#
Unraveling the Holomorphic Twist:
Central Charges

Pieter Bomans and Jingxiang Wu
Mathematical Institute, University of Oxford
Andrew Wiles Building, Radcliffe Observatory Quarter
Woodstock Road, Oxford, OX2 6GG, U.K.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
0000-0002-0907-9830 https://orcid.org/0000-0001-6867-1407

###### Abstract

The holomorphic twist provides a powerful framework to study minimally
protected sectors in supersymmetric quantum field theories. We investigate the
algebraic structure underlying the holomorphic twist of $\mathcal{N}=1$
superconformal field theories in four dimensions. In particular, in
holomorphically twisted theories the flavour and conformal symmetry algebras
are enhanced to infinite-dimensional higher Kac Moody and higher Virasoro
symmetry algebras respectively. We explicitly compute the binary and ternary
$\lambda$-brackets and clarify their relation with the underlying infinite-
dimensional symmetry algebra. Doing so we show that the central extensions of
said symmetry algebras precisely encode the conformal anomalies $a$ and $c$ as
well as the flavour central charges of the physical four-dimensional theory.
This parallels the familiar story in two dimensions where the conformal
anomaly $c$ is encoded in the central extension of the Virasoro algebra.

###### Contents

  1. 1 Introduction and summary
  2. 2 Infinite-dimensional symmetry in four dimensions
    1. 2.1 Infinite-dimensional symmetry and the $\lambda$-bracket
    2. 2.2 Higher Kac-Moody algebra
    3. 2.3 Higher Virasoro algebra
    4. 2.4 Non-conformal theories
  3. 3 Conformal anomalies from integrated correlators
    1. 3.1 Flavour current multiplets
    2. 3.2 Supercurrent multiplet
  4. 4 Examples
    1. 4.1 Free vector multiplet
    2. 4.2 $N_{f}$ free chirals
    3. 4.3 $\mathcal{N}=4$ SYM
    4. 4.4 Fundamental $\mathrm{SU}(2)$ SQCD
    5. 4.5 Adjoint $\mathrm{SU}(2)$ SQCD
  5. A 4d $\mathcal{N}=1$ superconformal algebra
    1. A.1 $\mathcal{N}=1$ superspace
    2. A.2 Lie algebra conventions
  6. B Residues, heat kernels and integrated correlators
    1. B.1 Higher-dimensional residues
    2. B.2 Heat kernel regularisation
    3. B.3 $\lambda$-bracket from integrated correlators
    4. B.4 $\lambda$-brackets from Feynman integrals
  7. C Basic aspects of homological algebra
  8. D (Higher) Kac Moody and Virasoro algebras
    1. D.1 Kac Moody algebra
    2. D.2 Higher Kac-Moody algebra
    3. D.3 Virasoro algebra
    4. D.4 Higher Virasoro algebra
    5. D.5 Proof of equation (D.23)
    6. D.6 Associativity relations of $\lambda$-brackets

## 1 Introduction and summary

The operator product expansion (OPE) lies at the heart of any non-perturbative
treatment of quantum field theory. Despite extensive research endeavours,
unravelling the detailed structure of the OPE in generic quantum field
theories remains an elusive intellectual pursuit. This challenge is
highlighted by the scarcity of tools accessible beyond the framework of
perturbation theory.

In order to resolve this impasse it proves fruitful to explore theories with
additional symmetries. Indeed, the OPE is best understood in conformal field
theories (CFTs) due to the extra constraints imposed by conformal invariance.
This becomes most evident in two dimensions, where for a large class of
theories, known as rational CFTs, the associativity of the OPE combined with
the Virasoro symmetry algebra allows one to obtain exact solutions for the
correlation functions of local operators [1]. The key factor in achieving this
success lies in the enhancement of the two-dimensional (global) conformal
algebra to the infinite-dimensional Virasoro algebra. The presence of such
infinite-dimensional algebras is tied to the two-dimensional setup, and at
first sight, it is not clear how to lift these tools to higher dimensions.

A second situation in which the OPE is under better control arises in the
context of supersymmetric quantum field theories (SQFTs). In four-dimensional
$\mathcal{N}=1$ theories, one can employ the power of holomorphy [2] and show
that certain observables such as the partition function [3] or correlation
functions in a protected sector of the theory, known as the chiral ring [4],
only depend holomorphically on the complex structure moduli. In both examples,
the holomorphicity in a protected sector greatly simplifies computations and
allows for a precise and detailed understanding of the OPE.

These two developments were neatly combined in 4d $\mathcal{N}=2$
superconformal field theories (SCFTs), where it was shown that the correlation
functions in a subsector of the full theory behave precisely like those of a
two-dimensional CFT [5]. By passing to the cohomology of a supercharge of the
schematic form $\mathbf{Q}\simeq\mathcal{Q}+\mathcal{S}$ one obtains a vertex
operator algebra (VOA) encoding the OPE of so-called Schur operators when
restricted to a complex plane.111Alternatively, one can obtain this sector by
performing a topological-holomorphic twist [6] followed by a
$\Omega$-deformation in the topological directions [7, 8, 9]. This example
demonstrates a first instance in which one can leverage the power of infinite-
dimensional symmetry in a higher-dimensional context.

In this work, our focus lies on four-dimensional $\mathcal{N}=1$ SCFTs
instead, where the supersymmetry algebra does not contain an appropriate
supercharge needed to obtain a 2d VOA as its cohomology. However, as
demonstrated in [10] and further expanded upon in [11, 12, 13], one can obtain
a similar infinite-dimensional symmetry algebra in four-dimensional
$\mathcal{N}=1$ QFTs.222Early hints of the presence of such infinite-
dimensional symmetry algebras in $\mathcal{N}=1$ theories were already
described in [14, 15, 16, 17, 18, 19] as well as in [20] and subsequent works.
The power of holomorphy can be seen in its full glory by passing to the
cohomology of a single supercharge $\mathbf{Q}\simeq\mathcal{Q}$. In the
holomorphically twisted sector, the algebraic structure underlying the OPE is
a four-dimensional generalisation of a VOA, a higher VOA [21, 22]. Analogous
to 2d CFTs, in the holomorphically twisted theory, the supersymmetry algebra
is enhanced to an infinite-dimensional higher Virasoro or higher Kac-Moody
algebra.

Operators in the holomorphically twisted theory are annihilated by a single
supercharge and for that reason, we call them _semi-chiral_ operators. The
semi-chiral sector receives contributions from all chiral (semi-)short
multiplets and therefore captures the same Hilbert space as the full
superconformal index. In particular, in contrast to the Schur sector of
$\mathcal{N}=2$ SCFTs [23], this cohomology captures states dual to black
holes. Moreover, the construction of the holomorphic twist does not rely on
conformal symmetry and therefore can be extended to any non-conformal
$\mathcal{N}=1$ QFT. Understanding the algebraic structure of this twisted
sector, and exploiting this infinite-dimensional symmetry algebra therefore
provides us with a window into obtaining a deeper understanding of various
aspects of $\mathcal{N}=1$ SCFTs such as confinement [13], $a$-theorems, RG
flows and the structure of black hole states in said theories.

This work aims to clarify the role of this infinite-dimensional symmetry
algebra, and in particular relate observables in the holomorphically twisted
theory to observables in the physical 4d $\mathcal{N}=1$ theory. In
particular, we show how the central extensions of the symmetry algebra encode
the conformal anomalies and flavour central charges of the original theory and
demonstrate this through a series of examples.

#### Warm up: two-dimensional VOAs

The higher VOAs we encounter after performing the holomorphic twist are very
similar to the familiar two-dimensional VOAs encountered in the context of
two-dimensional CFTs. Before describing their more exotic four-dimensional
cousins, let us briefly review some facts about VOAs in two dimensions. The
material below is well-known but phased in a slightly unfamiliar way which
generalises easily to higher dimensions. For more details on the two-
dimensional setting see for example [24] and references therein.

The algebraic structure underlying a two-dimensional CFT is given by a vertex
operator algebra. A VOA consists of the following data:333Additionally, the
OPE has to satisfy an appropriate locality axiom, which is crucial to
guarantee convergence of the OPE.

  * •

A vector space $V$; the state space.

  * •

A linear map $Y$; the state-operator map,

$Y\,\,:\,\,V\rightarrow(\text{End}\,V)[\\![z]\\!]\,\,:\,\,\mathbf{O}\mapsto\mathbf{O}(z)=\sum_{n\in\mathbf{Z}}\frac{\left\\{\mathbf{O}\,,\bullet\right\\}_{n}}{z^{n+1}}\,.$
(1.1)

  * •

The translation operator $T$; a derivation satisfying the translation axiom
$\left(T\mathbf{O}\right)(z)=\partial_{z}\mathbf{O}(z)$.

  * •

The vacuum vector $\left|{0}\right\rangle\in V$; satisfying
$\left|{0}\right\rangle(z)=\text{Id}_{V}$ and
$\mathbf{O}(z)\left|{0}\right\rangle=\mathbf{O}+(T\mathbf{O})z+\mathcal{O}(z^{2})$.

A VOA is naturally equipped with an infinite number of products represented by
the brackets $\left\\{\bullet\,,\bullet\right\\}_{n}$ in terms of which the
operator product expansion can be expressed as

$\mathbf{O}_{1}(z)\mathbf{O}_{2}(w)=\sum_{n\in\mathbb{Z}}\frac{\left\\{\mathbf{O}_{1}\,,\mathbf{O}_{2}\right\\}_{n}(w)}{(z-w)^{n+1}}\,.$
(1.2)

Conversely, the brackets $\left\\{\bullet\,,\,\bullet\right\\}_{n}$ provide us
with a convenient way to unpack the OPE data of the VOA, which can be
extracted as

$\left\\{\mathbf{O}_{1}\,,\mathbf{O}_{2}\right\\}_{n}(0)=\oint_{S^{1}}\frac{{\rm
d}z}{2\pi{\rm i}}z^{n}\mathbf{O}_{1}(z)\mathbf{O}_{2}(0),\quad
n\in\mathbb{Z}\,,$ (1.3)

This equation has a very transparent physical meaning. Any holomorphic
operator $\mathbf{O}(z)$ can be thought of as a current for a global symmetry
with conservation law $\bar{\partial}\mathbf{O}(z)=0$. This conservation law
continues to hold when we insert any $\bar{\partial}$-closed function $\rho$
to construct another current $\rho\,\mathbf{O}(z)$. Such functions are
elements of the Dolbeault cohomology,

$\rho\,\in\,H^{0,0}_{\bar{\partial}}\left(\mathbb{C}\backslash\\{0\\}\right)=\mathbb{C}[z,z^{-1}]\,,$
(1.4)

and the collection of all holomorphic currents naturally forms an infinite-
dimensional symmetry algebra.

Phrased differently, (1.3) is nothing but the action of the topological line
operator $\left\\{\mathbf{O},\bullet\right\\}_{n}$ constructed by integrating
the current $z^{n}\mathbf{O}(z)$ over the codim-1 manifold $S^{1}$. For any
holomorphic operator, the topological line operators therefore arise in
infinite towers labelled by $n\in\mathbb{Z}$.

The non-negative modes ($n\geq 0$) capture the singular part of the OPE. These
modes can be conveniently collected in a generating function called the
$\lambda$-bracket,

$\left\\{{\mathbf{O}_{1}\,}_{\lambda}\,{\mathbf{O}_{2}}\right\\}\equiv\oint_{S^{1}}\frac{{\rm
d}z}{2\pi{\rm i}}\mathrm{e}^{\lambda
z}\mathbf{O}_{1}(z)\mathbf{O}_{2}(0)=\sum_{n\geq
0}\frac{\lambda^{n}}{n!}\left\\{\mathbf{O}_{1}\,,\,\mathbf{O}_{2}\right\\}_{n}\,,$
(1.5)

This $\lambda$-bracket naturally generalises to higher dimensional situations
and will be a central object in this work. A vector space $V$, equipped with a
$\lambda$-bracket and a translation operator $T$ defines a Lie conformal
algebra. Any vertex operator algebra automatically defines a Lie conformal
algebra simply by forgetting about the regular part of the OPE. To get more
familiar with this structure we refer the interested reader to Appendix D
where we give some more explicit details on the Kac Moody algebra and Virasoro
algebra as Lie conformal algebras.

#### Higher VOA from higher dimensional holomorphic field theories

After this two-dimensional intermezzo, we are ready to discuss the case of
four-dimensional holomorphically twisted theories. After the twist operators
only depend holomorphically on the coordinates of $\mathbb{C}^{2}$ and we can
proceed entirely analogous to the two-dimensional case.

For any holomorphic operator $\mathbf{O}(z_{1},z_{2},\dots,z_{d})$ in
$\mathbb{C}^{d}$, $\rho\,\mathbf{O}$ continues to be holomorphic as long as
$\rho$ is $\bar{\partial}$-closed. Analogous to the above we naturally run
into an infinite-dimensional symmetry algebra. Such $\bar{\partial}$-closed
$\rho$ are elements of the Dolbeault cohomology,

$\rho\,\in\,H^{0,\bullet}_{\bar{\partial}}\left(\mathbb{C}^{d}\backslash\\{0\\}\right)\,.$
(1.6)

In higher dimensions, the Dolbeault cohomology
$H^{0,\bullet}_{\bar{\partial}}\left(\mathbb{C}^{d}\backslash\\{0\\}\right)$
is known to concentrate on degree $(0,0)$ and $(0,d-1)$ and is given as

$H^{0,i}_{\bar{\partial}}\left(\mathbb{C}^{d}\backslash\\{0\\}\right)=\left\\{\begin{array}[]{ll}\mathbb{C}\left[z_{1},\ldots,z_{d}\right]&i=0\,,\\\
\mathbb{C}\left[\partial_{1},\ldots,\partial_{d}\right]\,\omega_{\rm
BM}&i=d-1\,,\\\ 0&\text{otherwise}\,,\\\ \end{array}\right.$ (1.7)

where $\omega_{\rm BM}$ is the Bochner-Martinelli kernel introduced in more
detail in Appendix B. The degree $(0,0)$ cohomology provides us with a direct
generalisation of the positive modes in two dimensions, cf. (1.3),

$\left\\{\mathbf{O}_{1}\,,\mathbf{O}_{2}\right\\}_{n_{1},n_{2},\dots,n_{d}}(0)=\oint_{S^{2d-1}}\frac{{\rm
d}^{d}z}{(2\pi{\rm i})^{n}}\,z_{1}^{n_{1}}z_{2}^{n_{2}}\cdots
z_{d}^{n_{d}}\,\mathbf{O}_{1}(z)\mathbf{O}_{2}(0)\,,\qquad n_{i}\geq 0\,.$
(1.8)

Conversely, the negative modes can be generalised by picking $\rho$’s from the
degree $(0,d-1)$ cohomology,

$\left\\{\mathbf{O}_{1}\,,\mathbf{O}_{2}\right\\}_{n_{1},n_{2},\dots,n_{d}}(0)=\oint_{S^{2d-1}}\frac{{\rm
d}^{d}z}{(2\pi{\rm
i})^{d}}\,\partial_{z_{1}}^{-n_{1}-1}\partial_{z_{2}}^{-n_{2}-1}\cdots\partial_{z_{d}}^{-n_{d}-1}\omega_{\rm
BM}\,\mathbf{O}_{1}(z)\mathbf{O}_{2}(0)\,,\qquad n_{i}\leq-1\,.$ (1.9)

Note that in two dimensions this definition precisely reproduces the negative
modes in equation (1.3). Following the analogy with two dimensions, one can
construct infinitely many codimension-one topological surface operators by
integrating $\rho\,\mathbf{O}_{1}$ over the $(2d-1)$-sphere, $S^{2d-1}$. For
example, we can act on another local operator $\mathbf{O}_{2}(0)$ by wrapping
the topological surface operator on the $S^{2d-1}$ linking the origin,

$\oint_{S^{2d-1}}\frac{{\rm d}^{d}z}{(2\pi{\rm i})^{d}}\ \rho\
\mathbf{O}_{1}(z)\mathbf{O}_{2}(0)\,.$ (1.10)

Collecting the non-negative modes (1.8) we define the $\lambda$-bracket as the
following generating function,

$\left\\{\mathbf{O}_{1}\,{}_{\lambda}\,\mathbf{O}_{2}\right\\}=\oint_{S^{2d-1}}\frac{{\rm
d}^{d}z}{(2\pi{\rm i})^{d}}\mathrm{e}^{\lambda\cdot
z}\mathbf{O}_{1}(z)\mathbf{O}_{2}(0)\,.$ (1.11)

Unlike in one complex dimension where the brackets generate a Lie conformal
algebra and therefore satisfy the Jacobi identity, in higher dimensions, the
brackets only satisfy the Jacobi identity homotopically. Higher homotopies are
captured by a collection of higher ($\lambda$-)brackets. The definitions of
the higher brackets are well-known in $E_{d}$ algebra using the integration
cycles in the configuration space of points [25]. While we expect the same is
true for holomorphic theories, it is impractical to deal with configuration
space of more than three points. For this reason we follow an alternative
definition of the higher brackets as proposed in [12, 13], which we carefully
introduce in Section 2.1. The collection of all local operators together with
the $n$-ary ($\lambda$-)brackets form a holomorphic factorisation algebra,
which can be thought of as providing a free field realisation of the higher-
VOA on $\mathbb{C}^{d}$.

#### Summary of the results

This higher VOA structure naturally arises in the context of holomorphically
twisted four-dimensional $\mathcal{N}=1$ theories [10, 11, 12, 13]. In this
case, the holomorphic operators of interest are given by semi-chiral
superfields $\mathbf{O}$, which are in one-to-one correspondence with
$\mathbf{Q}$-cohomology classes. In particular, whenever the theory preserves
some flavour symmetry, the twisted theory contains an associated semi-chiral
superfield $\mathbf{J}$. Similarly, any (local) unitary $\mathcal{N}=1$ SCFT
contains a stress tensor which gives rise to a semi-chiral superfield
$\mathbf{S}_{\dot{\alpha}}$ in the twisted theory.

In two-dimensional VOAs, it is well-known that the modes of a conserved
current $J_{n}=\left\\{\mathbf{J}\,,\bullet\right\\}_{n}$ and the modes of the
holomorphic stress tensor $L_{n}=\left\\{\mathbf{S}\,,\bullet\right\\}_{n+1}$
generate respectively a Kac-Moody algebra and Virasoro algebra. The central
charges of these algebras encode important physical information about the
theory. The Virasoro central charge $c$ encodes the conformal anomaly of the
2d CFT, while the central charge of the Kac-Moody algebra equals the flavour
central charge $k_{G}$. In the two-dimensional context, these central charges
arise in the most singular terms of respectively the $\mathbf{S}\mathbf{S}$ or
$\mathbf{J}\mathbf{J}$ OPE, where $\mathbf{S}$ and $\mathbf{J}$ denote the
holomorphic stress tensor and the holomorphic current. Similarly, these
central charges can be read off from the $\lambda$-brackets

$\left\\{\mathbf{S}\,{}_{\lambda}\,\mathbf{S}\right\\}=\left(\lambda^{3}\frac{c}{12}+2\lambda+\partial\right)\mathbf{S}\,,\qquad\qquad\left\\{\mathbf{J}^{a}\,{}_{\lambda}\,\mathbf{J}^{b}\right\\}=f^{ab}{}_{c}\,\mathbf{J}^{c}+\delta^{ab}k_{G}\lambda\,.$
(1.12)

In four-dimensional holomorphically twisted theories the modes of conserved
current and stress tensor semi-chiral superfields are given by
$\mathbf{J}^{a}_{m,n}=\left\\{\mathbf{J}^{a}\,,\bullet\right\\}_{m,n}$ and
$\mathbf{L}^{\dot{\alpha}}_{m,n}=\left\\{\mathbf{S}_{{\dot{\alpha}}}\,,\bullet\right\\}_{m+1,n+1}$.
Analogous to the two-dimensional setup their modes now generate a higher Kac-
Moody and higher Virasoro symmetry algebra. It is then natural to expect that
the central charges of these higher VOAs encode physical information about the
original untwisted theory. We make this expectation precise by explicitly
computing binary and ternary $\lambda$-brackets of conserved current and
stress tensor semi-chiral superfields. The binary $\lambda$-brackets are given
by

$\left\\{\mathbf{S}_{\dot{\alpha}}\,{}_{\lambda}\mathbf{S}_{\dot{\beta}}\right\\}=\lambda_{\dot{\alpha}}\mathbf{S}_{\dot{\beta}}+\lambda_{\dot{\beta}}\,\mathbf{S}_{\dot{\alpha}}+\partial_{\dot{\alpha}}\mathbf{S}_{\dot{\beta}}\,,\qquad\qquad\left\\{\mathbf{J}^{a}\,{}_{\lambda}\,\mathbf{J}^{b}\right\\}=f^{ab}{}_{c}\,\mathbf{J}^{c}\,,$
(1.13)

where $f^{ab}{}_{c}$ are the anti-symmetric structure constants of the flavour
symmetry algebra. Unlike in two dimensions, these expressions do not contain
any information about the central charges. However, the main result of this
paper lies in the computation of the ternary $\lambda$-brackets. The
$\lambda$-bracket for three stress tensor semi-chiral superfields is

$\left\\{\mathbf{S}_{\dot{\alpha}}\,{}_{\lambda_{1}}\mathbf{S}_{\dot{\beta}}\,{}_{\lambda_{2}}\mathbf{S}_{\dot{\gamma}}\right\\}=\frac{4}{3}\,(3c-2a)\,\Lambda_{\mathcal{N}=4}+16\,(c-a)\,\Lambda_{\rm
vector}\,,$ (1.14)

where $\Lambda_{\mathcal{N}=4}$ and $\Lambda_{\rm vector}$ are theory-
independent kinematic factors which are defined in the main text in (3.15).
The coefficients in this expression are precisely given by the $a$ and $c$
conformal anomalies of the untwisted theory! We compute this $\lambda$-bracket
in all generality as an integrated three-point function of (descendants of)
stress tensors of the original untwisted theory and check our results in a
wide variety of examples. Similarly, we compute the $\lambda$-bracket of the
conserved current semi-chiral superfields resulting in the following
expression,

$\left\\{\mathbf{J}^{a}\,{}_{\lambda_{1}}\,\mathbf{J}^{b}\,{}_{\lambda_{2}}\,\mathbf{J}^{c}\right\\}=k_{G}\,d^{abc}(\lambda_{1},\lambda_{2})\,.$
(1.15)

In this equation $d^{abc}$ are the symmetric structure constants of the
flavour symmetry group while $k_{G}$ encodes the flavour central charge of the
untwisted theory. These results therefore present a clear connection between
the central charges in the holomorphic theory obtained after performing the
holomorphic twist and observables in the original untwisted theory.

#### Outlook

The study of the holomorphic twist is still in its early infancy and there are
many open questions. Although mathematical results are steadily emerging (see
[26, 27, 28, 29, 30, 31, 32] for a necessarily incomplete selection) it
remains a challenge to connect observables in the twisted theory to physical
observables in the untwisted theory such as correlation functions of local
operators. Our work presents a first step towards better understanding the
physics behind the holomorphic twist and opens a window into making progress
on a variety of fronts.

The first direction to make progress lies in better understanding the
$L_{\infty}$ structure of the modes and its representation theory. In this
work we compute $\lambda$-brackets but the map to the ternary $L_{\infty}$
3-bracket is not immediate [33]. We plan to come back to this question in the
near future. A more challenging question is to understand the $L_{\infty}$
modules and null vectors. A similar understanding in two dimensions gives rise
to the discovery of minimal models [1] and allows one to exactly solve the
OPE. The discovery of a similar structure in the holomorphic twist would
present a major breakthrough in our understanding of supersymmetric quantum
field theories.

Although in this work we focus on conformal theories, many aspects of our
story continue to hold for non-conformal $\mathcal{N}=1$ SQFTs. In this case,
the semi-chiral superfield corresponding to the stress tensor is no longer
given by $\mathbf{S}_{\dot{\alpha}}$ but rather by
$\widetilde{\mathbf{S}}=\partial^{\dot{\alpha}}\mathbf{S}_{\dot{\alpha}}$. The
modes of this semi-chiral superfield generate an infinite dimensional symmetry
algebra distinct from the higher Virasoro algebra. It would be interesting to
better understand the structure of this algebra and find out in which shape a
similar relation between conformal anomalies and central charges persists in
this setup. Understanding this mechanism would provide us with an elegant
framework to prove for example the $a$-theorem [34] in a similar fashion as
the original $c$-theorem in two-dimensions [35] and provide us with a deeper
understanding of RG flows in $\mathcal{N}=1$ SCFTs.

Finally, holomorphic theories arise in a variety of other examples. For
example, minimal twists of six-dimensional theories with at least
$\mathcal{N}=(1,0)$ supersymmetry give rise to six-dimensional higher VOAs.
Increasing the supersymmetry to six-dimensional $\mathcal{N}=(2,0)$ SCFTs,
there is an alternative twist analogous to $\mathcal{Q}+\mathcal{S}$ twist in
four dimensions [5, 36, 37] giving rise to a two-dimensional VOA. On the other
hand, for six-dimensional $\mathcal{N}=(1,1)$ SQFTs there is holomorphic-
topological twist [26] giving rise to an associated four-dimensional higher
VOA. It would be very interesting to better understand this unexplored
protected sector and the physical observables it encodes.

#### Structure of this paper

The remainder of this paper is organised as follows. In Section 2, we present
the holomorphic twist of four-dimensional $\mathcal{N}=1$ QFTs, discuss the
resulting infinite-dimensional symmetry algebras and carefully define the
$\lambda$-brackets. Subsequently, in section 3, we compute the binary and
ternary $\lambda$-bracket of stress tensors and conserved currents and relate
them to the conformal anomalies and flavour central charges of the original
theory. In section 4, we conclude with a series of examples demonstrating the
general formulae derived in the previous sections. Appendix A summarises our
conventions and notation and introduces the $\mathcal{N}=1$ superconformal
algebra. Appendices B-D contain various details omitted in the main text.

## 2 Infinite-dimensional symmetry in four dimensions

Consider a four-dimensional $\mathcal{N}=1$ superconformal theory in Euclidean
space.444For most of this work we specialise to superconformal theories but
the twisting procedure does not demand superconformal symmetry and many of our
results continue to hold in a non-conformal setup with minimal modifications.
The superconformal algebra contains (at least) two spinor supercharges
$\mathcal{Q}_{\alpha}$ and $\widetilde{\mathcal{Q}}_{\dot{\alpha}}$, subject
to the following commutation relations,555For the complete set of commutation
relations we refer the reader to Appendix A.

$\left\\{{\mathcal{Q}_{\alpha}},{\widetilde{\mathcal{Q}}_{\dot{\alpha}}}\right\\}=\mathcal{P}_{\alpha\dot{\alpha}}\,,$
(2.1)

where $\mathcal{P}_{\alpha{\dot{\alpha}}}$ are the translation generators and
$\alpha,\dot{\alpha}=\pm$ are fundamental
$\mathrm{SU}(2)_{L}\times\mathrm{SU}(2)_{R}\simeq\mathrm{SO}(4)$ indices. In
this work we are interested in the holomorphic twist666We take a modern point
of view where twisting is defined by passing to the cohomology of a nilpotent
supercharge. See Appendix F of [13] for a disambiguation of the terminology.
with respect to a nilpotent supercharge $\mathbf{Q}$, obtained by passing to
the $\mathbf{Q}$-cohomology. Without loss of generality, we pick
$\mathbf{Q}=\mathcal{Q}_{-}$ as our twisting supercharge. This choice selects
a complex structure such that
$\left\\{z^{\dot{\alpha}},\bar{z}^{\dot{\alpha}}\right\\}=\left\\{x^{+\dot{\alpha}},x^{-\dot{\alpha}}\right\\}$
are (anti-)holomorphic coordinates on $\mathbb{C}^{2}$.777In contrast to [13],
we follow standard conventions and refrain from interchanging dotted and
dotless indices. Moreover, we immediately notice that

$\left\\{{\mathbf{Q}},{\widetilde{\mathcal{Q}}_{\dot{\alpha}}}\right\\}=\partial_{\bar{z}^{\dot{\alpha}}}\,,$
(2.2)

and therefore the twisted theory is holomorphic in the cohomological sense. As
already alluded to in the introduction, this holomorphic theory behaves in
many ways similar to a two-dimensional vertex operator algebra. We follow this
analogy and describe how an infinite-dimensional symmetry algebra emerges in
this protected sector and define infinitely many $n$-ary operations encoding
the symmetry algebra.

The underlying vector space of this higher VOA is defined as the
$\mathbf{Q}$-cohomology of local operators. The inclusion of non-local
operators in this framework presents an interesting extension we hope to
address in future work. Since operators in the cohomology are annihilated by
one chiral supercharge we call them semi-chiral operators. The harmonic
representatives of the cohomology are characterised by the following condition
on the quantum numbers

$\left\\{{\mathbf{Q}},{\mathbf{Q}^{\dagger}}\right\\}=\Delta+\frac{3}{2}r+\frac{1}{2}j_{L}=0\,,$
(2.3)

where $\Delta$ is the conformal dimension, $r$ is the $r$-charge and $j_{L}$
denotes the eigenvalue of the Cartan generator of $\mathrm{SU}(2)_{L}$. In
superconformal theories, the operators are grouped in superconformal
multiplets where short multiplets satisfy certain shortening conditions.888We
follow the conventions of [38] to label superconformal multiplets. One can
easily see that cohomology classes are in one-to-one correspondence with
multiplets with a chiral shortening condition. In more detail:

  * •

$A_{i=1,2}$ shortening (where $i=2$ if and only if $j_{L}=0$): An
$A_{i}\overline{X}[j_{L},j_{R}]^{(r)}_{\Delta}$ multiplet contains the
following semi-chiral operator,

$\mathbf{O}^{(0)}_{\dot{\alpha}_{1}\cdots\dot{\alpha}_{\bar{j}}}=D_{+}\mathcal{O}_{+\cdots+\dot{\alpha}_{1}\cdots\dot{\alpha}_{\bar{j}}}\Big{|}_{\theta_{i}=\bar{\theta}_{i}=0}\,,$
(2.4)

where $\mathcal{O}$ is the corresponding superfield and $\overline{X}$ can
either be long or denote an anti-chiral shortening condition.

  * •

$B_{1}$ shortening: A $B_{1}\overline{X}[0,j_{R}]^{(r)}_{\Delta}$ multiplet
contains the following semi-chiral superfield,

$\mathbf{O}^{(0)}_{\dot{\alpha}_{1}\cdots\dot{\alpha}_{\bar{j}}}=\mathcal{O}_{\dot{\alpha}_{1}\cdots\dot{\alpha}_{\bar{j}}}\Big{|}_{\theta_{i}=\bar{\theta}_{i}=0}\,.$
(2.5)

Note that only for the $B_{1}$-type shortening condition the semi-chiral
operator is given by a superconformal primary, the subsector of such operators
forms the chiral ring [4]. In the twisted theory, the barred fermionic
coordinates $\bar{\theta}^{\dot{\alpha}}$ transform as anti-holomorphic one-
forms and it will be useful to replace them with the differentials ${\rm
d}\bar{z}^{\dot{\alpha}}$. Doing so we introduce the reduced superfield,

$\mathbf{O}=\mathrm{e}^{{\rm
d}\bar{z}^{\dot{\alpha}}\widetilde{\mathcal{Q}}_{\dot{\alpha}}}\mathbf{O}^{(0)}=\mathbf{O}^{(0)}+\mathbf{O}^{(1)}+\mathbf{O}^{(2)}\,,$
(2.6)

Following [13] we call such a superfield a semi-chiral superfield999See
Appendix A for our superspace conventions and the definition of $D_{-}$. if it
satisfies

$D_{-}\mathbf{O}=\left(\mathbf{Q}+\bar{\partial}\right)\mathbf{O}=0\,,$ (2.7)

where $\bar{\partial}=\frac{\partial}{\partial\bar{z}^{{\dot{\alpha}}}}{\rm
d}\bar{z}^{{\dot{\alpha}}}$. If $\mathbf{O}^{(0)}$ is a scalar operator, its
descendants $\mathbf{O}^{(i)}$ are $(0,i)$ forms on $\mathbb{C}^{2}$. For any
semi-chiral operator $\mathbf{O}^{(0)}$, the higher-form terms satisfy the
holomorphic descent relation,

$\mathbf{Q}\mathbf{O}^{(k)}+\bar{\partial}\mathbf{O}^{(k-1)}=0\,.$ (2.8)

In particular, the first relation
$\bar{\partial}\mathbf{O}^{(0)}=\mathbf{Q}\mathbf{O}^{(1)}$ states that
$\mathbf{O}^{(0)}$ is a holomorphic operator up to $\mathbf{Q}$ exact terms.
Note that similar to chiral ring operators, the OPE between two semi-chiral
operators is completely regular. This is a consequence of Hartog’s theorem,
which essentially states that there cannot be isolated singularities on
$\mathbb{C}^{2}$. Note however that the OPE between a semi-chiral operator and
its descendants, or between two descendants, can be singular. In order to
observe more refined algebraic structures in the holomorphically twisted
theory it is therefore imperative to carefully keep track of the descendants.
Doing so will allow us to extract a plethora of exact detailed OPE data.

### 2.1 Infinite-dimensional symmetry and the $\lambda$-bracket

The main upshot of the previous discussion is that the holomorphic twist, i.e.
the semi-chiral sector (i.e. the $\tfrac{1}{4}$-BPS sector) of any
$\mathcal{N}=1$ supersymmetric theory, can be described by a holomorphic field
theory on $\mathbb{C}^{2}$. In particular, the descent relations satisfied by
the components of a semi-chiral superfield can be interpreted as conservation
laws in the twisted theory.

Each semi-chiral operator $\mathbf{O}_{1}$ gives rise to infinitely many
conservation laws. Given any $\bar{\partial}$-closed $(0,\bullet)$ form
$\rho$, the operator $\rho\,\mathbf{O}_{1}$ is also conserved. More generally,
every operator $\mathbf{O}_{1}(z)$ in the twisted theory gives rise to
infinitely many symmetry generators $\\{\mathbf{O}_{1}\,,\bullet\\}_{m,n}$,
for $m,n\in\mathbb{Z}$, acting on operators at the origin as

$\left\\{\mathbf{O}_{1}\,,\mathbf{O}_{2}\right\\}_{m,n}(0)=\oint_{S^{3}}{\rm
d}^{2}z\,\rho_{m,n}\ \mathbf{O}_{1}(z)\mathbf{O}(0)\,.$ (2.9)

As already explained in the introduction, the forms $\rho$ are elements of the
Dolbeault cohomology
$H^{0,\bullet}\left(\mathbb{C}^{2}\backslash\\{0\\}\right)$, which is known to
concentrate in degree $(0,0)$ and $(0,1)$ [11, 10],

$\rho_{m,n}=\left\\{\begin{array}[]{lll}(z^{\dot{+}})^{m}(z^{\dot{-}})^{n}&\in\Omega^{0,0}\left(\mathbb{C}^{2}\backslash\\{0\\}\right)&m,n\geq
0\,,\\\
\partial_{{\dot{+}}}^{-m-1}\partial_{{\dot{-}}}^{-n-1}\omega_{BM}(z)&\in\Omega^{0,1}\left(\mathbb{C}^{2}\backslash\\{0\\}\right)&m,n\leq-1\,,\\\
\phantom{asdf}0&&\text{otherwise}\,.\end{array}\right.$ (2.10)

where $\omega_{\rm BM}(z)$ is the Bochner-Martinelli kernel,

$\omega_{\rm BM}(z)=\frac{1}{(2\pi{\rm i})^{2}}\frac{(\bar{z},{\rm
d}\bar{z})}{|z|^{4}}\in\Omega^{0,1}\left(\mathbb{C}^{2}\backslash\\{0\\}\right)\,,$
(2.11)

and we used the shorthand notation
$\partial_{\dot{\alpha}}=\frac{\partial}{\partial z^{{\dot{\alpha}}}}$ and
similarly
$\bar{\partial}_{\dot{\alpha}}=\frac{\partial}{\partial\bar{z}^{{\dot{\alpha}}}}$.
Analogous to two dimensions, the positive modes encode the singular part of
the OPE of $\mathbf{O}_{1}$ with another operator $\mathbf{O}_{2}$, while the
negative modes capture the regularised products
$\partial_{{\dot{+}}}^{m}\partial_{{\dot{-}}}^{n}\mathbf{O}_{1}\,\mathbf{O}_{2}$.
It is convenient to collect all the positive modes in a $\lambda$-bracket,

$\left\\{\mathbf{O}_{1}\,{}_{\lambda}\,\mathbf{O}_{2}\right\\}=\oint_{S^{3}}\frac{{\rm
d}^{2}z}{(2\pi i)^{2}}\ e^{\lambda\cdot
z}\,\mathbf{O}_{1}(z)\mathbf{O}_{2}(0),$ (2.12)

where $\lambda\cdot z=\lambda_{\dot{\alpha}}z^{\dot{\alpha}}$. Unlike in one
complex dimension where the $\lambda$-bracket satisfies a Jacobi identity, in
higher dimensions the Jacobi identity is only satisfied
homotopically.101010The simplest examples of such structures arise in
$L_{\infty}$ algebras which are a generalization of Lie algebras. For a brief
summary of the relevant properties of $L_{\infty}$ algebras see Appendix C. In
two dimensions, the triplet
$\left(V,T,\left\\{\bullet\,{}_{\lambda}\,\bullet\right\\}\right)$ satisfies
the axioms of a Lie conformal algebra. In higher dimensions, with the addition
of higher brackets one could call the triplet
$\left(V,T,\left\\{\bullet\,{}_{\lambda_{1}}\,\cdots\bullet{}_{\lambda_{i-1}}\,\bullet\right\\}_{i=2,3,\dots}\right)$
an L∞ conformal algebra. See related [39]. Higher homotopies are captured by
the higher brackets which roughly capture the extent to which the Jacobi
identity is violated. The resulting algebras are holomorphic generalizations
of $E_{d}$, or $d$-disc, algebras and the factorisation algebras discussed in
[20, 28, 29, 40, 25] in which case the higher $n$-ary brackets are defined
through integrals (over specific integration cycles) in the configuration
space of $n$ points. While the same is expected to hold for the holomorphic
theories under discussion, it is prohibitively impractical to deal with
configuration space of more than three points. For this reason, in
perturbation theory, it is more convenient to use an alternative definition of
$(n+1)$-ary $\lambda$-brackets as proposed in [12, 13].111111The equivalence
of this definitions with the definition through integrals over the
configuration space of $n$-points is non-trivial and will be explained in more
detail in [41].

$\left\\{\mathbf{O}_{1}\,{}_{\lambda_{1}}\,\cdots\,\mathbf{O}_{n}\,{}_{\lambda_{n}}\,\mathbf{O}_{0}\right\\}\equiv\mathbf{Q}\left(\prod_{k=1}^{n}\int_{\mathbb{C}^{2}}\frac{d^{2}z_{k}}{(2\pi
i)^{2}}e^{\lambda_{k}\cdot
z_{k}}:\mathbf{O}_{1}(z_{1},\bar{z}_{1})\cdots\mathbf{O}_{n}(z_{n},\bar{z}_{n})\mathbf{O}_{0}(0,0):\right)\,,$
(2.13)

where the operators $\mathbf{O}_{i}$ are semi-chiral superfields and $::$
refers to the operation of performing all possible Wick contractions. All
fields that are not Wick contracted will be collected and shifted to the
origin so that the right-hand side of (2.13) is a Feynman integral multiplying
an operator inserted at origin and never involves integrating operators. See
[12] for more a careful explanation of this definition.

For $n=1$, it is clear that this definition coincides with the definition of
the binary $\lambda$-bracket as in (2.12) by employing the defining property
of semi-chiral superfields (2.7),

$\displaystyle\left\\{\mathbf{O}_{1}\,{}_{\lambda_{1}}\,\mathbf{O}_{0}\right\\}$
$\displaystyle\equiv\mathbf{Q}\int_{\mathbb{C}^{2}}\frac{d^{2}z_{1}}{(2\pi{\rm
i})^{2}}e^{\lambda_{1}\cdot
z_{1}}:\mathbf{O}_{1}(z_{1})\mathbf{O}_{0}(0):=\int_{\mathbb{C}^{2}}\frac{d^{2}z_{1}}{(2\pi{\rm
i})^{2}}e^{\lambda_{1}\cdot
z_{1}}\bar{\partial}:\mathbf{O}_{1}(z_{1})\mathbf{O}_{0}(0):$ (2.14)
$\displaystyle=\oint_{S^{3}}\frac{{\rm d}^{2}z_{1}}{(2\pi{\rm i})^{2}}\
e^{\lambda_{1}\cdot z_{1}}\ :\mathbf{O}_{1}(z_{1})\mathbf{O}_{0}(0):\,,$

where the third equality follows from Stokes’s theorem. Continuing to $n=2$,
the ternary $\lambda$-bracket is given by

$\left\\{\mathbf{O}_{1}\,{}_{\lambda_{1}}\,\mathbf{O}_{2}\,{}_{\lambda_{2}}\,\mathbf{O}_{0}\right\\}\equiv\mathbf{Q}\int_{(\mathbb{C}^{2})^{2}}\frac{{\rm
d}^{2}z_{1}{\rm d}^{2}z_{2}}{(2\pi{\rm i})^{4}}\mathrm{e}^{\lambda_{1}\cdot
z_{1}+\lambda_{2}\cdot
z_{2}}:\mathbf{O}_{1}(z_{1})\mathbf{O}_{2}(z_{2})\mathbf{O}_{0}(0):\,.$ (2.15)

Similar to the $n$-ary $\lambda$-brackets defined above, one can define higher
brackets capturing the negative modes [41]. We leave a detailed discussion of
such brackets for future work.

Two important instances of ternary $\lambda$-brackets that play a central role
in this work are those containing three conserved current semi-chiral
operators, $\mathbf{J}^{a}$, or three stress tensors semi-chiral fields,
$\mathbf{S}_{\dot{\alpha}}$. In free theories, or free field realisations,
both these operators are quadratic in free fields. Hence, all fields are Wick
contracted in (2.15) and the resulting expression consists of a Feynman
integral multiplying the identity operator. In this case the expression
$\mathbf{Q}:\mathbf{O}_{1}(z_{1})\mathbf{O}_{2}(z_{2})\mathbf{O}_{0}(0):$
simply reduces to the anti-holomorphic differential $\bar{\partial}$ acting on
a three-point correlation function. For this reason, we can rewrite the
ternary $\lambda$-bracket as

$\left\\{\mathbf{O}_{1}\,{}_{\lambda_{1}}\,\mathbf{O}_{2}\,{}_{\lambda_{2}}\,\mathbf{O}_{0}\right\\}\equiv\int_{(\mathbb{C}^{2})^{2}}\frac{{\rm
d}^{2}z_{1}{\rm d}^{2}z_{2}}{(2\pi{\rm i})^{4}}\mathrm{e}^{\lambda_{1}\cdot
z_{1}+\lambda_{2}\cdot
z_{2}}\bar{\partial}\langle\mathbf{O}_{1}(z_{1})\mathbf{O}_{2}(z_{2})\mathbf{O}_{0}(0)\rangle\mathbb{1}\,,$
(2.16)

where we explicitly included the identity operator $\mathbb{1}$ to emphasize
the fact ternary $\lambda$-brackets
$\\{\mathbf{J}^{a}\,{}_{\lambda_{1}}\,\mathbf{J}^{b}\,{}_{\lambda_{2}}\,\mathbf{J}^{c}\\}$
and
$\\{\mathbf{S}_{\dot{\alpha}}\,{}_{\lambda_{1}}\,\mathbf{S}_{\dot{\beta}}\,{}_{\lambda_{2}}\,\mathbf{S}_{\dot{\gamma}}\\}$
are both proportional to the identity operator with coefficients being
functions of $\lambda$ parameters. In particular, they are central and encode
the central extensions of the higher VOA. More generally, this is not always
the case and ternary $\lambda$-brackets involving other operators generically
contain terms other than the identity.

The goal of this work is to compute these $\lambda$-brackets in interacting
$\mathcal{N}=1$ superconformal field theories, possibly with additional
flavour symmetry $\mathfrak{g}$. It is a non-trivial statement that in an
interacting theory (2.16) still holds when inserting three supercurrent semi-
chiral operators. We will use this as the definition for the $\lambda$-bracket
of three conserved (super)currents, in particular, this definition satisfies
the axioms of higher $\lambda$-brackets, see Appendix D. Before explicitly
computing the ternary $\lambda$-brackets we discuss some general aspects of
the expected higher Kac-Moody and higher Virasoro symmetry algebras arising in
the holomorphically twisted theory. More details can be found in Appendix D.

### 2.2 Higher Kac-Moody algebra

In a two-dimensional VOA with continuous global symmetry $\mathfrak{g}$, the
symmetry is enhanced to an infinite-dimensional symmetry algebra known as the
Kac-Moody algebra (or the Heisenberg algebra, when
$\mathfrak{g}=\mathfrak{u}(1)$). For any such symmetry, the theory contains
conserved holomorphic currents $\mathbf{J}^{a}(z)$, where $a$ is an adjoint
$\mathfrak{g}$ index. The Kac-Moody algebra is then generated by the following
action on local operators121212 For more details on the two-dimensional Kac-
Moody algebra see Appendix D.

$\mathbf{J}^{a}_{n}\cdot O(0)=\oint\frac{{\rm d}z}{2\pi{\rm
i}}z^{n}\mathbf{J}^{a}(z)\mathbf{O}(0),\quad n\in\mathbb{Z}\,.$ (2.17)

Similarly, in four-dimensional $\mathcal{N}=1$ theories, whenever the theory
enjoys a continuous global symmetry $\mathfrak{g}$, we find an infinite-
dimensional symmetry enhancement in the twisted theory. Every such continuous
global symmetry necessarily comes with an associated short current multiplet
$A_{2}\overline{A}_{2}[0,0]_{2}^{(0)}$ in the spectrum. As discussed above,
this multiplet contains a semi-chiral primary $\mathbf{J}=D_{+}\mathcal{J}$,
where $\mathcal{J}$ denotes the current superfield. Analogous to the two-
dimensional setting, the modes of $\mathbf{J}$ give rise to a higher Kac-Moody
symmetry algebra in the holomorphically twisted theory. In this case, the
modes can be defined as

$\mathbf{J}^{a}_{m,n}\cdot\mathbf{O}(0)\equiv\left\\{\mathbf{J}^{a}\,\,,\mathbf{O}\right\\}_{m,n}=\mathbf{Q}\int_{\mathbb{C}^{2}}{\rm
d}^{2}z\,\rho_{m,n}\,\mathbf{J}^{a}(z)\mathbf{O}(0)\,,\qquad
m,n\in\mathbb{Z}\,.$ (2.18)

We can obtain the commutation relations of the higher Kac-Moody algebra
through the associativity relations of the $\lambda$-bracket,

$\left\\{\mathbf{J}^{a}\,{}_{\lambda_{1}}\left\\{\mathbf{J}^{b}\,{}_{\lambda_{2}}\,\mathbf{O}\right\\}\right\\}-\left\\{\mathbf{J}^{b}\,{}_{\lambda_{2}}\left\\{\mathbf{J}^{a}\,{}_{\lambda_{1}}\,\mathbf{O}\right\\}\right\\}-\left\\{\left\\{\mathbf{J}^{a}\,{}_{\lambda_{1}}\,\mathbf{J}^{b}\right\\}\,{}_{\lambda_{1}+\lambda_{2}}\mathbf{O}\right\\}=0\,.$
(2.19)

The binary $\lambda$-bracket,
$\left\\{\mathbf{J}^{a}\,{}_{\lambda}\,\mathbf{J}^{b}\right\\}$ can be easily
computed, as will be done explicitly in the next section. However, note that
$\\{\mathbf{J}^{a}\,{}_{\lambda}\mathbf{J}^{b}\\}$ has to be $\mathrm{SU}(2)$
invariant as well as anti-symmetric in the adjoint $\mathfrak{g}$ indices
$a,b$. Therefore the only possible expression for the $\lambda$-bracket one
can write down is

$\left\\{\mathbf{J}^{a}\,{}_{\lambda}\,\mathbf{J}^{b}\right\\}=f^{ab}{}_{c}\mathbf{J}^{c}\,,$
(2.20)

where $f^{ab}{}_{c}$ are the structure constants of the flavour symmetry group
$\mathfrak{g}$ and the normalisation coefficient on the right-hand side has
been absorbed in the normalisation of the current operator. Expanding both
sides of the associativity relation (2.19) in $\lambda_{1,\dot{\pm}}$ and
$\lambda_{2,\dot{\pm}}$ we find the commutation relations,

$\left[\mathbf{J}_{m,n}^{a}\,,\mathbf{J}_{kl}^{b}\right]=f^{ab}{}_{c}\,\mathbf{J}^{c}_{m+k,n+l}\,.$
(2.21)

The ternary $\lambda$-bracket on can similarly be constrained through its
symmetries. First of all, the only $\mathrm{SU}(2)$ invariant polynomials of
$\lambda_{1,\dot{\pm}}$ and $\lambda_{2,\dot{\pm}}$ are given by powers of the
pairing $(\lambda_{1},\lambda_{2})$. Moreover, the graded commutativity of the
$\lambda$-bracket,

$\left\\{\mathbf{J}^{a}\,{}_{\lambda_{2}}\,\mathbf{J}^{b}\,{}_{\lambda_{1}}\,\mathbf{J}^{c}\right\\}=-\left\\{\mathbf{J}^{b}\,{}_{\lambda_{1}}\,\mathbf{J}^{a}\,{}_{\lambda_{2}}\,\mathbf{J}^{c}\right\\}=-\left\\{\mathbf{J}^{a}\,{}_{\lambda_{2}}\,\mathbf{J}^{c}\,{}_{-\lambda_{1}-\lambda_{2}-\partial}\,\mathbf{J}^{b}\right\\}\,$
(2.22)

constrains this to a single possible expression,

$\left\\{\mathbf{J}^{a}\,{}_{\lambda_{2}}\,\mathbf{J}^{b}\,{}_{\lambda_{1}}\,\mathbf{J}^{c}\right\\}=k\,d^{abc}(\lambda_{1},\lambda_{2})\,,$
(2.23)

where $d^{abc}$ is the totally symmetric invariant tensor on $\mathfrak{g}$
and the coefficients $k$ is a theory dependent constant. In the next section,
we will show that $k$ is given by the flavour central charge $k=k_{G}$.

In contrast to the two-dimensional Kac-Moody symmetry algebra, the higher Kac-
Moody algebra has a non-trivial ternary $L_{\infty}$-bracket due to the non-
trivial ternary $\lambda$-brackets, encoding its central extensions. The
ternary $L_{\infty}$-bracket is given by

$\left[\mathbf{J}_{n,m}^{a},\mathbf{J}_{k,l}^{b},\mathbf{J}_{r,s}^{c}\right]=K\,d^{abc}(ks-\ell
r)\delta_{k+r+n,0}\delta_{\ell+s+m,0}\,,$ (2.24)

where $K$ is proportional to $k_{G}$. This ternary bracket should again be
related to the ternary $\lambda$-bracket but the precise relationship is not
immediate. A proper derivation including the relation between $k$ and $K$ is
left to the future.

### 2.3 Higher Virasoro algebra

In a two-dimensional (conformal) VOA, the conformal $\mathfrak{sl}(2)$
symmetry is enhanced to the infinite-dimensional Virasoro symmetry algebra.
The theory contains a holomorphic stress tensor $\mathbf{S}(z)$ whose modes
generate holomorphic diffeomorphisms and act on local operators as131313In
Appendix D we review the two-dimensional Virasoro algebra in more detail as
well as list the standard commutation relations.

$\mathbf{L}_{n}\cdot\mathbf{O}(0)=\oint\frac{{\rm d}z}{2\pi{\rm
i}}z^{n}\mathbf{S}(z)\mathbf{O}(0),\quad n\in\mathbb{Z}\,.$ (2.25)

Similarly, in an $\mathcal{N}=1$ superconformal field theory the supercurrent
multiplet is a short $A_{1}\overline{A_{1}}[1,1]_{3}^{(0)}$ multiplet
containing a semi-chiral operator $\mathbf{S}_{\dot{\alpha}}\equiv
D_{+}\mathcal{S}_{+{\dot{\alpha}}}$, where
$\mathcal{S}_{\alpha{\dot{\alpha}}}$ denotes the supercurrent superfield.
Following the discussion above we can define the modes of
$\mathbf{S}_{\dot{\alpha}}$ through the brackets

$\mathbf{L}^{\dot{\alpha}}_{m,n}\cdot\mathbf{O}(0)=\left\\{\mathbf{S}_{\dot{\alpha}},\mathcal{O}\right\\}_{m,n}=\mathbf{Q}\int_{\mathbb{C}^{2}}{\rm
d}^{2}z\,\rho_{m,n}\,\mathbf{S}_{\dot{\alpha}}(z)\mathbf{O}(0)\,.$ (2.26)

More precisely, the negative modes, $m,n\leq-1$, act as

$\displaystyle\mathbf{L}^{{\dot{\alpha}}}_{m,n}\cdot\mathbf{O}$
$\displaystyle=\frac{(-1)^{-m-1}}{(-m-1)!}\frac{(-1)^{-n-1}}{(-n-1)!}\oint_{S^{3}}{\rm
d}^{2}z\ \partial_{\dot{+}}^{-m-1}\partial_{\dot{-}}^{-n-1}\omega_{\rm BM}\
\mathbf{S}_{\dot{\alpha}}\mathbf{O}$ (2.27)
$\displaystyle\equiv\frac{1}{(-m-1)!}\frac{1}{(-n-1)!}\partial_{\dot{+}}^{-m-1}\partial_{\dot{-}}^{-n-1}\mathbf{S}_{\dot{\alpha}}\mathbf{O}\,,$

which is nothing but the regularised product of derivatives of
$\mathbf{S}_{\dot{\alpha}}$ with the operator $\mathbf{O}$. The positive modes
generate holomorphic diffeomorphisms,

$\mathbf{L}^{\dot{\alpha}}_{m,n}\cdot\mathbf{O}=-l_{m,n}^{\dot{\alpha}}\mathbf{O}\,,$
(2.28)

where $l_{m,n}^{\alpha}$ are defined as the following vector fields

$l^{\dot{\alpha}}_{m,n}=-(z^{{\dot{+}}})^{m}(z^{{\dot{-}}})^{n}\partial_{{\dot{\alpha}}},\qquad
m,n\geq 0\,,$ (2.29)

Following the same logic as above, we can constrain the form of the binary
$\lambda$-bracket using the associativity relations for the $\lambda$-bracket
and demanding $\mathrm{SU}(2)$ covariance. Absorbing all normalisation factors
in $\mathbf{S}_{\dot{\alpha}}$ and $\lambda$ the following expression arises
as the unique possibility,

$\left\\{\mathbf{S}_{\dot{\alpha}}\,{}_{\lambda}\,\mathbf{S}_{{\dot{\beta}}}\right\\}=\lambda_{{\dot{\alpha}}}\mathbf{S}_{{\dot{\beta}}}+\lambda_{{\dot{\beta}}}\mathbf{S}_{{\dot{\alpha}}}+\partial_{\dot{\alpha}}\mathbf{S}_{{\dot{\beta}}}\,,$
(2.30)

Using this expression we can again expand (2.19) to obtain the commutation
relations of the higher Virasoro algebra [11, 21, 42],

$\displaystyle\left[{\mathbf{L}^{\dot{+}}_{m,n}},{\mathbf{L}^{\dot{+}}_{k,l}}\right]$
$\displaystyle=(m-k)\,\mathbf{L}^{\dot{+}}_{m+k-1,n+l}\,,\quad$
$\displaystyle\left[{\mathbf{L}^{\dot{+}}_{m,n}},{\mathbf{L}^{\dot{-}}_{k,l}}\right]$
$\displaystyle=n\,\mathbf{L}^{\dot{+}}_{m+k,n+l-1}-k\,\mathbf{L}^{\dot{-}}_{m+k-1,n+l}\,,$
(2.31)
$\displaystyle\left[{\mathbf{L}^{\dot{-}}_{m,n}},{\mathbf{L}^{\dot{-}}_{k,l}}\right]$
$\displaystyle=(n-l)\,\mathbf{L}^{\dot{-}}_{m+k,n+l-1}\,,\quad$
$\displaystyle\left[{\mathbf{L}^{\dot{-}}_{m,n}},{\mathbf{L}^{\dot{+}}_{k,l}}\right]$
$\displaystyle=-l\,\mathbf{L}^{\dot{+}}_{m+k,n+l-1}+m\,\mathbf{L}^{\dot{-}}_{m+k-1,n+l}\,.$

for $m,n,k,l\geq 0$, or $-k>m\geq 0$, $-l>n\geq 0$. In other cases the binary
brackets vanish. More details on the precise derivation of these commutation
relations are deferred to Appendix B.

The ternary $\lambda$-bracket can again be constrained through its symmetries.
It has to be central, i.e. it is a multiple of the identity operator with
coefficients proportional to the parameters $\lambda_{1,2}$. It has to
transform covariantly under $\mathrm{SU}(2)$ and has to be totally anti-
symmetric upon interchanging any pair of indices. Finally, since both sides of
the equation have to scale similarly the right-hand side has to be a quintic
polynomial in $\lambda_{12}$. These constraints completely determine the
ternary $\lambda$-bracket, up to two theory-dependent constants,

$\left\\{\mathcal{S}_{\dot{\alpha}}\,{}_{\lambda_{2}}\,\mathcal{S}_{\dot{\beta}}\,{}_{\lambda_{1}}\,\mathcal{S}_{\dot{\gamma}}\right\\}=A\,\Lambda_{\mathrm{vector}}+C\,\Lambda_{\mathcal{N}=4}\,,$
(2.32)

where $A$ and $C$ are independent of $\lambda_{1,2}$ and the polynomials
$\Lambda_{\rm vector}$ and $\Lambda_{\mathcal{N}=4}$ are defined as

$\displaystyle\Lambda_{\mathrm{vector}}$
$\displaystyle=-\frac{1}{24\pi^{2}}(\lambda_{1},\lambda_{2})^{2}\Big{[}\lambda_{1\dot{\alpha}}\epsilon_{\dot{\beta}\dot{\gamma}}+\lambda_{2\dot{\beta}}\epsilon_{\dot{\rho}\dot{\alpha}}-(\lambda_{1\dot{\gamma}}+\lambda_{2\dot{\gamma}})\epsilon_{\dot{\alpha}\dot{\beta}}\Big{]}$
(2.33) $\displaystyle\Lambda_{\mathcal{N}=4}$
$\displaystyle=\frac{1}{3\pi^{2}}\lambda_{1\dot{\alpha}}\lambda_{2\dot{\beta}}(\lambda_{1}+\lambda_{2})_{\dot{\gamma}}(\lambda_{1},\lambda_{2})\,.$
(2.34)

where
$(\lambda_{1},\lambda_{2})\equiv\lambda_{1\dot{+}}\lambda_{2\dot{-}}-\lambda_{1\dot{-}}\lambda_{2\dot{+}}$.
The naming for the polynomials might seem arbitrary at this point but will
become clear when we consider some examples. In the next section, we will
explicitly compute the $\lambda$-bracket involving three stress tensor semi-
chiral superfields and show that $A$ and $C$ are simple linear combinations of
the familiar conformal anomalies $a$ and $c$.

In terms of the modes $\mathbf{L}_{m,n}^{\dot{\alpha}}$, the non-trivial
ternary $\lambda$-brackets due to central extensions again forces us to have a
non-trivial ternary $L_{\infty}$-bracket, which takes the schematic form

$\left[\mathbf{L}_{m,n}^{\dot{\alpha}},\mathbf{L}_{k,l}^{\dot{\beta}},\mathbf{L}_{r,s}^{\dot{\gamma}}\right]=\tilde{A}\psi_{1}+\tilde{C}\psi_{2}\,,$
(2.35)

where $\psi_{1,2}$ are two theory-independent kinematic factors. The precise
expression is given in (D.39). A proper derivation of this bracket from the
$\lambda$-brackets, relating $\\{\tilde{A}\,,\,\tilde{C}\\}$ and
$\\{A\,,\,C\\}$ is left to future work.

### 2.4 Non-conformal theories

In the above, we considered the situation when the physical four-dimensional
$\mathcal{N}=1$ theory is conformal. Conformal symmetry however is not
necessary in order to define the holomorphic twist. Therefore a natural
question is to what extent the above discussion generalises in the non-
conformal setting. On the level of conserved current multiplets, nothing
changes, and we expect to observe a similar infinite-dimensional Kac-Moody
algebra in non-conformal theories.

On the other hand, as could be expected, the Virasoro symmetry is specific to
the conformal case. In non-conformal theories, the superconformal supercurrent
multiplet is not well-defined and instead, we need to use a more general
multiplet, the $\mathcal{S}$ multiplet [43, 44]. This multiplet exists for any
supersymmetric theory and satisfies the constraint

$\overline{D}^{\dot{\alpha}}\mathcal{S}_{\alpha\dot{\alpha}}=\chi_{\alpha}+\mathcal{Y}_{\alpha}\,,$
(2.36)

where $\chi_{\alpha}$ is a chiral multiplet and
$D_{(\alpha}\mathcal{Y}_{\beta)}=\overline{D}^{2}\mathcal{Y}_{\alpha}=0$.
Whenever $\mathcal{Y}_{\alpha}$ vanishes, the constraint implies that the
bottom component of the $\mathcal{S}$ multiplet is a conserved current and for
the remainder of this work we restrict ourselves to this situation.141414In
this case the $\mathcal{S}$ multiplet is not irreducible but decomposes in a
smaller multiplet known as the $\mathcal{R}$ multiplet [45, 46, 47, 48]. The
$\mathcal{S}$ multiplet contains a semi-chiral operator but without conformal
symmetry it is given by
$\widetilde{\mathbf{S}}=\partial^{\dot{\alpha}}D_{+}\mathcal{S}_{+{\dot{\alpha}}}$.
This semi-chiral operator still generates an infinite-dimensional symmetry
algebra in the twisted theory, but in contrast with the conformal case, the
positive modes only generate the sub-algebra of complex symplectomorphisms. We
leave a detailed study of this infinite-dimensional symmetry algebra for
future work.

Whenever the theory has additional $\mathrm{U}(1)$ flavour symmetries along
the RG flow they might mix with the UV R-symmetry to form the IR R-symmetry.
From the point of view of the $\mathcal{N}=1$ (non-conformal) supersymmetry
algebra, all such R-symmetries are allowed but only one of those is consistent
with superconformal symmetry, i.e. there is a unique IR (superconformal)
R-symmetry which can be selected using $a$-maximisation [49]. This ambiguity
is reflected in the fact that we can always improve the
$\mathcal{S}$-multiplet by adding improvement terms

$\displaystyle\mathcal{S}^{\prime}_{\alpha\dot{\alpha}}=$
$\displaystyle\mathcal{S}_{\alpha\dot{\alpha}}+\sum_{i}\zeta_{i}\left[{D_{\alpha}},{\bar{D}_{\dot{\alpha}}}\right]\mathcal{J}_{i}\,,$
(2.37) $\displaystyle\chi^{\prime}_{\alpha}=$
$\displaystyle\chi_{\alpha}+\sum_{i}\zeta_{i}\,\frac{3}{2}\,\bar{D}^{2}D_{\alpha}\mathcal{J}_{i}\,,$

where $\mathcal{J}_{i}$ are real linear multiplets containing the
$\mathrm{U}(1)_{i}$ flavour currents. In some of the examples in Section 4 we
find ourselves exactly in this situation and will explicitly see how this
phenomenon manifests itself in the framework of the holomorphic twist.

## 3 Conformal anomalies from integrated correlators

As introduced above, the symmetry algebra of any four-dimensional
$\mathcal{N}\geq 1$ superconformal theory gets enhanced to an infinite-
dimensional symmetry algebra after passing to the subsector obtained by
performing the holomorphic twist. Continuing the analogy with two-dimensional
VOAs we expect the central charges of the four-dimensional untwisted theory to
be encoded in the central extensions of the associated higher-VOA. In this
section, we make this statement precise by explicitly computing the binary and
ternary $\lambda$-brackets involving conserved (super)current semi-chiral
superfields. Doing so we recover the expressions obtained in the previous
section allowing us to fix the constant coefficients in terms of the conformal
anomalies $a$ and $c$ and flavour central charges.

To compute the two- and three-point functions in this section we made use of
the Mathematica packages CFTs4d [50] and SpinorAlgebra [51]. Although
conceptually straightforward, extracting the relevant descendant three-point
function from superspace expressions, see [52] and [53, 54, 51], is a most
tedious affair. In order not to obscure the main message, we defer the details
of the computations to Appendix B.

### 3.1 Flavour current multiplets

Whenever an $\mathcal{N}=1$ superconformal theory contains a continuous global
symmetry $\mathfrak{g}$, this symmetry is enhanced in the holomorphically
twisted theory to an infinite-dimensional four-dimensional higher Kac-Moody
symmetry $\hat{\mathfrak{g}}_{2,k}$. To see how the central charge $k$ of this
algebra can be related to the flavour central charge $k_{G}$ of the original
theory we proceed to explicitly compute the binary and ternary
$\lambda$-brackets containing resp. two and three conserved current semi-
chiral operators.

The conserved current $j^{a}_{\mu}$ is embedded in a real linear multiplet
$\mathcal{J}^{a}$ satisfying

$D^{2}\mathcal{J}^{a}=\overline{D}^{2}\mathcal{J}^{a}=0\,,$ (3.1)

where $a$ is an adjoint index of the flavour symmetry $\mathfrak{g}$. It is
straightforward to see that the associated semi-chiral superfield is given by
$\mathbf{J}^{a}=D_{+}\mathcal{J}^{a}$. More explicitly we have

$\mathbf{J}^{a}={\rm i}j^{a}_{+}+\left(j^{a}_{+{\dot{\alpha}}}+\frac{{\rm
i}}{2}\partial_{\dot{\alpha}}J^{a}\right){\rm
d}\bar{z}^{{\dot{\alpha}}}-\frac{1}{2}\partial_{\dot{\alpha}}\bar{j}^{a\dot{\alpha}}{\rm
d}^{2}\bar{z}\,,$ (3.2)

where $j^{\alpha}$ and $\bar{j}^{\dot{\alpha}}$ are chiral and anti-chiral
spinors respectively and $J$ is a scalar and the constraints (3.1) guarantee
that the current is conserved, $\partial^{\mu}j_{\mu}=0$. With the explicit
semi-chiral superfield at hand, we can compute the binary and ternary
$\lambda$-bracket following the prescription laid out in equation (2.13) and
(2.16) above. The binary $\lambda$-bracket is computed as follows,

$\displaystyle\left\\{\mathbf{J}^{a}\,{}_{\lambda}\,\mathbf{J}^{b}\right\\}=$
$\displaystyle\,\int_{\mathbb{C}^{2}}\frac{{\rm d}^{2}z}{(2\pi{\rm
i})^{2}}\,\mathrm{e}^{\lambda\cdot
z}\,\bar{\partial}\left[\mathbf{J}^{(1)a}(z)\mathbf{J}^{(0)b}\right]$ (3.3)
$\displaystyle=$ $\displaystyle\,\int_{\mathbb{C}^{2}}\frac{{\rm
d}^{2}z}{(2\pi{\rm i})^{2}}\,\mathrm{e}^{\lambda\cdot
z}\,\bar{\partial}\left[\frac{\left(\bar{z},{\rm
d}\bar{z}\right)}{|z|^{4}}\,f^{ab}{}_{c}\mathbf{J}^{(0)c}+\cdots\right]$
$\displaystyle=$ $\displaystyle\,f^{ab}{}_{c}\,\mathbf{J}^{(0)c}(0)\,.$

To advance from the first to the second line we used the OPE between
$\mathbf{J}^{(1)}$ and $\mathbf{J}^{(0)}$. Unlike for a general multiplet,
where the OPE of the descendants is not guaranteed to be fixed in terms of the
OPE of the superconformal primary, the semi-chiral shortening conditions
precisely allow us to do so. For this reason we can obtain the
$\mathbf{J}^{(1)}\mathbf{J}^{(0)}$ from the $\mathcal{J}\mathcal{J}$ OPE, see
for example [52, 53]. To compute the integral over $\mathbb{C}^{2}$ we
recognise the Bochner-Martinelli kernel allowing us to use a higher-
dimensional residue formula.151515See Appendix B for more details on the
Bochner-Martinelli kernel and higher dimensional residues. The
$\mathbf{J}^{(1)}\mathbf{J}^{(0)}$ OPE contains many more terms indicated by
the $\cdots$. However, these are all less singular and therefore will not
contribute to the integral. Note that the binary $\lambda$-bracket is
proportional to the anti-symmetric structure constant $f^{ab}{}_{c}$ of the
flavour symmetry algebra $\mathfrak{g}$. From this, we immediately see that
for abelian symmetries the binary $\lambda$-bracket is trivial, in line with
the previous discussion.

Next, we compute the ternary $\lambda$-bracket of three flavour current semi-
chiral operators. Following (2.16) the ternary $\lambda$-bracket is defined as

$\left\\{\mathbf{J}^{a}\,{}_{\lambda_{1}}\,\mathbf{J}^{b}\,{}_{\lambda_{2}}\,\mathbf{J}^{c}\right\\}=\int_{(\mathbb{C}^{2})^{2}}\frac{{\rm
d}^{2}z_{1}{\rm d}^{2}z_{2}}{(2\pi{\rm i})^{4}}\mathrm{e}^{\lambda_{1}\cdot
z_{1}+\lambda_{2}\cdot
z_{2}}\bar{\partial}\left\langle{\mathbf{J}^{a(2)}(z_{1})\mathbf{J}^{b(1)}(z_{2})\mathbf{J}^{c(0)}(0)}\right\rangle\,.$
(3.4)

Purely on kinematic grounds, the three-point function in this expression takes
the following form [50],

$\left\langle{\mathbf{J}^{a(2)}(z_{1})\mathbf{J}^{b(1)}(z_{2})\mathbf{J}^{c(0)}(0)}\right\rangle=\frac{c_{1}\,T^{1}_{23}U^{13}T^{1}_{13}+c_{2}\,T^{1}_{23}U^{12}U^{23}+c_{3}\,U^{12}U^{13}U^{21}}{|z_{12}|^{6}|z_{1}|^{4}|z_{2}|^{2}}\,,$
(3.5)

where the tensor structures $T^{i}_{jk}$ and $U^{ij}$ are defined as

$U^{ij}=s_{j\alpha}x_{ij}^{\alpha{\dot{\alpha}}}\bar{s}_{i{\dot{\alpha}}}\,,\qquad
T^{i}_{jk}=\frac{|x_{ij}|^{2}|x_{ik}|^{2}}{|x_{jk}|^{2}}\left(\frac{s_{j\alpha}x_{ij}^{\alpha{\dot{\alpha}}}\bar{s}_{i{\dot{\alpha}}}}{|x_{ij}|^{2}}-\frac{s_{k\alpha}x_{ik}^{\alpha{\dot{\alpha}}}\bar{s}_{i{\dot{\alpha}}}}{|x_{ik}|^{2}}\right)\,,$
(3.6)

where we defined $x_{ij}=x_{i}-x_{j}$ and contracted all indices with
auxiliary spinors $s_{i}$ and $\bar{s}_{j}$. Written as such this expression
is not entirely correct and one needs to contract the indices in line with the
index structure of the components of the semi-chiral superfield. The constants
$c_{1,2,3}$ can be fixed by imposing conservation of the current and relating
it to the supersymmetric three-point function of conserved current multiplets.
Doing so uniquely fixed the three coefficients to be $c_{1,2,3}=k_{G}d^{abc}$.
Acting on the resulting three-point function with $\bar{\partial}$ results in

$\bar{\partial}\left\langle{\mathbf{J}^{(2)a}(z_{1},\bar{z}_{1})\mathbf{J}^{(1)b}(z_{2},\bar{z}_{2})\mathbf{J}^{(0)c}(0)}\right\rangle=-k_{G}\,d^{abc}\,\frac{(\bar{z}_{1},\bar{z}_{2})}{|z_{12}|^{4}|z_{1}|^{4}|z_{2}|^{4}}\left(1+2\frac{(\bar{z}_{12},z_{1})}{|z_{12}|^{2}}\right)\,,$
(3.7)

where $d^{abc}=\mathrm{Tr}\,(T^{a}\left\\{{T^{b}},{T^{c}}\right\\})$ are the
symmetric structure constants of the flavour symmetry group, and we introduced
the notation $z_{ij}=z_{i}-z_{j}$ and analogously
$\bar{z}_{ij}=\bar{z}_{i}-\bar{z}_{j}$. Now all that is left is to integrate
this function over $(\mathbb{C}^{2})^{2}$. This integral is rather non-trivial
to perform but can be greatly simplified by employing heat kernel
regularisation. In particular, we find that the integral over the heat kernel
time is convergent making the result manifestly scheme independent. The
details of this procedure are explained and applied to the integral in
question in Appendix B.2 and B.3. Following that appendix, we obtain the
following result for the ternary $\lambda$-bracket,

$\left\\{\mathbf{J}^{a}\,{}_{\lambda_{1}}\,\mathbf{J}^{b}\,{}_{\lambda_{2}}\,\mathbf{J}^{c}\right\\}=k_{G}\,d^{abc}\,(\lambda_{1},\lambda_{2})\,.$
(3.8)

This matches with the expression in the previous section starting from the
mode algebra and rigorously demonstrates that indeed the central charge of the
higher Kac-Moody symmetry algebra encodes the flavour central charge of the
physical $\mathcal{N}=1$ theory. In the next section, we will confirm this
more explicitly in various examples.

### 3.2 Supercurrent multiplet

Next, we proceed analogously to compute the binary and ternary
$\lambda$-bracket for supercurrent semi-chiral superfields. Although the
computation is identical in spirit, the details are quite a bit more involved
and most of the explicit details are referred to Appendix B.3. Whenever a
theory has $\mathcal{N}=1$ superconformal symmetry, we observed that after the
holomorphic twist, this symmetry is enhanced to an infinite-dimensional higher
Virasoro symmetry algebra. The cohomology classifying the central extensions
is two-dimensional [42] which fits with the expectation that the central
charges should be related to the $a$ and $c$ conformal anomaly of the physical
theory.

The higher Virasoro symmetry algebra is generated by the modes of the semi-
chiral operator contained in the supercurrent multiplet
$\mathcal{S}_{\alpha{\dot{\alpha}}}$. There are various supercurrent
multiplets, appropriate in different situations. The most general supercurrent
multiplet is the $\mathcal{S}$ multiplet, which can be defined for any
$\mathcal{N}=1$ supersymmetric theory [43, 44]. In the context of
superconformal $\mathcal{N}=1$ theories there necessarily exists a non-
anomalous $\mathrm{U}(1)_{R}$ symmetry and the Kähler form on the target space
is always globally well-defined so we can use a simpler conformal supercurrent
multiplet.161616Whenever the theory has a non-anomalous $\mathrm{U}(1)_{R}$
symmetry or the Kähler form on the target space is globally well-defined there
are exist also consistent intermediate multiplets called respectively the
$\mathcal{R}$-multiplet [45, 46, 47, 48] or the Ferrara-Zumino multiplet [55].
This is a real multiplet satisfying

$D^{\alpha}\mathcal{S}_{\alpha{\dot{\alpha}}}=\overline{D}^{{\dot{\alpha}}}\mathcal{S}_{\alpha{\dot{\alpha}}}=0\,.$
(3.9)

In components, this multiplet is given by

$\displaystyle\mathcal{S}_{\mu}=$ $\displaystyle\,j_{\mu}+{\rm
i}\,\theta^{\alpha}S_{\mu\alpha}-{\rm
i}\,\bar{\theta}^{\dot{\alpha}}\bar{S}_{\mu\dot{\alpha}}+(\theta\sigma^{\nu}\bar{\theta})\left(2T_{\mu\nu}+\frac{{\rm
i}}{2}\epsilon_{\mu\nu\rho\sigma}\partial^{\rho}j^{\sigma}\right)$ (3.10)
$\displaystyle\,-\frac{1}{2}\theta^{2}\bar{\theta}^{\dot{\alpha}}(\bar{\sigma}^{\nu})_{\dot{\alpha}\alpha}\partial_{\nu}S_{\mu}^{\alpha}+\frac{1}{2}\bar{\theta}^{2}\theta^{\alpha}(\sigma^{\nu})_{\alpha\dot{\alpha}}\partial_{\nu}\bar{S}_{\mu}^{\dot{\alpha}}-\frac{1}{4}\theta^{2}\bar{\theta}^{2}\partial^{2}j_{\mu}\,,$

where the constraints (3.9) impose that the $R$-symmetry current, stress
tensor and the supersymmetry currents are conserved,
$\partial^{\mu}j_{\mu}=\partial^{\mu}T_{\mu\nu}=\partial^{\mu}S_{\mu\alpha}=\partial^{\mu}\bar{S}_{\mu\dot{\alpha}}=0$,
as well as the trace relations
$T^{\mu}{}_{\mu}=(\sigma^{\mu})_{\beta\dot{\beta}}\bar{S}_{\mu}^{\dot{\beta}}=(\bar{\sigma}^{\mu})_{\dot{\beta}\beta}S_{\mu}^{\beta}=0$.
As already noted in Section 2 above, the supercurrent multiplet is a
$\overline{A}_{2}A_{2}[1,1]_{3}^{(0)}$ superconformal multiplet which contains
a semi-chiral operator
$\mathbf{S}_{{\dot{\alpha}}}=D_{+}\mathcal{S}_{+{\dot{\alpha}}}$ given as

$\mathbf{S}_{\dot{\alpha}}=\,{\rm i}\,S_{+\dot{\alpha}+}+{\rm
d}\bar{z}^{\dot{\beta}}\,\left(T_{{\dot{\alpha}}{\dot{\beta}}}+\frac{{\rm
i}}{2}\left(\star{\rm
d}j\right)_{{\dot{\alpha}}{\dot{\beta}}}\right)+\frac{1}{2}{\rm
d}\bar{z}^{2}\,\partial_{{\dot{\beta}}}\bar{S}_{+\dot{\alpha}}^{\dot{\beta}}\,.$
(3.11)

where $T_{{\dot{\alpha}}{\dot{\beta}}}$ and $\left(\star{\rm
d}j\right)_{{\dot{\alpha}}{\dot{\beta}}}$ are the holomorphic part of the
stress tensor and dual R-symmetry field strength. Having introduced the semi-
chiral superfield $\mathbf{S}_{\dot{\alpha}}$ we proceed to compute the binary
and ternary $\lambda$-brackets. To compute the binary bracket we proceed
identically as above. We first use the OPE [52] after which we compute the
binary bracket by using the higher-dimensional residue theorem

$\displaystyle\left\\{\mathbf{S}_{{\dot{\alpha}}}\,{}_{\lambda}\,\mathbf{S}_{{\dot{\beta}}}\right\\}=$
$\displaystyle\,\int_{\mathbb{C}^{2}}\frac{{\rm d}^{2}z}{(2\pi{\rm
i})^{2}}\,\mathrm{e}^{\lambda\cdot
z}\,\bar{\partial}\left[\mathbf{S}_{\dot{\alpha}}^{(1)}(z)\mathbf{S}_{\dot{\beta}}^{(0)}(0)\right]$
(3.12) $\displaystyle=$ $\displaystyle\,\int_{\mathbb{C}^{2}}\frac{{\rm
d}^{2}z}{(2\pi{\rm i})^{2}}\,\mathrm{e}^{\lambda\cdot
z}\,\bar{\partial}\left[\frac{{\rm
d}\bar{z}_{{\dot{\alpha}}}}{|z|^{4}}\mathbf{S}_{{\dot{\beta}}}^{(0)}(0)+\frac{(\bar{z},{\rm
d}\bar{z})}{|z|^{4}}\partial_{{\dot{\alpha}}}\mathbf{S}_{{\dot{\beta}}}^{(0)}(0)+\cdots\right]$
$\displaystyle=$
$\displaystyle\,\lambda_{{\dot{\alpha}}}\mathbf{S}_{{\dot{\beta}}}^{(0)}(0)+\lambda_{{\dot{\beta}}}\mathbf{S}_{{\dot{\alpha}}}^{(0)}(0)+\partial_{{\dot{\alpha}}}\mathbf{S}_{{\dot{\beta}}}^{(0)}(0)\,.$

In line with the discussion in the previous section. Next, we proceed to the
ternary bracket,

$\left\\{\mathbf{S}_{{\dot{\alpha}}}\,{}_{\lambda_{1}}\,\mathbf{S}_{{\dot{\beta}}}\,{}_{\lambda_{2}}\,\mathbf{S}_{\dot{\gamma}}\right\\}=\int_{(\mathbb{C}^{2})^{2}}\frac{{\rm
d}^{2}z_{1}}{(2\pi{\rm i})^{2}}\frac{{\rm d}^{2}z_{2}}{(2\pi{\rm
i})^{2}}\mathrm{e}^{\lambda_{1}\cdot z_{1}+\lambda_{2}\cdot
z_{2}}\bar{\partial}\left\langle{\mathbf{S}_{{\dot{\alpha}}}^{(2)}(z_{1},\bar{z}_{1})\mathbf{S}_{\dot{\beta}}^{(1)}(z_{2},\bar{z}_{2})\mathbf{S}_{\dot{\gamma}}^{(0)}(0)}\right\rangle\,.$
(3.13)

As shown in [52] the three-point function is completely determined as a sum
over tensor structures, where the only theory-dependent data is contained in
the prefactors which are linear functions of the conformal anomalies $a$ and
$c$. In particular, we can use these results to compute the three-point
function with the specific components as indicated above. In comparison with
the expression above for conserved currents, in this case, there are eleven
possible tensor structures appearing as explicitly listed in Appendix B. In
Appendix B we explicitly list the tensor structures and determine the correct
coefficients. Since the explicit resulting expression for the integrand is
rather lengthy we refrain from listing it here. The evaluation of the
remaining integral in (3.13) is very tedious and the details are relegated to
Appendix B.3. Note however that just like for the conserved currents, the
integration over the heat kernel time is completely convergent. Note however
that the condition for convergence crucially depends on some non-trivial
cancellations between different tensor structures. Carefully performing all
the steps outlined above we arrive at the final result for the stress tensor
ternary $\lambda$-bracket,

$\left\\{\mathbf{S}_{\dot{\alpha}}\,{}_{\lambda_{1}}\,\mathbf{S}_{\dot{\beta}}\,{}_{\lambda_{2}}\,\mathbf{S}_{\dot{\gamma}}\right\\}=\frac{4}{3}\,\left(3c-2a\right)\,\Lambda_{\mathcal{N}=4}+16\,\left(a-c\right)\,\Lambda_{\mathrm{vector}}\,.$
(3.14)

This expression contains the (theory independent) kinematic tensors
$\Lambda_{\mathcal{N}=4}$ and $\Lambda_{\rm vector}$ given by

$\displaystyle\Lambda_{\mathrm{vector}}$
$\displaystyle=-\frac{1}{24\pi^{2}}(\lambda_{1},\lambda_{2})^{2}\Big{[}\lambda_{1\dot{\alpha}}\epsilon_{\dot{\beta}\dot{\gamma}}+\lambda_{2\dot{\beta}}\epsilon_{\dot{\rho}\dot{\alpha}}-(\lambda_{1\dot{\gamma}}+\lambda_{2\dot{\gamma}})\epsilon_{\dot{\alpha}\dot{\beta}}\Big{]}\,,$
(3.15) $\displaystyle\Lambda_{\mathcal{N}=4}$
$\displaystyle=\frac{1}{3\pi^{2}}\lambda_{1\dot{\alpha}}\lambda_{2\dot{\beta}}(\lambda_{1}+\lambda_{2})_{\dot{\gamma}}(\lambda_{1},\lambda_{2})\,.$
(3.16)

All theory-dependent data is contained in the two prefactors as linear
combinations of the conformal anomalies $a$ and $c$. This computation
therefore presents us with a precise understanding of how the conformal
anomalies $a$ and $c$ are encoded in central extensions of the infinite-
dimensional higher Virasoro symmetry algebra.

## 4 Examples

Having established our main results, (3.8) and (3.14), we continue with a
variety of explicit examples to demonstrate their validity. Our examples
consist of Lagrangian $\mathcal{N}=1$ SCFTs for which an explicit free field
realisation in terms of coupled $\mathbf{b}\mathbf{c}$-$\beta$$\gamma$ system
has been formulated [20, 56, 27, 26, 13]. More concretely, consider an
$\mathcal{N}=1$ supersymmetric gauge theory with a vector multiplet for the
gauge Lie algebra $\mathfrak{g}$ and a collection of chiral multiplets
transforming in the representations $R_{i}$ of $\mathfrak{g}$ and possibly a
non-trivial superpotential $W$, which is a gauge invariant holomorphic
function of the chiral superfields. It was shown in [56] that in the BV-BRST
formalism, by adding the supercharge $\mathbf{Q}$ to the BRST charge
$Q_{\mathrm{BRST}}$, one obtains a holomorphically twisted theory equivalent
to the following holomorphic field theory with BV action

$S_{\rm BV}=\int_{\mathbb{C}^{2}}{\rm d}^{2}z\
\mathrm{Tr}\,\mathbf{b}\left(\bar{\partial}\mathbf{c}-\frac{1}{2}[\mathbf{c},\mathbf{c}]\right)+\sum_{i}{\text{\boldmath$\beta$}}_{i}\left(\bar{\partial}{\text{\boldmath$\gamma$}}^{i}+[\mathbf{c},{\text{\boldmath$\gamma$}}^{i}]\right)+W({\text{\boldmath$\gamma$}}),$
(4.1)

where $\bar{\partial}={\rm d}\bar{z}^{\dot{+}}\bar{\partial}_{\dot{+}}+{\rm
d}\bar{z}^{\dot{-}}\bar{\partial}_{\dot{-}}$ is the Dolbeault differential on
$\mathbb{C}^{2}$ and

$\displaystyle\mathbf{b}$
$\displaystyle=\mathbf{b}^{(0)}+\mathbf{b}^{(1)}+\mathbf{b}^{(2)}\in\Omega^{0,\bullet}(\mathbb{C}^{2},\mathfrak{g}^{\vee}),$
$\displaystyle\mathbf{c}$
$\displaystyle=\mathbf{c}^{(0)}+\mathbf{c}^{(1)}+\mathbf{c}^{(2)}\in\Omega^{0,\bullet}(\mathbb{C}^{2},\mathfrak{g})[1]\,,$
(4.2) $\displaystyle{\text{\boldmath$\beta$}}_{i}$
$\displaystyle={\text{\boldmath$\beta$}}_{i}^{(0)}+{\text{\boldmath$\beta$}}^{(1)}_{i}+{\text{\boldmath$\beta$}}_{i}^{(2)}\in\Omega^{0,\bullet}(\mathbb{C}^{2},R_{i}^{\vee})[1],$
$\displaystyle{\text{\boldmath$\gamma$}}^{i}$
$\displaystyle={\text{\boldmath$\gamma$}}^{i(0)}+{\text{\boldmath$\gamma$}}^{i(1)}+{\text{\boldmath$\gamma$}}^{i(2)}\in\Omega^{0,\bullet}(\mathbb{C}^{2},R_{i})\,,$

The action (4.1) in terms of the superfields should be thought of as
substituting (4.2) and selecting the correct form degree, i.e. the
$(2,2)$-form part, necessary to integrate over $\mathbb{C}^{2}$. In
particular, the kinetic (BF) terms define Wick contractions between
$\mathbf{b}^{(1)}$ and $\mathbf{c}^{(0)}$, $\mathbf{b}^{(0)}$ and
$\mathbf{c}^{(1)}$ and similarly between ${\text{\boldmath$\beta$}}^{(1)}$ and
${\text{\boldmath$\gamma$}}^{(0)}$ and ${\text{\boldmath$\beta$}}^{(1)}$ and
${\text{\boldmath$\gamma$}}^{(0)}$ where each contraction uses the Green’s
function pairing resulting in a factor $\omega_{\rm BM}$, where $\omega_{\rm
BM}$ is the kernel for $\bar{\partial}$ as reviewed in Appendix B. We will not
review the precise derivation of this action but content ourselves with
presenting an intuitive picture explaining the origin of the various parts of
the resulting BV action. For more details, we refer the reader to the
references above.

Each free chiral multiplet in the untwisted theory gives rise to two semi-
chiral superfields. A chiral multiplet corresponds to a superfield $\chi$
satisfying $D_{\alpha}\chi=0$ and the equation of motion
$\overline{D}^{2}\chi=0$. We immediately recognise the bosonic semi-chiral
superfield

${\text{\boldmath$\gamma$}}=\chi\,.$ (4.3)

The complex conjugate superfield $\bar{\chi}$ contains a fermionic semi-chiral
superfield given by

${\text{\boldmath$\beta$}}=D_{+}\bar{\chi}\,.$ (4.4)

These two superfields form a $\beta$$\gamma$-system very similar to the more
familiar two-dimensional $\beta\gamma$-system. In addition, one can add a
superpotential $W(\chi)$ changing the equations of motion to
$\overline{D}^{2}\chi=\partial_{\chi}W(\chi)$ which appears unchanged in the
BV action (4.1).

When the $\mathcal{N}=1$ theory contains gauge fields, the corresponding
holomorphic gauge theory has been identified with a holomorphic version of the
$\mathbf{b}\mathbf{c}$ BF theory. The elementary fields in the twisted theory
are a bosonic superfield $\mathbf{b}$ and a fermionic superfield $\mathbf{c}$
with ghost degree $1$. The identification of the twisted fields with the
$\mathbf{Q}$-cohomology of the original theory is non-trivial as it involves
various simplifications which modify the field content without changing the
cohomology. The vector multiplet BPS letters with respect to the supercharge
$\mathbf{Q}$ are given by the self-dual $(2,0)$ part of the field strength,
$F_{++}$, and the gaugino $\bar{\lambda}_{{\dot{\alpha}}}$. Roughly speaking
these fields can be identified with the bottom components of the superfields
${\rm d}^{2}z\,\mathbf{b}^{(0)}$ and $\partial_{\dot{\alpha}}\mathbf{c}^{(0)}$
respectively. For a more extended discussion of the correspondence to the
physical fields see for example [13]. Finally, entirely analogous to the 2d
$\beta\gamma$-system, the gauging of a flavour symmetry proceeds by coupling
the $\beta$$\gamma$ system to an adjoint valued $\mathbf{b}\mathbf{c}$ system
describing the gauge sector.

Throughout this section we choose the following normalisation for the
$\mathbf{b}\mathbf{c}$ and $\beta$$\gamma$ systems,

$\left\\{\mathbf{b}_{A}\,{}_{\lambda}\,\mathbf{c}^{B}\right\\}=\delta_{A}^{B}\,,\qquad\qquad\\{{\text{\boldmath$\beta$}}_{ia}\,{}_{\lambda}\,{\text{\boldmath$\gamma$}}^{jb}\\}=\delta_{a}^{b}\delta_{i}^{j}\,,$
(4.5)

where $A,B$ denote adjoint indices, $i,j$ label the different chiral
multiplets and $a,b$ are indices for the representation $R_{i}$. When the
theory contains a flavour symmetry, the conserved currents in the twisted
theory take the simple form
$\mathbf{J}\propto{\text{\boldmath$\beta$}}_{i}{\text{\boldmath$\gamma$}}^{j}$
where all gauge indices are contracted and the normalisation is fixed by
demanding that the binary $\lambda$-bracket takes the following form

$\left\\{\mathbf{J}^{a}\,{}_{\lambda}\,\mathbf{J}^{b}\right\\}=f^{ab}{}_{c}\,\mathbf{J}^{c}\,.$
(4.6)

A first guess for the stress tensor would be given by the sum
$\mathbf{S}_{\dot{\alpha},0}={\text{\boldmath$\beta$}}_{i}\partial_{\dot{\alpha}}{\text{\boldmath$\gamma$}}^{i}-\mathbf{b}\partial_{\dot{\alpha}}\mathbf{c}$.
However, this naive expression is not $\mathbf{Q}$-closed. Since the the
stress tensor is quadratic in free fields we only have to require
$\mathbf{S}_{\dot{\alpha}}$ to be annihilated by the tree level and one loop
supercharge,

$\mathbf{Q}\ \mathbf{S_{\dot{\alpha}}}^{(0)}=(Q_{0}+Q_{1})\
\mathbf{S_{\dot{\alpha}}}^{(0)}=0,\,.$ (4.7)

where the supercharges $Q_{0}$ and $Q_{1}$ are defined in terms of the binary
and ternary $\lambda$-brackets as [13],

$Q_{0}\mathbf{S}_{\dot{\alpha}}^{(0)}\equiv\\{\mathbf{I}\,{}_{0}\,\mathbf{S}\\}\,,\qquad\qquad
Q_{1}\mathbf{S}\equiv\\{\mathbf{I}\,{}_{0}\,\mathbf{I}\,{}_{0}\,\mathbf{S}^{(0)}\\}\,,$
(4.8)

where $\mathbf{I}$ is given by the interaction terms in (4.1) and all
$\lambda$ parameters in brackets are put to zero. Demanding that the stress
tensor defines a semi-chiral superfield fixes its form to

$\mathbf{S}_{\dot{\alpha}}\propto\mathbf{S}_{{\dot{\alpha}},0}-\zeta\,\partial_{\dot{\alpha}}\mathbf{J}\,,$
(4.9)

where $\mathbf{J}$ is the superconformal R-symmetry current superfield and the
coefficient $\zeta$ is fixed by the $r$-charges of the matter fields. Whenever
the theory is defined as the IR fixed of an RG flow, along the flow the
R-symmetry might mix with abelian flavour symmetries. At the IR fixed point,
however, there is a unique superconformal R-symmetry which can be selected
through $a$-maximisation. This step is crucial to reproduce the correct
symmetry algebra. Similar to the current superfield above, we fix the
normalisation of the supercurrent semi-chiral superfield by demanding that the
binary $\lambda$-bracket takes the following universal form, (2.30),

$\left\\{\mathbf{S}_{{\dot{\alpha}}}\,{}_{\lambda}\,\mathbf{S}_{{\dot{\beta}}}\right\\}=\lambda_{{\dot{\alpha}}}\mathbf{S}_{{\dot{\beta}}}+\lambda_{{\dot{\beta}}}\mathbf{S}_{{\dot{\alpha}}}+\partial_{{\dot{\alpha}}}\mathbf{S}_{{\dot{\beta}}}\,.$
(4.10)

Having introduced the general properties of the twisted theories, let us
proceed to some explicit examples.

### 4.1 Free vector multiplet

As a first example we consider the free vector multiplet. Its holomorphic
twist is given by a $\mathbf{b}\mathbf{c}$ system with BV action

$S_{\rm BV}=\int_{\mathbb{C}^{2}}{\rm d}^{2}z\
\mathbf{b}\bar{\partial}\mathbf{c}\,.$ (4.11)

The stress tensor satisfying the closure condition, (4.7), is given by

$\mathbf{S}_{\dot{\alpha}}=-\mathbf{b}\partial_{\dot{\alpha}}\mathbf{c}\,,$
(4.12)

and we used the binary $\lambda$-bracket to fix the normalisation. With this
stress tensor at hand, we are ready to compute the ternary $\lambda$-bracket.
Following the definition (2.15) above we compute the $\lambda$-bracket by
performing all the Wick rotations and integrating the remaining product of
propagators, where as mentioned above each propagator is given by a Bochner-
Martinelli kernel. The computation is somewhat involved but entirely
algorithmic and was explained in all generality in [12]. For the details of
this particular computation, we refer the reader to section 4.3 of that paper.
In Appendix B we provide an explicit map from the expression in that paper to
the ternary $\lambda$-bracket. The resulting expression is given by

$\displaystyle\left\\{\mathbf{S}_{\dot{\alpha}}\,{}_{\lambda_{1}}\,\mathbf{S}_{\dot{\beta}}\,{}_{\lambda_{2}}\,\mathbf{S}_{\dot{\gamma}}\right\\}=$
$\displaystyle\,-\frac{1}{24\pi^{2}}(\lambda_{1},\lambda_{2})^{2}\Big{[}\lambda_{1\dot{\alpha}}\epsilon_{\dot{\beta}\dot{\gamma}}+\lambda_{2\dot{\beta}}\epsilon_{\dot{\rho}\dot{\alpha}}-(\lambda_{1\dot{\gamma}}+\lambda_{2\dot{\gamma}})\epsilon_{\dot{\alpha}\dot{\beta}}\Big{]}$
(4.13) $\displaystyle=$ $\displaystyle\,\Lambda_{\mathrm{vector}}\,,$ (4.14)

explaining the origin of the naming in (3.14). Comparing with this equation we
find the correct values for the central charges of a single free vector,

$a=\frac{3}{16}\,,\qquad\qquad c=\frac{1}{8}\,.$ (4.15)

### 4.2 $N_{f}$ free chirals

As a second example, consider a theory of $N_{f}$ free chiral multiplets. The
BV action of the holomorphically twisted theory is given by

$S_{\rm BV}=\int_{\mathbb{C}^{2}}{\rm
d}^{2}z\sum_{i=1}^{N_{f}}{\text{\boldmath$\beta$}}_{i}\bar{\partial}{\text{\boldmath$\gamma$}}^{i}\,.$
(4.16)

The theory has a global $\mathrm{U}(N_{f})$ symmetry rotating the chirals,
with an associated conserved current

$\mathbf{J}_{i}^{j}={\text{\boldmath$\beta$}}_{i}{\text{\boldmath$\gamma$}}^{j},\quad
i,j=1,\dots,N_{f}\,,$ (4.17)

where the trace $\mathrm{Tr}\,\,\mathbf{J}$ generates the $\mathrm{U}(1)_{R}$
symmetry and the remaining generators generate an $\mathrm{SU}(N_{f})$ flavour
symmetry. In this section, it is more convenient to work with fundamental
indices but one can straightforwardly convert the results to adjoint indices
using the relevant intertwining matrices. To compute the binary
$\lambda$-brackets of two currents we first compute the bracket of a current
with (derivatives of) free fields,

$\displaystyle\left\\{\mathbf{J}_{i}^{j}\,{}_{\lambda}\,\partial_{\dot{+}}^{n_{1}}\partial_{\dot{-}}^{n_{2}}{\text{\boldmath$\gamma$}}^{k}\right\\}$
$\displaystyle=+\delta_{i}^{k}\sum_{m_{1}=0}^{n_{1}}\sum_{m_{2}=0}^{n_{2}}\left(\begin{array}[]{c}n_{1}\\\
m_{1}\end{array}\right)\left(\begin{array}[]{c}n_{2}\\\
m_{2}\end{array}\right)\lambda_{\dot{+}}^{m_{1}}\lambda_{\dot{-}}^{m_{2}}\partial_{\dot{+}}^{n_{1}-m_{1}}\partial_{\dot{-}}^{n_{2}-m_{2}}{\text{\boldmath$\gamma$}}^{j}$
(4.18)
$\displaystyle=+\delta_{i}^{k}(\partial_{\dot{+}}+\lambda_{\dot{+}})^{n_{1}}(\partial_{\dot{-}}+\lambda_{\dot{-}})^{n_{2}}{\text{\boldmath$\gamma$}}^{j}$
$\displaystyle\\{\mathbf{J}_{i}^{j}\,{}_{\lambda}\,\partial_{\dot{+}}^{n_{1}}\partial_{\dot{-}}^{n_{2}}{\text{\boldmath$\beta$}}_{k}\\}$
$\displaystyle=-\delta_{k}^{j}\sum_{m_{1}=0}^{n_{1}}\sum_{m_{2}=0}^{n_{2}}\left(\begin{array}[]{c}n_{1}\\\
m_{1}\end{array}\right)\left(\begin{array}[]{c}n_{2}\\\
m_{2}\end{array}\right)\lambda_{\dot{+}}^{m_{1}}\lambda_{\dot{-}}^{m_{2}}\partial_{\dot{+}}^{n_{1}-m_{1}}\partial_{\dot{-}}^{n_{2}-m_{2}}{\text{\boldmath$\beta$}}_{i}$
$\displaystyle=$
$\displaystyle-\delta_{k}^{j}(\partial_{\dot{+}}+\lambda_{\dot{+}})^{n_{1}}(\partial_{\dot{-}}+\lambda_{\dot{-}})^{n_{2}}{\text{\boldmath$\beta$}}_{i}$

Combining these equations we can straightforwardly compute the binary
$\lambda$-bracket,

$\left\\{\mathbf{J}_{i}^{j}\,{}_{\lambda}\,\mathbf{J}_{k}^{l}\right\\}=\delta_{i}^{l}\mathbf{J}_{k}^{j}-\delta_{k}^{j}\mathbf{J}_{i}^{l}\,,$
(4.19)

to confirm that this current indeed has the correct normalisation to reproduce
the universal form (4.6). Following similar steps as above, we compute the
ternary $\lambda$-bracket of three currents resulting in

$\left\\{\mathbf{J}_{i}^{j}\,{}_{\lambda_{1}}\,\mathbf{J}_{k}^{l}\,{}_{\lambda_{2}}\,\mathbf{J}_{m}^{n}\right\\}=d_{ij,kl,mn}\lambda_{1}\wedge\lambda_{2}$
(4.20)

where the coefficients are precisely the totally symmetric tensor for
$U(N_{f})$, which matches precisely with equation (3.8).

As explained above, in the presence of additional abelian flavour symmetries
the stress tensor can mixed with the derivatives of the current. In this case,
the only available symmetry is the overall
$\mathrm{U}(1)\subset\mathrm{U}(N_{f})$ which in fact can be identified with
the superconformal R-symmetry. The stress tensor semi-chiral superfield then
takes the form of a one-parameter family,

$\mathbf{S}_{\dot{\alpha}}=\sum_{i=1}^{N_{f}}{\text{\boldmath$\beta$}}_{i}\partial_{{\dot{\alpha}}}{\text{\boldmath$\gamma$}}^{i}-\zeta\partial_{{\dot{\alpha}}}\mathbf{J}_{i}^{i}$
(4.21)

with an a priori arbitrary coefficient $\zeta$. Computing the ternary
$\lambda$-bracket results in,

$\displaystyle\left\\{\mathbf{S}_{\dot{\alpha}}\,{}_{\lambda_{1}}\,\mathbf{S}_{\dot{\beta}}\,{}_{\lambda_{2}}\,\mathbf{S}_{\dot{\gamma}}\right\\}=\frac{3}{2}N_{f}\zeta(1-\zeta)(1-2\zeta)\Lambda_{\mathcal{N}=4}-N_{f}(1-2\zeta)\Lambda_{\mathrm{vector}}$
(4.22)

Comparing this equation to (3.14), we can identify the central charges

$\displaystyle a$
$\displaystyle=-\frac{3}{16}N_{f}\left(1-8\zeta+18\zeta^{2}-12\zeta^{3}\right)\,,$
(4.23) $\displaystyle c$
$\displaystyle=-\frac{1}{8}N_{f}\left(1-11\zeta+27\zeta^{2}-18\zeta^{3}\right)\,,$

Note that $a$ is maximised at $\zeta=\frac{1}{3}$, which is precisely
$\frac{1}{2}r$, where $r=\frac{2}{3}$ is the $r$-charge of a free chiral.
Plugging this back into the equation above we reproduce the correct values of
the central charges for a collection of $N_{f}$ free chirals,

$a=\frac{N_{f}}{48},\qquad\qquad c=\frac{N_{f}}{24}\,.$ (4.24)

### 4.3 $\mathcal{N}=4$ SYM

Viewed as a $\mathcal{N}=1$ SCFT, $\mathcal{N}=4$ super Yang-Mills (SYM) is a
gauge theory with $N_{f}=3$ chiral multiplets in the adjoint representation of
the gauge Lie algebra $\mathfrak{g}$. The BV action for the twisted theory is
given by

$S_{\rm BV}=\int_{\mathbb{C}^{2}}d^{2}z\
\mathrm{Tr}\,\mathbf{b}\left(\bar{\partial}\mathbf{c}-\frac{1}{2}[\mathbf{c},\mathbf{c}]\right)+{\text{\boldmath$\beta$}}_{i}\bar{\partial}{\text{\boldmath$\gamma$}}^{i}+\mathbf{c}[{\text{\boldmath$\beta$}}_{i},{\text{\boldmath$\gamma$}}^{i}]+{\text{\boldmath$\gamma$}}^{1}[{\text{\boldmath$\gamma$}}^{2},{\text{\boldmath$\gamma$}}^{3}]\,.$
(4.25)

The choice of $\mathcal{N}=1$ subalgebra of $\mathcal{N}=4$ breaks the
$\mathrm{SU}(4)_{R}$ symmetry into $\mathrm{U}(1)_{R}\times\mathrm{SU}(3)_{f}$
where the latter is a global symmetry from an $\mathcal{N}=1$ point of view.
The current operator of the $\mathrm{SU}(3)_{f}$ flavour symmetry is given by

$\mathbf{J}^{a}=\sum_{i,j}\beta_{i}(t^{a})^{i}_{j}\gamma^{j},\quad
a=1,\cdots,9$ (4.26)

where $t^{a}$ are orthonormal basis of $SU(3)$, given explicitly in the
appendix A.2. The computation proceeds very similarly as above and we find the
binary $\lambda$-brackets of the current as

$\left\\{\mathbf{J}^{a},\mathbf{J}^{b}\right\\}=f^{ab}{}_{c}J^{c}\,,$ (4.27)

matching with the general form (4.6). Next, we compute the ternary
$\lambda$-bracket of currents, resulting in (for $\mathrm{SU}(2)$ gauge group)

$\left\\{\mathbf{J}^{a}\,{}_{\lambda_{1}}\,\mathbf{J}^{b}\,{}_{\lambda_{2}}\,\mathbf{J}^{c}\right\\}=\dim_{\mathfrak{g}}\,d^{abc}\,\left(\lambda_{1},\lambda_{2}\right)\,.$
(4.28)

The stress tensor, satisfying (4.7), takes a form very similar to the examples
above,

$\mathbf{S}_{\dot{\alpha}}=-\mathbf{b}\partial_{\dot{\alpha}}\mathbf{c}+\sum_{i=1}^{N_{f}=3}{\text{\boldmath$\beta$}}_{i}\partial_{\dot{\alpha}}{\text{\boldmath$\gamma$}}^{i}-\frac{1}{3}\sum_{i=1}^{N_{f}=3}\partial_{\dot{\alpha}}({\text{\boldmath$\beta$}}_{i}{\text{\boldmath$\gamma$}}^{i})$
(4.29)

where gauge indices are contracted. Note that the coefficient in front of the
$R$-symmetry current matches with half the $r$-charge $r=\frac{2}{3}$ of the
chiral multiplets in $\mathcal{N}=4$ SYM. After checking that this stress
tensor indeed is correctly normalised we compute the ternary $\lambda$-bracket
with the following result,

$\displaystyle\left\\{\mathbf{S}_{\dot{\alpha}}\,{}_{\lambda_{1}}\,\mathbf{S}_{\dot{\beta}}\,{}_{\lambda_{2}}\,\mathbf{S}_{\dot{\gamma}}\right\\}=$
$\displaystyle\,\frac{\dim_{\mathfrak{g}}}{9\pi^{2}}\lambda_{1\dot{\alpha}}\lambda_{2\dot{\beta}}(\lambda_{1}+\lambda_{2})_{\dot{\gamma}}(\lambda_{1},\lambda_{2})$
(4.30) $\displaystyle=$
$\displaystyle\,\frac{1}{3}\dim_{\mathfrak{g}}\Lambda_{\mathcal{N}=4}$ (4.31)

where $\dim_{\mathfrak{g}}$ is the dimension of the lie algebra
$\mathfrak{g}$. Comparing this to (3.14), we find the following values of the
central charges,

$a=c=\frac{1}{4}\dim_{\mathfrak{g}}\,,$ (4.32)

which precisely matches with the expected values for the $\mathcal{N}=4$ SYMg
theory. This example thus clarifies our choice of naming for the second
kinematic tensor in (3.14).

### 4.4 Fundamental $\mathrm{SU}(2)$ SQCD

As our next example, we consider $\mathrm{SU}(2)$ SQCD with $N_{f}$ pairs of
fundamental chiral multiplet and anti-fundamental chiral multiplets and
vanishing superpotential. The BV action for the twisted theory is

$S_{\rm BV}=\int_{\mathbb{C}^{2}}{\rm d}^{2}z\
\mathrm{Tr}\,\mathbf{b}\left(\bar{\partial}\mathbf{c}-\frac{1}{2}[\mathbf{c},\mathbf{c}]\right)+\sum_{i=1}^{N_{f}}{\text{\boldmath$\beta$}}_{i}\bar{\partial}{\text{\boldmath$\gamma$}}^{i}+{\text{\boldmath$\beta$}}_{i}\mathbf{c}{\text{\boldmath$\gamma$}}^{i}+\tilde{{\text{\boldmath$\beta$}}}^{i}\bar{\partial}\tilde{{\text{\boldmath$\gamma$}}}_{i}+\tilde{{\text{\boldmath$\gamma$}}}_{i}\mathbf{c}\tilde{{\text{\boldmath$\beta$}}}^{i}\,.$
(4.33)

Since the fundamental representation of $\mathrm{SU}(2)$ is pseudoreal, the
global symmetry is enhanced from
$\mathrm{SU}(N_{f})\times\mathrm{SU}(N_{f})\times\mathrm{U}(1)_{b}$ to
$\mathrm{SO}(4N_{f})\times\mathrm{U}(1)_{b}$, where $U(1)_{b}$ is the baryonic
symmetry under which ${\text{\boldmath$\gamma$}}^{i}$ has charge 1 and
$\tilde{{\text{\boldmath$\gamma$}}}_{i}$ has charge -1. The current operators
for the $\mathrm{SU}(N_{f})\times\mathrm{SU}(N_{f})$ symmetry are given by

$\mathbf{J}_{i}^{j}={\text{\boldmath$\beta$}}_{i}{\text{\boldmath$\gamma$}}^{j},\qquad\qquad\tilde{\mathbf{J}}^{i}_{j}=\tilde{{\text{\boldmath$\beta$}}}^{i}\tilde{{\text{\boldmath$\gamma$}}}_{j}\,.$
(4.34)

By combining these currents with additional currents with one tilded field we
find the currents for the full $\mathrm{SO}(4N_{f})$ flavour symmetry. As in
the previous examples we can compute the $\lambda$-brackets containing three
currents and read off the flavour central charge $k_{G}=2$, independent of
$N_{f}$. The semi-chiral superfield corresponding to the baryonic $U(1)_{b}$
current is

$\mathbf{J}_{b}={\text{\boldmath$\beta$}}_{i}{\text{\boldmath$\gamma$}}^{i}-\tilde{{\text{\boldmath$\beta$}}}^{i}\tilde{{\text{\boldmath$\gamma$}}}_{i}\,,$
(4.35)

which has vanishing ternary $\lambda$-bracket.

The correctly normalised stress tensor satisfying (4.7) is given by

$\mathbf{S}_{\alpha}=-\mathbf{b}\partial_{\dot{\alpha}}\mathbf{c}+\sum_{i=1}^{N_{f}}{\text{\boldmath$\beta$}}_{i}\partial_{\dot{\alpha}}{\text{\boldmath$\gamma$}}^{i}+\sum_{i=1}^{N_{f}}\tilde{{\text{\boldmath$\beta$}}}_{i}\partial_{\dot{\alpha}}\tilde{{\text{\boldmath$\gamma$}}}^{i}-\frac{N_{f}-2}{2N_{f}}\sum_{i=1}^{N_{f}}\partial_{\dot{\alpha}}\left({\text{\boldmath$\gamma$}}_{i}{\text{\boldmath$\beta$}}^{i}+\tilde{{\text{\boldmath$\gamma$}}}_{i}\tilde{{\text{\boldmath$\beta$}}}^{i}\right)\,.$
(4.36)

where again the coefficient matches with half the $r$ charge of the chirals.
With this expression at hand, we can compute the ternary $\lambda$-bracket,

$\displaystyle\left\\{\mathbf{S}_{\dot{\alpha}}\,{}_{\lambda_{1}}\,\mathbf{S}_{\dot{\beta}}\,{}_{\lambda_{2}}\,\mathbf{S}_{\dot{\gamma}}\right\\}=3\left(1-\frac{4}{N_{f}^{2}}\right)\Lambda_{\mathcal{N}=4}-5\Lambda_{\mathrm{vector}}$
(4.37)

Comparing with (3.14), we find the value of the central charges to be

$a=\frac{21}{16}-\frac{9}{N_{f}^{2}},\quad
c=\frac{13}{8}-\frac{9}{N_{f}^{2}}\,,$ (4.38)

again reproducing the expected result. In this case, there is no free
parameter and no need to perform $a$-maximisation since baryonic symmetries
cannot mix with the $R$-symmetry. In the next example, however, we consider
adjoint SQCD where $a$-maximisation will be necessary.

### 4.5 Adjoint $\mathrm{SU}(2)$ SQCD

As our final example, we continue with $\mathrm{SU}(2)$ SQCD but introduce an
additional chiral multiplet $\chi$ in the adjoint representation of the gauge
algebra, which couples to the fundamental chirals with vanishing
superpotential. In the range $3\geq N_{f}\geq 6$ this theory is believed to
flow to an $\mathcal{N}=1$ SCFT in the IR. To distinguish the $\beta$$\gamma$
system originating from the adjoint chiral multiplet from the others we call
the fields

${\text{\boldmath$\beta$}}_{\rm
adjoint}={\text{\boldmath$\psi$}}\,,\qquad\qquad{\text{\boldmath$\gamma$}}_{\rm
adjoint}={\text{\boldmath$\phi$}}\,.$ (4.39)

The BV action of the twisted theory is given by

$S_{\rm BV}=\int_{\mathbb{C}^{2}}d^{2}z\
\mathrm{Tr}\,\mathbf{b}\left(\bar{\partial}\mathbf{c}-\frac{1}{2}[\mathbf{c},\mathbf{c}]\right)+{\text{\boldmath$\psi$}}(\bar{\partial}{\text{\boldmath$\phi$}}+[\mathbf{c},{\text{\boldmath$\phi$}}])+\sum_{i=1}^{N_{f}}{\text{\boldmath$\beta$}}_{i}\bar{\partial}{\text{\boldmath$\gamma$}}^{i}+{\text{\boldmath$\beta$}}_{i}\mathbf{c}{\text{\boldmath$\gamma$}}^{i}+\tilde{{\text{\boldmath$\beta$}}}^{i}\bar{\partial}\tilde{{\text{\boldmath$\gamma$}}}_{i}+\tilde{{\text{\boldmath$\gamma$}}}_{i}\mathbf{c}\tilde{{\text{\boldmath$\beta$}}}^{i}\,.$
(4.40)

By adding the additional adjoint chiral, the global symmetry is broken to
$\mathrm{SU}(N_{f})\times\mathrm{U}(1)_{b}\times\mathrm{U}(1)_{c}$ where the
last $\mathrm{U}(1)_{c}$ acts on the adjoint chiral with charge $1$ and on the
(anti-)fundamental chirals with charge $\frac{1}{2N_{f}}$. The surviving
$\mathrm{SU}(N_{f})$ symmetry is the diagonal subgroup of the two
$\mathrm{SU}(N_{f})$s,

$\mathbf{J}_{\rm diag,i}^{j}=\mathbf{J}^{j}_{i}+\tilde{\mathbf{J}}^{j}_{i}\,,$
(4.41)

Computing the $\mathbf{J}\mathbf{J}\mathbf{J}$ $\lambda$-bracket we can read
off the flavour central charge which in this case is given by $k_{G}=4$.

In this case, the R-symmetry can mix with the additional non-baryonic
$\mathrm{U}(1)_{c}$ symmetry. For this reason, we find a one-parameter family
of (correctly normalised) stress tensors that satisfy the $\mathbf{Q}$-closure
condition (4.7),

$\displaystyle\mathbf{S}_{{\dot{\alpha}}}=\,-\mathbf{b}\partial_{\dot{\alpha}}\mathbf{c}$
$\displaystyle+{\text{\boldmath$\psi$}}\partial_{\dot{\alpha}}{\text{\boldmath$\phi$}}+\sum_{i=1}^{N_{f}}\left({\text{\boldmath$\beta$}}_{i}\partial_{\dot{\alpha}}{\text{\boldmath$\gamma$}}^{i}+\tilde{{\text{\boldmath$\beta$}}}_{i}\partial_{\dot{\alpha}}\tilde{{\text{\boldmath$\gamma$}}}^{i}\right)$
(4.42)
$\displaystyle-\frac{\zeta}{2}\partial_{\dot{\alpha}}\left({\text{\boldmath$\phi$}}{\text{\boldmath$\psi$}}\right)-\left(\frac{1}{2}-\frac{\zeta}{N_{f}}\right)\sum_{i=1}^{N_{f}}\partial_{\dot{\alpha}}({\text{\boldmath$\gamma$}}_{i}{\text{\boldmath$\beta$}}^{i}+\tilde{{\text{\boldmath$\gamma$}}}_{i}\tilde{{\text{\boldmath$\beta$}}}^{i})$

With this stress tensor at hand, we once more compute the ternary
$\lambda$-bracket,

$\displaystyle\left\\{\mathbf{S}_{\dot{\alpha}}\,{}_{\lambda_{1}}\,\mathbf{S}_{\dot{\beta}}\,{}_{\lambda_{2}}\,\mathbf{S}_{\dot{\gamma}}\right\\}=\frac{3}{8}\zeta\left(14-9\zeta+\left(3-\frac{32}{N_{f}^{2}}\right)\zeta^{2}\right)\Lambda_{\mathcal{N}=4}-5\zeta\Lambda_{\mathrm{vector}}\,.$
(4.43)

Comparing this with (3.14), we find the values of the central charges to be

$\displaystyle a=$
$\displaystyle\,\frac{3}{32}\zeta\left(32-27\zeta+\left(9-\frac{96}{N_{f}^{2}}\right)\zeta^{2}\right)\,,$
(4.44) $\displaystyle c=$
$\displaystyle\,\frac{1}{32}\zeta\left(106-81\zeta+\left(27-\frac{288}{N_{f}^{2}}\right)\zeta^{2}\right)\,.$

We find the superconformal $R$-symmetry maximising the central charge $a$ at

$\zeta_{*}=\frac{N_{f}\left(9N_{f}-\sqrt{1024-15N_{f}^{2}}\right)}{-96+9N_{f}^{2}}\,.$
(4.45)

At this value of the parameter we find the central charges to be

$\displaystyle a_{*}$
$\displaystyle=\frac{N_{f}\left(1024\sqrt{1024-15N_{f}^{2}}+3N_{f}\left(-4608+189N_{f}^{2}-5N_{f}\sqrt{1024-15N_{f}^{2}}\right)\right)}{48\left(32-3N_{f}^{2}\right)^{2}}\,,$
(4.46) $\displaystyle c_{*}$
$\displaystyle=\frac{N_{f}\left(592\sqrt{1024-15N_{f}^{2}}+3N_{f}\left(-2544+117N_{f}^{2}-5N_{f}\sqrt{1024-15N_{f}^{2}}\right)\right)}{24\left(32-3N_{f}^{2}\right)^{2}}\,,$

In line with results from the literature [49, 57]. With these examples, we
have covered a wide variety of different theories and phenomena firmly
establishing the validity of our results. Turning on the superpotential
$W={\text{\boldmath$\gamma$}}{\text{\boldmath$\phi$}}\tilde{\text{\boldmath$\gamma$}}$
triggers a relevant deformation towards $\mathcal{N}=2$ SQCD. For $N_{f}=4$,
we find that with this superpotential turned on, our result reproduces the
correct conformal anomalies as well as the flavour central charges. This is
the only $\mathcal{N}=2$ case in which the resulting IR theory is conformal
and indeed we find that the stress tensor $\mathbf{S}_{\dot{\alpha}}$ is not
semi-chiral for other values of $N_{f}$.

Acknowledgements

It is with pleasure that we thank Christopher Beem, Davide Gaiotto and Brian
Williams for useful and inspiring discussions. We also gratefully acknowledge
Andreas Stergiou for pointing out useful Mathematica packages which
facilitated some of the computations done in this work. PB and JW are grateful
to the organizers of the Pollica Summer Workshop, supported by the Regione
Campania, Universitá degli Studi di Salerno, Universitá degli Studi di Napoli
“Federico II”, the Physics Department “Ettore Pancini” and “E. R. Caianiello”,
and Istituto Nazionale di Fisica Nucleare, where part of this work was
performed. The contributions of PB were made possible through the support of
grant No. 494786 from the Simons Foundation and the ERC Consolidator Grant No.
864828, titled “Algebraic Foundations of Supersymmetric Quantum Field Theory”
(SCFTAlg). The work of JW is supported by the UKRI Frontier Research Grant,
underwriting the ERC Advanced Grant titled “Generalized Symmetries in Quantum
Field Theory and Quantum Gravity” and by the Simons Foundation Collaboration
on “Special Holonomy in Geometry, Analysis, and Physics”, Award ID: 724073,
Schafer-Nameki.

##  Appendix A 4d $\mathcal{N}=1$ superconformal algebra

In this appendix, we introduce the four-dimensional $\mathcal{N}=1$
superconformal algebra while simultaneously establishing our notation and
conventions. We exhibit the commutation relations for the complexified algebra
but throughout this work, we always specialise to Euclidean signature.

The spacetime symmetry algebra for $\mathcal{N}=1$ superconformal field
theories is the superalgebra $\mathfrak{sl}(4|1)$. The maximal bosonic
subalgebra is $\mathfrak{so}(6,\mathbf{C})\times\mathfrak{gl}(1,\mathbf{C})$.
The four-dimensional conformal algebra $\mathfrak{so}(6,\mathbf{C})$ is
generated by translations, special conformal transformations, rotations, and
dilatations. The generators for these transformations are given by

$\mathcal{P}_{\alpha{\dot{\alpha}}}\,,\qquad\mathcal{K}^{{\dot{\alpha}}\alpha}\,,\qquad{\mathcal{M}_{\alpha}}^{\beta}\,,\qquad{\mathcal{M}^{{\dot{\alpha}}}}_{{\dot{\beta}}}\,,\qquad\mathcal{H}\,,$
(A.1)

where $\alpha,\beta=\pm$ and ${\dot{\alpha}},{\dot{\beta}}=\dot{\pm}$ are
fundamental $\mathrm{SU}(2)_{L}\times\mathrm{SU}(2)_{R}$ indices. Adding 4
Poincaré supercharges $\mathcal{Q}_{\alpha}$,
$\widetilde{\mathcal{Q}}_{\dot{\alpha}}$ and 4 conformal supercharges
$\mathcal{S}^{\alpha}$, $\widetilde{\mathcal{S}}^{\dot{\alpha}}$ to this
algebra we obtain the full 4d $\mathcal{N}=1$ superalgebra
$\mathfrak{sl}(4|1)$. These supercharges are acted upon by the $\mathrm{U}(1)$
R-symmetry with generator $\hat{r}$.171717We put the hat to distinguish from
the charge $r$ under this symmetry.

The commutation relations for the $\mathfrak{so}(6,\mathbf{C})$ conformal
algebra are

$\displaystyle\left[{{\mathcal{M}}_{\alpha}}^{\beta},{{\mathcal{M}}_{\gamma}}^{\delta}\right]$
$\displaystyle=c_{1}\left(\delta_{\gamma}^{\beta}{{\mathcal{M}}_{\alpha}}^{\delta}-\delta_{\alpha}^{\delta}{{\mathcal{M}}_{\gamma}}^{\beta}\right)\,,$
$\displaystyle\left[{\mathcal{M}^{\dot{\alpha}}}_{\dot{\beta}},{\mathcal{M}^{\dot{\gamma}}}_{\dot{\delta}}\right]$
$\displaystyle=c_{1}\left(\delta^{\dot{\alpha}}_{\dot{\delta}}{\mathcal{M}^{\dot{\gamma}}}_{\dot{\beta}}-\delta^{\dot{\gamma}}_{\dot{\beta}}{\mathcal{M}^{\dot{\alpha}}}_{\dot{\delta}}\right)\,,$
(A.2)
$\displaystyle\left[{{\mathcal{M}}_{\alpha}}^{\beta},{{\mathcal{P}}_{\gamma\dot{\gamma}}}\right]$
$\displaystyle=c_{1}\left(\delta_{\gamma}^{\beta}{\mathcal{P}}_{\alpha\dot{\gamma}}-\frac{1}{2}\delta_{\alpha}^{\beta}{\mathcal{P}}_{\gamma\dot{\gamma}}\right)\,,$
$\displaystyle\left[{\mathcal{M}^{\dot{\alpha}}}_{\dot{\beta}},{{\mathcal{P}}_{\gamma\dot{\gamma}}}\right]$
$\displaystyle=c_{1}\left(\delta^{\dot{\alpha}}_{\dot{\gamma}}\mathcal{P}_{\gamma\dot{\beta}}-\frac{1}{2}\delta^{\dot{\alpha}}_{\dot{\beta}}{\mathcal{P}}_{\gamma\dot{\gamma}}\right)\,,$
$\displaystyle\left[{{\mathcal{M}}_{\alpha}}^{\beta},{{\mathcal{K}}^{\dot{\gamma}\gamma}}\right]$
$\displaystyle=-c_{1}\left(\delta_{\alpha}^{\gamma}{\mathcal{K}}^{\dot{\gamma}\beta}-\frac{1}{2}\delta_{\alpha}^{\beta}{\mathcal{K}}^{\dot{\gamma}\gamma}\right)\,,$
$\displaystyle\left[{\mathcal{M}^{\dot{\alpha}}}_{\dot{\beta}},{{\mathcal{K}}^{\dot{\gamma}\gamma}}\right]$
$\displaystyle=-c_{1}\left(\delta^{\dot{\gamma}}_{\dot{\beta}}\mathcal{K}^{\dot{\alpha}\gamma}-\frac{1}{2}\delta^{\dot{\alpha}}_{\dot{\beta}}{\mathcal{K}}^{\dot{\gamma}\gamma}\right)\,,$
$\displaystyle\left[\mathcal{H},\mathcal{P}_{\alpha\dot{\alpha}}\right]$
$\displaystyle=\mathcal{P}_{\alpha\dot{\alpha}}\,,$
$\displaystyle\left[\mathcal{H},\mathcal{K}^{\dot{\alpha}\alpha}\right]$
$\displaystyle=-\mathcal{K}^{\dot{\alpha}\alpha}\,,$
$\displaystyle\left[\mathcal{K}^{\dot{\alpha}\alpha},\mathcal{P}_{\beta\dot{\beta}}\right]$
$\displaystyle=4c_{2}^{2}\left(\delta_{\beta}^{\alpha}\delta^{\dot{\alpha}}_{\dot{\beta}}\mathcal{H}+\frac{1}{c_{1}}\delta_{\beta}^{\alpha}{\mathcal{M}^{\dot{\alpha}}}_{\dot{\beta}}+\frac{1}{c_{1}}\delta^{\dot{\alpha}}_{\dot{\beta}}{{\mathcal{M}}_{\beta}}^{\alpha}\right)\,.$

The non-vanishing commutators between the supercharges are

$\displaystyle\left\\{\mathcal{Q}_{\alpha},\widetilde{\mathcal{Q}}_{\dot{\alpha}}\right\\}$
$\displaystyle=c_{3}\mathcal{P}_{\alpha\dot{\alpha}}\,,$ (A.3)
$\displaystyle\left\\{\widetilde{\mathcal{S}}^{\dot{\alpha}},\mathcal{S}^{\alpha}\right\\}$
$\displaystyle=c_{3}\mathcal{K}^{\dot{\alpha}\alpha}\,,$
$\displaystyle\left\\{\mathcal{Q}_{\alpha},\mathcal{S}^{\beta}\right\\}$
$\displaystyle=c_{2}c_{3}\left(\delta_{\alpha}^{\beta}\mathcal{H}+\frac{2}{c_{1}}{\mathcal{M}_{\alpha}}^{\beta}-\frac{3}{2q}\delta_{\alpha}^{\beta}\,\hat{r}\right)\,,$
$\displaystyle\left\\{\widetilde{\mathcal{S}}^{\dot{\alpha}},\widetilde{\mathcal{Q}}_{\dot{\beta}}\right\\}$
$\displaystyle=c_{2}c_{3}\left(\delta^{\dot{\alpha}}_{\dot{\beta}}\mathcal{H}+\frac{2}{c_{1}}{\mathcal{M}^{\dot{\alpha}}}_{\dot{\beta}}+\frac{3}{2q}\delta^{\dot{\alpha}}_{\dot{\beta}}\,\hat{r}\right)\,,$

Finally, the bosonic generators act on the supercharges as

$\displaystyle\left[{\mathcal{M}_{\alpha}}^{\beta},\mathcal{Q}_{\gamma}\right]$
$\displaystyle=c_{1}\left(\delta_{\gamma}^{\beta}\mathcal{Q}_{\alpha}-\frac{1}{2}\delta_{\alpha}^{\beta}\mathcal{Q}_{\gamma}\right)\,,$
$\displaystyle\left[{\mathcal{M}^{\dot{\alpha}}}_{\dot{\beta}},\widetilde{\mathcal{Q}}_{\dot{\gamma}}\right]$
$\displaystyle=c_{1}\left(\delta_{\dot{\gamma}}^{\dot{\alpha}}\widetilde{\mathcal{Q}}_{\dot{\beta}}-\frac{1}{2}\delta^{\dot{\alpha}}_{\dot{\beta}}\widetilde{\mathcal{Q}}_{\dot{\gamma}}\right)\,,$
(A.4)
$\displaystyle\left[{\mathcal{M}_{\alpha}}^{\beta},\mathcal{S}^{\gamma}\right]$
$\displaystyle=-c_{1}\left(\delta_{\alpha}^{\gamma}\mathcal{S}^{\beta}-\frac{1}{2}\delta_{\alpha}^{\beta}\mathcal{S}^{\gamma}\right)\,,\quad$
$\displaystyle\left[{\mathcal{M}^{\dot{\alpha}}}_{\dot{\beta}},\widetilde{\mathcal{S}}^{\dot{\gamma}}\right]$
$\displaystyle=-c_{1}\left(\delta^{\dot{\gamma}}_{\dot{\beta}}\widetilde{\mathcal{S}}^{\mathcal{I}\dot{\alpha}}-\frac{1}{2}\delta^{\dot{\alpha}}_{\dot{\beta}}\widetilde{\mathcal{S}}^{\dot{\gamma}}\right)\,,$
$\displaystyle\left[\mathcal{H},\mathcal{Q}_{\alpha}\right]$
$\displaystyle=\frac{1}{2}\mathcal{Q}_{\alpha}\,,$
$\displaystyle\left[\mathcal{H},\widetilde{\mathcal{Q}}_{\dot{\alpha}}\right]$
$\displaystyle=\frac{1}{2}\widetilde{\mathcal{Q}}_{\dot{\alpha}}\,.$
$\displaystyle\left[\mathcal{H},\mathcal{S}^{\alpha}\right]$
$\displaystyle=-\frac{1}{2}\mathcal{S}^{\alpha}\,,$
$\displaystyle\left[\mathcal{H},\widetilde{\mathcal{S}}^{\dot{\alpha}}\right]$
$\displaystyle=-\frac{1}{2}\widetilde{\mathcal{S}}^{\dot{\alpha}}\,.$
$\displaystyle\left[{\mathcal{K}^{\dot{\alpha}\alpha}},\mathcal{Q}_{\beta}\right]$
$\displaystyle=2c_{2}\delta_{\beta}^{\alpha}\widetilde{\mathcal{S}}^{\dot{\alpha}}\,,$
$\displaystyle\left[{\mathcal{K}^{\dot{\alpha}\alpha}},\widetilde{\mathcal{Q}}_{\dot{\beta}}\right]$
$\displaystyle=2c_{2}\delta_{\dot{\beta}}^{\dot{\alpha}}\mathcal{S}^{\alpha}\,,$
$\displaystyle\left[{\mathcal{P}_{\alpha\dot{\alpha}}},\mathcal{S}^{\beta}\right]$
$\displaystyle=-2c_{2}\delta_{\alpha}^{\beta}\widetilde{\mathcal{Q}}_{\dot{\alpha}}\,,$
$\displaystyle\left[{\mathcal{P}_{\alpha\dot{\alpha}}},\widetilde{\mathcal{S}}^{\dot{\beta}}\right]$
$\displaystyle=-2c_{2}\delta^{\dot{\beta}}_{\dot{\alpha}}\mathcal{Q}_{\alpha}\,,$
$\displaystyle\left[\hat{r},\mathcal{Q}_{\alpha}\right]$
$\displaystyle=q\mathcal{Q}_{\alpha}\,,$
$\displaystyle\left[\hat{r},\widetilde{\mathcal{Q}}_{\dot{\alpha}}\right]$
$\displaystyle=-q\widetilde{\mathcal{Q}}_{\dot{\alpha}}\,,$
$\displaystyle\left[\hat{r},\mathcal{S}^{\alpha}\right]$
$\displaystyle=-q\mathcal{S}^{\alpha}\,,$
$\displaystyle\left[\hat{r},\widetilde{\mathcal{S}}^{\dot{\alpha}}\right]$
$\displaystyle=q\widetilde{\mathcal{S}}^{\dot{\alpha}}\,.$

All other commutators vanish. In order to easily compare with other references
we left explicit the $r$-charge of the supercharges as well as the
normalisation of the translation, special conformal and rotation generators.
In the main text, we pick our preferred choice of constants,

$c_{1}=-2\,,\qquad\qquad c_{2}=c_{3}=1\qquad\qquad\text{and}\qquad\qquad
q=-1\,.$ (A.5)

In radial quantization, the various generators satisfy the following
hermiticity conditions

$\displaystyle\mathcal{H}^{\dagger}=\mathcal{H}\,,$
$\displaystyle\qquad(\mathcal{P}_{\alpha\dot{\alpha}})^{\dagger}=\mathcal{K}^{\dot{\alpha}\alpha}\,,\qquad({{\mathcal{M}}_{\alpha}}^{\beta})^{\dagger}={{\mathcal{M}}_{\beta}}^{\alpha}\,,\qquad({{\mathcal{M}}^{\dot{\alpha}}}_{\dot{\beta}})^{\dagger}={{\mathcal{M}}^{\dot{\beta}}}_{\dot{\alpha}}\,,$
(A.6)
$\displaystyle\hat{r}^{\dagger}=\hat{r}\,,\qquad({\mathcal{Q}_{\alpha}})^{\dagger}={\mathcal{S}^{\alpha}}\,,\qquad({{\widetilde{\mathcal{Q}}}_{\dot{\alpha}}})^{\dagger}={{\widetilde{\mathcal{S}}}^{\dot{\alpha}}}\,.$

In the notation of [38], the supercharges transform in the representations,
$\mathcal{Q}\in[1,0]_{\frac{1}{2}}^{(q)}$ and
$\widetilde{\mathcal{Q}}\in[0,1]_{\frac{1}{2}}^{(-q)}$, where the
representation $[L]^{(q)}_{\Delta}$ has Lorentz representation $L$ specified
by the Dynkin labels of the two $\mathfrak{su}(2)$s, $\Delta$ is the scaling
dimension and $q$ is the $\mathrm{U}(1)_{r}$ charge.

In our conventions fundamental $\mathrm{SU}(2)$ indices are raised and lowered
as follows,

$\phi^{\alpha}=\epsilon^{\alpha\beta}\,\phi_{\beta}\,,\qquad\phi_{\alpha}=\epsilon_{\alpha\beta}\,\phi^{\beta}\,,$
(A.7)

where the anti-symmetric $\mathrm{SU}(2)$ invariant tensor $\epsilon$ is
defined as

$\epsilon_{12}=\epsilon^{21}=1\,.$ (A.8)

To contract $\mathrm{SU}(2)$ tensors we use the following notation. For two
covariant or two contravariant $\mathrm{SU}(2)$ vectors we define the
contraction as follows,

$(x,y)=x^{\dot{\alpha}}\,y^{\dot{\beta}}\,\epsilon_{{\dot{\alpha}}{\dot{\beta}}}=\epsilon^{{\dot{\beta}}{\dot{\alpha}}}\,x_{\dot{\alpha}}\,y_{\dot{\beta}}\,.$
(A.9)

On the other hand, for one covariant and one contravariant vector, we define
the contraction as

$x\cdot y=x_{\dot{\alpha}}\,y^{\dot{\alpha}}\,.$ (A.10)

### A.1 $\mathcal{N}=1$ superspace

In some computations in this paper, we employ $\mathcal{N}=1$ superspace. In
order to accommodate for our conventions of the superconformal algebra
introduced above, our conventions differ slightly from the usual conventions
of Wess and Bagger [58]. In particular, we define the supercovariant
derivatives as

$\displaystyle D_{\alpha}=$
$\displaystyle\,\frac{\partial}{\partial\theta^{\alpha}}+\frac{{\rm
i}}{2}\sigma^{\mu}_{\alpha{\dot{\alpha}}}\bar{\theta}^{\dot{\alpha}}\frac{\partial}{\partial
x^{\mu}}\,,$ (A.11) $\displaystyle\overline{D}_{\dot{\alpha}}=$
$\displaystyle\,-\frac{\partial}{\partial\bar{\theta}^{\dot{\alpha}}}-\frac{{\rm
i}}{2}\theta^{\alpha}\sigma^{\mu}_{\alpha{\dot{\alpha}}}\frac{\partial}{\partial
x^{\mu}}\,.$

The differential operators for the supercharges on the other hand are given by

$\displaystyle Q_{\alpha}=$
$\displaystyle\,\frac{\partial}{\partial\theta^{\alpha}}-\frac{{\rm
i}}{2}\sigma^{\mu}_{\alpha{\dot{\alpha}}}\bar{\theta}^{\dot{\alpha}}\frac{\partial}{\partial
x^{\mu}}\,,$ (A.12) $\displaystyle\overline{Q}_{\dot{\alpha}}=$
$\displaystyle\,-\frac{\partial}{\partial\bar{\theta}^{\dot{\alpha}}}+\frac{{\rm
i}}{2}\theta^{\alpha}\sigma^{\mu}_{\alpha{\dot{\alpha}}}\frac{\partial}{\partial
x^{\mu}}\,.$

These differential operators satisfy the following anti-commutation relations,

$\displaystyle\left\\{{D_{\alpha}},{\overline{D}_{\dot{\alpha}}}\right\\}=$
$\displaystyle\,{\rm
i}\sigma^{\mu}_{\alpha{\dot{\alpha}}}\frac{\partial}{\partial
x^{\mu}}=P_{\alpha{\dot{\alpha}}}\,,$ (A.13)
$\displaystyle\left\\{{Q_{\alpha}},{\overline{Q}_{\dot{\alpha}}}\right\\}=$
$\displaystyle\,-{\rm
i}\sigma^{\mu}_{\alpha{\dot{\alpha}}}\frac{\partial}{\partial
x^{\mu}}=-P_{\alpha{\dot{\alpha}}}\,,$
$\displaystyle\left\\{{D_{\alpha}},{D_{\beta}}\right\\}=$
$\displaystyle\left\\{{\overline{D}_{\dot{\alpha}}},{\overline{D}_{\dot{\beta}}}\right\\}=\left\\{{Q_{\alpha}},{Q_{\beta}}\right\\}=\left\\{{\overline{Q}_{\dot{\alpha}}},{\overline{Q}_{\dot{\beta}}}\right\\}=0\,,$
$\displaystyle\left\\{{D_{\alpha}},{Q_{\beta}}\right\\}=$
$\displaystyle\left\\{{D_{\alpha}},{\overline{Q}_{\dot{\beta}}}\right\\}=\left\\{{\overline{D}_{\dot{\alpha}}},{Q_{\beta}}\right\\}=\left\\{{\overline{D}_{\dot{\alpha}}},{\overline{Q}_{\dot{\beta}}}\right\\}=0\,.$

At first sight, the sign in the first two equations might seem off but this is
consistent with the supersymmetry algebra (A.3) due to the definition of the
differential operators involving an additional minus sign. See for example
Appendix A of [53] for a detailed discussion of this issue.

### A.2 Lie algebra conventions

We work with orthonormal basis $t^{a}$, $a=1,\cdots,\dim N^{2}$ of
$\mathfrak{gl}(N)$, so that adjoint indices can be raised and lowered
freely.181818The elements are defined as are $\frac{\sqrt{2}}{2}$ times the
generalised Gell-Mann matrices. For explicit expressions, see for example
[59]. Our normalisation is such that the Killing form is given by,

$\mathrm{Tr}\,t^{a}\,t^{b}=\delta^{ab}\,,$ (A.14)

while the commutation relation are given by

$\left[{t^{a}},{t^{b}}\right]=f^{ab}{}_{c}\,t^{c}\,.$ (A.15)

We define the completely anti-symmetric and symmetric structure constants as
follows,

$\displaystyle
f^{abc}=\mathrm{Tr}\,\left(\left[{t^{a}},{t^{b}}\right]\,t^{c}\right)\,,\qquad\qquad
d^{abc}=\mathrm{Tr}\,\left(\left\\{{t^{a}},{t^{b}}\right\\}\,t^{c}\right)\,.$
(A.16)

With this definition, the anti-symmetric structure constants satisfy the
following identity,

$f^{abc}f_{ab}{}^{d}=2h^{\vee}\delta^{cd}\,,$ (A.17)

where $h^{\vee}$ denotes the dual Coxeter number.

Another basis for $\mathfrak{sl}(N)$ that is useful in some application is
given by $\\{E_{i,j}|i\neq j\\}\cup\\{E_{i,i}-E_{1,1}|i=2,\dots N\\}$ where
$E_{i,j}$ denotes the coordinate matrix with all entries zero except one $1$
at position $\\{i,j\\}$. The commutation relations for the nilpotent elements
are,

$\left[E_{i,j},E_{k,l}\right]=\delta_{k,j}E_{i,l}-\delta_{i,l}E_{k,j}$ (A.18)

These are related to the Chevalley generators $F^{\pm}_{i}$ for
$\mathfrak{n}_{\pm}$ as $F^{\pm}_{i}=E_{i\pm 1,i}$.

##  Appendix B Residues, heat kernels and integrated correlators

In this appendix, we collect various technical details that were omitted from
the main text. We start with a discussion of complex analysis in higher
dimensions and in particular introduce a higher-dimensional residue theorem.
This will be a crucial tool for many of the calculations in this work. We
proceed by introducing some general aspects of heat kernel regularisation,
after which we provide a detailed computation of the ternary $\lambda$-bracket
both in the general case as derived in Section 3 as well as for the various
examples discussed in Section 4.

### B.1 Higher-dimensional residues

Cauchy’s integral formula for a disc $D_{1}\subset\mathbb{C}$, states that for
any smooth function $f:D_{1}\rightarrow\mathbb{C}$ and a point
$z\in\mathbb{C}$, we have

$f(w)=\frac{1}{2\pi{\rm i}}\oint_{\partial D_{1}}\frac{{\rm
d}z}{z-w}f(z)+\frac{1}{2\pi{\rm i}}\int_{D_{1}}\frac{{\rm
d}z}{z-w}\wedge\bar{\partial}f\,.$ (B.1)

If $f$ is a holomorphic function the last term drops out and we find the well-
known residue formula. The goal of this subsection is to introduce a
generalisation of this result to higher dimensions.

A special role in (B.1) is played by the integration kernel $\frac{1}{2\pi{\rm
i}}\frac{{\rm d}z}{z-w}$. We can generalise Cauchy’s formula to $d$ complex
dimensions by introducing the Bochner-Martinelli kernel,

$\omega_{\rm BM}(z,w)=\frac{(-1)^{d}(d-1)!}{(2\pi{\rm
i})^{d}|z-w|^{2d}}\sum_{i=1}^{d}(\bar{z}_{i}-\bar{w}_{i})\wedge{\rm
d}\bar{z}_{1}\wedge\cdots\wedge\widehat{{\rm
d}\bar{z}}_{i}\wedge\cdots\wedge{\rm d}\bar{z}_{d}\,.$ (B.2)

where the $i$th anti-holomorphic differential is omitted in the sum and
$|z|^{2}=\sum_{i=1}^{d}(z_{i},\bar{z}_{i})$. This is a differential form of
type $(0,d-1)$ on $\mathbb{C}^{d}\backslash\\{w\\}$ which upon setting $d=1$
precisely reduces to the ’Cauchy kernel’ introduced above. The Bochner-
Martinelli kernel is a Green’s function for the Dolbeault differential
$\bar{\partial}$, i.e.

$\bar{\partial}\left({\rm d}^{d}z\,\omega_{\rm
BM}(z,0)\right)=\delta^{(2d)}(z,\bar{z}){\rm d}^{d}z\wedge{\rm
d}^{d}\bar{z}\,,$ (B.3)

and therefore is closely related to the harmonic $(n,n)$-form on
$\mathbb{C}^{d}$. Having introduced this kernel, we can generalise the Cauchy
integration formula as follows for any smooth function
$f:D_{d}\rightarrow\mathbb{C}$ on a $d$-dimensional disc,

$f(w)=\oint_{\partial D_{d}}{\rm d}^{d}z\wedge\omega_{\rm
BM}(z,w)f(z)+\int_{D_{d}}{\rm d}^{d}z\wedge\omega_{\rm
BM}(z,w)\wedge\bar{\partial}f(z)\,.$ (B.4)

For holomorphic functions, the second term vanishes, while for cohomologically
holomorphic functions it is $\mathbf{Q}$-exact.

In this work, we are mainly interested in the case $d=2$ for which the
Bochner-Martinelli kernel takes the form,

$\omega_{\rm BM}(z,w)=\frac{1}{(2\pi{\rm
i})^{2}}\frac{\left(\bar{z}-\bar{w},{\rm d}\bar{z}\right)}{|z-w|^{4}}\,.$
(B.5)

### B.2 Heat kernel regularisation

The integrated correlators encountered in this work are often rather
complicated to compute directly. However, we can make progress by introducing
heat kernel regularisation. Recall the definition of the heat kernel in $d$
complex dimensions,

$K_{t}(x,y)=\frac{1}{(\pi
t)^{d}}e^{-\frac{|x-y|^{2d}}{t}}\,,\qquad\qquad\text{where
}\,x,y\in\mathbb{C}^{d}\,,$ (B.6)

which satisfies the heat equation,

$\nabla_{x}^{2}K_{t}(x,y)=\partial_{t}K_{t}(x,y)\,,$ (B.7)

where $\nabla_{x}^{2}=\sum_{i=1}^{d}\partial_{x_{i}}\partial_{\bar{x}_{i}}$
for all $t>0$, with the initial condition

$\lim\limits_{t\rightarrow 0}K_{t}(x,y)=\delta(x-y)\,.$ (B.8)

From now on we specify to $d=2$ where we can use the heat kernel to regularise
the Bochner-Martinelli kernel as follows,

$\omega_{{\rm BM},\epsilon}(z,0)=\frac{(\bar{z},{\rm d}\bar{z})}{(2{\rm
i})^{2}}\int_{\epsilon}^{\infty}\frac{{\rm d}t}{t}\ K_{t}(z)\,,$ (B.9)

where we have introduced a UV regulator $\epsilon$. Sending
$\epsilon\rightarrow 0$, we recover the standard Bochner-Martinelli kernel as
introduced in (B.5). Acting on the regularised Bochner-Martinelli kernel with
$\bar{\partial}$ we find,

$\bar{\partial}\omega_{{\rm BM},\epsilon}(z)=\frac{({\rm d}\bar{z},{\rm
d}\bar{z})}{(2{\rm i})^{2}}K_{\epsilon}(z),$ (B.10)

where we recognise the regularised $\delta$-function
$\delta^{(4)}_{\epsilon}(z)=K_{\epsilon}(z)$. More generally, for arbitrary
powers of $z_{e}$, we can rewrite inverse powers as follows

$\frac{1}{|z_{e}|^{2\Delta}}=\lim\limits_{\epsilon\rightarrow
0}\frac{\pi^{2}}{\Gamma(\Delta)}\int_{\epsilon}^{\infty}\frac{{\rm
d}t}{t^{\Delta-1}}K_{t}(z)\,.$ (B.11)

The integrals resulting from the $\lambda$-brackets in Section 3 are always of
the form,

$I\left(\Gamma_{\Delta_{e}}\right)=\int_{\mathbb{C}^{2n}}\left(\prod_{v=1}^{n}\frac{{\rm
d}^{2}z_{v}{\rm d}^{2}\bar{z}_{v}}{(2\pi{\rm
i})^{2}}\mathrm{e}^{\lambda_{v}\cdot
z_{v}}\right)\left(\prod_{e}\frac{1}{|z_{e}|^{2\Delta_{e}}}\right)f(z_{v},\bar{z}_{v})\,,$
(B.12)

where $f$ is a homogeneous function,
$f(az_{v}\,,\,a\bar{z}_{v})=a^{\Lambda}f(z_{v},\bar{z}_{v})$, and $n$
indicates that the integral results from an $n$-ary bracket. These integrals
can be associated with a (decorated) graph $\Gamma_{\Delta_{e}}$ with $n$
vertices $v$ and a collection of edges $e$ connecting them. For example, for
the ternary $\lambda$-bracket the graph is given by a triangle as indicated in
Figure 1. In this case $v=0,1,2$ and the edges take value
$e\in\left\\{01\,,\,02\,,\,12\right\\}$. The $\Delta_{e}$ in the diagram
indicate the inverse power appearing in the integrand.

Figure 1: A triangle diagram encoding the integrand (B.12). Every vertex is
associated with a variable while every edge is associated with the difference
between two vertices. Every edge variable appears in the integrand with an
inverse power $2\Delta_{e}$.

Using the formula (B.11) we can rewrite this integral as

$I\left(\Gamma_{\Delta_{e}}\right)=\int\left(\prod_{e}\frac{{\rm
d}t_{e}}{\Gamma(\Delta_{e})t^{\Delta_{e}+1}}\right)\int_{\mathbb{C}^{2n}}\left(\prod_{v=1}^{n}\frac{{\rm
d}^{2}z_{v}{\rm d}^{2}\bar{z}_{v}}{(2\pi{\rm
i})^{2}}\right)\mathrm{e}^{-S}f(z_{v},\bar{z}_{v})\,.$ (B.13)

where the function $S$ in the exponent is given by

$S=z_{v}A^{vv^{\prime}}\bar{z}_{v^{\prime}}-\lambda^{v}z_{v}=z_{v}A^{vv^{\prime}}\left(\bar{z}_{v^{\prime}}-(A^{-1}\lambda)_{v^{\prime}}\right)\,,$
(B.14)

with the matrix $A$ defined as

$A^{vv^{\prime}}=\sum_{e}l^{v,e}l^{v^{\prime},e}\,,$ (B.15)

and $l^{v,e}$ is defined as follows. If the edge $e$ runs from $v_{1}$ to
$v_{2}$, the vector $l^{v,e}$ takes value $1$ at $v=v_{1}$ and $-1$ if
$v=v_{2}$. Following these rules in the example of the triangle in Figure 1 we
find the matrix

$A^{vv^{\prime}}=\begin{pmatrix}\frac{1}{t_{01}}+\frac{1}{t_{12}}&-\frac{1}{t_{12}}\\\
-\frac{1}{t_{12}}&\frac{1}{t_{02}}+\frac{1}{t_{12}}\end{pmatrix}\,.$ (B.16)

After changing coordinates,
$\bar{z}_{v}=\bar{w}_{v}+\left(A^{-1}\lambda\right)_{v}$, the integrals over
$z_{v}$ and $\bar{w}_{v}$ reduce to Gaussian integrals and can
straightforwardly be performed. In particular, the only contributions come
from terms where $z_{v}$ and $\bar{w}_{v}$ are pairwise combined. In that
case, we have the ’propagator’

$\left\langle{z_{v\alpha}\bar{w}_{v^{\prime}\beta}}\right\rangle=\left(A^{-1}\right)_{vv^{\prime}}\epsilon_{\alpha\beta}\,.$
(B.17)

Having done so we are left with a set of scale invariant integrals over the
heat kernel parameters $t_{e}$. These integrals are invariant under rescaling
$t_{e}\rightarrow\Lambda t_{e}$ and therefore should be divided by the volume
$\mathbf{V}(\mathbb{R}_{+})=\int_{\mathbb{R}_{+}}\frac{{\rm d}t}{t}$ in order
to account for this gauge redundancy and render the integral finite.

### B.3 $\lambda$-bracket from integrated correlators

Having introduced the heat kernel regularisation procedure, we can proceed to
work out the integrals arising from the integrated correlators in Section 3.
In all such cases, the graph takes the same form as in Figure 1.

#### Flavour current semi-chiral superfields

First, let us discuss the ternary $\lambda$-bracket of three flavour current
semi-chiral superfields. As discussed in the main text, there are three
possible tensor structures appearing in the
$\left\langle{\mathbf{J}^{(2)}\mathbf{J}^{(1)}\mathbf{J}^{(0)}}\right\rangle$
three-point function. Supersymmetry and current conservation fix the
coefficients and after acting with $\bar{\partial}$ only a single tensor
structure remains, see (3.7). The integral we are left to compute is therefore
given by,

$I_{F}=\int_{(\mathbb{C}^{2})^{2}}\frac{{\rm d}^{2}z_{1}}{(2\pi{\rm
i})^{2}}\frac{{\rm d}^{2}z_{2}}{(2\pi{\rm i})^{2}}\mathrm{e}^{\lambda_{1}\cdot
z_{1}+\lambda_{2}\cdot
z_{2}}\frac{(\bar{z}_{1},\bar{z}_{2})}{|z_{12}|^{4}|z_{1}|^{4}|z_{2}|^{4}}\left(1+2\frac{(\bar{z}_{12},z_{1})}{|z_{12}|^{2}}\right)\,.$
(B.18)

Introducing the heat kernel regularisation we find

$I_{F}=\int\left(\frac{{\rm d}t_{01}\,{\rm d}t_{02}\,{\rm
d}t_{12}}{t_{01}^{3}t_{02}^{3}t_{12}^{3}}\right)\int_{(\mathbb{C}^{2})^{2}}\frac{{\rm
d}^{2}z_{1}}{(2\pi{\rm i})^{2}}\frac{{\rm d}^{2}z_{2}}{(2\pi{\rm
i})^{2}}\mathrm{e}^{-S}\left((\bar{z}_{1},\bar{z}_{2})+\frac{(\bar{z}_{1},\bar{z}_{2})(\bar{z}_{12},z_{1})}{t_{12}}\right)\,.$
(B.19)

Following the steps above we immediately see that this integral only gets
contributions at $\mathcal{O}(\lambda^{2})$. Indeed, after changing
coordinates as specified above we can perform the integration along $z_{i}$
and $\bar{w}_{i}$ to obtain the following integral over $t_{e}$,

$\displaystyle I_{F}^{(1)}=$ $\displaystyle\,\int\left(\frac{{\rm
d}t_{01}\,{\rm d}t_{02}\,{\rm
d}t_{12}}{t_{01}^{3}t_{02}^{3}t_{12}^{3}}\right)\frac{\epsilon^{{\dot{\alpha}}{\dot{\beta}}}}{\left(\det
A\right)^{2}}\left(A^{-1}\lambda\right)_{1{\dot{\alpha}}}\left(A^{-1}\lambda\right)_{2{\dot{\beta}}}\,,$
(B.20) $\displaystyle=$ $\displaystyle\,-\int_{X}{\rm d}t_{01}\,{\rm
d}t_{02}\,{\rm
d}t_{12}\frac{1}{\left(t_{01}+t_{02}+t_{12}\right)^{3}}\left(\lambda_{1},\lambda_{1}\right)\,,$
$\displaystyle I_{F}^{(2)}=$ $\displaystyle\,\int\left(\frac{{\rm
d}t_{01}\,{\rm d}t_{02}\,{\rm
d}t_{12}}{t_{01}^{3}t_{02}^{3}t_{12}^{4}}\right)\frac{\epsilon^{{\dot{\alpha}}{\dot{\beta}}}\epsilon^{\dot{\gamma}\dot{\delta}}}{\left(\det
A\right)^{2}}$ (B.21)
$\displaystyle\qquad\bigg{[}\epsilon_{\dot{\gamma}\dot{\delta}}\left((A^{-1})_{12}-(A^{-1})_{22}\right)\left(A^{-1}\lambda\right)_{1{\dot{\alpha}}}\left(A^{-1}\lambda\right)_{2{\dot{\beta}}}$
$\displaystyle\qquad\,\,\,+\epsilon_{\dot{\gamma}{\dot{\beta}}}\left((A^{-1})_{12}-(A^{-1})_{22}\right)\left(A^{-1}\lambda\right)_{1{\dot{\alpha}}}\left(A^{-1}\lambda\right)_{2\dot{\delta}}$
$\displaystyle\qquad\,\,\,+\epsilon_{\dot{\gamma}{\dot{\alpha}}}\left((A^{-1})_{11}-(A^{-1})_{21}\right)\left(A^{-1}\lambda\right)_{2{\dot{\beta}}}\left(A^{-1}\lambda\right)_{2\dot{\delta}}\bigg{]}$
$\displaystyle=$ $\displaystyle\,-\int_{X}{\rm d}t_{01}\,{\rm d}t_{02}\,{\rm
d}t_{12}\frac{3t_{02}}{\left(t_{01}+t_{02}+t_{12}\right)^{4}}\left(\lambda_{1},\lambda_{1}\right)\,.$

Changing coordinates to $T_{1}=\frac{t_{02}}{t_{01}}$ and
$T_{2}=\frac{t_{12}}{t_{01}}$ and dividing by the volume
$\mathbf{V}(\mathbb{R}_{+})$ we can simply integrate the resulting expressions
with the final result,

$I_{F}=I_{F}^{(1)}+I_{F}^{(2)}=\left(-\frac{1}{2}-\frac{1}{2}\right)\left(\lambda_{1},\lambda_{2}\right)=-\left(\lambda_{1},\lambda_{2}\right)\,.$
(B.22)

Substituting this in the expression (3.4) we find the result presented in
(3.8) in the main text. Let us stretch the fact that this integral is
completely UV and IR finite and there are no divergences which have to be
regulated. For this reason the result presented is completely scheme
independent.

#### Supercurrent semi-chiral superfields

The computation of the ternary $\lambda$-bracket containing supercurrent semi-
chiral superfields is entirely analogous to the computation for conserved
currents. Technically, however, it is much more involved due to the higher
conformal dimension and the appearance of free indices. In particular, the
(non-supersymmetric) three-point function,
$\left\langle{\mathbf{S}^{(2)}\mathbf{S}^{(1)}\mathbf{S}^{(0)}}\right\rangle$
allows for twelve tensor structures,

$\left\langle{\mathbf{S}_{\dot{\alpha}}^{(2)}(x_{1})\mathbf{S}_{\dot{\beta}}^{(1)}(x_{2})\mathbf{S}_{\dot{\gamma}}^{(0)}(x_{3})}\right\rangle=\frac{1}{(|x_{12}|^{2})^{4}(|x_{13}|^{2})^{3}(|x_{23}|^{2})^{2}}\sum_{i=1}^{12}c_{i}I_{i}\,,$
(B.23)

where the tensor structures are given by

$\displaystyle I_{1}=$
$\displaystyle\left(T^{1}_{23}\right)^{2}\,U^{12}\,T^{2}_{13}\,U^{23}\,U^{32}\,,\qquad\qquad$
$\displaystyle I_{2}=$
$\displaystyle\,T^{1}_{23}\,\left(U^{13}\right)^{2}\,\left(T^{2}_{13}\right)^{2}\,U^{31}\,,$
(B.24) $\displaystyle I_{3}=$
$\displaystyle\,T^{1}_{23}\,\left(U^{13}\right)^{2}\,U^{21}\,T^{2}_{13}\,U^{32}\,,$
$\displaystyle I_{4}=$
$\displaystyle\,\left(T^{1}_{23}\right)^{2}\,U^{12}\,\left(U^{23}\right)^{2}\,U^{32}\,,$
$\displaystyle I_{5}=$
$\displaystyle\,T^{1}_{23}\,U^{12}\,U^{13}\,T^{2}_{13}\,U^{23}\,U^{31}\,,$
$\displaystyle I_{6}=$
$\displaystyle\,T^{1}_{23}\,U^{12}\,U^{13}\,U^{21}\,U^{23}\,U^{32}\,,$
$\displaystyle I_{7}=$
$\displaystyle\,U^{12}\,\left(U^{13}\right)^{2}\,U^{21}\,T^{2}_{13}\,U^{31}\,,$
$\displaystyle I_{8}=$
$\displaystyle\,U^{12}\,\left(U^{13}\right)^{2}\,\left(U^{21}\right)^{2}\,U^{32}\,,$
$\displaystyle I_{9}=$
$\displaystyle\,T^{1}_{23}\,\left(U^{12}\right)^{2}\,\left(U^{23}\right)^{2}\,U^{31}\,,$
$\displaystyle I_{10}=$
$\displaystyle\,T^{1}_{23}\,\left(U^{12}\right)^{2}\,U^{21}\,U^{23}\,T^{3}_{12}\,,$
$\displaystyle I_{11}=$
$\displaystyle\,\left(U^{12}\right)^{2}\,U^{13}\,U^{21}\,U^{23}\,U^{31}\,,$
$\displaystyle I_{12}=$
$\displaystyle\,\left(U^{12}\right)^{2}\,U^{13}\,\left(U^{21}\right)^{2}\,T^{3}_{12}\,.$

The basic building blocks $T^{i}_{jk}$ and $U^{ij}$ are defined in (3.6),
which we repeat here for convenience,

$U^{ij}=s_{j\alpha}x_{ij}^{\alpha{\dot{\alpha}}}\bar{s}_{i{\dot{\alpha}}}\,,\qquad\qquad
T^{i}_{jk}=\frac{|x_{ij}|^{2}|x_{ik}|^{2}}{|x_{jk}|^{2}}\left(\frac{s_{j\alpha}x_{ij}^{\alpha{\dot{\alpha}}}\bar{s}_{i{\dot{\alpha}}}}{|x_{ij}|^{2}}-\frac{s_{k\alpha}x_{ik}^{\alpha{\dot{\alpha}}}\bar{s}_{i{\dot{\alpha}}}}{|x_{ik}|^{2}}\right)\,,$
(B.25)

In the expressions above we used index-free notation where all indices are
contracted with commuting spinors $s_{i\alpha}$ and
$\bar{s}_{i{\dot{\alpha}}}$. Supersymmetry fixes all the constants in this
expression purely in terms of the conformal anomalies $a$ and $c$ [52]. In
particular, we find

$\displaystyle c_{1}=$
$\displaystyle\,c_{2}=2c_{3}=c_{4}=c_{4}=c_{10}=2c-a\,,$ (B.26) $\displaystyle
c_{6}=$ $\displaystyle\,c_{7}=2c_{8}=c_{9}=2c_{11}=c_{10}=2a\,.$

Expanding out the expressions of the tensor structures and contracting the
appropriate indices we note that they can all be written as linear
combinations of the following basic building blocks

$X_{(ij)(kl)(mn)(pq){\dot{\alpha}}{\dot{\beta}}\dot{\gamma}}=\frac{\left(\bar{z}_{12},\bar{z}_{13}\right)\left(\bar{z}_{ij}\right)_{\dot{\alpha}}\left(\bar{z}_{kl}\right)_{\dot{\beta}}\left(\bar{z}_{mn}\right)_{\dot{\gamma}}}{|x_{12}|^{2\Delta_{12}}|x_{13}|^{2(9-\Delta_{12}-\Delta_{23})}|x_{23}|^{2\Delta_{23}}}\left(\bar{z}_{pq},{\rm
d}\bar{z}_{2}\right){\rm d}^{2}\bar{z}_{1}$ (B.27)

In the integrand of the ternary $\lambda$-bracket we are interested in the
${\rm d}^{2}\bar{z}_{1}{\rm d}^{2}\bar{z}_{2}$ component of
$\bar{\partial}X_{(ij)(kl)(mn)(pq){\dot{\alpha}}{\dot{\beta}}\dot{\gamma}}$
and will always put $z_{3}=0$. The resulting expression for
$\bar{\partial}X_{(ij)(kl)(mn)(pq){\dot{\alpha}}{\dot{\beta}}\dot{\gamma}}$ is
given by

$\displaystyle\bar{\partial}$ $\displaystyle
X_{(ij)(kl)(mn)(pq){\dot{\alpha}}{\dot{\beta}}\dot{\gamma}}=\,\frac{1}{|x_{12}|^{2\Delta_{12}}|x_{13}|^{2(9-\Delta_{12}-\Delta_{23})}|x_{23}|^{2\Delta_{23}}}\Bigg{[}\left(\bar{z}_{1},\bar{z}_{pq}\right)\left(\bar{z}_{ij}\right)_{\dot{\alpha}}\left(\bar{z}_{kl}\right)_{\dot{\beta}}\left(\bar{z}_{pq}\right)_{\dot{\gamma}}$
(B.28)