimensional representation on which
$U(\mathfrak{g}_{XYZ}[[z]])$ acts by zero. We recall the well known Chevalley-
Eilenberg cochain complex that provides a resolution of $\mathbb{C}$

$U(\mathfrak{g}_{NDD}[[z]])\otimes\bigwedge\nolimits^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}\mathfrak{g}_{NDD}[[z]]\to\mathbb{C}\,.$
(6.23)

By tensoring with $\mathcal{M}_{vac}$ we find a free resolution of
$\mathcal{M}_{vac}$ as a $U(\mathfrak{g}_{NDD}((z)))$ module,

$\widetilde{\mathcal{M}}_{vac}=U(\mathfrak{g}_{NDD}((z)))\otimes\bigwedge\nolimits^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}\mathfrak{g}_{NDD}[[z]]$
(6.24)

Explicitly we adjoin to $\oint\mathcal{V}_{\partial}$ infinite many fermionic
variables $\eta_{X,n}$ and bosonic variables $y_{n},z_{n}$ for $n\geq 0$. The
Chevalley-Eilenberg differential can be written as

$d=\sum_{n\geq
0}X_{n}\partial_{\eta_{X,n}}+\psi_{Y,n}\partial_{y_{n}}+\psi_{Z,n}\partial_{z_{n}}+\sum_{n,m\geq
0}\eta_{X,n+m}\partial_{y_{n}}\partial_{z_{m}}\,.$ (6.25)

Here by $\psi_{Y,n}$, we mean a left multiplication by $\psi_{Y,n}$, and
similarly for $X_{n},\psi_{Z,n}$.

The $R\mathrm{Hom}$ of $\mathcal{M}_{vac}$ to itself is then the complex of
maps of $\oint\mathcal{V}_{\partial}$-module from
$\widetilde{\mathcal{M}}_{vac}$ to $\mathcal{M}_{vac}$:

$\begin{split}R\mathrm{Hom}_{\oint\mathcal{V}_{\partial}}(\mathcal{M}_{vac},\mathcal{M}_{vac})&=\mathrm{Hom}_{\oint\mathcal{V}_{\partial}}(\widetilde{\mathcal{M}}_{vac},\mathcal{M}_{vac})\\\
&=\mathrm{Hom}_{\mathbb{C}}(\bigwedge\nolimits^{\\!*}\mathfrak{g}_{NDD}[[z]],\mathcal{M}_{vac})\,.\end{split}$
(6.26)

This complex is
$\mathcal{M}_{vac}\otimes\mathbb{C}[\\{\eta_{X,n}^{*},y_{n}^{*},z_{n}^{*}\\}_{n\geq
0}]$, where the generators $\eta_{X,n}^{*},y_{n}^{*},z_{n}^{*}$ are linear
dual of $\eta_{X,n},y_{n},z_{n}$, respectively. The differential on this
complex is inherited from the Chevalley-Eilenberg differential 6.25. It is
given by

$d=\sum_{n,m\geq
0}y_{n}^{*}z_{m}^{*}\frac{\partial}{\partial\eta_{X,n+m}^{*}}+\sum_{n\geq
0}y_{n}^{*}\psi_{Y,n}+z_{n}^{*}\psi_{Z,n}\,.$ (6.27)

Since all the negative mode in the boundary charges commute with each other,
we can write the vacuum module as a polynomial algebra

$\mathbb{C}[\\{X_{n},\psi_{Y,n},\psi_{Z,n}\\}_{n<0}]$

Then left multiplication by $\psi_{Y,n}$ for $n\geq 0$ is identified with the
operator $\sum_{-n<m<0}X_{n+m}\partial_{\psi_{Z,m}}$ and left multiplication
by $\psi_{Z,n}$ for $n\geq 0$ is identified with
$\sum_{-n<m<0}X_{n+m}\partial_{\psi_{Y,m}}$. Hence the differential becomes

$d=\sum_{n,m\geq 0}y_{n}^{*}z_{m}^{*}\partial_{\eta_{X,n+m}}+\sum_{n\geq
0,-n<m<0}X_{n+m}y_{n}^{*}\partial_{\psi_{Z,m}}+X_{n+m}z_{n}^{*}\partial_{\psi_{Y,m}}\,.$
(6.28)

To make things clear we can change the notation
$\partial^{n}\psi_{X}=\eta_{X,n}^{*},\partial^{n}Y=y_{n}^{*},\partial^{n}Z=z_{n}^{*}$
and
$\partial^{n}X=X_{-n-1},\partial^{n}\psi_{Y}=\psi_{Y,-n-1},\partial^{n}\psi_{Z}=\psi_{Z,-n-1}$
for $n\geq 0$. After this change of notation the complex computing
self-$R\mathrm{Hom}$ is generated by
$\partial^{n}X,\partial^{n}Y,\partial^{n}Z$,
$\partial^{n}\psi_{X},\partial^{n}\psi_{Y},\partial^{n}\psi_{Z}$ for $n\geq
0$. One can easily check that the differential is exactly the same as 4.3.

$Q\partial^{n}\psi_{X}=\sum_{0\leq l\leq
n}\partial^{n-l}Y\partial^{l}Z,\;Q\partial^{n}\psi_{Y}=\sum_{0\leq l\leq
n}\partial^{n-l}Z\partial^{l}X,\;Q\partial^{n}\psi_{Z}=\sum_{0\leq l\leq
n}\partial^{n-l}X\partial^{l}Y\,.$ (6.29)

This is indeed the bulk algebra of XYZ model.

####  6.1.3 XYZ model with NNN b.c.

If we choose Neumann boundary condition for all the bulk chiral multiplets,
the superpotential $W=\mathbf{X}\mathbf{Y}\mathbf{Z}$ does not vanish at the
boundary. As we discussed in Section 5.1, we must add boundary fermions with
appropriate $E$ and $J$ terms to factorize the superpotential. In our case, a
convenient choice is by adding one boundary Fermi multiplet
$(\mathbf{\Gamma}(z),\widetilde{\mathbf{\Gamma}}(z))$ and taking
$E=\mathbf{X}$, $J=\mathbf{Y}\mathbf{Z}$. With this choice, the boundary
chiral algebra is generated by
$X(z),Y(z),Z(z),\Gamma(z),\widetilde{\Gamma}(z)$ with boundary SUSY/BRST
differential

$Q\Gamma(z)=X(z),\;\;Q\widetilde{\Gamma}(z)=YZ(z)\,,$ (6.30)

and a nontrivial OPE

$\Gamma(z)\widetilde{\Gamma}(0)\sim\frac{1}{z}\,.$ (6.31)

The algebra of charges is generated by
$\\{X_{n},Y_{n},Z_{n},\Gamma_{n},\widetilde{\Gamma}_{n}\\}_{n\in\mathbb{Z}}$
where

$X_{n}=\oint z^{n}X(z)dz\,,$ (6.32)

and similarly for $Y_{n},Z_{n},\Gamma_{n},\widetilde{\Gamma}_{n}$. The OPE
gives us the following anti-commutation relation on the algebra of charges

$[\Gamma_{n},\widetilde{\Gamma}_{m}]=\delta_{n+m,-1}\,.$ (6.33)

The boundary SUSY/BRST differential becomes the following differential

$d\Gamma_{n}=X_{n},\;\;d\widetilde{\Gamma}_{n}=\sum_{m+l=n-1}Y_{m}Z_{l}\,.$
(6.34)

The (differential graded) vacuum module $\mathcal{M}_{vac}$ is generated by a
vacuum vector $|0\rangle$ annihilated by
$X_{n},Y_{n},Z_{n},\Gamma_{n},\widetilde{\Gamma}_{n}$ for $n\geq 0$. A free
resolution is obtained by adjoining to the algebra
$\oint\mathcal{V}_{\partial}$ infinite many fermionic variables
$\\{\eta_{X,n},\eta_{Y,n},\eta_{Z,n}\\}_{n\geq 0}$ and bosonic variables
$\\{\sigma_{n},\widetilde{\sigma}_{n}\\}_{n\geq 0}$. Based on our previous
construction, a naive choice of Koszul differential is

$d\eta_{X,n}=X_{n},\;\;d\eta_{Y,n}=Y_{n},\;\;d\eta_{Z,n}=Z_{n},\;\;d\sigma_{n}=\Gamma_{n},\;\;d\widetilde{\sigma}_{n}=\widetilde{\Gamma}_{n}\,,$
(6.35)

where, as before, $X_{n}$ represent left multiplication by $X_{n}$ and so on.
However, this differential is not compatible with the original differential
6.34 on $\oint\mathcal{V}_{\partial}$. For example,
$d^{2}\sigma_{n}=d\Gamma_{n}=X_{n}\neq 0$. To remedy this problem we modify
our differential on the variable
$\\{\sigma_{n},\widetilde{\sigma}_{n}\\}_{n\geq 0}$ as follows

$\begin{split}d\sigma_{n}&=\Gamma_{n}-\eta_{X,n}\\\
d\widetilde{\sigma}_{n}&=\widetilde{\Gamma}_{n}-\sum_{0\leq
m<n}(Y_{n-1-m}\eta_{Z,m}+Z_{n-1-m}\eta_{Y,m})\\\ &-\sum_{m\geq
n}(Y_{n-1-m}\eta_{Z,m}+Z_{n-1-m}\eta_{Y,m})\,.\end{split}$ (6.36)

One can easily check that this modified $d$ satisfy $d^{2}=0$ and is a valid
differential. We denote this complex by $(\widetilde{\mathcal{M}}_{vac},d)$.
We can also prove that this complex is quasi-isomorphic to the vacuum module.
To show this we consider a double complex whose horizontal differential
$d_{h}$ is given by 6.36 only, and whose vertical differential is
$d_{v}=d-d_{h}$. Consider the associated spectral sequence whose first page
has differential $d_{h}$. Then we note that the complex
$(\widetilde{\mathcal{M}}_{vac},d_{h})$ is actually the standard Koszul
resolution with respect to a regular sequence. The cohomology of the first
page then sit completely in degree $0$, and we have

$H^{0}(\widetilde{\mathcal{M}}_{vac},d_{h})\cong\mathcal{M}_{vac}\,.$ (6.37)

The vertical differential on $\mathcal{M}_{vac}$ is exactly the boundary
SUSY/BRST differential on the vacuum module. Therefore the total complex
$(\widetilde{\mathcal{M}}_{vac},d)$ is the resolution of $\mathcal{M}_{vac}$
that we are looking for.

After obtaining the right resolution, we can proceed to compute the
self-$R\mathrm{Hom}$. The complex is given by
$\mathcal{M}_{vac}\otimes\mathbb{C}[\\{\eta_{X,n}^{*},\eta_{Y,n}^{*},\eta_{Z,n}^{*},\sigma^{*}_{n},\widetilde{\sigma}_{n}^{*}\\}_{n\geq
0}]$ with differential

$\begin{split}&d=\sum_{n<0}X_{n}\partial_{\Gamma_{n}}+\sum_{n,m<0}Y_{n}Z_{m}\partial_{\widetilde{\Gamma}_{n+m+1}}\\\
&-\sum_{n\geq 0}\left(\sigma_{n}^{*}\partial_{\eta_{X,n}^{*}}+\sum_{0\leq
m\leq
n}\widetilde{\sigma}_{m}^{*}Z_{m-n-1}\partial_{\eta_{Y,n}^{*}}+\widetilde{\sigma}_{m}^{*}Y_{m-n-1}\partial_{\eta_{Z,n}^{*}}-\sigma_{n}^{*}\Gamma_{n}-\widetilde{\sigma}_{n}^{*}\widetilde{\Gamma}_{n}\right)\,.\end{split}$
(6.38)

As before, we can identify the vacuum module with the polynomial algebra

$\mathbb{C}[\\{X_{n},Y_{n},Z_{n},\Gamma_{n},\widetilde{\Gamma}_{n}\\}_{n<0}\,.$
(6.39)

Hence left multiplication by $\Gamma_{n}$ is identified with
$\partial_{\widetilde{\Gamma}_{-n-1}}$ for $n\geq 0$ and left multiplication
by $\widetilde{\Gamma}_{n}$ is identified with $\partial_{\Gamma_{-n-1}}$ for
$n\geq 0$. After this identification, we can rewrite the differential as

$\begin{split}d=&\sum_{n<0}\widetilde{\sigma}_{-n-1}^{*}\partial_{\Gamma_{n}}+\sigma^{*}_{-n-1}\partial_{\widetilde{\Gamma}_{n}}+\sum_{n<0}X_{n}\partial_{\Gamma_{n}}+\sum_{n,m<0}Y_{n}Z_{m}\partial_{\widetilde{\Gamma}_{n+m+1}}\\\
&-\sum_{n\geq 0}\left(\sigma_{n}^{*}\partial_{\eta_{X,n}^{*}}+\sum_{0\leq
m\leq
n}\widetilde{\sigma}_{m}^{*}Z_{m-n-1}\partial_{\eta_{Y,n}^{*}}+\widetilde{\sigma}_{m}^{*}Y_{m-n-1}\partial_{\eta_{Z,n}^{*}}\right)\,.\end{split}$
(6.40)

This is a very large complex, but we can use a spectral sequence to eliminate
the variable $\sigma_{n},\widetilde{\sigma}_{n}$ for $n\geq 0$ and
$\Gamma_{n},\widetilde{\Gamma}_{n}$ for $n<0$. We prove in Appendix C that
this complex has the same cohomology as a complex freely generated by
$\\{X_{-n-1},Y_{-n-1},Z_{-n-1},\eta_{X,n}^{*},\eta_{Y,n}^{*},\eta_{Z,n}^{*}\\}_{n\geq
0}$ with differential

$d=\sum_{n\geq 0}\sum_{0\leq m\leq
n}Y_{-m-1}Z_{m-n-1}\partial_{\eta_{X,n}^{*}}+X_{-m-1}Z_{m-n-1}\partial_{\eta_{Y,n}^{*}}+X_{-m-1}Y_{m-n-1}\partial_{\eta_{Z,n}^{*}}\,.$
(6.41)

We can observe that, by a change of notation, this is indeed the complex
computing bulk algebra of the XYZ model.

####  6.1.4 XYZ model with DDD b.c.

In this section, we choose Dirichlet boundary conditions for all three chiral
multiplets in the XYZ model. If we naively work with the vertex algebra model,
we are left with fields $\psi_{X}(z),\psi_{Y}(z),\psi_{Z}(z)$ at the boundary.
The boundary chiral algebra neglecting all higher operations is trivial. There
are no boundary differential and non trivial OPE. Thus the self-Ext
computation will gives us the algebra of three free chiral multiplets, which
is not correct. To give the right bulk algebra we need to encode the
superpotential $\partial_{X}\partial_{Y}\partial_{Z}W=1$ into the structure of
boundary algebra. To do this we use the flipping technique, which provides us
a dg vertex model for the yet known $A_{\infty}$ vertex algebra at the
boundary.

We couple a boundary fermi multiplet $(\Gamma(z),\widetilde{\Gamma}(z))$ with
the NDD boundary condition boundary algebra $(X(z),\psi_{Y}(z),\psi_{Z}(z))$
with coupling $E=X$. This dg vertex algebra has a boundary differential

$Q\Gamma(z)=X(z),\;\;Q\widetilde{\Gamma}(z)=0\,,$ (6.42)

and boundary OPE

$\begin{split}\psi_{Y}(z)\psi_{Z}(0)&\sim\frac{1}{z}X(0)\,,\\\
\Gamma(z)\widetilde{\Gamma}(0)&\sim\frac{1}{z}\,.\end{split}$ (6.43)

This dg vertex algebra is expected to provide us the right model of the
”$A_{\infty}$” boundary algebra with DDD b.c. The algebra of charges of this
dg vertex algebra can be described as follows. We define
$\mathfrak{g}_{NDDF}:=\mathrm{span}_{\mathbb{C}}\\{\Gamma,\widetilde{\Gamma},X,\psi_{Y},\psi_{Z}\\}$
and

$\hat{\mathfrak{g}}_{NDDF}:=\mathfrak{g}_{NDDF}((z))\otimes\mathbb{C}K\,.$
(6.44)

We have the Lie bracket

$\displaystyle=X\otimes z^{n+m}\,,$ (6.45) $\displaystyle[\Gamma\otimes
z^{n},\Gamma\otimes z^{m}]$ $\displaystyle=K\delta_{n+m,-1}\,,$

and a differential

$d\Gamma\otimes z^{n}=X\otimes z^{n}$ (6.46)

Then the algebra of charges of the boundary algebra can be written as

$\oint\mathcal{V}_{\partial}[DDD]\cong\oint\mathcal{V}_{\partial}[NDD+\text{fermi}]=U(\hat{\mathfrak{g}}_{NDDF})/(K-1)$
(6.47)

To make the relation between $\hat{\mathfrak{g}}_{NDDF}$ and the DDD boundary
algebra more clear, we can compute the cohomology of
$(\hat{\mathfrak{g}}_{NDDF},d)$. Its easy to find that

$H^{\mathbin{\vbox{\hbox{{$\bullet$}}}}}(\hat{\mathfrak{g}}_{NDDF},d)=\hat{\mathfrak{g}}_{DDD}:=\mathfrak{g}_{DDD}((z))\otimes\mathbb{C}K\,,$
(6.48)

where
$\mathfrak{g}_{DDD}=\mathrm{span}_{\mathbb{C}}\\{\tilde{\Gamma},\psi_{Y},\psi_{Z}\\}$.
We can identify $\tilde{\Gamma}$ with $\psi_{X}$ in the boundary algebra of
DDD boundary condition. The bracket 6.45 on $\hat{\mathfrak{g}}_{NDDF}$
induced a trivial bracket on $\hat{\mathfrak{g}}_{DDD}$. This is expected
because the vertex algebra of DDD boundary condition has trivial OPE. However,
interesting information is encoded in the higher operations. Homotopy transfer
theorem [60] tell us that there is an $L_{\infty}$ structure on
$\hat{\mathfrak{g}}_{DDD}$ such that there is an $L_{\infty}$ quasi-
isomorphism between $(\hat{\mathfrak{g}}_{DDD},\\{l_{k}\\}_{k\geq 3})$ and the
dg Lie algebra $(\hat{\mathfrak{g}}_{NDDF},d,[-,-])$. We can explicitly
construct the higher operations $\\{l_{k}\\}_{k\geq 3}$ through the trees of
[61] or by using homological perturbation theory [62]. Consider the following
data of a homotopy retract

$h\leavevmode\hbox to24.27pt{\vbox
to26.4pt{\pgfpicture\makeatletter\hbox{\hskip 23.05507pt\lower-9.78426pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{ {}{{}}{}{{}{}{}{{}}{{{{}{}{}{}}}
{{{}{}{}{}}} {{{}{}{}{}}} {{}{}{}{}}}}{} {} {} {}
{}{}{}{}{}{}{{}}\pgfsys@moveto{0.0pt}{9.10509pt}\pgfsys@curveto{-3.1427pt}{14.54842pt}{-10.10332pt}{16.41351pt}{-15.54665pt}{13.27081pt}\pgfsys@curveto{-20.98997pt}{10.12811pt}{-22.85507pt}{3.1675pt}{-19.71237pt}{-2.27583pt}\pgfsys@curveto{-16.56967pt}{-7.71916pt}{-9.60905pt}{-9.58426pt}{-4.16573pt}{-6.44156pt}\pgfsys@curveto{-2.43575pt}{-5.44267pt}{-0.99889pt}{-4.00581pt}{-0.2pt}{-2.62222pt}\pgfsys@stroke\pgfsys@invoke{
}{{}{{}}{}{}{{}}{{{}}{{{}}{\pgfsys@beginscope\pgfsys@invoke{
}\pgfsys@transformcm{0.50003}{0.86601}{-0.86601}{0.50003}{-0.09999pt}{-2.44904pt}\pgfsys@invoke{
}\pgfsys@invoke{ \lxSVG@closescope }\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope}}{{}}}} } \pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}(\hat{\mathfrak{g}}_{NDDF},d)\overset{p}{\underset{i}{\rightleftarrows}}(\hat{\mathfrak{g}}_{DDD},d=0)$
(6.49)

where $i$ and $p$ are the natural inclusion and projection respectively, and
$h$ is a homotopy satisfying

$dh+hd=1-p\circ i\,.$ (6.50)

Explicitly, $h$ is given by $h(X\otimes z^{n})=\Gamma\otimes z^{n}$ and maps
all other elements to zero. The operations $l_{3}$ of
$\hat{\mathfrak{g}}_{DDD}$ can be constructed via the following tree

$l_{3}$$=\sum_{\text{perm}}$$p$$i$$i$$h$$i$$-$$p$$i$$i$$i$$h$

We find the following $l_{3}$ on $\hat{\mathfrak{g}}_{DDD}$

$\begin{split}l_{3}(\psi_{Y}\otimes z^{n},\psi_{Z}\otimes
z^{m},\psi_{X}\otimes z^{l})&=[h([\psi_{Y}\otimes z^{n},\psi_{Y}\otimes
z^{m}]),\tilde{\Gamma}\otimes z^{l}]\\\ &=K\delta_{n+m+l,-1}\end{split}$
(6.51)

Indeed, this looks like a ”triple OPE”
$\psi_{Y}\psi_{Z}\psi_{X}\sim\partial_{X}\partial_{Y}\partial_{Z}W$ that
encode the $3$rd derivative of $W$ at the boundary.

We can proceed to compute self-Ext in two ways. We can either work directly
with the $L_{\infty}$ algebra $\hat{\mathfrak{g}}_{DDD}$ or we can work with