$\displaystyle\qquad+\left(\bar{z}_{1},\bar{z}_{2}\right)\left(\delta_{i}^{2}-\delta_{j}^{2}\right)\left(\bar{z}_{pq}\right)_{\dot{\alpha}}\left(\bar{z}_{kl}\right)_{\dot{\beta}}\left(\bar{z}_{mn}\right)_{\dot{\gamma}}+\left(\bar{z}_{1},\bar{z}_{2}\right)\left(\delta_{k}^{2}-\delta_{l}^{2}\right)\left(\bar{z}_{ij}\right)_{\dot{\alpha}}\left(\bar{z}_{pq}\right)_{\dot{\beta}}\left(\bar{z}_{mn}\right)_{\dot{\gamma}}$
$\displaystyle\qquad+\left(\bar{z}_{1},\bar{z}_{2}\right)\left(\delta_{m}^{2}-\delta_{n}^{2}\right)\left(\bar{z}_{ij}\right)_{\dot{\alpha}}\left(\bar{z}_{kl}\right)_{\dot{\beta}}\left(\bar{z}_{pq}\right)_{\dot{\gamma}}+\left(\bar{z}_{1},\bar{z}_{2}\right)\left(\delta_{p}^{2}-\delta_{q}^{2}\right)\left(\bar{z}_{ij}\right)_{\dot{\alpha}}\left(\bar{z}_{kl}\right)_{\dot{\beta}}\left(\bar{z}_{mn}\right)_{\dot{\gamma}}$
$\displaystyle\qquad+\left(\bar{z}_{1},\bar{z}_{2}\right)\left(\bar{z}_{ij}\right)_{\dot{\alpha}}\left(\bar{z}_{kl}\right)_{\dot{\beta}}\left(\bar{z}_{mn}\right)_{\dot{\gamma}}\left(\Delta_{1}\frac{(\bar{z}_{pq},z_{12})}{|z_{12}|^{2}}-\Delta_{2}\frac{(\bar{z}_{pq},z_{2})}{|z_{2}|^{2}}\right)\Bigg{]}{\rm
d}^{2}\bar{z}_{1}{\rm d}^{2}\bar{z}_{2}\,.$

Following the same steps as above, we can again use the heat kernel
regularisation method introduced above to compute the integrated correlators.
The full expression is long and not particularly illuminating, so we refrain
from explicitly stating it here. Note however that in this case, the integral
over the heat kernel time is not always finite. Indeed, considering a generic
linear combination of basic building blocks will generically result in IR
divergences of the integral and to obtain a finite result one has to put a
cut-off $R\gg 1$ and carefully renormalise the result. Luckily we will not
have to concern ourselves with this subtlety since as it will turn out the
specific linear combination we care about produce finite integrals, both in
the UV and IR, similar to the integrals presented above for the
$\mathbf{J}\mathbf{J}\mathbf{J}$ $\lambda$-bracket. Computing the integral for
the specific linear combination introduced above results in the final
expression presented in equation (3.14).

### B.4 $\lambda$-brackets from Feynman integrals

For the readers’ convenience, we provide an explicit connection of the
$\lambda$-brackets computed in this work to the universal Feynman integrals of
[12]. The main object of interest we study in this article is the following
$(n+1)$-ary $\lambda$-bracket,

$\left\\{\mathbf{O}_{1}\,{}_{\lambda_{1}}\,\cdots\,\mathbf{O}_{n}\,{}_{\lambda_{n}}\,\mathbf{O}_{0}\right\\}\equiv\mathbf{Q}\left(\prod_{k=1}^{n}\int_{\mathbb{C}^{2}}\frac{d^{2}z_{k}}{(2\pi
i)^{2}}e^{\lambda_{k}\cdot
z_{k}}:\mathbf{O}_{1}(z_{1},\bar{z}_{1})\cdots\mathbf{O}_{n}(z_{n},\bar{z}_{n})\mathbf{O}_{0}(0,0):\right)$
(B.29)

which receives contributions from $(n-1)$-loop diagrams, as explained
carefully in section 5 of [12].

Recall that all the universal Feynman integrals
$\mathcal{I}_{\Gamma}[\lambda;\delta]$ can be readily “bootstrapped” using
quadratic identities. The relation of these integrals to the
$\lambda$-brackets is as follows:

  1. 1.

Suppose there are no derivatives in any $\mathbf{O}_{i}$. We formally shift
the holomorphic coordinate of $\mathbf{O}_{i}$ by $\delta_{i}$, i.e.
$\mathbf{O}_{i}(z_{i},\bar{z}_{i})\rightarrow\mathbf{O}_{i}(z_{i}+\delta_{i},\bar{z}_{i})$.

  2. 2.

Each Wick contraction gives a propagator $\omega_{{\rm BM},e}$, and a set of
contractions assemble a Feynman graph $\Gamma$ with the operators at the
vertices and the contractions as the edges. If there are uncontracted fields
left over, they will be collected at the origin. We denote those fields as
$\widehat{\mathbf{O}}_{i}(\delta_{i})$. The evaluation of a bracket is then
determined as the sum over all possible $(n+1)$-loop graphs.

  3. 3.

For each graph $\Gamma$, we then have the following schematic structure,

$(-1)^{\bullet}\left(\int\bar{\partial}\prod_{e\in\Gamma}\omega_{e}\right):\widehat{\mathbf{O}}_{1}(\delta_{1})\cdots\widehat{\mathbf{O}}_{n}(\delta_{n})\widehat{\mathbf{O}}_{0}(0):$
(B.30)

where $(-1)^{\bullet}$ denotes possible signs due to the Grassmannian nature
of the propagator and $\bar{\partial}$ arises from $\mathbf{Q}$ acting on the
semi-chiral superfield. This gives us a contribution

$(-1)^{\bullet}\mathcal{I}_{\Gamma}[\partial_{\delta_{i}}+\lambda_{i};\delta_{i}]:\widehat{\mathbf{O}}_{1}(\delta_{1})\cdots\widehat{\mathbf{O}}_{n}(\delta_{n})\widehat{\mathbf{O}}_{0}(0):|_{\delta_{i}=0},$
(B.31)

where $\mathcal{I}_{\Gamma}[\lambda_{i};\delta_{i}]$ are universal Feynman
integrals in [12] with $\lambda_{i}$ replaced by
$\partial_{\delta_{i}}+\lambda_{i}$, acting appropriately on the corresponding
operator $\widehat{\mathbf{O}}_{i}(\delta_{i})$.

  4. 4.

If there are derivatives in $\mathbf{O}_{i}$, they can be accounted for by
thinking of the fields $\mathbf{f}(\delta_{i})$ as a generating series

$\mathbf{f}(\delta_{i})=\sum_{n}\frac{\delta_{i}^{n}}{n!}\partial^{n}\mathbf{f}$
(B.32)

and expanding (B.31) in $\delta_{i}$ to the desired order before putting
$\delta_{i}$ to zero.

For more details on this prescription as well as details on how to compute the
universal Feynman integrals, we refer the reader to [12].

##  Appendix C Basic aspects of homological algebra

$L_{\infty}$ algebras are a generalisation of Lie algebras where the Jacobi
identities are only satisfied up to homotopy. See e.g. [60, 61] for an
introduction. Such algebras have wide-ranging applications in physics, ranging
from (closed) string field theory to QFTs and holography [62, 63, 64, 65].

Let $\mathfrak{g}$ be a $\mathbb{Z}$-graded vector space
$\mathfrak{g}=\bigoplus_{n\in\mathbb{Z}}\mathfrak{g}_{n}$. An $L_{\infty}$
structure on $\mathfrak{g}$ is a collection of multi-linear maps
$[\bullet,\dots,\bullet]_{k}:\mathfrak{g}^{\otimes k}\rightarrow\mathfrak{g}$
of degree $2-k$, $k\geq 1$, that are graded skew-symmetric

$\left[x_{\sigma(1)},\ldots,x_{\sigma(n)}\right]_{n}=(-1)^{\mathrm{sgn}\,\sigma}\epsilon(\sigma,x)\left[x_{1},\ldots,x_{n}\right]_{n}\,.$
(C.1)

where $\epsilon(\sigma,x)$ is the Koszul sign of the permutation $\sigma$. In
addition, these maps have to satisfy the following set of generalised Jacobi
identities,

$\sum_{k=1}^{n}(-1)^{k}\sum_{\sigma\in\mathop{\rm
Unsh}\nolimits(k,n-k)}(-1)^{\sigma}\epsilon(\sigma,x)\left[\left[x_{\sigma(1)},\ldots,x_{\sigma(k)}\right]_{k},x_{\sigma(k+1)},\ldots,x_{\sigma(n)}\right]_{n-k+1}=0\,,$
(C.2)

where $\mathrm{Unsh}(k,n-k)$ is a the subset of the permutation group $S_{n}$
called $(k,n-k)$-unshuffle defined such that

$\sigma(1)<\sigma(2)<\cdots<\sigma(k),\quad\sigma(k+1)<\cdots<\sigma(n)\,.$
(C.3)

For small values of $n$ such identities take the form

  * •

$n=1$: It is customary to denote the 1-bracket $\left[\bullet\right]_{1}$ by
${\rm d}$. The $n=1$ Jacobi identity is simply ${\rm d}^{2}=0$ so that ${\rm
d}$ is a differential on $\mathfrak{g}$.

  * •

$n=2$: In this case (C.2) reduces to

${\rm d}\left[x_{1},x_{2}\right]_{2}=\left[{\rm
d}x_{1},x_{2}\right]+(-1)^{\left|x_{1}\right|}\left[x_{1},{\rm d}x_{2}\right]$
(C.4)

which states that the differential ${\rm d}$ satisfies a graded Leibniz rule
with respect to the 2-bracket $[\bullet,\bullet]_{2}$. In the main text, we
omit the subscript $2$ and simply denote the 2-bracket as $[\bullet,\bullet]$.

  * •

$n=3$: In this case the generalised Jacobi identity (C.2) yields

$\displaystyle\left[\left[x_{1},x_{2}\right]_{2},x_{3}\right]_{2}+(-1)^{\left(\left|x_{1}\right|+\left|x_{2}\right|\right)\left|x_{3}\right|}\left[\left[x_{3},x_{1}\right]_{2},x_{2}\right]_{2}+(-1)^{\left(\left|x_{2}\right|+\left|x_{3}\right|\right)\left|x_{1}\right|}\left[\left[x_{2},x_{3}\right]_{2},x_{1}\right]_{2}$
(C.5) $\displaystyle\qquad={\rm
d}\left[x_{1},x_{2},x_{3}\right]_{3}+\left[{\rm
d}x_{1},x_{2},x_{3}\right]_{3}+(-1)^{\left|x_{1}\right|}\left[x_{1},{\rm
d}x_{2},x_{3}\right]_{3}+(-1)^{\left|x_{1}\right|+\left|x_{2}\right|}\left[x_{1},x_{2},{\rm
d}x_{3}\right]_{3}$

which states that the binary $L_{\infty}$-bracket satisfies the usual Jacobi
identity up to homotopy, where the deviation from the Jacobi identity for Lie
algebras is given by the ternary
$L_{\infty}$-bracket$[\bullet,\bullet,\bullet]_{3}$. In the main text, we omit
the subscript $3$ and simply denote the ternary $L_{\infty}$-bracket as
$[\bullet,\bullet,\bullet]$.

Heuristically, an $L_{\infty}$ algebra is a natural structure combining the
notion of a Lie algebra and that of a chain complex, such that Jacobi
identities are satisfied up to an infinite tower of relations. To give some
more intuition into this structure let us consider some familiar special
cases.

#### Ordinary Lie algebra

A Lie algebra $(\mathfrak{g}_{0},[-,-]_{2})$ is an $L_{\infty}$ algebra for
which $\mathfrak{g}$ is concentrated in degree zero, i.e.
$\mathfrak{g}=\mathfrak{g}_{0}$, and the only non-trivial $L_{\infty}$-bracket
is the binary bracket. In this case the Jacobi identities (C.2) reduce to the
familiar expression

$\left[\left[x_{1},x_{2}\right]_{2},x_{3}\right]_{2}+\left[\left[x_{3},x_{1}\right]_{2},x_{2}\right]_{2}+\left[\left[x_{2},x_{3}\right]_{2},x_{1}\right]_{2}=0\,,$
(C.6)

while all other generalised identities vanish identically.

#### dg Lie algebra

A differential graded Lie algebra (dgla) $(\mathfrak{g},{\rm
d},[\bullet,\bullet]_{2})$ is an $L_{\infty}$ algebra for which only the 1-
and 2-bracket are non-trivial.

#### Lie 2-algebra

A Lie 2-algebra is to a Lie 2=group as a Lie algebra is to a 2-group. A Lie
2-algebra is an $L_{\infty}$ algebra with generators concentrated in the
lowest two degrees, $\mathfrak{g}=\mathfrak{g}_{0}\oplus\mathfrak{g}_{1}$ with
the only non-trivial $L_{\infty}$ brackets being the 1-, 2- and 3-bracket. The
1-bracket encodes the differential ${\rm
d}:\mathfrak{g}_{1}\rightarrow\mathfrak{g}_{0}$, the binary bracket when
restricted to degree zero is defines a Lie bracket. The ternary bracket
$[\bullet,\bullet,\bullet]_{3}:\mathfrak{g}_{0}\otimes\mathfrak{g}_{0}\otimes\mathfrak{g}_{0}\rightarrow\mathfrak{g}_{1}$
is called the Jacobiator.

##  Appendix D (Higher) Kac Moody and Virasoro algebras

In this appendix we provide additional details on the construction and
commutation relations of the infinite-dimensional symmetry algebras
encountered in the main text. This appendix is largely based on the set of
lecture notes [66] to which we refer the reader for more details. In the main
text we extensively use the $\lambda$-bracket of the higher Kac-Moody and
Virasoro algebra. To complete the one-complex-dimensional story we here
include a discussion of the $\lambda$-bracket in this context.

### D.1 Kac Moody algebra

The one-dimensional affine Kac-Moody algebra associated with the Lie algebra
$\mathfrak{g}$ is defined as a central extension of the loop algebra
$L\mathfrak{g}\equiv\mathfrak{g}[z,z^{-1}]$,

$0\rightarrow\mathbb{C}K\rightarrow\hat{\mathfrak{g}}\rightarrow
L\mathfrak{g}\rightarrow 0\,.$ (D.1)

Such central extensions are classified by $H^{2}(L\mathfrak{g})$. As vector
spaces, the above sequence splits as $\hat{\mathfrak{g}}\simeq
L\mathfrak{g}\oplus\mathbb{C}K$. The commutation relations are given as
follows,

$\displaystyle\left[{A\otimes f(z)},{B\otimes g(z)}\right]$
$\displaystyle=\left[{A},{B}\right]\otimes
f(z)g(z)+K\varphi_{\kappa}\left({A\otimes f(z)},{B\otimes g(z)}\right)\,,$
(D.2) $\displaystyle\left[{K},{\ \bullet\ }\right]$ $\displaystyle=0\,.$

where $A,B\in\mathfrak{g}$ and $\left[{A},{B}\right]$ is the Lie bracket in
$\mathfrak{g}$. $\varphi_{\kappa}\in H^{2}(L\mathfrak{g})$ is a 2-cocycle that
takes the form

$\varphi_{\kappa}\left({A\otimes f(z)},{B\otimes
g(z)}\right)=-\kappa(A,B)\oint\frac{dz}{2\pi i}f(z)\partial g(z)\,,$ (D.3)

where $\kappa(\bullet,\bullet)$ is a quadratic invariant polynomial on
$\mathfrak{g}$.

The simplest example is the Heisenberg algebra $\mathbf{H}$ for which
$\mathfrak{g}=\mathfrak{u}(1)$. We define a basis for $\mathbf{H}$,

$\rho_{n}=z^{n},\quad n\in\mathbb{Z}\,,$ (D.4)

and the 2-cocycle is $\phi(z^{n},z^{m})=n\delta_{m+n}$. From (D.2) we can read
off the commutation relations,

$\left[{\rho_{n}},{\rho_{m}}\right]=n\,\delta_{m+n}\,\mathbf{1}\,.$ (D.5)

When $\mathfrak{g}$ is a simple Lie algebra, there is a unique quadratic
invariant polynomial given by the Killing form, which in our conventions is
given by $\mathrm{Tr}\,t^{a}t^{b}=\delta^{ab}$ where
$\\{{t}^{a}\\}_{a=1,\dots,\dim\,\mathfrak{g}}$ is a basis of $\mathfrak{g}$.
We then define the fields
$\mathbf{J}^{a}(z)=\sum_{n\in\mathbf{Z}}\frac{\mathbf{J}^{a}_{n}}{z^{n+1}}=\sum_{n\in\mathbf{Z}}\frac{t^{a}\otimes\mathbf{b}_{n}}{z^{n+1}}$,
whose modes satisfy the following commutation relations

$\left[{\mathbf{J}^{a}_{n}},{\mathbf{J}^{b}_{m}}\right]=\left[{t^{a}},{t^{b}}\right]\otimes\rho_{m+n}+n\,\delta^{ab}\delta_{m+n}\,K\,.$
(D.6)

As introduced in the introduction Lie conformal algebras present a useful way
to rephrase the Kac-Moody algebra which easily generalises to higher
dimensions. Formally, a Lie algebra is a vector space equipped with a bracket

$[\bullet,\bullet]:\mathfrak{g}\otimes\mathfrak{g}\rightarrow\mathfrak{g}\,,$
(D.7)

that is bilinear, skew-symmetric and satisfies the Jacobi identity. A Lie
conformal algebra in turn is a $\mathbb{C}[\partial]$-module $R$ endowed with
an $\mathbb{C}$-bilinear $\lambda$-bracket

$[\bullet\,\lambda\,\bullet]:R\otimes R\rightarrow\mathbb{C}[\lambda]\otimes
R\,,$ (D.8)

which is bilinear, skew-symmetric and satisfies the Jacobi identity in a
slightly generalised way. A very nice exposition can be found in [24]. We will
not review all the proper definition but use the examples of Kac Moody algebra
(and Virasoro algebra in the next section) to demonstrate that the
$\lambda$-bracket is a useful and convenient way to repackage the singular OPE
data.

A Kac Moody algebra defines a Lie conformal algebra
($\mathbb{C}[\partial]\otimes\mathfrak{g}+\mathbb{C}K$) as follows. Given two
fields $a=a(w)$ and $b=b(w)\in\mathfrak{g}[\\![w,w^{-1}]\\!]$ their OPE is
given as follows,

$a(z)b(w)=\sum_{n\in\mathbb{Z}}\frac{\left\\{a\,,b\right\\}_{n}(w)}{(z-w)^{n+1}}\,.$
(D.9)

where the products $\left\\{a\,,b\right\\}_{n}$ are sometimes denoted as
$a_{(n)}b$. The linear operators $a_{m}=\left\\{a\,,\bullet\right\\}_{m}$ are
referred to as modes of $a$. We define the $\lambda$-bracket as the generating
function

$\left\\{a\,{}_{\lambda}\,b\right\\}(w)\equiv\sum_{n\geq
0}\frac{\lambda^{n}}{n!}\left\\{a\,,b\right\\}_{n}(w)\in\mathfrak{g}[\\![w,w^{-1}]\\!][\\![\lambda]\\!]$
(D.10)

Going back to the Kac-Moody algebra, the familiar OPE between the Kac-Moody
currents reads

$\mathbf{J}^{a}(z)\mathbf{J}^{b}(0)\sim
k_{G}\,\frac{\delta^{ab}}{z^{2}}+if^{ab}{}_{c}\,\frac{\mathbf{J}^{c}(0)}{z}\,,$
(D.11)

which gives rise to the $\lambda$-bracket

$\left\\{\mathbf{J}^{a}\,{}_{\lambda}\,\mathbf{J}^{b}\right\\}=k_{G}\,\delta^{ab}\lambda+f^{ab}{}_{c}\,\mathbf{J}^{c}\,.$
(D.12)

### D.2 Higher Kac-Moody algebra

To construct the two-complex-dimensional higher Kac-Moody algebra we proceed
entirely analogous to the one-dimensional case above. The definition of a
higher Kac Moody algebra was given [21]. We will not review the precise
definitions but rather give an operational review of the construction of such
higher algebras.

The two-dimensional higher Kac-Moody algebra is defined as a $L_{\infty}$
central extension of $\mathfrak{g}\otimes A_{2}$,

$0\rightarrow\mathbb{C}K\rightarrow\hat{\mathfrak{g}}_{2,K}\rightarrow\mathfrak{g}\otimes
A_{2}\rightarrow 0\,,$ (D.13)

where
$A_{2}=\left(H^{2,\bullet}\left(\mathbb{C}^{2}\backslash\\{0\\}\right),\bar{\partial}\right)$
denotes the cochain complex modelling punctured affine space considered as an
abelian dg Lie algebra. Note that in one complex dimension this simply reduces
to the loop group $L\mathfrak{g}$. The commutation relation are given as
follows,

$\displaystyle\left[{A\otimes f},{B\otimes g}\right]_{2}$
$\displaystyle=\left[{A},{B}\right]\otimes fg\,,$ (D.14)
$\displaystyle\left[{K},{\bullet}\right]_{2}$ $\displaystyle=0\,,$
$\displaystyle\left[A\otimes f,B\otimes g,C\otimes h\right]_{3}$
$\displaystyle=K\varphi_{\kappa}\left(A\otimes f,B\otimes g,C\otimes
h\right)\,.$

where $A,B,C\in\mathfrak{g}$ and $f,g,h\in A_{2}$. The central extension
$\varphi_{\kappa}\in H^{2}(\mathfrak{g}\otimes A_{2})$ is defined by a
2-cocycle,

$\varphi_{\kappa}\left(A\otimes f,B\otimes g,C\otimes
h\right)=\kappa(A,B,C)\mathop{\rm Res}\nolimits_{z=0}\left(f\partial
g\cdots\partial h\right)\,,$ (D.15)

which is a tri-linear map
$\hat{\mathfrak{g}}_{2,K}\times\hat{\mathfrak{g}}_{2,K}\times\hat{\mathfrak{g}}_{2,K}\rightarrow\mathbb{C}$.
Therefore, in contrast to the one-dimensional case, centrally extended higher
Kac-Moody algebra is not a Lie algebra. Instead, we need to generalise the
notion of Lie algebra by including a ternary map resulting in an $L_{\infty}$
algebra, as described in Appendix C above.191919In fact this is known on
general ground [10, 21]. In the spirit of radial quantisation, mode expansions
are best understood as ‘sphere algebra’ $\mathrm{Map}(S^{3},\mathfrak{g})$ and
such centrally extended sphere algebras are naturally modelled by $L_{\infty}$
algebras.

The simplest example is the higher Heisenberg algebra where we take
$\mathfrak{g}=\mathfrak{u}(1)$. In this case we can choose a basis of modes
where the positive modes are a direct generalisation of (D.4),

$\rho_{mn}=-z_{1}^{m}z_{2}^{n}\,,\qquad\qquad m,n\geq 0\,,$ (D.16)

while the negative modes are defined as

$\rho_{mn}=-\hat{\partial}_{\dot{+}}^{-m-1}\hat{\partial}_{\dot{-}}^{-n-1}\omega_{\rm
BM}\,,\qquad\qquad m,n\leq-1\,,$ (D.17)

where
$\hat{\partial}_{\dot{\alpha}}^{-n-1}=\frac{(-1)^{-n-1}}{(-n-1)!}\partial_{\dot{\alpha}}^{-n-1}$.
The remaining mixed modes are vanishing by definition. Note that this
precisely corresponds to the degree zero and degree one part of the cohomology
$H^{2,\bullet}(\mathbb{C}\backslash\\{0\\})$ respectively. In this simple
example all the generators commute with each other so we immediately find

$\left[{\rho_{mn}},{\rho_{kl}}\right]_{2}=0\,,$ (D.18)

indicating that the higher Heisenberg algebra is an abelian infinite
dimensional algebra. The central extension does not appear in the binary
bracket, but plugging the basis elements in (D.15) we find the following non-
trivial three-bracket

$\left[{\rho_{mn}},{\rho_{kl}},{\rho_{rs}}\right]_{3}=(ks-
lr)\delta_{m+k+r}\delta_{n+l+s}\,.$ (D.19)

For higher Heisenberg algebra, the 3-bracket is the only non-trivial bracket.

The higher Kac-Moody algebra is very similar to the higher Heisenberg algebra
and can be obtained simply by taking the tensor product of the basis elements
with a Lie algebra element. Indeed, we define the modes of the Kac-Moody
algebra as

$\mathbf{J}^{a}_{mn}=t^{a}\otimes\rho_{mn}\,.$ (D.20)

As introduced in Section 2, the two-bracket is non-trivial and is determined
in terms of the anti-symmetric structure constants of the Lie algebra
$\mathfrak{g}$,

$\left[{\mathbf{J}^{a}_{m,n}},{\mathbf{J}^{b}_{k,l}}\right]_{2}=f^{ab}{}_{c}\mathbf{J}^{c}_{m+k,n+l}\,,$
(D.21)

when both $m,n$ and $k,l$ are non-negative or when $-k-1\geq m\geq 0$ and is
vanishing otherwise. It is straightforward to derive this commutator when both
modes are non-negative. The brackets involving both positive and negative
modes need some extra work, since the product of positive modes and negative
modes

$z_{\dot{+}}^{m_{1}}z_{\dot{-}}^{n_{2}}\partial_{\dot{+}}^{m_{2}}\partial_{\dot{-}}^{n_{2}}\omega$
(D.22)

in general, is not a linear combination of
$z_{\dot{+}}^{k_{1}}z_{\dot{-}}^{k_{2}}$ and
$\partial_{\dot{+}}^{l_{1}}\partial_{\dot{-}}^{l_{2}}\omega$ unless one work
in the cohomology
$H^{0,\bullet}_{\bar{\partial}}(\mathbb{C}^{2}\backslash{0})$. More precisely,
we can show that there is a nice module structure of positive modes acting on
the negative modes in $\bar{\partial}$-cohomology

$\displaystyle\left[z_{1}^{m}z_{2}^{n}\times\hat{\partial}_{1}^{m^{\prime}}\hat{\partial}_{2}^{n^{\prime}}\omega_{\rm
BM}\right]$
$\displaystyle=\left[\hat{\partial}_{1}^{m^{\prime}-m}\hat{\partial}_{2}^{n^{\prime}-n}\omega_{\rm
BM}\right]\,,\qquad$ $\displaystyle m^{\prime}\geq m\geq 0\,,\quad
n^{\prime}\geq n\geq 0$ (D.23)
$\displaystyle\left[z_{1}^{m}z_{2}^{n}\times\hat{\partial}^{m^{\prime}}_{1}\hat{\partial}^{n^{\prime}}_{2}\omega_{\rm
BM}\right]$ $\displaystyle=\left[0\right],\qquad$ otherwise.

A proof of this relation can be found in Appendix D.5. Using this we can
straightforwardly derive the commutation relations (D.21). This restriction on
the modes in the commutator is very different from the one-complex-dimensional
case and is a first indication that there might be some higher structure
present. Similar to the 2-Heisenberg case, the central extension appears in a
ternary bracket. The central extensions are classified by
$H^{2}(\mathfrak{g}\otimes A_{2})$ which for simple Lie algebras
$\mathfrak{g}$ is one-dimensional. Plugging the modes (D.20) in (D.15) we find
the following ternary bracket,

$\left[\mathbf{J}^{a}_{m,n},\mathbf{J}^{a}_{k,l},\mathbf{J}^{a}_{r,s}\right]_{3}=K\,d^{abc}\,(ks-
lr)\delta_{m+k+r}\delta_{n+l+s}\,,$ (D.24)

where $d^{abc}$ is the symmetric invariant of the Lie algebra $\mathfrak{g}$.
For a choice of $K$ we denote the two-dimensional higher Kac-Moody algebra as
$\hat{\mathfrak{g}}_{2,K}$.

### D.3 Virasoro algebra

One can think of the Virasoro algebra, $\mathop{\rm
Vir}\nolimits=\mathbb{C}[z,z^{-1}]\otimes\partial_{z}+\mathbb{C}\,c$, as a
central extension of the Witt algebra, $\mathop{\rm
Witt}\nolimits=\mathbb{C}[z,z^{-1}]\partial_{z}$, the algebra of vector fields
on a punctured disk, with the Lie bracket given by the Lie bracket of vector
fields,

$\left[{f(z)\partial_{z}},{g(z)\partial_{z}}\right]=\left(f(z)\partial_{z}g(z)-g(z)\partial_{z}f(z)\right)\partial_{z}\,.$
(D.25)

Choosing the basis
$\left\\{\mathbf{L}_{m}=-z^{m+1}\partial_{m}\right\\}_{m\in\mathbf{Z}}$, we
find the commutation relations

$\left[\mathbf{L}_{n},\mathbf{L}_{m}\right]=(n-m)\mathbf{L}_{n+m}\,.$ (D.26)

The central extension $\varphi\in H^{2}(\mathop{\rm Witt}\nolimits)$ is
defined by a 2-cocycle and can be written as the residue,

$\varphi(\mathbf{L}_{n},\mathbf{L}_{m})=\frac{1}{12}\oint\frac{{\rm
d}z}{2\pi{\rm i}}\partial_{z}f(z)\partial^{2}_{z}g(z)\,.$ (D.27)

so that

$\varphi(\mathbf{L}_{n},\mathbf{L}_{m})=\frac{1}{12}n(n^{2}-1)\delta_{n+m,0}\,.$
(D.28)

Altogether, we recover the familiar form for the commutation relation of the
Virasoro algebra

$\left[\mathbf{L}_{n},\mathbf{L}_{m}\right]=(n-m)\mathbf{L}_{n+m}+\frac{c}{12}n\left(n^{2}-1\right)\delta_{n+m,0}\,.$
(D.29)

In some scenarios it can be useful to regard the Virasoro algebra as a Lie
conformal algebra. In physical application the Virasoro algebra is obtained as
the mode algebra of the holomorphic stress tensor $T(z)$. Note that in physics
literature the numbering of modes is usually shifted from the maths notation
so that it is customary to define
$\mathbf{L}_{n}=\left\\{T\,,\bullet\right\\}_{n+1}$. We can then define the
$\lambda$-bracket of the stress tensor with a primary operator as follows,

$\displaystyle\left\\{{T(z)\,}_{\lambda}\mathcal{O}(0)\right\\}=$
$\displaystyle\oint_{S^{1}}\frac{{\rm d}z}{2\pi{\rm i}}\,\mathrm{e}^{\lambda
z}\,T(z)\mathcal{O}(0)$ (D.30) $\displaystyle=$ $\displaystyle
L_{-1}\,\mathcal{O}(0)+\lambda\,L_{0}\,\mathcal{O}(0)+L_{1}\,\mathcal{O}(0)+\cdots$
$\displaystyle=$
$\displaystyle\partial\mathcal{O}(0)+h\,\lambda\,\mathcal{O}(0)\,,$

where $h$ is the conformal weight of the primary operator $\mathcal{O}$. From
this expression, we can straightforwardly read off the singular part of the
OPE

$T(z)\mathcal{O}(0)\sim\frac{h\,\mathcal{O}(0)}{z}+\partial\mathcal{O}(0)\,.$
(D.31)

Similarly, the we compute the $\lambda$-bracket of two stress tensors as

$\left\\{T\,{}_{\lambda}\,T\right\\}=\Big{(}{\lambda^{3}\frac{c}{12}}+2\lambda+\partial\Big{)}T\,,$
(D.32)

which again encodes the singular part of the OPE

$T(z)T(0)\sim\frac{c/2}{z^{4}}+\frac{2T}{z^{2}}+\frac{\partial T}{z}\,.$
(D.33)

### D.4 Higher Virasoro algebra

Similar to the above, we now generalise this to the higher Virasoro algebra in
two complex dimensions. The higher Virasoro algebra can be obtained as a
central extension of the higher $\mathop{\rm Witt}\nolimits_{2}$ algebra which
is defined as the cochain complex
$\left(A_{2}\otimes\mathbb{C}\\{\partial_{\dot{\alpha}}\\},\bar{\partial}\right)$.
The commutation relations are defined through the Lie bracket of the vector
fields

$\left[A\otimes\partial_{{\dot{\alpha}}},B\otimes\partial_{{\dot{\beta}}}\right]=A\,\mathcal{L}_{\partial_{{\dot{\alpha}}}}(B)\otimes\partial_{{\dot{\beta}}}-(-1)^{|A||B|}B\,\mathcal{L}_{\partial_{{\dot{\beta}}}}(A)\otimes\partial_{{\dot{\alpha}}}\,,$
(D.34)

where $\mathcal{L}_{\partial_{z}}$ is the Lie derivative. Analogous to the
higher Kac-Moody algebra, we can define a basis,

$\mathbf{L}_{m,n}^{\dot{\alpha}}=\rho_{m,n}\otimes\partial_{\dot{\alpha}}\,.$
(D.35)

From this expression we can straightforwardly read of the commutation
relations for the non-negative modes while the commutation relations between
negative modes vanish. The mixed commutation relations for positive and
negative modes need some more work but can be derived using the identity in
cohomology (D.23). In the end, the resulting non-vanishing commutators are
given by,

$\displaystyle\left[{L_{m,n}^{1}},{L_{m^{\prime},n^{\prime}}^{1}}\right]$
$\displaystyle=(m-m^{\prime})\,L^{1}_{m+m^{\prime}-1,n+n^{\prime}}\,,$ (D.36)
$\displaystyle\left[{L_{m,n}^{2}},{L_{m^{\prime},n^{\prime}}^{2}}\right]$
$\displaystyle=(n-n^{\prime})\,L^{2}_{m+m^{\prime},n+n^{\prime}-1}\,,$
$\displaystyle\left[{L_{m,n}^{1}},{L_{m^{\prime},n^{\prime}}^{2}}\right]$
$\displaystyle=n\,L^{1}_{m+m^{\prime},n+n^{\prime}-1}-m^{\prime}\,L^{2}_{m+m^{\prime}-1,n+n^{\prime}}\,,$
$\displaystyle\left[{L_{m,n}^{2}},{L_{m^{\prime},n^{\prime}}^{1}}\right]$
$\displaystyle=m\,L^{2}_{m+m^{\prime}-1,n+n^{\prime}}-n^{\prime}\,L^{1}_{m+m^{\prime},n+n^{\prime}-1}\,.$

The central extensions of the higher Witt algebra are classified by Gelfand-
Fuks cohomology classes, $\varphi\in H^{2}(\mathrm{Witt}_{2})$, which is known
to be two-dimensional [42, 11]. Given an element
$\xi=\sum\alpha^{\dot{\alpha}}\partial_{\dot{\alpha}}$ in the higher Witt
algebra, its Jacobian is defined as

$[\mathop{\rm
Jac}\nolimits(\xi)]^{\dot{\beta}}_{\dot{\alpha}}=\partial_{\dot{\alpha}}\alpha^{\dot{\beta}}$
(D.37)

The following expressions then define representatives for the two non-trivial
2-cocycles,

$\displaystyle\psi_{1}\left(\xi_{0},\xi_{1},\xi_{2}\right)$
$\displaystyle=\,\mathop{\rm Res}\nolimits_{z=0}\mathrm{Tr}\,\left(\mathop{\rm
Jac}\nolimits\left(\xi_{0}\right)\partial\mathop{\rm
Jac}\nolimits\left(\xi_{1}\right)\partial\mathop{\rm
Jac}\nolimits\left(\xi_{2}\right)\right)\,,$ (D.38)
$\displaystyle\psi_{2}\left(\xi_{0},\xi_{1},\xi_{2}\right)$
$\displaystyle=\,\mathop{\rm Res}\nolimits_{z=0}\mathrm{Tr}\,\left(\mathop{\rm
Jac}\nolimits\left(\xi_{0}\right)\right)\mathop{\rm
Tr}\nolimits\left(\partial\mathop{\rm
Jac}\nolimits\left(\xi_{1}\right)\partial\mathop{\rm
Jac}\nolimits\left(\xi_{2}\right)\right)\,.$

Assuming $m,n<0$ and $k,l,r,s\geq 0$, we then find the following ternary
$L_{\infty}$-bracket,

$\left[\mathbf{L}_{m,n}^{\dot{\alpha}},\mathbf{L}_{k,l}^{\dot{\beta}},\mathbf{L}_{r,s}^{\dot{\gamma}}\right]=\tilde{A}\,\psi_{1}+\tilde{C}\,\psi_{2},$
(D.39)

with

$\displaystyle\psi_{1}=$ $\displaystyle
K_{1}\delta_{m+k+r,3}\delta_{n+l+s,0}\delta^{\dot{\alpha}}_{\dot{+}}\delta^{\dot{\beta}}_{\dot{+}}\delta^{\dot{\gamma}}_{\dot{+}}+K_{2}\delta_{m+k+r,0}\delta_{n+l+s,3}\delta^{\dot{\alpha}}_{\dot{-}}\delta^{\dot{\beta}}_{\dot{-}}\delta^{\dot{\gamma}}_{\dot{-}}$
$\displaystyle+\Bigg{[}K_{3}\delta^{\dot{\alpha}}_{\dot{+}}\delta^{\dot{\beta}}_{\dot{-}}\delta^{\dot{\gamma}}_{\dot{-}}+K_{2}\delta^{\dot{\alpha}}_{\dot{-}}\delta^{\dot{\beta}}_{\dot{-}}\delta^{\dot{\gamma}}_{\dot{+}}+K_{4}\delta^{\dot{\alpha}}_{\dot{-}}\delta^{\dot{\beta}}_{\dot{+}}\delta^{\dot{\gamma}}_{\dot{-}}\Bigg{]}\delta_{m+k+r,1}\delta_{n+l+s,2}$
(D.40)
$\displaystyle+\Bigg{[}K_{3}\delta^{\dot{\alpha}}_{\dot{+}}\delta^{\dot{\beta}}_{\dot{-}}\delta^{\dot{\gamma}}_{\dot{+}}+K_{1}\delta^{\dot{\alpha}}_{\dot{+}}\delta^{\dot{\beta}}_{\dot{+}}\delta^{\dot{\gamma}}_{\dot{-}}+K_{4}\delta^{\dot{\alpha}}_{\dot{-}}\delta^{\dot{\beta}}_{\dot{+}}\delta^{\dot{\gamma}}_{\dot{+}}\Bigg{]}\delta_{m+k+r,2}\delta_{n+l+s,1}\,,$

and

$\displaystyle\psi_{2}=$ $\displaystyle
K_{1}\delta_{m+k+r,3}\delta_{n+l+s,0}\delta^{\dot{\alpha}}_{\dot{+}}\delta^{\dot{\beta}}_{\dot{+}}\delta^{\dot{\gamma}}_{\dot{+}}+K_{2}\delta_{m+k+r,0}\delta_{n+l+s,3}\delta^{\dot{\alpha}}_{\dot{-}}\delta^{\dot{\beta}}_{\dot{-}}\delta^{\dot{\gamma}}_{\dot{-}}$
$\displaystyle+\Bigg{[}K_{3}\delta^{\dot{\alpha}}_{\dot{-}}\delta^{\dot{\beta}}_{\dot{-}}\delta^{\dot{\gamma}}_{\dot{+}}+K_{2}\delta^{\dot{\alpha}}_{\dot{+}}\delta^{\dot{\beta}}_{\dot{-}}\delta^{\dot{\gamma}}_{\dot{-}}+K_{4}\delta^{\dot{\alpha}}_{\dot{-}}\delta^{\dot{\beta}}_{\dot{+}}\delta^{\dot{\gamma}}_{\dot{-}}\Bigg{]}\delta_{m+k+r,1}\delta_{n+l+s,2}$
(D.41)
$\displaystyle+\Bigg{[}K_{3}\delta^{\dot{\alpha}}_{\dot{+}}\delta^{\dot{\beta}}_{\dot{-}}\delta^{\dot{\gamma}}_{\dot{+}}+K_{1}\delta^{\dot{\alpha}}_{\dot{-}}\delta^{\dot{\beta}}_{\dot{+}}\delta^{\dot{\gamma}}_{\dot{+}}+K_{4}\delta^{\dot{\alpha}}_{\dot{+}}\delta^{\dot{\beta}}_{\dot{+}}\delta^{\dot{\gamma}}_{\dot{-}}\Bigg{]}\delta_{m+k+r,2}\delta_{n+l+s,1}\,,$

where the coefficients $K_{i}$ are

$\displaystyle K_{1}$ $\displaystyle=kr(s-l+lr-ks),$ $\displaystyle\quad
K_{2}$ $\displaystyle=ls(k-r+lr-ks)\,,$ (D.42) $\displaystyle K_{3}$
$\displaystyle=ks(-1+k+s+lr-ks),$ $\displaystyle\quad K_{4}$
$\displaystyle=lr(1-l-r+lr-ks)\,.$

In particular, the difference between the two terms is simple

$\displaystyle\psi_{1}-\psi_{2}=$
$\displaystyle\,(K_{1}-K_{4})\delta_{m+k+r,2}\delta_{n+l+s,1}\left(\delta^{\dot{\alpha}}_{\dot{+}}\delta^{\dot{\beta}}_{\dot{+}}\delta^{\dot{\gamma}}_{\dot{-}}-\delta^{\dot{\alpha}}_{\dot{-}}\delta^{\dot{\beta}}_{\dot{+}}\delta^{\dot{\gamma}}_{\dot{+}}\right)$
(D.43)
$\displaystyle+(K_{2}-K_{3})\delta_{m+k+r,1}\delta_{n+l+s,2}\left(\delta^{\dot{\alpha}}_{\dot{-}}\delta^{\dot{\beta}}_{\dot{-}}\delta^{\dot{\gamma}}_{\dot{+}}-\delta^{\dot{\alpha}}_{\dot{+}}\delta^{\dot{\beta}}_{\dot{-}}\delta^{\dot{\gamma}}_{\dot{-}}\right)$

The coefficient $\tilde{A}$ and $\tilde{C}$ are expected to be a linear
combination of the conformal anomalies $a$ and $c$. We plan to come back to
this and their relation to the $\lambda$-brackets computed in the main text in
the future.

### D.5 Proof of equation (D.23)

In this subsection we provide a simple proof of (D.23). First of all, observe
that

$\hat{\partial}_{\dot{\alpha}}^{m}\omega_{\rm
BM}=(m+1)\bar{\mathbf{Z}}_{\dot{\alpha}}^{m}\omega_{\rm BM}\,,$ (D.44)

where we defined

$\bar{\mathbf{Z}}_{{\dot{\alpha}}}=\frac{\bar{z}_{{\dot{\alpha}}}}{|z|^{2}}\,,$
(D.45)

which satisfies the constraint
$z^{\dot{\alpha}}\bar{\mathbf{Z}}_{\dot{\alpha}}=1$. The identity (D.23) only
holds in cohomology and thus it is important to understand which terms are
$\bar{\partial}$-exact. For example, noting that

$\bar{\partial}(\bar{\mathbf{Z}}_{{\dot{\alpha}}})=-(2\pi{\rm
i})^{2}\epsilon_{\dot{\alpha}\dot{\beta}}z^{\dot{\beta}}\omega_{\rm BM}\,,$
(D.46)

we immediately see that expressions of the form

$\left[(z^{\dot{+}})^{n}(z^{\dot{-}})^{m}\omega_{\rm BM}\right]=[0],\qquad
n+m>0\,,$ (D.47)

are $\bar{\partial}$-exact and thus vanish in the cohomology. Using the above
we can derive a second useful cohomological identity,

$\epsilon_{{\dot{\alpha}}\dot{\gamma}}\left[z^{\dot{\gamma}}\,\bar{\mathbf{Z}}_{\dot{\beta}}\,\omega_{\rm
BM}\right]=-\frac{1}{(2\pi{\rm
i})^{2}}\left[\bar{\mathbf{Z}}_{\dot{\beta}}\,\bar{\partial}\bar{\mathbf{Z}}_{\dot{\alpha}}\right]=\frac{1}{(2\pi{\rm
i})^{2}}\left[\bar{\mathbf{Z}}_{\dot{\alpha}}\,\bar{\partial}\bar{\mathbf{Z}}_{\dot{\beta}}\,\right]=-\epsilon_{{\dot{\beta}}\dot{\gamma}}\left[z^{\dot{\gamma}}\,\bar{\mathbf{Z}}_{\dot{\alpha}}\,\omega_{\rm
BM}\right]\,.$ (D.48)

From these identities we find that

$\left[\omega_{\rm
BM}\right]=2\left[z^{\dot{+}}\bar{\mathbf{Z}}_{\dot{+}}\omega_{\rm
BM}\right]=2\left[z^{\dot{-}}\bar{\mathbf{Z}}_{\dot{-}}\omega_{\rm
BM}\right]\,.$ (D.49)

More generally we have202020Note that
$(z^{\dot{\alpha}})^{n}[\alpha]=[(z^{\dot{\alpha}})^{n}\alpha]$ but we cannot
multiply $\bar{\mathbf{Z}}_{i}$, i.e.
$\bar{\mathbf{Z}}_{\dot{\alpha}}[\alpha]\neq[\bar{\mathbf{Z}}_{\dot{\alpha}}\alpha]$

$\left[\omega_{\rm
BM}\right]=(m+1)\left[(z^{\dot{+}}\bar{\mathbf{Z}}_{\dot{+}})^{m}\omega_{\rm
BM}\right]=(m+1)\left[(z^{\dot{-}}\bar{\mathbf{Z}}_{\dot{-}})^{m}\omega_{\rm
BM}\right]\,.$ (D.50)

Using this we see that

$\left[(z^{\dot{\pm}})^{m}\hat{\partial}_{\dot{\pm}}^{m}\omega_{\rm
BM}\right]=\left[\omega_{\rm BM}\right]\,.$ (D.51)

Similarly, using the identity

$\bar{\partial}\left(\bar{\mathbf{Z}}_{\dot{\alpha}}\bar{\mathbf{Z}}_{\dot{\beta}}\right)=-(2\pi{\rm
i})^{2}\epsilon_{({\dot{\beta}}|\dot{\gamma}|}\hat{\partial}_{{\dot{\alpha}})}\omega_{\rm
BM}\,,$ (D.52)

where the symmetrisation is only over ${\dot{\alpha}}$ and ${\dot{\beta}}$, we
find that

$\left[z^{\dot{-}}\hat{\partial}_{\dot{+}}\omega_{\rm
BM}\right]=\left[z^{\dot{+}}\hat{\partial}_{\dot{-}}\omega_{\rm
BM}\right]=[0]\,.$ (D.53)

Repeating the above steps in the case of multiple derivatives and/or higher
powers of $z_{\dot{\alpha}}$ straightforwardly results in (D.23).

### D.6 Associativity relations of $\lambda$-brackets

Finally, we finish this appendix by listing the various associativity
relations the Higher $\lambda$-brackets satisfy. These relations were
instrumental in constraining the allowed expressions for the
$\lambda$-brackets in Section 2. The whole set of associativity relations for
general $n$-ary $\lambda$-brackets remains an open question. However, for the
purposes of this work we only need a subset of relations derived in [13, 41].
The $n$-ary $\lambda$-brackets are graded symmetric under permutations of
their arguments and with the $\lambda$ parameters. The last entry of the
$\lambda$-brackets behaves in a slightly different manner from the others but
if we define $\lambda_{n}\equiv-\sum_{i=1}^{n-1}\lambda_{i}-\partial$, with
the latter symbol acting outside the bracket they are graded symmetric under
permutation of all arguments.

The binary $\lambda$-bracket $\\{\bullet\,_{\lambda}\bullet\\}$ is best under
control. In this case the bracket has to satisfy the following set of axioms:

  1. 1.

$\\{\partial A\,{}_{\lambda}\,B\\}=-\lambda\\{A\,{}_{\lambda}\,B\\}$ ,

  2. 2.

$\partial\\{A\,{}_{\lambda}\,B\\}=-\lambda\\{A\,{}_{\lambda}\,B\\}+\\{A\,{}_{\lambda}\,\partial
B\\}$ , which implies $\\{A\,{}_{\lambda}\,\partial
B\\}=(\partial+\lambda)\\{A\,{}_{\lambda}\,B\\}$ ,

  3. 3.

$\\{A\,{}_{\lambda}\,B\\}=(-1)^{|A||B|}\\{B\,{}_{-\lambda-\partial}\,A\\}$ ,

  4. 4.

$\\{A\,{}_{\lambda_{1}}\\{B\,{}_{\lambda_{2}}\,C\\}\\}-(-1)^{(|A|+1)(|B|+1)}\\{B\,{}_{\lambda_{2}}\\{A\,{}_{\lambda_{1}}\,C\\}\\}+(-1)^{|A|}\\{\\{A\,{}_{\lambda_{1}}\,B\\}\,{}_{\lambda_{1}+\lambda_{2}}C\\}=0$
,

where in the equations above $|A|$ and $|B|$ denote the fermion parity of the
operators.

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