$\mathfrak{z}(\mathfrak{g})$ Frenkel:2007 . Indeed, as $\mathbf{I}_{G}$ is
defined as the two-sided Drinfel’d—Sokolov reduction of
$\mathcal{D}^{ch}_{G,\kappa_{c}}$, it has a subalgebra isomorphic to
$\mathfrak{z}(\mathfrak{g})$ (cf. Arakawa:2018egx ; Beem:2022mde  for
discussion of Feigin–Frenkel centres in chiral algebras of class $\SS$), so
there is a map $\mathfrak{z}(\mathfrak{g})\rightarrow\mathbf{I}_{G}$, the
image of which should appear in our free field realisation as a Feigin–Frenkel
center subalgebra in $\Pi_{G}$.

To construct this Feigin–Frenkel subalgebra, we specify an Ansatz by asserting
a chiral version of the map $\beta$. This Ansatz will be a quantum corrected
version of the expressions for the generators of $\mathbb{C}[S_{f}]$, which
arise from the ring map
$\beta^{\\#}:\mathbb{C}[S_{f}]\rightarrow\mathbb{C}[\mathrm{KT}_{G}]$.

We make the following observation: the morphism $\beta$ comes from the
restriction of $\mathbb{C}[\mathfrak{g}^{*}]^{G}$ to
$\mathrm{KT}_{\mathfrak{g}}$, so in particular the functions on
$\mathrm{im}\,\beta$ extend smoothly to the closure
$\overline{\mathrm{KT}}_{\mathfrak{g}}=\chi+\mathfrak{t}^{*}+\bigoplus_{\alpha\in\Delta}\mathfrak{g}_{\alpha}$.
Therefore, the expressions for the generators will only feature positively
graded generators of $\mathbb{C}[\mathrm{KT}_{\mathfrak{g}}]$. Furthermore,
the classical map $\beta$ does not require passage to $\mathrm{KT}_{G}$, so
the expressions for $\beta^{\\#}(P_{i})$ only feature positive integer powers
of the $\gamma_{i}$ and $b_{i}$.

We can construct a chiral analogue of $\beta$ in the following manner. Let
$V^{0}(\mathfrak{b})$ be the universal affine vertex algebra associated to the
Borel subalgebra $\mathfrak{b}$ at level zero. This algebra contains a
descending chain of vertex ideals, which are chiral analogues of the lower
central series of $\mathfrak{n}$, given by

$V^{0}(\mathfrak{b})\supset V(\mathfrak{n})\supset
V([\mathfrak{n},\mathfrak{n}])\supset
V([\mathfrak{n},[\mathfrak{n},\mathfrak{n}]])\supset\dots\supset 0~{},$ (136)

where the chain terminates in finitely many steps since $\mathfrak{n}$ is
solvable. The quotient $V^{0}(\mathfrak{b})/V([\mathfrak{n},\mathfrak{n}])$ is
a chiral quantisation of
$\overline{\mathrm{KT}}_{\mathfrak{g}}\cong(\mathfrak{b}/[\mathfrak{n},\mathfrak{n}])^{*}$,
and there is a corresponding inclusion
$V^{0}(\mathfrak{b})/V([\mathfrak{n},\mathfrak{n}])\hookrightarrow\Pi_{\mathfrak{g}}$,
where we can view $\Pi_{\mathfrak{g}}$ as a localisation of
$V^{0}(\mathfrak{b})/V([\mathfrak{n},\mathfrak{n}])$.

Starting with $V^{\kappa}(\mathfrak{g})$, the universal affine vertex algebra
of $\mathfrak{g}$ at level $\kappa$, principal Drinfel’d–Sokolov reduction
returns the (universal) principal ${\cal W}$-algebra ${\cal
W}_{prin}(\mathfrak{g},\kappa)$. This ${\cal W}$-algebra can be realised as a
subalgebra of the semi-infinite BRST complex that computes the reduction, with
explicit expressions for the generators of this embedded subalgebra computed
by the tic-tac-toe algorithm of deBoer:1993iz . In particular, the generators
of ${\cal W}_{prin}(\mathfrak{g},\kappa)$ can be expressed entirely in terms
of the so-called _hatted current subalgebra_ , which itself is isomorphic to
$V^{\kappa-\kappa_{c}}(\mathfrak{b})$—note the shift by the critical level
$\kappa_{c}$. It was also observed in loc. cit. that passing to the quotient
$V^{\kappa-\kappa_{c}}(\mathfrak{b})/V(\mathfrak{n})\cong
V^{\kappa-\kappa_{c}}(\mathfrak{t})$ produces the usual quantum Miura
transformation, expressing generators of ${\cal
W}_{prin}(\mathfrak{g},\kappa)$ in terms of $\mathrm{rk}\,\mathfrak{g}$ many
Abelian currents.

At the critical level, the principal ${\cal W}$-algebra is isomorphic to the
Feigin–Frenkel centre, i.e., ${\cal
W}_{prin}(f,\kappa_{c})\cong\mathfrak{z}(\mathfrak{g})$. One therefore has an
embedding $\mathfrak{z}(\mathfrak{g})\hookrightarrow V^{0}(\mathfrak{b})$.
Since $V([\mathfrak{n},\mathfrak{n}])$ is a vertex ideal, this map can be
extended to the quotient giving a map
$\mathfrak{z}(\mathfrak{g})\hookrightarrow
V^{0}(\mathfrak{b})\twoheadrightarrow
V^{0}(\mathfrak{b})/V([\mathfrak{n},\mathfrak{n}])$, which gives rise to a
commutative vertex subalgebra—that is isomorphic to the Feigin–Frenkel
centre—contained in $V^{0}(\mathfrak{b})/V([\mathfrak{n},\mathfrak{n}])$. Our
candidate chiralisation of $\beta$ is then given by the further composition

$\beta^{ch}:\mathfrak{z}(\mathfrak{g})\hookrightarrow
V^{0}(\mathfrak{b})/V([\mathfrak{n},\mathfrak{n}])\hookrightarrow\Pi_{\mathfrak{g}}~{}.$
(137)

Concretely,
$\beta^{ch}(P_{i})=(P_{i}^{t})|_{\mathfrak{b}/[\mathfrak{n},\mathfrak{n}]}$,
where $P_{i}^{t}$ are the tic-tac-toed expressions for the generators of
${\cal W}_{prin}(\mathfrak{g},\kappa_{c})$, as detailed in deBoer:1993iz  and
$|_{\mathfrak{b}/[\mathfrak{n},\mathfrak{n}]}$ denotes the projection
$V^{0}(\mathfrak{b})\twoheadrightarrow
V^{0}(\mathfrak{b})/V([\mathfrak{n},\mathfrak{n}])$. In a sense, $\beta^{ch}$
is a deformed version of the Miura transformation. Of course, this
construction also applies away from the critical level, giving rise to an
embedding $\beta^{ch}:{\cal W}_{prin}(\mathfrak{g},\kappa)\hookrightarrow
V^{\kappa-\kappa_{c}}(\mathfrak{b})$.

######  Remark 4.1.

The above construction is actually just one in a series of embedded
realisations of principal ${\cal W}$-algebras. Let $\mathfrak{n}_{i}$ be the
lower central series of $\mathfrak{n}$, i.e.,

$\mathfrak{n}_{j}=\underbrace{[\mathfrak{n},[\mathfrak{n},[\mathfrak{n},[\dots,\mathfrak{n}]]\dots]]}_{j}~{},$
(138)

for $j=1,\dots,d_{\mathfrak{g}}$, where $\mathfrak{n}_{d_{\mathfrak{g}}}$ is
the last non-zero term. There is a system of embeddings $\beta^{ch}_{j}:{\cal
W}_{prin}(\mathfrak{g},\kappa)\hookrightarrow
V^{\kappa-\kappa_{c}}(\mathfrak{b})/V(\mathfrak{n}_{k})$ for
$j=1,\dots,d_{\mathfrak{g}}+1$, which maps generators according to

$\beta^{ch}_{j}(P_{i})=P_{i}^{t}|_{\mathfrak{b}/\mathfrak{n}_{k}}~{},$ (139)

where $|_{\mathfrak{b}/\mathfrak{n}_{j}}$ is the projection
$V^{0}(\mathfrak{b})\twoheadrightarrow
V^{0}(\mathfrak{b})/V(\mathfrak{n}_{j})$. The usual Miura transform is given
by $\beta^{ch}_{1}$, our embedding corresponds to $\beta^{ch}_{2}$, and
$\beta^{ch}_{d_{\mathfrak{g}}+1}$ is the usual tic-tac-toe embedding from
deBoer:1993iz .

It would be interesting to know whether these generalised Miura maps have a
more geometric interpretation, viz., the relation of the Miura map to opers in
the critical case.

In practice, an alternative strategy for calculating these generators would be
to write down the most general correction to the classical expression,
$\beta^{\\#}(P_{i})$, and then fix free coefficients by imposing that all OPEs
amongst the generators are regular. In principle, one might worry that the
“most general correction” may have an infinite number of unfixed parameters,
because the vertex algebra $\mathbf{I}_{G}$ is not conical. However, based on
our discussion of $\beta^{ch}$ we should only look for corrections which
contain positive powers of the
$(\gamma_{i})_{i=1}^{\mathrm{rk}\,\mathfrak{g}}$—of which there are only
finitely many. In the examples in the following sections, the expressions that
one arrives at by imposing these regularity constraints are precisely the ones
obtained by actually computing $\beta^{ch}$ with the tic-tac-toe algorithm.

With a realisation of the Feigin–Frenkel centre established, we turn to the
additional generators of $\mathbf{I}_{G}$. We shall proceed by defining a
free-field expression for $X$ that arises from the finite setting in (112).
Suppose that $G$ is not simply-connected of type $D_{2N}$, then $V_{F}$ is
irreducible. Let $(a_{1},a_{2},\dots_{,}a_{\mathrm{rk}\,\mathfrak{g}})$ be the
components of the highest weight of $V_{F}$ in the simple root basis. We then
define

$X\coloneqq\prod_{j=1}^{\mathrm{rk}\,\mathfrak{g}}\gamma_{j}^{-a_{j}}~{}.$
(140)

Similarly, if $G$ is ${\rm Spin}(2N)$ we define $X_{\rm s}$ and $X_{\rm c}$
analogously to (140) using the highest weights of $V_{\rm s}$ and $V_{\rm c}$
respectively. We now formulate the chiral version of Conjecture 2.4:

######  Conjecture 4.2.

Suppose $G$ is not $\mathrm{Spin}(2N)$. Then the vertex algebra
$\mathbf{I}_{G}$ is isomorphic to the vertex subalgebra of $\Pi_{G}$ that is
weakly generated by the generators
$(\beta^{ch}(P_{i}))_{i=1}^{\mathrm{rk}\,\mathfrak{g}})$ of the Feigin–Frenkel
centre and by the generator $X$.

If $G$ is $\mathrm{Spin}(2N)$, then $\mathbf{I}_{G}$ is isomorphic to the
vertex subalgebra of $\Pi_{G}$ that is weakly generated by the generators
$(\beta^{ch}(P_{i}))_{i=1}^{\mathrm{rk}\,\mathfrak{g}})$ of the Feigin–Frenkel
centre and by the generators $X_{\rm s}$ and $X_{\rm c}$.

Again inspired by the finite dimensional setting we have beliefs on the set of
strong generators of $\mathbf{I}_{G}$.

######  Conjecture 4.3.

Suppose $G$ is not $\mathrm{Spin}(2N)$. Then $\mathbf{I}_{G}$ is strongly
generated by the generators $(P_{i})_{i=1}^{\mathrm{rk}\,\mathfrak{g}}$ of the
Feigin–Frenkel centre and generators $(\hat{g}_{i})_{i=1}^{{\rm dim}V_{F}}$ in
one-to-one correspondence with the weight vectors of $V_{F}^{*}$.

If $G$ is $\mathrm{Spin}(2N)$, then $\mathbf{I}_{G}$ is strongly generated by
by the generators $(P_{i})_{i=1}^{\mathrm{rk}\,\mathfrak{g}}$ of the
Feigin–Frenkel centre and generators, $(\hat{g}_{{\rm s}i})_{i=1}^{{\rm
dim}\,V_{\rm s}}$ and $(\hat{g}_{{\rm c}i})_{i=1}^{{\rm dim}\,V_{\rm c}}$ in
one-to-one correspondence with the weight vectors of $V_{\rm s}^{*}$ and
$V_{\rm c}^{*}$ respectively.

Unlike the finite dimensional setting, we will not attempt a proof of these
conjectures. This should potentially be dealt with in terms of the
(micro-)localisation of $\mathbf{I}_{G}$ over its associated variety. Such a
localisation theory is still in its early stages (see, e.g., Arakawa:2015loc ;
Kuwabara:2021hyp ) but we hope that a geometric strategy will prove fruitful.
For our purposes here, we will be content to consider examples for the low
rank cases of ${\rm SL}_{2}$ and ${\rm SL}_{3}$, and leave more extensive
analysis of examples to future investigation.

It is known Arakawa:2018egx  that $\mathbf{I}_{G}$ is conformal—with critical
central charge given by

$c_{\mathbf{I}_{G}}=2\mathrm{rk}\,\mathfrak{g}+(\rho,\rho^{\vee})~{},$ (141)

where $\rho$ is the half sum of positive roots of $\mathfrak{g}$—i.e., the
Weyl vector—and $\rho^{\vee}$ is the half sum of positive coroots. After
imposing the $\mathrm{rk}\,\mathfrak{g}$ many constraints on (133), which
fixed $\Delta_{\gamma_{i}}=2$ we are left with $\mathrm{rk}\,\mathfrak{g}$
many unfixed background charges. These are fixed by requiring that the
quadratic Feigin–Frenkel generator, $P_{1}$ is quasiprimary. The residue
$T_{(2)}P_{1}$ is linear in the background charges and setting it to zero
produces $2\mathrm{rk}\,\mathfrak{g}$ many linear equations for the background
charges. However, these equations are not linearly independent. The expression
for $P_{1}$ is unchanged under the symmetry
$\partial\phi\mapsto-\partial\delta$ and $\partial\delta\mapsto\partial\phi$.
Thus, the swaps $\alpha_{i}\mapsto-\beta_{i}$ and
$\beta_{i}\mapsto-\alpha_{i}$ leaves the system of linear equations invariant.
This symmetry cuts the system down to $\mathrm{rk}\,\mathfrak{g}$ many linear
equations for $\mathrm{rk}\,\mathfrak{g}$ many unknowns. In the ${\rm SL}_{N}$
case it is schematically clear that these equations are independent, leading
to a unique assignment of background charges. For other choices of $G$ we
expect the same argument to hold, but we have not attempted a careful
analysis. This assignment of background charges, in the ${\rm SL}_{N}$ case,
fixes the central charge to agree with (141) without additional input, and the
general $G$ case should behave similarly.

The expression (133), with the correctly chosen values of
$\alpha_{i},\beta_{i}$, gives a free field realisation of the conformal vector
of $\mathbf{I}_{G}$. On general grounds, we do not expect this to be a strong
generator of $\mathbf{I}_{G}$, but rather it should arise as a composite of
the other strong generators. We will verify this explicitly in the case
$G=SL_{2}$ below.

###  4.3 Free field realisation for $\mathbf{I}_{{\rm SL}_{2}}$

Let us work through the ${\rm SL}_{2}$ case as an illustrative example. The
chiralisation of $\mathrm{KT}_{G}$ is $\Pi_{{\rm SL}_{2}}$, which is strongly
generated by

$\gamma^{\pm\frac{1}{2}}\equiv e^{\pm\frac{1}{2}(\phi+\delta)}~{},\quad
b\equiv\frac{1}{2}\partial(\delta-\phi)~{}.$ (142)

We first find the expression for $S=\beta^{ch}(P_{1})$, the generator of the
Feigin–Frenkel centre. Our prescription above yields (cf. (deBoer:1993iz, ,
Eqn. (3.4)), with some change of conventions),

$S=bb+\gamma+\partial b~{}.$ (143)

We note here that rather than using this deformed Miura map, we could adopt a
more bootstrap-inspired approach. The non-chiral image $\beta(P_{1})$ is
generated by $S=b^{2}+\gamma$, and we should allow for quantum corrections as
the self-OPE of $bb+\gamma$ is not regular. The most general corrected Ansatz
is

$S=bb+\gamma+\alpha\partial b~{},$ (144)

for some $\alpha\in\mathbb{C}$. The normalisation is such that the Fourier
mode $S_{(1)}$ acts via the generalised eigenvalue
$\frac{1}{4}\lambda(2\lambda+1)$ on the Weyl module $\mathbb{V}_{\lambda}$.
Imposing regularity of the $S\times S$ OPE fixes $\alpha=1$ and so fixes the
corrected $S$ to be precisely of the form returned by the deformed Miura map.

We expect only one additional strong generator, corresponding to the function
$Y$ on $\mathfrak{Z}_{{\rm SL}_{2}}$. This operator appears in the simple pole
in the $S\times X$ OPE, which is the chiral version of the extraction of the
non-chiral $Y$ from the Poisson bracket $\\{S,X\\}$. Indeed, defining $Y$ as
such, the OPEs close and we find a strongly generated vertex algebra with the
following generators,

$\begin{split}S&=bb+\gamma+\partial b~{},\\\ X&=\gamma^{-\frac{1}{2}}~{},\\\
Y&=-b\gamma^{-\frac{1}{2}}~{}.\end{split}$ (145)

Comparing with (56), the expression for $Y$ (and $X$) in terms of $\Pi_{G}$
fields matches precisely the expression in terms of $\mathrm{KT}_{G}$
previously with no quantum corrections, but this is probably not particularly
meaningful. Indeed, $Y$ defined here is not quasiprimary, but rather
$Y+\frac{3}{4}X^{\prime}$ is instead, and if we worked in terms of that
quasiprimary instead there would indeed be quantum corrections.

The OPEs between these strong generators are given by

$\begin{split}S(z)X(w)&\sim\frac{\frac{3}{4}X(w)}{(z-w)^{2}}+\frac{Y(w)}{z-w}~{},\\\
S(z)Y(w)&\sim\frac{\frac{3}{4}Y(w)}{(z-w)^{2}}+\frac{(SX)(w)}{z-w}~{},\\\
Y(z)X(w)&\sim\frac{\frac{1}{2}(XX)(w)}{z-w}~{},\\\
Y(z)Y(w)&\sim\frac{-\frac{1}{4}(XX)(w)}{(z-w)^{2}}-\frac{\frac{1}{4}(X\partial
X)(w)}{z-w}~{}.\end{split}$ (146)

The determinantal relation $D_{{\rm SL}_{2}}=SX^{2}-Y^{2}\overset{!}{=}1$ is
chiralised here to the relation

$\hat{D}_{{\rm SL}_{2}}=SXX-YY-\tfrac{3}{2}\partial YX+\tfrac{3}{2}Y\partial
X-\tfrac{3}{4}\partial X\partial
X-\tfrac{1}{8}X\partial^{2}X\overset{!}{=}\mathbbm{1}~{}.$ (147)

The generators $S$, $X$, and $Y$, together with the OPEs and relation above
define a genuine strongly generated vertex algebra, which is a vertex
subalgebra of $\Pi_{{\rm SL}_{2}}$.

As expected, the free-field conformal vector (with background charges
$\alpha=\frac{3}{2}$ and $\beta=-\frac{1}{2}$) can be expressed as the
following composite operator in terms of the strong generators,

$\begin{split}T_{0}=2S(\partial YX-Y\partial X)+3S\partial X\partial
X+\tfrac{1}{2}Y\partial^{3}X-\partial SYX+\tfrac{5}{2}\partial S\partial
XX-2\partial Y\partial Y&\\\ +\tfrac{3}{2}\partial
Y\partial^{2}X-\tfrac{3}{4}\partial^{2}X\partial^{2}X-\tfrac{3}{2}\partial^{2}Y\partial
X-\tfrac{7}{12}\partial^{3}X\partial
X-\tfrac{1}{2}\partial^{3}YX-\tfrac{1}{24}\partial^{4}XX&~{}.\end{split}$
(148)

The central charge is $c=26$, which agrees with (141).

The equivariant affine ${\cal W}$-algebra for $\mathfrak{g}=\mathfrak{sl}_{2}$
was presented in terms of strong generators and their OPEs in Arakawa:2018egx
. By performing the Drinfel’d–Sokolov reduction, which is still tractable in
this case, one can show that the the resulting vertex algebra is isomorphic to
our realisation. We will not review this computation here.

###  4.4 Interlude—a more universal centraliser

It is interesting to observe that the singular OPEs in (146) define,
abstractly, a strongly generated vertex algebra that we denote
$\widetilde{\mathbf{I}}_{{\rm SL}_{2}}$ in which the vertex ideal generated by
$\hat{D}_{{\rm SL}_{2}}-\mathbbm{1}$ is not quotiented out. That is to say,
the OPEs satisfy the Jacobi identities without removing any null states. The
(now nonvanishing) operator $\hat{D}_{{\rm SL}_{2}}$ is not only null, but is
actually central in $\widetilde{\mathbf{I}}_{{\rm SL}_{2}}$. Consequently,
there is a one parameter family of analogous chiral universal centralisers
given by the quotient of $\widetilde{\mathbf{I}}_{{\rm SL}_{2}}$ by the ideal
$\hat{D}_{{\rm SL}_{2}}-\xi\mathbbm{1}$ for $\xi\in\mathbb{C}$. For $\xi\neq
0$, these quotients are all isomorphic to $\mathbf{I}_{{\rm SL}_{2}}$, with
the isomorphism realised by rescaling the strong generators $X$ and $Y$.
However, the centreless specialisation $\hat{D}_{{\rm SL}_{2}}=0$ is not
isomorphic to $\mathbf{I}_{{\rm SL}_{2}}$. This centreless specialisation
seems to be something of an oddity; in particular it is not a chiral
quantisation of $\mathfrak{Z}_{{\rm SL}_{2}}$ since the determinant is set to
zero.

The conformal vector $T_{0}$ does not lift to one in
$\widetilde{\mathbf{I}}_{{\rm SL}_{2}}$. For example, the cubic pole in the
$T_{0}\times T_{0}$ OPE is proportional to $\partial\hat{D}_{{\rm SL}_{2}}$
(and the other poles are also polluted by the appearence of
$\partial\hat{D}_{{\rm SL}_{2}}$ and its derivatives). The Virasoro OPE only
holds in the quotient. Indeed, $\widetilde{\mathbf{I}}_{{\rm SL}_{2}}$ does
not have a conformal vector. Any candidate conformal vector must be built out
of the $S,X,Y$, and the OPEs amongst these generators are homogeneous in the
number of $X$ and $Y$ fields appearing (i.e., there is an outer $U(1)$
symmetry of this vertex algebra under which $X$ and $Y$ have charge one and
$S$ has charge zero). Charge conservation then prevents any candidate
conformal vector from realising the Virasoro OPE.

This outer $U(1)$ action leads to a family of gradings on
$\widetilde{\mathbf{I}}_{{\rm SL}_{2}}$—the OPEs are compatible with any
grading that assign $|S|=2$, $|X|=\Delta$ and $|Y|=\Delta+1$ for any
$\Delta\in\mathbb{Z}$. The simple quotient $\hat{D}_{{\rm
SL}_{2}}-\xi\mathbbm{1}$, for $\xi\neq 0$, violates this grading for all
choices except $\Delta=-1$. The centreless specialisation, $\xi=0$, however,
is compatible with this grading for any $\Delta\in\mathbb{Z}$.

From a geometric point of view, this one parameter family arises from a closed
embedding of ${\rm SL}_{N}$ into the ambient space of ${\rm GL}_{N}$. The
cotangent bundle $T^{\ast}{\rm GL}_{N}\cong\mathfrak{gl}_{N}^{*}\times{\rm
GL}_{N}$ has a splitting

$T^{*}{\rm GL}_{N}\cong\mathfrak{gl}_{N}^{*}\times{\rm
GL}_{N}\cong(\mathfrak{sl}_{N}^{*}\times\mathfrak{gl}^{*}_{1})\times{\rm
GL}_{1}\times{\rm SL}_{N}\cong(T^{*}\mathbb{C}^{\times})\times(T^{*}{\rm
SL}_{N})~{},$ (149)

arising from the splitting
$\mathfrak{gl}_{N}\cong\mathfrak{gl}_{1}\oplus\mathfrak{sl}_{N}$ and ${\rm
GL}_{N}\cong{\rm SL}_{N}\rtimes{\rm GL}_{1}$. Consider the Poisson subalgebra
of $\mathbb{C}[T^{*}{\rm GL}_{N}]$ generated by the
$\mathbb{C}[\mathfrak{sl}_{N}^{*}]\otimes\mathbb{C}[{\rm GL}_{N}]$ factors in
the splitting. We denote its spectrum by $\widetilde{T}^{*}{\rm SL}_{N}$. As
varieties, we have that $\widetilde{T}^{*}{\rm SL}_{N}\cong{{\rm
GL}_{1}}\times T^{*}{\rm SL}_{N}\cong\mathfrak{sl}_{N}^{*}\times{\rm GL}_{N}$.

This thickened space inherits Hamiltonian left and right action of ${\rm
SL}_{N}$, and we can perform the corresponding two-sided Kostant–Whittaker
reduction of this ambient space. The ${\rm GL}_{1}$ factor of the splitting is
$N_{L}\times N_{R}$ invariant, and lies in the (intersection)
$\overline{\mu}_{L}^{-1}(\chi)\cap\overline{\mu}_{R}^{-1}(\chi)$. Therefore,
the two-sided reduction will be isomorphic to
$\mathbb{C}^{\times}\times\mathfrak{Z}_{{\rm SL}_{N}}$, and we denote this
space by $\widetilde{\mathfrak{Z}}_{{\rm SL}_{N}}$. It is naturally a scheme
over $S_{f}$ by extending the structure morphism of $\mathfrak{Z}_{{\rm
SL}_{N}}$. The cotangent bundle $T^{*}{\rm SL}_{N}$ is a closed, Poisson
subvariety of $\widetilde{T}^{*}{\rm SL}_{N}$, so by functoriality of
reduction, $\mathfrak{Z}_{{\rm SL}_{N}}$ is a closed, Poisson, subvariety of
$\widetilde{\mathfrak{Z}}_{{\rm SL}_{N}}$.

There exist a family of $\mathbb{C}^{\times}$ actions on $T^{*}{\rm GL}_{N}$,
which acts as uniform rescaling of the vector space $\mathfrak{gl}_{N}^{*}$
and as left multiplication by a constant diagonal matrix on ${\rm GL}_{N}$.
Under these $\mathbb{C}^{\times}$ actions, the linear functions on
$\mathfrak{gl}_{N}^{*}$ can be assigned unit weight, in which case the Poisson
bracket has degree $-1$, while the weights of the matrix entries, $g_{ij}$,
which generate $\mathbb{C}[{\rm GL}_{N}]$ are unfixed. Any assignment
$|g_{ij}|=\Delta$ for all $i,j=1,\dots,N$ is compatible with the Poisson
structure. The fundamental invariants of $\mathfrak{sl}_{N}$,
$(P_{i})_{i=1}^{N}$ will have weight $|P_{i}|=i+1$ for $i=1,\dots,N$, i.e.,
the weight is equal to their degree. The thickened space
$\widetilde{T}^{*}{\rm SL}_{N}$ inherits this grading.

Similarly, the fibres of $\widetilde{\mathfrak{Z}}_{{\rm SL}_{N}}$ will
inherit the grading on $\widetilde{T}^{*}{\rm SL}_{N}$. Under
Kostant–Whittaker reduction the $\mathbb{C}^{\times}$ weights are shifted by
the weight associated to the principal $\mathfrak{sl}_{2}$-embedding.
Consider, specifically, the bottow row $g_{Nj}$ whose weights under the
$\mathbb{C}^{\times}$ action must satisfy $|g_{Nj}|=|g_{N1}|+(j-1)$. Denoting
$g_{N1}$ by $X$, as is our convention, and denoting its weight by
$|X|=\Delta$, any grading where $|g_{Nj}|=\Delta+(j-1)$ is compatible with the
Poisson bracket.

Picking out a closed subvariety by imposing the determinantal relation
$D_{{\rm GL}_{N}}\overset{!}{=}1$ is incompatible with this grading—unless
$\Delta=1-N$. Note that any such relation $D_{{\rm GL}_{N}}\overset{!}{=}\xi$,
for $\xi\in\mathbb{C}^{\times}$ forces $\Delta=1-N$. The closed subvariety
satisfying $D_{{\rm GL}_{N}}=1$ gives a closed embedding $\mathfrak{Z}_{{\rm
SL}_{N}}\hookrightarrow\widetilde{\mathfrak{Z}}_{{\rm SL}_{N}}$. In fact, any
subvariety $D_{{\rm GL}_{N}}=\xi$ gives a closed embedding of
$\mathfrak{Z}_{\rm SL_{N}}$ inside $\widetilde{\mathfrak{Z}}_{{\rm SL}_{N}}$
as schemes over $S_{f}$.

The thickened space, $\widetilde{T}^{*}{\rm SL}_{N}$ has a closure isomorphic
to $\mathbb{C}\times{\rm SL}_{N}$. Performing Kostant–Whittaker reduction on
this space gives a closure of $\widetilde{\mathfrak{Z}}_{{\rm SL}_{N}}$. In
this closure, we have an additional subvariety with vanishing ideal $D_{{\rm
GL}_{N}}$. This subvariety of determinant zero matrices is not isomorphic to
$\mathfrak{Z}_{{\rm SL}_{N}}$ and seems fairly exotic.

The geometric construction above reflects many of the properties we have seen
in $\widetilde{\mathbf{I}}_{{\rm SL}_{2}}$, namely the auxiliary grading and
one-parameter family of quotients. Let us present a schematic chiral version
of this construction that should explain the existence of this more general
vertex algebra.

Let $\mathcal{D}^{ch}_{{\rm GL}_{N},-N}$ be CDOs on ${\rm GL}_{N}$ at level
$-N$; these admit a splitting $\mathcal{D}^{ch}_{{\rm
GL}_{N},-N}\cong\mathcal{D}^{ch}(\mathbb{C}^{\times})\otimes\mathcal{D}^{ch}_{{\rm
SL}_{N},-N}$. This is the chiral version of the splitting (149). Functions on
$T^{*}\mathbb{C}^{\times}$ are generated by $D_{{\rm GL}_{N}}^{\pm}$ and the
fibre coordinate $p$, with Poisson bracket $\\{p,D_{{\rm GL}_{N}}^{\pm}\\}=\pm
D_{{\rm GL}_{N}}^{\pm}$. Abusing notation, the CDOs on $\mathbb{C}^{\times}$
are strongly generated by $D_{{\rm GL}_{N}}^{\pm}$ and $p$ with OPE

$p(z)D_{{\rm GL}_{N}}(w)^{\pm}\sim\frac{D_{{\rm GL}_{N}}^{\pm}(w)}{z-w}~{}.$
(150)

Let $\overline{\mathcal{D}^{ch}}_{SL_{N},-N}\subset\mathcal{D}^{ch}_{{\rm
GL}_{N},-N}$ be the vertex subalgebra corresponding to
$\mathrm{ker}~{}p_{(0)}$. It is strongly generated by the strong generators of
$\mathcal{D}^{ch}_{{\rm SL}_{N},-N}$ and by $D_{{\rm GL}_{N}}$. To recover
$\mathcal{D}^{ch}_{{\rm SL}_{N},-N}$ we quotient out by the vertex ideal
generated by $D_{{\rm GL}_{N}}-\mathbbm{1}$. Under Drinfel’d–Sokolov
reduction, the $\mathcal{D}^{ch}_{{\rm SL}_{N},-N}$ subalgebra reduces to
$\mathbf{I}_{{\rm SL}_{N}}$. Meanwhile, the generator $D_{{\rm GL}_{N}}$ gets
corrected to $\hat{D}_{{\rm GL}_{N}}$, which is the composite expression for
the determinant inside $\mathbf{I}_{{\rm SL}_{N}}$.

Thus, the Drinfel’d–Sokolov reduction of $\overline{D}_{{\rm SL}_{N},-N}$
yields $\mathbf{I}_{{\rm SL}_{N}}\otimes\langle\hat{D}_{{\rm GL}_{N}}\rangle$,
where $\langle\hat{D}_{{\rm GL}_{N}}\rangle$ is a commutative vertex algebra
generated by $\hat{D}_{{\rm GL}_{N}}$ and its inverse. The reduction has a
one-parameter family of quotients by setting $\hat{D}_{{\rm
GL}_{N}}=\xi\mathbbm{1}$ for $\xi\in\mathbb{C}^{\times}$. Each of these
quotients is isomorphic to $\mathbf{I}_{{\rm SL}_{N}}$, and we can identify
this reduction as the more universal centraliser $\widetilde{\mathbf{I}}_{{\rm
SL}_{N}}$.

###  4.5 Free field realisation for $\mathbf{I}_{{\rm SL}_{3}}$

We return to another example, this time in rakn two. Once again, our starting
point is to find expressions for the Feigin–Frenkel generators $S$ and $W$.
Like in the ${\rm SL}_{2}$ case, we can utilise our deformed Miura approach to
produce the right expressions. For $G={\rm SL}_{3}$, the restrictions of the
generators of the Harish-Chandra centre to $S_{f}$ are given, up to a
convention for overall scaling, by

$\begin{split}S&=\frac{1}{2}(b_{1}^{2}+b_{2}^{2}-b_{1}b_{2}+\gamma_{1}+\gamma_{2})~{},\\\
W&=\frac{1}{3}(b_{1}b_{2}(b_{1}-b_{2})+\gamma_{1}b_{2}-\gamma_{2}b_{1}~{})~{},\end{split}$
(151)

and by performing the tic-tac-toe procedure as in (deBoer:1993iz, , Eqn.
(3.10)), we arrive at chiral versions given by

$\begin{split}S&=\frac{1}{2}(b_{1}^{2}+b_{2}^{2}-b_{1}b_{2}+\gamma_{1}+\gamma_{2})+\frac{1}{2}\partial(b_{1}+b_{2})~{},\\\
W&=\frac{1}{3}\left(b_{1}b_{2}(b_{1}-b_{2})+b_{2}\gamma_{1}-b_{1}\gamma_{2}\right)+\frac{1}{6}\partial(\gamma_{1}-\gamma_{2})+\frac{1}{3}(b_{1}\partial
b_{1}-b_{2}\partial b_{2})\\\ &\phantom{=}+\frac{1}{6}(b_{2}\partial
b_{1}-b_{1}\partial
b_{2})+\frac{1}{6}\partial^{2}(b_{1}-b_{2})~{}.\end{split}$ (152)

Alternatively, we can start with an Ansatz based on the finite-dimensional
expression and add quantum corrections while demanding regularity of the OPEs.
The appropriate Ansätze are

$\begin{split}S&=\frac{1}{2}(b_{1}^{2}+b_{2}^{2}-b_{1}b_{2}+\gamma_{1}+\gamma_{2})+\alpha_{1}(\partial
b_{1}+\partial b_{2})~{},\\\
W&=\frac{1}{3}(b_{1}b_{2}(b_{1}-b_{2})+b_{2}\gamma_{1}-b_{1}\gamma_{2})+\alpha_{2}\partial(\gamma_{1}-\gamma_{2})+\alpha_{3}(b_{1}\partial
b_{1}-b_{2}\partial b_{2})\\\ &\phantom{=}+\alpha_{4}(b_{2}\partial
b_{1}-b_{1}\partial b_{2})+\alpha_{5}\partial^{2}(b_{1}-b_{2})~{},\end{split}$
(153)

where $\alpha_{i}\in\mathbb{C}$. The corrections are constrained to satisfy
the requirement that $S$ be invariant under the outer-automorphism
$b_{1}\mapsto b_{2}$, $e_{1}\mapsto e_{2}$ while $W$ is negated. Requiring
regularity of the $S\times S$ OPE fixes the parameter $\alpha_{1}=1/2$, and
regularity of $W\times W$ and $W\times S$ fixes the remaining coefficients as
follows,

$\alpha_{2}=\frac{1}{6}~{},\qquad\alpha_{3}=\frac{1}{3}~{},\qquad\alpha_{4}=\frac{1}{6}~{},\qquad\alpha_{5}=\frac{1}{6}~{},$
(154)

which agrees exactly with the expressions arising from $\beta^{ch}$.

To complete the construction, we start with the strong generator
$X=-\gamma^{-2/3}_{1}\gamma^{-1/3}_{2}$, and identify additional strong
generators $Y$ and $Z$ by analysing the simple poles in the $S\times X$ and
$S\times Y$ OPEs, respectively (here we again define $Y$ and $Z$ to not be
quasi-primaries, which could be rectified by adding descendants of other
generators). The complete set of strong generators then takes the form

$\begin{split}S&=\frac{1}{2}(b_{1}^{2}+b_{2}^{2}-b_{1}b_{2}+\gamma_{1}+\gamma_{2})+\frac{1}{2}\partial(b_{1}+b_{2})~{},\\\
W&=\frac{1}{3}\left(b_{1}b_{2}(b_{1}-b_{2})+b_{2}\gamma_{1}-b_{1}\gamma_{2}\right)+\frac{1}{6}\partial(\gamma_{1}-\gamma_{2})+\frac{1}{3}(b_{1}\partial
b_{1}-b_{2}\partial b_{2})\\\ &\phantom{=}+\frac{1}{6}(b_{2}\partial
b_{1}-b_{1}\partial b_{2})+\frac{1}{6}\partial^{2}(b_{1}-b_{2})~{},\\\
X&=-\gamma^{-2/3}_{1}\gamma^{-1/3}_{2}~{},\\\
Y&=-\frac{1}{2}b_{1}(\gamma^{-2/3}_{1}\gamma^{-1/3}_{2})~{},\\\
Z&=-\frac{1}{2}\big{(}\gamma_{1}-\gamma_{2}+b_{1}b_{1}-b_{2}b_{2}+b_{1}b_{2}-\partial
b_{2}\big{)}\gamma_{1}^{-2/3}\gamma_{2}^{-1/3}~{}.\end{split}$ (155)

The OPEs between the Feigin–Frenkel generators and the others are given by

$\begin{split}S(z)X(w)&\sim\frac{\frac{4}{3}X}{(z-w)^{2}}+\frac{-Y}{z-w}~{},\\\
S(z)Y(w)&\sim\frac{\frac{4}{3}Y}{(z-w)^{2}}+\frac{-Z-\frac{1}{2}(SX)}{z-w}~{},\\\
S(z)Z(w)&\sim\frac{\frac{4}{3}Z}{(z-w)^{2}}+\frac{-\frac{1}{2}(SY)-(WX)+\frac{1}{6}(\partial
SX)}{z-w}~{},\\\
W(z)X(w)&\sim\frac{-\frac{20}{27}X}{(z-w)^{3}}+\frac{\frac{5}{6}Y}{(z-w)^{2}}+\frac{-Z+\frac{1}{6}(SX)}{z-w}~{},\\\
W(z)Y(w)&\sim\frac{-\frac{20}{27}Y}{(z-w)^{3}}+\frac{\frac{5}{6}Z+\frac{5}{12}(SX)}{(z-w)^{2}}+\frac{-(WX)-\frac{1}{3}(SY)+\frac{1}{6}(\partial
SX)}{z-w}~{},\\\
W(z)Z(w)&\sim\frac{-\frac{20}{27}Z}{(z-w)^{3}}+\frac{\frac{5}{6}(WX)+\frac{5}{12}(SY)-\frac{5}{36}(\partial
SX)}{(z-w)^{2}}\\\ &+\frac{-(WY)+\frac{1}{6}(SZ)+\frac{1}{6}(\partial
SY)-\frac{1}{4}(SSX)}{z-w}~{},\end{split}$ (156)

where we have suppressed the dependence of the residues on the coordinate $w$
for clarity. Similarly, for the OPEs amongst $X$, $Y$, and $Z$, we have

$\displaystyle Y(z)X(w)$ $\displaystyle\sim\frac{-\frac{2}{3}(XX)}{z-w}~{},$
(157) $\displaystyle Y(z)Y(w)$
$\displaystyle\sim\frac{-\frac{4}{9}(XX)}{(z-w)^{2}}+\frac{-\frac{4}{9}(X\partial
X)}{z-w}~{},$ (158) $\displaystyle Z(z)X(w)$
$\displaystyle\sim\frac{\frac{1}{9}(XX)}{(z-w)^{2}}+\frac{-\frac{5}{6}(YX)-\frac{4}{9}(X\partial
X)}{z-w}~{},$ (159) $\displaystyle Z(z)Y(w)$
$\displaystyle\sim\frac{\frac{2}{27}(XX)}{(z-w)^{3}}+\frac{-\frac{4}{9}(YX)-\frac{4}{18}(X\partial
X)}{(z-w)^{2}}$ (160)
$\displaystyle+\frac{-\frac{1}{3}(YY)+\frac{1}{6}(ZX)-\frac{1}{4}(SXX)-\frac{4}{9}(\partial
YX)+\frac{5}{36}(Y\partial
X)-\frac{4}{27}(X\partial^{2}X)+\frac{5}{54}(\partial X\partial X)}{z-w}~{},$
(161) $\displaystyle Z(z)Z(w)$
$\displaystyle\sim\frac{\frac{1}{81}(XX)}{(z-w)^{4}}+\frac{\frac{1}{81}(X\partial
X)}{(z-w)^{3}}$ (162)
$\displaystyle+\frac{-\frac{13}{36}(YY)-\frac{1}{9}(ZX)-\frac{1}{6}(SXX)-\frac{1}{9}(\partial
YX)+\frac{1}{54}(Y\partial
X)-\frac{8}{81}(X\partial^{2}X)+\frac{1}{81}(\partial X\partial
X)}{(z-w)^{2}}$ (163)
$\displaystyle-\frac{\frac{1}{18}(\partial(ZX)+\partial(\partial
YX)-\frac{1}{6}\partial(Y\partial
X)+\frac{3}{2}\partial(SXX)+\frac{2}{3}X\partial^{3}X+\frac{13}{2}Y\partial
Y)}{z-w}~{}.$ (164)

The determinantal null ideal is chiralised to

$\begin{split}&D_{{\rm
SL}_{3}}=-\tfrac{1}{8}SSSXXX+WWXXX+\tfrac{1}{2}WSYXX+\tfrac{1}{2}SSYYX+WYYY-\tfrac{1}{4}SSZX-3WZYX\\\
&-SZYY+\tfrac{1}{2}SZZX+ZZZ+\tfrac{9}{8}SSYX^{\prime}X+\tfrac{83}{144}SSX^{\prime}X^{\prime}X-\tfrac{11}{8}SSY^{\prime}XX-\tfrac{1}{32}SSX^{\prime\prime}XX\\\
&-\tfrac{107}{144}SYYX^{\prime\prime}-\tfrac{1091}{432}SYX^{\prime\prime}X^{\prime}-\tfrac{29}{108}SYX^{\prime\prime\prime}X-\tfrac{23}{12}SZYX^{\prime}-\tfrac{71}{72}SZX^{\prime}X^{\prime}+\tfrac{5}{12}SZY^{\prime}X\\\
&+\tfrac{19}{18}SZX^{\prime\prime}X-\tfrac{2}{3}SY^{\prime}YY-\tfrac{19}{36}SY^{\prime}YX^{\prime}+\tfrac{257}{216}SY^{\prime}X^{\prime}X^{\prime}-\tfrac{41}{24}SY^{\prime}Y^{\prime}X+\tfrac{511}{216}SY^{\prime}X^{\prime\prime}X+\tfrac{8}{3}SZ^{\prime}YX\\\
&+\tfrac{101}{36}SZ^{\prime}X^{\prime}X-\tfrac{649}{432}SX^{\prime\prime}X^{\prime}X^{\prime}+\tfrac{5765}{3456}SX^{\prime\prime}X^{\prime\prime}X+\tfrac{19}{16}SY^{\prime\prime}YX+\tfrac{11}{6}SY^{\prime\prime}X^{\prime}X-\tfrac{161}{144}SZ^{\prime\prime}XX\\\
&+\tfrac{1651}{2592}SX^{\prime\prime\prime}X^{\prime}X+\tfrac{131}{648}SY^{\prime\prime\prime}XX+\tfrac{14053}{62208}SX^{(4)}XX-\tfrac{1}{12}WSX^{\prime}XX+\tfrac{7}{2}WYYX^{\prime}+\tfrac{64}{9}WYX^{\prime}X^{\prime}\\\
&+\tfrac{7}{36}WYX^{\prime\prime}X-\tfrac{13}{2}WZX^{\prime}X-\tfrac{1}{2}WS^{\prime}XXX+\tfrac{199}{27}WX^{\prime}X^{\prime}X^{\prime}-7WY^{\prime}YX-\tfrac{263}{18}WY^{\prime}X^{\prime}X\\\
&+5WZ^{\prime}XX-\tfrac{137}{72}WX^{\prime\prime}X^{\prime}X+\tfrac{163}{72}WY^{\prime\prime}XX-\tfrac{19}{24}WX^{\prime\prime\prime}XX+\tfrac{7}{576}YYX^{(4)}-\tfrac{52799}{46656}YX^{\prime\prime\prime}X^{\prime\prime}\\\
&-\tfrac{61225}{186624}YX^{(4)}X^{\prime}-\tfrac{98117}{933120}YX^{(5)}X-\tfrac{71}{216}ZYX^{\prime\prime\prime}+4ZZY^{\prime}+\tfrac{71}{36}ZZX^{\prime\prime}+\tfrac{52}{9}ZY^{\prime}Y^{\prime}+\tfrac{667}{108}ZY^{\prime}X^{\prime\prime}\\\
&+\tfrac{3557}{2592}ZX^{\prime\prime}X^{\prime\prime}-\tfrac{25}{72}ZY^{\prime\prime}Y+\tfrac{59}{432}ZY^{\prime\prime}X^{\prime}+\tfrac{1493}{2592}ZX^{\prime\prime\prime}X^{\prime}+\tfrac{203}{216}ZY^{\prime\prime\prime}X+\tfrac{12823}{31104}ZX^{(4)}X\\\
&-\tfrac{5}{8}S^{\prime}SYXX-\tfrac{31}{24}S^{\prime}SX^{\prime}XX+\tfrac{1}{2}S^{\prime}YYY+\tfrac{11}{8}S^{\prime}YYX^{\prime}+\tfrac{37}{18}S^{\prime}YX^{\prime}X^{\prime}+\tfrac{137}{144}S^{\prime}YX^{\prime\prime}X-\tfrac{3}{4}S^{\prime}ZYX\\\
&+\tfrac{25}{12}S^{\prime}ZX^{\prime}X-\tfrac{3}{16}S^{\prime}S^{\prime}XXX+\tfrac{1205}{864}S^{\prime}X^{\prime}X^{\prime}X^{\prime}-\tfrac{9}{4}S^{\prime}Y^{\prime}YX+\tfrac{37}{9}S^{\prime}Y^{\prime}X^{\prime}X-\tfrac{5}{24}S^{\prime}Z^{\prime}XX\\\
&+\tfrac{1121}{144}S^{\prime}X^{\prime\prime}X^{\prime}X+\tfrac{23}{288}S^{\prime}Y^{\prime\prime}XX+\tfrac{4261}{5184}S^{\prime}X^{\prime\prime\prime}XX-\tfrac{1}{4}W^{\prime}SXXX+W^{\prime}YX^{\prime}X-\tfrac{3}{2}W^{\prime}ZXX\\\
&+\tfrac{41}{12}W^{\prime}X^{\prime}X^{\prime}X-\tfrac{7}{2}W^{\prime}Y^{\prime}XX-\tfrac{3}{4}W^{\prime}X^{\prime\prime}XX-\tfrac{547}{1296}Y^{\prime}YX^{\prime\prime\prime}+\tfrac{80}{27}Y^{\prime}Y^{\prime}Y^{\prime}+\tfrac{737}{144}Y^{\prime}Y^{\prime}X^{\prime\prime}\\\
&+\tfrac{63505}{31104}Y^{\prime}X^{\prime\prime}X^{\prime\prime}+\tfrac{60653}{46656}Y^{\prime}X^{\prime\prime\prime}X^{\prime}+\tfrac{128695}{186624}Y^{\prime}X^{(4)}X-\tfrac{323}{108}Z^{\prime}YX^{\prime\prime}-4Z^{\prime}ZY-\tfrac{46}{9}Z^{\prime}ZX^{\prime}\\\
&-\tfrac{40}{9}Z^{\prime}Y^{\prime}Y-\tfrac{140}{27}Z^{\prime}Y^{\prime}X^{\prime}+\tfrac{40}{9}Z^{\prime}Z^{\prime}X-\tfrac{6881}{864}Z^{\prime}X^{\prime\prime}X^{\prime}+\tfrac{197}{48}Z^{\prime}Y^{\prime\prime}X-\tfrac{9575}{7776}Z^{\prime}X^{\prime\prime\prime}X+\tfrac{1}{6}S^{\prime\prime}SXXX\\\
&-\tfrac{3}{16}S^{\prime\prime}YYX-\tfrac{2}{3}S^{\prime\prime}YX^{\prime}X+\tfrac{49}{48}S^{\prime\prime}ZXX-\tfrac{605}{1728}S^{\prime\prime}X^{\prime}X^{\prime}X+\tfrac{39}{16}S^{\prime\prime}Y^{\prime}XX+\tfrac{6841}{3456}S^{\prime\prime}X^{\prime\prime}XX\\\
&+\tfrac{1}{4}W^{\prime\prime}YXX+\tfrac{11}{12}W^{\prime\prime}X^{\prime}XX-\tfrac{8747}{15552}X^{\prime\prime}X^{\prime\prime}X^{\prime\prime}-\tfrac{323}{432}Y^{\prime\prime}YX^{\prime\prime}+\tfrac{7}{36}Y^{\prime\prime}Y^{\prime}Y+\tfrac{4571}{2592}Y^{\prime\prime}Y^{\prime}X^{\prime}\\\
&-\tfrac{913}{324}Y^{\prime\prime}X^{\prime\prime}X^{\prime}+\tfrac{1631}{1296}Y^{\prime\prime}Y^{\prime\prime}X-\tfrac{52315}{93312}Y^{\prime\prime}X^{\prime\prime\prime}X+\tfrac{14}{9}Z^{\prime\prime}YY+\tfrac{817}{216}Z^{\prime\prime}YX^{\prime}-\tfrac{77}{36}Z^{\prime\prime}ZX\\\
&+\tfrac{10307}{2592}Z^{\prime\prime}X^{\prime}X^{\prime}-\tfrac{97}{18}Z^{\prime\prime}Y^{\prime}X-\tfrac{2255}{5184}Z^{\prime\prime}X^{\prime\prime}X-\tfrac{1}{72}S^{\prime\prime\prime}YXX+\tfrac{395}{1296}S^{\prime\prime\prime}X^{\prime}XX+\tfrac{1}{72}W^{\prime\prime\prime}XXX\\\
&-\tfrac{63841}{23328}X^{\prime\prime\prime}X^{\prime\prime}X^{\prime}-\tfrac{90493}{209952}X^{\prime\prime\prime}X^{\prime\prime\prime}X-\tfrac{13}{108}Y^{\prime\prime\prime}YY-\tfrac{3317}{3888}Y^{\prime\prime\prime}YX^{\prime}-\tfrac{2327}{1728}Y^{\prime\prime\prime}X^{\prime}X^{\prime}+\tfrac{6929}{3888}Y^{\prime\prime\prime}Y^{\prime}X\\\
&+\tfrac{1999}{5832}Y^{\prime\prime\prime}X^{\prime\prime}X-\tfrac{1}{24}Z^{\prime\prime\prime}YX+\tfrac{1841}{3888}Z^{\prime\prime\prime}X^{\prime}X+\tfrac{463}{10368}S^{(4)}XXX-\tfrac{72935}{139968}X^{(4)}X^{\prime}X^{\prime}\\\
&+\tfrac{9155}{93312}X^{(4)}X^{\prime\prime}X-\tfrac{865}{15552}Y^{(4)}YX+\tfrac{9443}{186624}Y^{(4)}X^{\prime}X+\tfrac{1159}{15552}Z^{(4)}XX+\tfrac{2243}{349920}X^{(5)}X^{\prime}X\\\
&+\tfrac{244}{3645}Y^{(5)}XX+\tfrac{2237}{51840}X^{(6)}XX~{}.\end{split}$
(165)

There is, currently, no expression in the literature to which we can compare
our presentation of $\mathbf{I}_{SL_{3}}$. Nevertheless, our Ansatz has the
properties one would expect from computing a schematic two-sided
Drinfel’d—Sokolov reduction of $\mathcal{D}^{ch}_{SL_{3},-3}$.

##  5 Class $\SS$ free field realisations at genus zero

We now move on to the construction of free field realisations of chiral
algebras of class $\SS$ in the $\mathfrak{a}_{1}$ case. In Beem:2019tfp , the
authors described (amongst other things) a free field realisation of the
vertex algebra associated to the four-punctured sphere,
$\mathbf{V}_{\mathfrak{a}_{1},4}\cong L_{-2}(\mathfrak{d}_{4})$, which was
inspired by the geometry of the associated variety/Higgs branch (in this case,
the closure of the minimal nilpotent orbit in $\mathfrak{d}_{4}$). This
amounted to a free field realisation of $\mathbf{V}_{\mathfrak{a}_{1},4}$
inside $\mathbf{V}_{{\mathfrak{a}_{1}},3}\otimes\Pi_{{\rm SL}_{2}}$, with
$\mathbf{V}_{{\mathfrak{a}_{1}},3}$ being just a system of symplectic bosons.

More generally, in this section we find evidence for a general inductive free
field realisation where the number of punctures can be increased at the
expense of introducing extra lattice degrees of freedom,101010For a selection
of values of $s$, such relations were observed previously by Carlo Meneghelli
CMUnpublished .

$\mathbf{V}_{\mathfrak{a}_{1},s}\subset\mathbf{V}_{\mathfrak{a}_{1},s-1}\otimes\Pi_{{\rm
SL}_{2}}~{}.$ (166)

At the level of Higgs branches/associated varieties, this relation arises from
the existence of a Zariski open $U_{n}\subset X_{n}$ such that passing to an
étale $2$-cover $\widetilde{U}_{n}$ gives

$\mathbb{C}[\widetilde{U}_{n}]\cong\mathbb{C}[X_{s-1}]\otimes\mathbb{C}[T^{*}\mathbb{C}^{\times}]~{}.$
(167)

The realisations $\mathbf{V}_{{\rm SL}_{2},s}\hookrightarrow\mathbf{V}_{{\rm
SL}_{2},s-1}\otimes\Pi_{{\rm SL}_{2}}$ should then be thought of as being a
consequence of localising along $U_{n}\hookrightarrow X_{s}$ and passing to an
appropriate cover. Here we will not endeavour to make these ideas more precise
in the abstract, but instead proceed by example. Furthermore, we will skip
directly to the chiral case, from which the corresponding relations of finite-
dimensional symplectic varieties can be recovered by passing to the (reduced)
$C_{2}$-algebra $(R_{V})_{red}$.

In the rest of this section, we will construct genus zero chiral algebras of
class $\SS$ of type $\mathfrak{a}_{1}$ using these free field methods. In
principle such a program could exploit the fact that in type
$\mathfrak{a}_{1}$ the trinion theory is a theory of free hypermultiplets,
with associated vertex operator algebra given by symplectic bosons, and so use
$\mathbf{V}_{{\rm SL}_{2},3}$ as a starting point as in Beem:2019tfp . Such an
approach obscures some symmetry of the problem—indeed, by generalised
$S$-duality all (full) punctures should be treated equally in the associated
vertex algebra, while starting with the trinion VOA privileges the
corresponding three “initial” punctures. We instead take the chiral universal
centraliser as our starting point, or more precisely its free field
realisation from Section 4.3. We will therefore present these vertex algebras
in free field form in a manner that renders generalised $S$-duality completely
manifest.

###  5.1 General structure of genus zero free field realisations

We will realise $\mathbf{V}_{\mathfrak{a}_{1},s}$ inside
$\bigotimes_{i=0}^{s}\Pi^{i}_{{\rm SL}_{2}}$, where the superscript $i$ labels
the puncture associated to that lattice factor when $i\neq 0$, and the zeroth
copy will be used to construct $\mathbf{I}_{{\rm SL}_{2}}$. We can think of
$i$th copy for $i>0$ being used to construct $\mathbf{V}_{\mathfrak{a}_{1},i}$
in an inductive process, though at the end of the day all punctures will be
treated symmetrically. For $i\neq 0$ we denote the weight $\pm\frac{1}{2}$
generators by $e_{i}^{\pm\frac{1}{2}}$ and the weight one generator by
$p_{i}$. For the generators of $\Pi^{0}_{{\rm SL}_{2}}$ that are used in the
free field realisation of the sphere VOA, we continue to use
$\\{\gamma^{\pm\frac{1}{2}},b\\}$ as in Section 4.3.

For each $i>0$, we can realise a critical-level affine $\mathfrak{sl}_{2}$
subalgebra in $\Pi^{i}_{{\rm SL}_{2}}$, generated by the currents

$e_{i}=\left(e_{i}^{\frac{1}{2}}\right)^{2}~{},\quad
h_{i}=-2\partial\phi_{i}~{},\quad
f_{i}=(\partial^{2}\delta_{i}-(\partial\delta_{i})^{2})e_{i}^{-1}~{}.$ (168)

These currents generate the (simple quotient of the) universal current algebra
$V^{\kappa_{c}}(\mathfrak{sl}_{2})$, i.e., the quotient of
$V^{\kappa_{c}}(\mathfrak{g})$ where all elements of the Feigin–Frenkel centre
are set to zero. The universal $V^{\kappa_{c}}(\mathfrak{sl}_{2})$ is
recovered by reintroducing a commutative vertex algebra with generator $S$
that also commutes all of $\Pi^{i}_{{\rm SL}_{2}}$. This plays the role of the
strong generator of the Feigin–Frenkel centre, and is included in the
definition of the lowering operator,

$f_{i}=(S+\partial^{2}\delta_{i}-(\partial\delta_{i})^{2})e_{i}^{-1}~{}.$
(169)

In chiral algebras of class $\SS$, the Feigin–Frenkel generators for the
current algebras at each puncture are identified Arakawa:2018egx ;
Beem:2022mde , and this is accomplished in the free field realisation by
letting the same $S$ appear in each instance of
$V^{\kappa_{c}}(\mathfrak{sl}_{2})$. Indeed, we further identify this with the
Feigin–Frenkel generator of $\mathbf{I}_{{\rm SL}_{2}}\subset\Pi^{0}_{{\rm
SL}_{2}}$, which gives a realisation of $V^{\kappa_{c}}(\mathfrak{sl}_{2})$ as
a subalgebra of $\mathbf{I}_{G}\otimes\Pi^{i}_{{\rm SL}_{2}}$.

A conjectured accounting of strong generators for $\mathbf{V}_{{\rm
SL}_{2},s}$ has appeared previously in the literature (see (Beem:2014rza, ,
Conjectures 1 & 2)). In particular, there are expected to be $s$-many sets of
$\mathfrak{sl}_{2}$ currents, the Virasoro stress tensor, and generators
$X^{a_{1}a_{2}\dots a_{s}}$ which transform in the $s$-fold fundamental
representation $\mathbf{2}^{\otimes s}$ of the zero-mode algebra of the
$\mathfrak{sl}_{2}$ currents. Our approach will be to identify free-field
candidates for these generators, which can then be checked to strongly
generate closed vertex subalgebras.

We start of by constructing the $\mathfrak{sl}_{2}$ currents
$(e_{i},h_{i},f_{i})$ for each puncture $i=1,\dots,s$ as above. Next, we
identify the highest-weight of the $s$-fundamental operator,

$X^{\underbrace{++\dots++}_{s}}\coloneqq\gamma^{-\frac{1}{2}}\prod_{i=1}^{s}e^{\frac{1}{2}}_{i}~{},$
(170)

The remaining components $X^{a_{1}a_{2}\dots a_{s}}$ of the $s$-fundamental
representation are realised by acting with the $f_{i,(0)}$ zero modes. In
general, we have

$X^{a_{1}a_{2}\dots
a_{s}}=\left(\prod_{\\{i\,:\,a_{i}=-\\}}f_{i,(0)}\right)\,X^{++\dots+}~{},$
(171)

For $s\neq 3$ there are expected to be null relations amongst the composites
of the putative strong generators. For small $s$, we will explicitly identify
these null ideals, but in general their structure can be quite intricate.
However, the free field realisation constructs a quotiented vertex algebra, so
some (conjecturally all) of the the null relations will be automatically
satisfied in our realisations of the $\mathbf{V}_{{\rm SL}_{2},s}$. Finding
the exact form of the null relations in the free field realisation amounts to
a book-keeping exercise.

The vertex algebras $\mathbf{V}_{{\rm SL}_{2},s}$ are conformal for all
$s\in\mathbb{N}$ with central charge given by

$c_{s}=(s-(2s-4)h^{\vee}){\rm
dim}~{}\mathfrak{g}-(s-2)\mathrm{rk}~{}\mathfrak{g}=26-10s~{}.$ (172)

We previously identified the conformal vector of $\mathbf{V}_{{\rm
SL}_{2},0}\cong\mathbf{I}_{{\rm SL}_{2}}$ as a canonical free field conformal
vector with appropriate background charges chosen. This continues to be the
case after introducing punctures, and for $\mathbf{V}_{{\rm SL}_{2},s}$ we
define a conformal vector

$T_{s}=\sum_{i=0}^{s}\bigg{(}\partial\delta_{i}\partial\delta_{i}-\partial\phi_{i}\partial\phi_{i}-\alpha_{i}\partial^{2}\phi_{i}-\beta_{i}\partial^{2}\delta\bigg{)}~{}.$
(173)

Demanding that $\Delta_{e_{i}}=\Delta_{p}=1$ and fixing the central charge
forces the background charges to $\alpha_{i>0}=0$, and $\beta_{i>0}=1$. The
background charges $\alpha_{0}$ and $\beta_{0}$ have been set to $\frac{3}{2}$
and $-\frac{1}{2}$ respectively from our arguments in Section 4.2.

This conformal vector will generally be included as an additional strong
generator for $\mathbf{V}_{{\rm SL}_{2},s}$. For the special cases
$s=0,1,2,3,4$, the conformal vector is actually redundant, and can be written
as a composite of the remaining strong generators, so in these cases there
will be a null relation that allows $T_{s}$ to be identified with this
composite. In our free field realisations, this relation will be automatically
satisfied—this was shown above for the case of $s=0$.

To summarise, we present the following conjectures.

######  Conjecture 5.1.

The vertex algebra $\mathbf{V}_{{\rm SL}_{2},s}$, for $s\geqslant 1$, is
isomorphic to the subalgebra of $\Pi_{{\rm SL}_{2}}^{\otimes s+1}$ that is
weakly generated by the moment maps $e_{i},h_{i},f_{i}$ for $i=1,\dots,s$, the
highest weight state $X^{+++\dots+}$. This vertex subalgebra will be simple,
with all null relations in its strongly-generated presentation satisfied
identically.

######  Conjecture 5.2.

The subalgebra of $\Pi^{\otimes s+1}_{{\rm SL}_{2}}$ that is weakly generated
by $e_{i},h_{i},f_{i}$ for $i=1,\dots s$ and $X^{+++\dots+}$ is strongly
generated by

  * •

$e_{i},h_{i},f_{i}$, for $i=1,\dots,s$.

  * •

$X^{a_{1}a_{2}\dots a_{s}}$, for $a_{i}=+,-$.

  * •

$T_{s}$, the conformal vector.

Together, these conjectures imply an algorithmic approach to constructing the
strong generators—matching the expectation of Beem:2014rza —in a free-field
realisation of $\mathbf{V}_{{\rm SL}_{2},s}$ inside $\Pi^{\otimes s+1}_{{\rm
SL}_{2}}$. For $1\leqslant s\leqslant 4$, we will explicitly verify these
statements.

A corollary of these conjectures is that the free field realisations can be
performed recursively. Given $\mathbf{V}_{{\rm SL}_{2},s}$ as a strongly
generated vertex algebra, i.e. not as a free field realisation, one can
construct a realisation of $\mathbf{V}_{{\rm SL}_{2},s+1}$, inside
$\mathbf{V}_{{\rm SL}_{2},s}\otimes\Pi_{{\rm SL}_{2}}$. The vertex algebra
$\mathbf{V}_{{\rm SL}_{2},s}$ comes with affine currents $e_{i},h_{i},f_{i}$
for $i=1,\ldots,s$; let $S$ denote the generator of their common
Feigin–Frenkel centre. We can then construct $e_{s+1},h_{s+1},f_{s+1}$ inside
$\mathbf{V}_{{\rm SL}_{2},s}\otimes\Pi_{{\rm SL}_{2}}$ using that common $S$.
Then taking $X^{a_{1}a_{2}\dots a_{s}}$ to be the state in $\mathbf{V}_{{\rm
SL}_{2},s}$ with weight $\Delta=\frac{1}{2}(s-2)$, which transforms in the
$\mathbf{2}^{\otimes s}$ under the $\mathfrak{sl}_{2}$ actions provided by
$e_{i,(0)},h_{i,(0)},f_{i,(0)}$, we can define

$X^{a_{1}a_{2}\dots a_{s}+}\coloneqq X^{a_{1}a_{2}\dots
a_{s}}e_{s+1}^{\frac{1}{2}}~{},X^{a_{1}a_{2}\dots a_{s}-}\coloneqq
f_{s+1,(0)}X^{a_{1}a_{2}\dots a_{s}+}~{}.$ (174)

Letting $T_{s}$ be the unique conformal vector of $\mathbf{V}_{{\rm
SL}_{2},s}$ with central charge $c_{s}$, then we can define the conformal
vector

$T_{s+1}\coloneqq
T_{s}+\partial\delta_{s+1}\partial\delta_{s+1}-\partial\phi_{s+1}\partial\phi_{s+1}-\partial^{2}\delta_{s+1}~{},$
(175)

with central charge $c_{s+1}$. The vertex algebra $\mathbf{V}_{{\rm
SL}_{2},s+1}$ is isomorphic to the subalgebra of $\mathbf{V}_{{\rm
SL}_{2},s}\otimes\Pi_{{\rm SL}_{2}}$ that is strongly generated by
$e_{i},h_{i},f_{i}$, for $i=1,\dots s+1$; the $X^{a_{1}a_{2}\dots a_{s+1}}$;
and the conformal vector $T_{s+1}$.

Note also that this construction makes manifest the action of the symmetric
group $S_{s}$ on $\mathbf{V}_{{\rm SL}_{2},s}$. The symmetric group acts by
permuting the various affine current algebras as well as by the indices of
$X^{a_{1}a_{2}\dots a_{s}}$. The presence of this automorphism is a
consequence of generalised $S$-duality in four dimensions. The same symmetry
is manifest in the construction of Arakawa:2018egx , which uses $s$-identical
copies of $\mathbf{W}_{{\rm SL}_{2}}\equiv\mathbf{V}_{{\rm SL}_{2},1}$ as the
starting point of its construction of $\mathbf{V}_{{\rm SL}_{2},s}$.

It is an interesting feature of these free field realisations that the
prescription for iterative addition of punctures is a kind of inverse functor
to DS reduction—much like in Adamovic2019 . Given that such an inverse functor
has already been provided by the Feigin–Frenkel gluing functor
$\mathrm{H}^{\frac{\infty}{2}+0}({\cal Z},\mathbf{W}_{G}\otimes-)$ in
Arakawa:2018egx , it is a natural question whether the relation between the
two inverse operations can be clarified, and in particular whether the free
field approach can be understood as providing a solution to the semi-infinite
cohomology problem arising in the Feigin–Frenkel gluing.

###  5.2 Examples

In the remainder of this section we present details of the implementation of
this algorithm for some small values of $s$, leading to explicit constructions
of the corresponding $\mathbf{V}_{{\rm SL}_{2},s}$. We start with $s=1$ and
work up to $s=6$ iteratively.

####  5.2.1 The cap

The cap vertex algebra was dubbed the _equivariant affine ${\cal W}$-algebra_
in Arakawa:2018egx ; here we consider the $\mathfrak{a}_{1}$ case
$\mathbf{V}_{{\rm SL}_{2},1}\simeq\mathbf{W}_{{\rm SL}_{2}}$. A concrete
expression for the strong generators and OPEs of this algebra can be found in
the appendix of loc. cit., and our realisation will eventually reproduce those
results up to field redefinitions. First, however, the general scheme
presented above leads to an alternative presentation for $\mathbf{W}_{{\rm
SL}_{2}}$.

Let $e_{1}$, $h_{1}$, $f_{1}$ be the strong generators of the
$V^{\kappa_{c}}(\mathfrak{sl}_{2})$ subalgebra inside $\Pi^{0}_{{\rm
SL}_{2}}\otimes\Pi^{1}_{{\rm SL}_{2}}$ defined as in (168) and (169). The
highest weight state of the generator $X^{a}$ (transforming in the fundamental
representation of $\mathfrak{sl}_{2}$) is realised by adjoining
$e_{1}^{\frac{1}{2}}$ to the generator $X$ of $\mathbf{I}_{{\rm SL}_{2}}$,
while the lowest weight state is recovered by acting with $f_{1}$. This gives
the following strong generators,

$\begin{split}X^{+}&=\gamma^{-\frac{1}{2}}e_{1}^{\frac{1}{2}}~{},\\\
X^{-}&\coloneqq
f_{1,(0)}X^{+}=-(b+\partial\delta_{1})\gamma^{-\frac{1}{2}}e_{1}^{-\frac{1}{2}}~{}.\end{split}$
(176)

These generators have conformal weight $-\frac{1}{2}$ and have regular OPEs
between themselves. Together with the affine currents, they strongly generate
a vertex algebra with a nontrivial center, the generator $C_{cap}$ of which
takes the form

$C_{cap}=J^{(ab)}X_{a}X_{b}-2(\partial X^{a})X_{a}~{},$ (177)

where we have introduced a matrix $J_{1}^{ab}$ of $\mathfrak{sl}_{2}$ currents
according to

$J_{1}\coloneqq\begin{pmatrix}e_{1}&-\frac{1}{2}h_{1}\\\
-\frac{1}{2}h_{1}&-f_{1}\end{pmatrix}~{},$ (178)

and $X^{+}\equiv X^{1}$, $X^{-}\equiv X^{2}$, and where $\mathfrak{sl}(2)$
indices are lowered according to $\ast_{a}=\epsilon_{ab}\ast^{b}$, with
$\epsilon_{+-}=1$. The central quotient of this vertex algebra where we
enforce $C_{cap}=-\mathbbm{1}$ will be identified with $\mathbf{W}_{{\rm
SL}_{2}}$.

This presentation does not match that of Arakawa:2018egx , which is due to the
amusing fact that this VOA can be equivalently defined to be strongly
generated by a different subset of fields. Indeed, the naive expectation for
the strong generators of the VOA realised by (one-sided) quantum
Drinfel’d–Sokolov reduction from CDOs on ${\rm SL}_{2}$ is that there will be
weight $-\frac{1}{2}$ and a weight $+\frac{1}{2}$ fundamentals of the
remaining $\mathfrak{sl}_{2}$, along with a Feigin–Frenkel generator. The
extra weight $+\frac{1}{2}$ generators can be written as $Y^{a}=J^{ab}X_{b}$,
which in terms of free fields gives

$\begin{split}Y^{+}&=b\gamma^{-\frac{1}{2}}e_{1}^{\frac{1}{2}}~{}.\\\
Y^{-}&\coloneqq
f_{1,(0)}Y^{+}=b(b+\partial\delta_{1})\gamma^{-\frac{1}{2}}e_{1}^{-\frac{1}{2}}+\gamma^{\frac{1}{2}}e_{1}^{-\frac{1}{2}}~{}.\end{split}$
(179)

Meanwhile, the Feigin–Frenkel generator $S=J^{ab}J_{ab}$ is, of course, given
as in $\mathbf{I}_{{\rm SL}_{2}}$ by

$S=b^{2}+\gamma+\partial b~{}.$ (180)

The $X^{a}$, $Y^{a}$, and $S$ also strongly generate $\mathbf{W}_{{\rm
SL}_{2}}$, with their singular OPEs being given by

$\begin{split}S(z)X^{a}(w)&=\frac{\frac{3}{4}X^{a}}{(z-w)^{2}}+\frac{Y^{a}}{z-w}~{},\\\
S(z)Y^{a}(w)&=\frac{\frac{3}{4}Y^{a}}{(z-w)^{2}}+\frac{SX^{a}}{z-w}~{},\\\
Y^{a}(z)X^{b}(w)&=\frac{\frac{1}{2}(X^{a}X^{b})}{z-w}~{},\\\
Y^{a}(z)Y^{b}(w)&=-\frac{\frac{1}{4}(X^{a}X^{b})}{(z-w)^{2}}+\frac{\epsilon^{ab}\frac{1}{2}\big{(}Y^{c}X_{d}-\frac{1}{2}\partial
X^{c}X_{d}\big{)}-\frac{1}{4}\partial X^{a}X^{b}}{z-w}~{}.\end{split}$ (181)

The null relation in this case is a lift of (147) in $\mathbf{I}_{{\rm
SL}_{2}}$, and is given by

$(X^{a}Y_{a})+\frac{1}{2}\big{(}\partial X^{a}X_{a}\big{)}=\mathbbm{1}~{},$
(182)

which can also be thought of as the chiralised version of the determinantal
relation in the equivariant Slodowy slice ${\rm SL}_{2}\times S_{f}$.

In this presentation, the affine currents $e_{1}$, $h_{1}$, $f_{1}$ are
realised as composites,

$\begin{split}e_{1}=&SX^{+}X^{+}-Y^{+}Y^{+}-\frac{3}{2}\big{(}X^{+}\partial
Y^{+}-\partial X^{+}Y^{+})-\frac{3}{4}\partial X^{+}\partial
X^{+}-\frac{1}{8}X^{+}\partial^{2}X^{+}~{},\\\
h_{1}=&-2SX^{-}X^{+}+2Y^{-}Y^{+}+\frac{3}{2}(X^{-}\partial Y^{+}+X^{+}\partial
Y^{-}-\partial X^{-}Y^{+}-\partial X^{+}Y^{-})\\\ &+\frac{3}{2}\partial
X^{+}\partial X^{-}+\frac{1}{4}X^{+}\partial^{2}X^{-}~{},\\\
f_{1}=&-SX^{-}X^{-}+Y^{-}Y^{-}+\frac{3}{2}\big{(}X^{-}\partial Y^{-}-\partial
X^{-}Y^{-}\big{)}+\frac{3}{4}\partial X^{-}\partial
X^{-}+\frac{1}{8}X^{-}\partial^{2}X^{-}~{}.\end{split}$ (183)

Similarly, the stress tensor $T_{1}$ is composite and takes the form

$\begin{split}T_{1}&=-S\big{(}\partial X^{a}X_{a}\big{)}+\partial
Y^{a}Y_{a}+\frac{3}{2}\big{(}\partial X^{a}\partial
Y_{a}\big{)}+\frac{3}{8}\partial^{2}X^{a}\partial
X_{a}+\frac{1}{6}\partial^{3}X^{a}X_{a}~{}.\end{split}$ (184)

(The conformal vector can also be expressed in terms of just the currents and
the $X^{a}$, but the form is not particularly illuminating.) This second
presentation of the cap agrees precisely with the expressions appearing in the
appendix of Arakawa:2018egx  after making the identifications $S\mapsto-S$,
$X^{i}\mapsto-\begin{pmatrix}c\\\ a\end{pmatrix}$ and
$Y^{i}\mapsto\begin{pmatrix}d\\\ b\end{pmatrix}$.

####  5.2.2 The cylinder

Given $\mathbf{W}_{{\rm SL}_{2}}$, we can construct the cylinder VOA as a
subalgebra $\mathcal{D}^{ch}_{{\rm SL}_{2},-2}\subset\mathbf{W}_{{\rm
SL}_{2}}\otimes\Pi_{{\rm SL}_{2}}$. This involves first defining a new
$V^{\kappa_{c}}(\mathfrak{sl}_{2})$ subalgebra inside $\mathbf{W}_{{\rm
SL}_{2}}\otimes\Pi_{{\rm SL}_{2}}$—now with generators
$e_{2},h_{2},f_{2}$—which commutes with the previous affine current algebra.
We further define generators

$\begin{split}X^{a+}&\coloneqq X^{a}e_{2}^{\frac{1}{2}}~{},\\\
X^{a-}&\coloneqq f_{2,(0)}X^{a+}~{}.\end{split}$ (185)

Replacing the $X^{a}$ by their free field expressions from the previous
subsection, we have

$\begin{split}X^{++}&=\gamma^{-\frac{1}{2}}e_{1}^{\frac{1}{2}}e_{2}^{\frac{1}{2}}~{},\\\
X^{-+}&=-(b+\partial\delta_{1})\gamma^{-\frac{1}{2}}e_{1}^{-\frac{1}{2}}e_{2}^{\frac{1}{2}}~{},\\\
X^{--}&=\big{(}(b+\partial\delta_{1})(b+\partial\delta_{2})+\big{)}\gamma^{-\frac{1}{2}}e_{1}^{-\frac{1}{2}}e_{2}^{-\frac{1}{2}}~{}.\end{split}$
(186)

The expressions for $X^{-+}$ and $X^{+-}$ are related by performing the
permutation $1\leftrightarrow 2$ on all subscripts. This arises from the
manifest permutation symmetry of our construction.

The generators $X^{ab}$ live in the bifundamental representation of
$(\mathfrak{sl}_{2})_{1}\times(\mathfrak{sl}_{2})_{2}$, with action given by
the zero modes of $e_{i}$, $h_{i}$, $f_{i}$. The OPEs between the $X^{ab}$ are
regular; the only non-trivial OPEs involve $e_{i}$, $h_{i}$, $f_{i}$.

The (central) null ideal in the cap is now lifted to

$\begin{vmatrix}X^{++}&X^{+-}\\\
X^{-+}&X^{--}\end{vmatrix}=X^{++}X^{--}-X^{+-}X^{-+}\overset{!}{=}\mathbbm{1}~{},$
(187)

and the commutative subalgebra generated by the $X^{ab}$ is identified with
the commutative vertex subalgebra ${\cal O}(J_{\infty}{\rm SL}_{2})$, i.e.,
the chiralisation of ${\cal O}({\rm SL}_{2})$ inside $\mathcal{D}^{ch}_{{\rm
SL}_{2},-2}$. The $X^{ab}$ along with the currents, $e_{1},h_{1},f_{1}$, are
strong generators of $\mathcal{D}^{ch}_{{\rm SL}_{2},-2}$.

The conformal vector is again composite, now being given by

$T_{2}=J^{ab}\partial X_{a}\,\\!^{c}X_{cb}+\partial^{2}X^{ab}X_{ab}+2\partial
X^{ab}\partial X_{ab}~{}.$ (188)

Replacing all strong generators with their free field expressions, this
reduces exactly to the expression for the canonical $T_{2}$ in terms of
lattice bosons with background charges.

####  5.2.3 Three punctures

Now we reach the traditional starting point of the free field realisation of
Beem:2019tfp , the three punctured sphere. The vertex algebra
$\mathbf{V}_{{\rm SL}_{2},3}$ is isomorphic to the symplectic bosons
associated to the $\mathbf{2}\otimes\mathbf{2}\otimes\mathbf{2}$
representation of
$(\mathfrak{sl}_{2})_{1}\times(\mathfrak{sl}_{2})_{2}\times(\mathfrak{sl}_{2})_{3}$.
Repeating the same steps that are by now becoming familiar, we define a
$V^{\kappa_{c}}(\mathfrak{sl}_{2})$ subalgebra inside $\mathcal{D}^{ch}_{{\rm
SL}_{2},-2}\otimes\Pi_{{\rm SL}_{2}}$, generated by $e_{3}$, $h_{3}$, $f_{3}$.
We further define trifundamental generators

$X^{ab+}\coloneqq X^{ab}e_{3}^{\frac{1}{2}}~{},\quad X^{ab-}\coloneqq
f_{3,(0)}X^{ab+}~{},$ (189)

In terms of the full free field realisation, these generators are given by:

$\begin{split}X^{+++}&=\gamma^{-\frac{1}{2}}e_{1}^{\frac{1}{2}}e_{2}^{\frac{1}{2}}e_{3}^{\frac{1}{2}}~{},\\\
X^{-++}&=-(b+\partial\delta_{1})\gamma^{-\frac{1}{2}}e_{1}^{-\frac{1}{2}}e_{2}^{\frac{1}{2}}e_{3}^{\frac{1}{2}}~{},\\\
X^{--+}&=\big{(}(b+\partial\delta_{1})(b+\partial\delta_{2})+\gamma\big{)}\gamma^{-\frac{1}{2}}e_{1}^{-\frac{1}{2}}e_{2}^{-\frac{1}{2}}e_{3}^{\frac{1}{2}}~{},\\\
X^{---}&=-\big{(}(b+\partial\delta_{1})(b+\partial\delta_{2})(b+\partial\delta_{3})+(b+\partial\delta_{1}+\partial\delta_{2}+\partial\delta_{3})\gamma-\partial\gamma\big{)}\gamma^{-\frac{1}{2}}e_{1}^{-\frac{1}{2}}e_{2}^{-\frac{1}{2}}e_{3}^{-\frac{1}{2}}~{},\\\
\end{split}$ (190)

where the other generators are related to the ones displayed by permutations
of the lattice variables. The OPEs can be computed direclty, and now take the
(expected) exceptionally simple form,

$X^{abc}(z)X^{a^{\prime}b^{\prime}c^{\prime}}(w)\sim\frac{\epsilon^{aa^{\prime}}\epsilon^{bb^{\prime}}\epsilon^{cc^{\prime}}}{z-w}~{}.$
(191)

There are no non-trivial null relations amongst the $X^{abc}$, but the
relations in the cylinder now lift to relations identifying the various
current subalgebras as composites of the $X^{abc}$. In particular, we now have

$\begin{split}J^{ab}_{1}=\frac{1}{2}X^{ai_{1}j_{1}}X^{b}_{i_{1}j_{1}}~{},\\\
J^{ab}_{2}=\frac{1}{2}X^{i_{1}aj_{1}}X_{i_{1}\phantom{b}j_{1}}^{\phantom{i_{1}}b}~{},\\\
J^{ab}_{3}=\frac{1}{2}X^{i_{1}j_{1}a}X_{i_{1}j_{1}}^{\phantom{i_{1}j_{1}}b}~{}.\end{split}$
(192)

The stress tensor $T_{3}$ is just the standard symplectic boson stress tensor,
which again reduces to the canonical lattice conformal vector with background
charges.

####  5.2.4 Four Punctures

The free field realisation of $\mathbf{V}_{{\rm SL}_{2},4}$ inside
$\mathbf{V}_{{\rm SL}_{2},3}\otimes\Pi_{{\rm SL}_{2}}$ can be found in
Beem:2019tfp , but we will revisit the computation in the present context. We
again realise a new $V^{\kappa_{c}}(\mathfrak{sl}_{2})$ subalgebra inside
$\mathbf{V}_{{\rm SL}_{2},3}\otimes\Pi_{{\rm SL}_{2}}$, and as above, adopt a
basis for the currents $J^{ab}_{4}$.

The quadrifundamental generators take the usual form,

$X^{a_{1}a_{2}a_{3}+}\coloneqq X^{a_{1}a_{2}a_{3}}e_{4}^{\frac{1}{2}}~{},\quad
X^{a_{1}a_{2}a_{3}-}\coloneqq f_{4,(0)}X^{a_{1}a_{2}a_{3}+}~{}.$ (193)

which can be expressed entirely in terms of free fields as follows

$\begin{split}X^{++++}&=\gamma^{-\frac{1}{2}}e_{1}^{\frac{1}{2}}e_{2}^{\frac{1}{2}}e_{3}^{\frac{1}{2}}e_{4}^{\frac{1}{2}}~{},\\\
X^{-+++}&=-(b+\partial\delta_{1})\gamma^{-\frac{1}{2}}e_{1}^{-\frac{1}{2}}e_{2}^{\frac{1}{2}}e_{3}^{\frac{1}{2}}e_{4}^{\frac{1}{2}}~{},\\\
X^{--++}&=\big{(}(b+\partial\delta_{1})(b+\partial\delta_{2})+\gamma\big{)}\gamma^{-\frac{1}{2}}e_{1}^{-\frac{1}{2}}e_{2}^{-\frac{1}{2}}e_{3}^{\frac{1}{2}}e_{4}^{\frac{1}{2}}~{},\\\
X^{---+}&=-\big{(}(b+\partial\delta_{1})(b+\partial\delta_{2})(b+\partial\delta_{3})+(b+\partial\delta_{1}+\partial\delta_{2}+\partial\delta_{3})\gamma-\partial\gamma\big{)}\gamma^{-\frac{1}{2}}e_{1}^{-\frac{1}{2}}e_{2}^{-\frac{1}{2}}e_{3}^{-\frac{1}{2}}e_{4}^{\frac{1}{2}}~{},\\\
X^{----}&=\bigg{(}\prod_{i=1}^{4}(b+\partial\delta_{i})+\sum_{i<j}^{4}(b+\partial\delta_{i})(b+\partial\delta_{j})\gamma-2(b+\partial\delta_{1}+\partial\delta_{2}+\partial\delta_{3}+\partial\delta_{4})(b+\partial\gamma^{\frac{1}{2}}\gamma^{-\frac{1}{2}})\gamma\\\
&-2(bb+\partial
b)\gamma+b\partial\gamma+\gamma^{2}+\partial^{2}\gamma\bigg{)}\gamma^{-\frac{1}{2}}e_{1}^{-\frac{1}{2}}e_{2}^{-\frac{1}{2}}e_{3}^{-\frac{1}{2}}e_{4}^{-\frac{1}{2}}~{}.\end{split}$
(194)

Again one finds free-field expressions for the unlisted generators by applying
the appropriate permutation on subscripts. The OPE between the generators is

$X^{a_{1}a_{2}a_{3}a_{4}}(z)X^{b_{1}b_{2}b_{3}b_{4}}(w)\sim\frac{2\prod_{i=1}^{4}\epsilon^{a_{i}b_{i}}}{(z-w)^{2}}+\frac{\sum_{i=1}^{4}\prod_{j\neq
i}\epsilon^{a_{j}b_{j}}J^{(a_{i}b_{i})}(w)}{z-w}~{},$ (195)

and taken together with the three sets of $V^{\kappa_{c}}(\mathfrak{sl}_{2})$
currents these are strong generators of $L_{-2}(\mathfrak{d}_{4})$, the simple
quotient of $V^{-2}(\mathfrak{d}_{4})$. The null ideals of this vertex algebra
and their relation to Joseph’s minimal realisations Joseph:1974min  were
investigated in Beem:2019tfp .

####  5.2.5 Five Punctures

Continuing further, we come to a realisation of $\mathbf{V}_{{\rm SL}_{2},5}$.
We define the usual $e_{5},h_{5},f_{5}$ currents and the generators
$X^{a_{1}a_{2}a_{3}a_{4}a_{5}}$ as before. In terms of free-fields, the
expressions are somewhat burdensome to write down. (Additionally, it is only
the state $X^{-----}$ that is novel. All other states can be constructed (by
appealing to symmetry) from the free-field realisations we have already
constructed.)

The nontrivial OPEs are the OPEs of the pentafundamental generators
$X^{a_{1}a_{2}a_{3}a_{4}a_{5}}$ amongst themselves, and at this stage the OPEs
are very intricate and so to keep the expressions to a manageable length we
will suppress all conformal descendants, i.e. we only show contributions from
conformal primaries in the OPE. We then have

$\begin{split}X^{a_{1}a_{2}a_{3}a_{4}a_{5}}(z)X^{b_{1}b_{2}b_{3}b_{4}b_{5}}(w)&\sim\frac{4\prod_{i=1}^{5}\epsilon^{a_{i}b_{i}}}{(z-w)^{3}}+\frac{2\sum_{i=1}^{5}\prod_{j\neq
i}^{5}\epsilon^{a_{j}b_{j}}J_{i}^{(a_{i}b_{i})}(w)}{(z-w)^{2}}+\\\
&\frac{1}{z-w}\bigg{(}\sum_{i<j}^{5}\prod_{k\neq
i,j}^{5}\epsilon^{a_{k}b_{k}}(J_{i}^{(a_{i}b_{i})}J_{j}^{(a_{j}b_{j})})(w)-2\prod_{i=1}^{5}\epsilon^{a_{i}b_{i}}(S+\tfrac{1}{4}T_{5})(w)\bigg{)}~{}.\end{split}$
(196)

Here, we have the appearance of a new primary $S+\frac{1}{4}T_{5}$. In
general, one can show that the state $S-\frac{6}{c_{s}}$ is a primary for
$T_{s}$, with the appropriate value of the central charge.

Finding the exact expression for the generators of the null ideal is now
somewhat involved, though largely a book-keeping exercise using the free field
expressions. We will not reproduce them here or in the next example.

####  5.2.6 Six Punctures

We conclude with the realisation of $\mathbf{V}_{\mathfrak{a}_{1},6}$, which
has strong generators $(e_{i},h_{i},f_{i})$ for $i=1,2,\dots,6$, the
hexafundamental $X^{a_{1}a_{2}a_{3}a_{4}a_{5}a_{6}}$, and the stress tensor
$T_{6}$. The nontrivial OPE is the $X\times X$ self-OPE of the
hexafundamental, which takes the form

$\begin{split}X^{a_{1}a_{2}a_{3}a_{4}a_{5}a_{6}}(z)X^{b_{1}b_{2}b_{3}b_{4}b_{5}b_{6}}(w)&\sim\frac{8\prod_{i=1}^{6}\epsilon^{a_{i}b_{i}}}{(z-w)^{4}}+4\frac{\sum_{i}\prod_{j\neq
i}\epsilon^{a_{j}b_{j}}J^{(a_{i}b_{i})(w)}}{(z-w)^{3}}+\\\
&\frac{1}{(z-w)^{2}}\bigg{(}2\sum_{i<j}\prod_{k\neq
i,j}^{6}\epsilon^{a_{k}b_{k}}(J^{(a_{i}b_{i})}J^{(a_{j}b_{j})})(w)-6\prod_{i=1}^{6}\epsilon^{a_{i}b_{i}}(S+\tfrac{3}{17}T_{6})\bigg{)}+\\\
&\frac{1}{(z-w)}\bigg{(}\sum_{i<j<k}\prod_{l\neq
i,j,k}\epsilon^{a_{l}b_{l}}(J^{(a_{i}b_{i})}J^{(a_{j}b_{j})}J^{(a_{k}b_{k})})\\\
&-2\sum_{i}^{6}\prod_{k\neq
i}\epsilon^{a_{k}b_{k}}\bigg{(}(S+\tfrac{3}{17}T_{6})J^{(a_{i}b_{i})}+A^{a_{i}b_{i}}\bigg{)}\end{split}$
(197)

where there appear novel primary operators, $A^{a_{i}b_{i}}$, given by

$A_{i}^{a_{i}b_{i}}=\bigg{(}(J_{i})^{a_{i}}_{c}(\partial
J_{i})^{cb_{i}}-\frac{1}{8}T(J_{i})^{a_{i}b_{i}}-\frac{5}{48}(\partial^{2}J_{i})^{a_{i}b_{i}}\bigg{)}~{}.$
(198)

It may be noteworthy that there seems to be some numerology involved in the
residues appearing in the OPE. This is not entirely surprising, since the OPEs
in the free field realisation are computed by Wick contractions—which are
manifestly of a combinatorial nature. It would be interesting to look for a
more elegant combinatorial description of these OPEs that makes the
$n$-punctured generalisation immediate, but we will not pursue this for now.

##  6 Concluding remarks

###  6.1 Screening charges

In many instances, free-field realisations of vertex algebras are accompanied
by certain screening charges whose kernel recovers the subalgebra of interest
(cf. Adamovic2019 ; Adamovic2003115 ; deBoer:1996ti ; Frenkel:2007 ). One can
then often think of a free field realisation of a vertex algebra as a
resolution whose first differential is the screening charge.

In the free field realisation of Wakimoto modules, for example, such screening
charges exist and admit a geometric interpretation Frenkel:geo2008 . Namely,
they arise from the Cousin–Grothendieck spectral sequence associated to the
Bruhat stratification on the flag variety; the free-field realisation is a
chiral analogue of the BGG resolution for $\mathfrak{g}$-modules.

A slight modification of the arguments in loc. cit. can be applied to our
case. First, we fix some partial ordering $w_{0}\geq w_{1}\geq w_{2}\geq\dots
w_{N}$ on the Weyl group $W$, with $w_{0}$ the longest element as before. Then
$G$ has a stratification

$G=\underset{i}{\bigsqcup}\,G^{i}~{},$ (199)

where $G^{i}=Bw_{i}B$ and $G^{0}\equiv G^{*}$. Let $j_{i}:G^{i}\hookrightarrow
G$ denote the immersions of the strata. If $M$ is a $D$-module on $G$, then
the Cousin–Grothendieck spectral sequence tells us that $M$ has a resolution

$0\rightarrow M\hookrightarrow
j_{0*}j_{0}^{!}M\xrightarrow{\delta_{0}}j_{1*}j_{1}^{!}M\xrightarrow{\delta_{1}}\dots\xrightarrow{\delta_{N-1}}j_{N*}j_{N}^{!}M\rightarrow
0~{}.$ (200)

In particular, if $M={\cal D}_{G}$ is the sheaf of differential operators on
$G$, then $j_{0}^{!}{\cal D}_{G}$ can be identified with the sheaf of
differential operators on the big cell, ${\cal D}_{G^{*}}$. From
(Arakawa:tdo2011, , Theorem 5.2.), this lifts to a resolution of the sheaf of
(critical level) chiral differential operators on $G$,
$\mathcal{D}^{ch}_{G,\kappa_{c}}$. Taking global sections, we have a complex
of vertex algebras $(C^{\bullet},\delta_{\bullet})$, with
$C^{0}=\mathcal{D}^{ch}_{G^{*},\kappa_{c}}(G^{*})$.

To construct a resolution of the chiral universal centraliser, we can perform
the two-sided Drinfel’d–Sokolov reduction on $\mathcal{D}^{ch}_{G}(G)$. We
form the tri-complex $(C^{\bullet\bullet\bullet},\delta,d_{L},d_{R})$, where
$d_{L}$ and $d_{R}$ are the BRST differentials computing the left and right DS
reductions, respectively. Passing to $\delta$-cohomology and then $d_{L}$ and
$d_{R}$ cohomology will, by construction, recover the chiral universal
centraliser in cohomological degree zero.

Note that the DS reduction subcomplex is acyclic away from degree zero (see
Frenkel:lan2006  for more detail), so a spectral sequence argument implies
that we can algernatively compute the $d_{L}$ and $d_{R}$ cohomology first
before computing the $\delta$-cohomology and still realise the chiral
universal centraliser in degree zero. Since $G^{*}\cong N\times T\times N$,
the two-sided DS reduction gives $\mathcal{D}^{ch}_{T}(T)$ in degree zero. The
resulting complex, $(\tilde{C}^{\bullet},\delta)$ has zeroth term
$\tilde{C}^{0}=\mathcal{D}^{ch}_{T}(T)$, with

$H^{0}(\tilde{C},\delta)\equiv\mathrm{ker}~{}\delta_{0}\cong\mathbf{I}^{cl}_{G}~{}.$
(201)

Of course, $\mathcal{D}^{ch}_{T}(T)$ is isomorphic to the free field vertex
algebra $\Pi_{G}$, so the above presentation suggests a reinterpretation of
our free-field realisation along with some kind of screening charge
$\delta_{0}$, i.e., the restriction of the Cousin–Grothendieck differential.
However, the above analysis at this level of abstraction does not lead to an
explicit expression for a screening charge in the usual sense. We leave a more
detailed and concrete analysis for future work.

###  6.2 Monopoles and Abelianisation

Our realisation of the universal centraliser in terms of functions on the
Kostant–Toda lattice is somewhat evocative of Abelianisation-type
constructions, in that $\mathrm{KT}_{G}\cong
T^{*}(\mathbb{C}^{\times})^{\mathrm{rk}\,\mathfrak{g}}\cong T^{*}\check{T}$
and $T^{\ast}\mathbb{C}^{\times}$ may be identified as the Coulomb branch of
pure $U(1)$ $\mathcal{N}=4$ gauge theory in three dimensions. Indeed, the
construction of Braverman–Finkelberg–Nakajima for the Coulomb branches of
gauge theories comes with an Abelianisation map. Let $\check{T}$ be the
Langlands dual of the maximal torus $T$, and let
$\mathrm{Gr}_{\check{T}}=\check{T}({\cal K})/\check{T}({\cal O})$ be the
affine Grassmannian. Since $\check{T}$ is abelian, the affine Grassmannian is
just a collection of points, parameterised by the coweights of $\check{T}$.
There is a closed embedding
$i:\mathrm{Gr}_{\check{T}}\hookrightarrow\mathrm{Gr}_{\check{G}}$, which gives
rise to a dominant morphism Braverman:2016wma ,

$\mathfrak{Z}_{G}\cong\mathrm{Spec}~{}H^{\check{G}({\cal
O})}_{\bullet}(\mathrm{Gr}_{\check{G}})\rightarrow\mathrm{Spec}~{}H^{\check{T}({\cal
O})}_{\bullet}(\mathrm{Gr}_{\check{T}})/\\!\\!/W\cong(\mathfrak{t}\times\check{T})/\\!\\!/W~{}.$
(202)

This is actually an isomorphism away from the closed subspace
$\mathfrak{t}^{\circ}=\bigcup_{\alpha\in\Delta}\mathfrak{t}_{\alpha}$, where
$\mathfrak{t}_{\alpha}$ is the root hyperplane $\alpha=0$. As a result,
$\mathfrak{Z}_{G}$ and $(\mathfrak{t}\times\check{T})/\\!\\!/W$ are
birational. In Bezrukavnikov:2003 , this was used to construct the universal
centraliser in terms of a blow up of $(\mathfrak{t}\times\check{T})/\\!\\!/W$.

In the physics literature, $(\mathfrak{t}\times\check{T})/\\!\\!/W$ is just
the classical description of the Coulomb branch, which receives quantum
corrections from monopole operators Bullimore:2015lsa . It is tempting, then,
to attempt to interpret our free field realisations (and the corresponding
finite-dimensional constructions) as an instance of this Abelianisation
paradigm. However, such an interpretation is not immediately forthcoming, as
in particular for us the Poisson structure on the “Abelian” space
($\mathrm{KT}_{G}$) is _not corrected_ , by monopole operators or otherwise.

A construction of Darboux coordinates by Kato has appeared in Kato:2020jup ,
however this is for the $K$-theoretic Coulomb branch of pure gauge theory,
which is not the universal centraliser.

Nevertheless, it would be interesting to investigate the possibility of a link
between the two pictures. This could help frame our construction in terms that
are more in keeping with both physics and geometric representation theory, as
well as offering the possibility for generalisation away from the case of pure
gauge theory.

###  6.3 Open questions

In addition to the various conjectures outlined in the main body of the
papers, there are a number of issues that remain open to which we hope someone
(ourselves or otherwise) will return.

First of all, in starting our construction of chiral algebras of class $\SS$
with the universal centraliser (of type ${\mathrm{S}L}_{2}$), it is clear that
the vertex algebras $\mathbf{V}_{{\mathrm{S}L}_{2},s}$ are sensitive to the
choice of the Lie _group_. Indeed, one could replicate the construction,
starting with $\mathbf{I}_{{\mathrm{P}GL}_{2}}$, and recover a series of
vertex algebras $\mathbf{V}_{{\mathrm{P}GL}_{2},s}$ which are not isomorphic
to $\mathbf{V}_{{\mathrm{S}L}_{2},s}$. It would seem natural to expect the
$\mathbf{V}_{{\mathrm{P}GL}_{2},s}$ to be finite quotients of the
$\mathbf{V}_{{\mathrm{S}L}_{2},s}$ with respect to the
$\mathbb{Z}/2\mathbb{Z}$ action of the centre. However, it is unclear whether
passing to the $\mathbb{Z}/2\mathbb{Z}$ invariants in $\mathbf{V}_{SL_{2},s}$
commutes with passing to the Zhu’s $C_{2}$ algebra.

The most natural progression is to continue the free-field program for
$\mathbf{V}_{G,s}$ to groups of higher rank. For ${\rm SL}_{3}$, we already
have a concrete presentation for the chiral universal centraliser and one
could hope in this manner to obtain a presentation of the equivariant affine
${\cal W}$-algebra as a strongly generated vertex algebra. However, the
calculations for ${\rm SL}_{2}$ already become unwieldy at six punctures. To
establish the free field realisation on stronger footing we would need to
develop a more abstract formalism—rather than the algorithmic approach we have
taken so far. We hope to develop such a formalism in upcoming work BBS:unpub .

The free-field realisations naturally give rise to a construction of a class
of vertex algebra modules. Namely, there are natural restriction functors
arising from $\Pi_{{\rm SL}_{2}}^{\otimes s}$ modules to modules of
$\mathbf{V}_{{\rm SL}_{2},s}$. There is also, of course, a web of such
restrictions, e.g., from $\Pi_{{\rm SL}_{2}}\otimes\mathbf{V}_{{\rm
SL}_{2},s}$ to $\mathbf{V}_{{\rm SL}_{2},s+1}$ modules. One would hope that
the properties of these functors give some partial characterisation of the
modules of the chiral algebras of class $\SS$, which are of interest in both
four dimensional and three dimensional physics.

## Acknowledgments

The authors are grateful to Tomoyuki Arakawa, Dylan Butson, Carlo Meneghelli,
and Leonardo Rastelli for useful discussions and collaborations on related
topics. Christopher Beem’s work is supported in part by grant #494786 from the
Simons Foundation, by ERC Consolidator Grant #864828 “Algebraic Foundations of
Supersymmetric Quantum Field Theory” (SCFTAlg), and by the STFC consolidated
grant ST/T000864/1. Sujay Nair is supported in part by EPSRC studentship
#2272671.

##  Appendix A Proof of Theorem 2.2

In this appendix, we prove Theorem 2.2 regarding the Poisson structure of the
universal centraliser for $G={\rm SL}_{N}$. Namely, we show the following.

######  Theorem.

The Poisson algebra $I_{{\rm SL}_{N}}$ is Poisson generated by
$(P_{i})_{i=1}^{\mathrm{rk}\,\mathfrak{g}}$ and the generator $X=g_{N1}$.

###### Proof.

Throughout this appendix we adopt the notation $G={\rm SL}_{N}$ and
$\mathfrak{g}=\mathfrak{sl}_{N}$. The proof makes extensive use of the
realisation of $\mathfrak{Z}_{G}$ as the two-sided Kostant–Whittaker reduction
of $T^{*}G$. As such, we start with some preliminaries and also recall certain
technical details regarding Poisson (coisotropic) reduction in the present
setting.

###  A.1 Preliminaries

Consider the triple $\mathfrak{g}^{*}_{L}\times\mathfrak{g}^{*}_{R}\times G$,
where the subscripts simply distinguish the two $\mathfrak{g}^{*}$ factors.
Choose generators $\widetilde{g}_{ij}$—conjugate to the matrix elements of a
$G$ matrix—for
$\mathbb{C}[G]\subset\mathbb{C}[\mathfrak{g}_{L}^{*}\times\mathfrak{g}^{*}_{R}\times
G]$. This space is equipped with a Poisson structure by letting linear
functions in $\mathbb{C}[\mathfrak{g}^{*}_{R/L}]\cong\rm Sym~{}\mathfrak{g}$
act according to the left (right) $\mathfrak{g}$ action on the
$\widetilde{g}_{ij}$, which then organise themselves into the bifundamental
representation $V_{F}\otimes V_{F}^{*}$.

The cotangent bundle $T^{\ast}G$ is embedded in
$\mathfrak{g}^{*}_{L}\times\mathfrak{g}^{*}_{R}\times G$ as a closed, Poisson
subvariety,

$T^{*}G=\\{(x,y,g)\in\mathfrak{g}^{*}_{L}\times\mathfrak{g}^{*}_{R}\times
G~{}|~{}Ad^{*}_{g}x=y\\}~{}.$ (203)

Let ${\cal J}$ be the corresponding defining ideal for $T^{*}G$; it is linear
in the generators of $\mathbb{C}[\mathfrak{g}^{*}_{L/R}]$ and is degree $N$ in
the generators, $\widetilde{g}_{ij}$.

The moment maps for left and right $G$ actions on $T^{*}G$ are inherited from
the projections
$\pi_{L/R}:\mathfrak{g}^{*}_{L}\times\mathfrak{g}^{*}_{R}\times
G\rightarrow\mathfrak{g}^{\ast}_{L/R}$, respectively, which are the
$\mathfrak{g}_{L}$ and $\mathfrak{g}_{R}$ moment maps of the ambient space. We
will study the two-sided KW reduction of this ambient space and carefully
analyse the fate of the linear coordinate functions $\widetilde{g}_{Ni}$ in
particular.

Let ${\cal
X}\colonequals(\chi+\mathfrak{b}^{*})_{L}\times(\chi+\mathfrak{b}^{*})_{R}\times
G=\pi_{L}^{-1}(\chi+\mathfrak{b}^{*})\cap\pi_{R}^{-1}(\chi+\mathfrak{b})$,
which is a coisotropic subvariety of our ambient space. The two-sided KW
reduction is then equivalent to the quotient ${\cal W}\coloneqq{\cal
X}/\\!\\!/N_{L}\times N_{R}$. Let $\pi_{LR}$ be the corresponding projection
$\pi_{LR}:{\cal X}\rightarrow{\cal W}$.

To realise the two-sided KW reduction of $T^{*}G$, we first define its
intersection ${\cal Y}\colonequals T^{\ast}G\cap{\cal X}$. The reduction in
question is then given by ${\cal Y}/\\!\\!/N_{L}\times
N_{R}\cong\mathfrak{Z}_{G}$. Summarising, we have the following base
change,111111By a slight abuse of notation we also denote the natural
projection ${\cal Y}\rightarrow{\cal Y}/\\!\\!/N_{L}\times N_{R}$ by
$\pi_{LR}$.

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\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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}\pgfsys<EMAIL_ADDRESS>(204)

Each square is Cartesian; they both describe ${\cal Y}$ as a fibre product.
The arrows directed down and to the left in the diagram are all closed
embeddings. Moreover,
$T^{*}G\rightarrow\mathfrak{g}_{L}^{*}\times\mathfrak{g}_{R}^{*}\times G$ is
Poisson, so $\mathfrak{Z}_{G}\rightarrow{\cal W}$ is also Poisson by
functoriality.

The closed subvariety $S_{f}\times S_{f}\times
G\subset(\chi+\mathfrak{b}^{*})_{L}\times(\chi+\mathfrak{b}^{*})_{R}\times G$
is a global slice to the $N_{L}\times N_{R}$ action on ${\cal X}$, so ${\cal
W}$ can be identified with $S_{f}\times S_{f}\times G$ as varieties. Letting
${\cal I}$ denote the defining ideal of $S_{f}\times S_{f}\times G$ inside
${\cal X}$, the Poisson structure on ${\cal W}\simeq S_{f}\times S_{f}\times
G$ is then defined in the following (standard) manner.

For every $a,b\in\mathbb{C}[{\cal W}]$ we have canonical extensions
$\pi_{LR}^{\\#}(a),\pi_{LR}^{\\#}(b)\in\mathbb{C}[{\cal X}]$. Choose any
further extensions
$A,B\in\mathbb{C}[\mathfrak{g}^{*}_{L}\times\mathfrak{g}^{*}_{R}\times G]$.
Then the Poisson bracket on ${\cal W}$ is the unique bracket satisfying

$\pi_{LR}^{\\#}(\\{a,b\\}_{{\cal
W}})=\iota^{\\#}(\\{A,B\\}_{\mathfrak{g}_{L}^{*}\times\mathfrak{g}_{R}^{*}\times
G})~{}.$ (205)

In the present case, things are considerably simpler than in general. Since
$G\times S_{f}\times S_{f}$ is a slice, the following triangle commutes.

${{\cal X}}$${{\cal W}}$${S_{f}\times S_{f}\times
G}$$\scriptstyle{\pi_{LR}}$$\scriptstyle{\sim}$ (206)

On functions, this means that if $f\in\mathbb{C}[{\cal W}]$ is identified with
to $f^{\prime}\in\mathbb{C}[S_{f}\times S_{f}\times G]$ under the isomorphism,
then its extension $\pi^{\\#}_{LR}(f)$ has the form
$\pi^{\\#}_{LR}(f)=f^{\prime}+{\cal I}$. Consequently, determining the
extension $\pi_{LR}^{\\#}$ is equivalent to identifying an injective section

$\sigma:\mathbb{C}[S_{f}\times S_{f}\times G]\cong\mathbb{C}[{\cal X}]/{\cal
I}\rightarrow\mathbb{C}[{\cal X}]~{},$ (207)

whose image is isomorphic to $\mathbb{C}[{\cal X}]^{N_{L}\times N_{R}}$.

We are free to choose any further extension to
$\mathbb{C}[\mathfrak{g}^{*}_{L}\times\mathfrak{g}^{*}_{R}\times G]$, and in
particular we can choose these extensions to live in the Poisson subalgebra
$\mathbb{C}[\mathfrak{b}^{*}\times\mathfrak{b}^{*}\times G]^{N_{L}\times
N_{R}}$ of $\mathbb{C}[\mathfrak{g}^{*}\times\mathfrak{g}^{*}\times G]$, which
is identified as a ring with $\mathbb{C}[{\cal X}]^{N_{L}\times N_{R}}$, the
target of $\sigma$.

Summing up, for $a,b\in\mathbb{C}[S_{f}\times S_{f}\times G]$, their Poisson
bracket can be computed as

$\sigma\big{(}\\{a,b\\}_{S_{f}\times S_{f}\times
G}\big{)}=\\{\sigma(a),\sigma(b)\\}_{\mathfrak{g}^{*}_{L}\times\mathfrak{g}^{*}_{R}\times
G}~{},$ (208)

where we interpret $\sigma(a),\sigma(b)$ as elements of the Poisson subalgebra
$\mathbb{C}[\mathfrak{b}^{*}_{L}\times\mathfrak{b}^{*}_{R}\times
G]^{N_{L}\times N_{R}}$ as described above. With this technical detail out of
the way, we now move on to the proof of Theorem 2.2 itself.

###  A.2 Reducing coordinates

Let $\Phi$ be the set of positive roots of $\mathfrak{g}$; we want to
introduce some coordinates on $\mathfrak{g}^{*}$. For $\mathfrak{t}^{*}$ we
shall introduce the linear duals, $(h_{i})_{i=1}^{\mathrm{rk}\,\mathfrak{g}}$
in $\mathfrak{t}$ (as in the case of $\mathrm{KT}_{\mathfrak{g}}$). Now for
$\mathfrak{n}^{*}$, if $(e_{\alpha})_{\alpha\in\Phi}$ are a basis of
$\mathfrak{n}$ then $\big{(}(e_{\alpha},-)\big{)}_{\alpha\in\Phi}$ form a
basis of $\mathfrak{n}^{*}$. Therefore, the linear dual to $(e_{\alpha},-)$ is
$e_{-\alpha}\in\mathfrak{n}_{-}$. We choose $(e_{-\alpha})_{\alpha\in\Phi}$ to
be coordinates on $\mathfrak{n}^{*}$. The ideal ${\cal I}$ is generated by
$h_{i,L}$, $h_{i,R}$ for $i=1,\ldots N$ along with a linearly independent
basis of the differences,121212As previously, we append subscripts $L$ and
$R$ to distinguish between the generators of
$\mathbb{C}[(\chi+\mathfrak{b}^{*})_{L/R}]$.

$e_{-\alpha,L}-e_{-\beta,L}~{},e_{-\alpha,R}-e_{-\beta,R}~{},$ (209)

where $\alpha,\beta\in\Phi$ are two roots with the same height.

The generator $P_{1}\in\mathbb{C}[S_{f}]$ is extended to
$\mathbb{C}[\chi+\mathfrak{b}^{*}]$ as the restricted quadratic fundamental
invariant, so the associated left and right versions are realised according to

$\sigma(P_{1,L/R})=\kappa^{ij}h_{i,L/R}h_{j,L/R}+\sum_{\alpha\in\Delta}e_{-\alpha,L/R}~{},$
(210)

where we recall that $\Delta$ is the set of simple roots of $\mathfrak{g}$ and
$\kappa^{ij}$ are the components of the Killing form.

Next consider the coordinate functions on $G$. We will use $g_{ij}$ for the
generators of $\mathbb{C}[G]\subset\mathbb{C}[S_{f}\times S_{f}\times G]$, and
continue to use $\widetilde{g}_{ij}$ for the generators of
$\mathbb{C}[G]\subset{\cal X}$. The subalgebra generated by “bottom row”
entries $(\widetilde{g}_{Ni})_{i=1}^{N}$ is $N_{L}$-invariant, while that
generated by “left column” entries, $(\widetilde{g}_{i1})_{i=1}^{N}$ is
$N_{R}$-invariant. In particular, $\widetilde{g}_{N1}$ is itself $N_{L}\times
N_{R}$-invariant since it is the highest weight vector. So the function
$g_{1N}$ on $S_{f}\times S_{f}\times G$ extends in an $N_{L}\times
N_{R}$-invariant way to all of ${\cal X}$, and
$\sigma(g_{N1})=\widetilde{g}_{N1}$.

Consider now the function $\sigma(\\{P_{1,R},g_{N1}\\})\in\mathbb{C}[{\cal
X}]$—the Poisson bracket is computed in
$\mathbb{C}[\mathfrak{g}^{*}_{L}\times\mathfrak{g}^{*}_{R}\times G]$ using
(210), which gives

$\sigma\big{(}\\{P_{1,R},g_{N1}\\}\big{)}=\widetilde{g}_{N2}+2\sum_{i,j=1}^{\mathrm{rk}\,\mathfrak{g}}\kappa^{ij}h_{i,R}a_{i}(h_{j})g_{N1}~{},$
(211)

where $a_{i}$ are the components of the highest weight of the fundamental
representation of ${\rm SL}_{N}$. By construction, this Poisson bracket must
lie in $\mathbb{C}[{\cal X}]^{N_{L}\times N_{R}}$, and so lies in the image of
$\sigma$. Restricting the expression (211) to $S_{f}\times S_{f}\times G$
returns $g_{N2}$. Since $\sigma$ is injective, this fixes $\sigma(g_{N2})$ to
the above expression.

The sum of simple negative roots acts on the fundamental representation of
${\rm SL}_{N}$ as a lowering operator, Therefore, from (210)

$\\{\sigma(P_{1,R}),\widetilde{g}_{Ni}\\}=\widetilde{g}_{N,i+1}+c_{i}g_{N,i}~{},$
(212)

for $i=1,\ldots,N-1$ where $c_{i}$ are linear in the $(h_{j})_{j=1}^{N}$.

######  Lemma A.1.

For $i=1,\dots,N$, the extension of $g_{Ni}$ is of the form

$\sigma(g_{Ni})=\widetilde{g}_{N,i+1}+\sum_{j=1}^{i-1}K_{ij}\widetilde{g}_{Nj}~{},$
(213)

where $K_{ij}$ are homogeneous, degree $j$ polynomials in the generators of
${\cal I}$.

###### Proof.

We proceed by induction, noting that the case $i=1,2$ have been verified. Now
suppose $i<N$. From (210) and the inductive assumption, we have

$\\{\sigma(P_{1,R},\sigma(g_{Ni})\\}=\widetilde{g}_{N,i+1}+c_{i}g_{Ni}+\sum_{j=1}^{i-1}K_{ij}\\{P_{1,R},\widetilde{g}_{N,j}\\}+\sum_{j=1}^{i-1}\\{\sigma(P_{1,R}),K_{ij}\\}\widetilde{g}_{N,j}~{}.$
(214)

The subleading terms are all in ${\cal I}$, except for the final term which we
split as $\\{\sigma(P_{1,R}),K_{ij}\\}=F_{ij}+{\cal I}$, where $F_{ij}$ are
homogeneous polynomials of degree $j$ in the $(P_{k,R})_{k=1}^{N}$ (restricted
to $S_{f}$). Restricting to $S_{f}\times S_{f}\times G$, we have
$\\{P_{1,R},g_{Ni}\\}=g_{N,i+1}+\sum_{j}^{i-1}F_{ij}g_{Nj}$.

Therefore, the extension of $g_{N,i+1}$ is

$\sigma(g_{N,i+1})=\widetilde{g}_{N,i+1}+c_{i}g_{Ni}+\sum_{j=1}^{i-1}K_{j}\widetilde{g}_{N,j+1}~{}+\sum_{j=1}^{i-1}K_{j}c_{j}\widetilde{g}_{Nj}~{},$
(215)

which is in the desired form. ∎

In the proof of Lemma A.1 we argued that for $i=1,\dots,N-1$, if
$\sigma(g_{Ni})$ have the form of (213) then $\sigma(g_{N,i+1})$ can be
written as a linear combination of $\\{\sigma(P_{1,R}),\sigma(g_{N,i+1})\\}$
and the $(F_{ij}g_{Nj})_{j=1}^{i-1}$. By restricting to $S_{f}\times
S_{f}\times G$, we find an analogous result for the $(g_{Ni})_{i=1}^{N}$. By
induction, this means that each $g_{Ni}$ are Poisson generated by $g_{N1}$ and
the fundamental invariants, $(P_{i,R})_{i=1}^{N}$.

All that remains is to consider the restriction of these functions to
$\mathfrak{Z}_{G}\subset S_{f}\times S_{f}\times G$. The embedding
$\mathfrak{Z}_{G}\rightarrow S_{f}\times S_{f}\times G$ has image

$\\{(s,s^{\prime},g)\in S_{f}\times S_{f}\times
G~{}|~{}\mathrm{Ad}^{*}_{g}s=s^{\prime}\\}~{}.$ (216)

Since $S_{f}$ is transversal to the $G$-action, two distinct points on the
slice cannot be $G$-conjugate. Therefore, this embedding factors through the
closed embedding $\mathfrak{Z}_{G}\hookrightarrow G\times S_{f}$ with $G\times
S_{f}\subset G\times S_{f}\times S_{f}$ realised as the diagonal. This amounts
to enforcing the relation $P_{i,L}=P_{i,R}$ for each $i=1,\dots,N$. We know
from Proposition 2.1 that the matrix entries of the bottom row of $G$ are ring
generators. Thus, we conclude that $g_{N1}$ and $P_{i,R}$ Poisson generate
$I_{G}$. ∎

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