aainstitutetext: Krichever Center, Skolkovo Institute of Science and
Technology, 121205, Moscow, Russiabbinstitutetext: Landau Institute for
Theoretical Physics, 142432, Chernogolovka, Russiaccinstitutetext: National
Research University Higher School of Economics, Moscow, Russia

#  $(2,2p+1)$ minimal string and intersection theory I

A. Artemev a,c I. Chaban<EMAIL_ADDRESS>
###### Abstract

In view of the recent progress in studying matrix model-2D gravity duality, we
reexamine some features of $(2,2p+1)$ minimal string. After reviewing both
sides of the proposed correspondence in this case, a previously unnoted
identification between correlation numbers of tachyon operators in certain
domain of parameter space and ”$p$-deformed volumes”, which are certain
integral transforms of topological recursion data, is described and clarified.
This identification allows us to efficiently study correlation numbers at
finite matter central charge. In particular, we obtain an intersection-
theoretic formula and the simplest recurrent equations for them, analogous to
the ones recently derived for Virasoro minimal string. These formulas might be
useful in establishing a more thorough connection between worldsheet and
matrix model approaches.

###### Keywords:

CFT, Matrix Models, Liouville gravity

##  1 Introduction

Minimal string (MS) is one of exactly solvable models of non-critical string
theory, or two-dimensional gravity, extensively studied since the 1980s. An
interesting thing about it is a conjectured duality to a double-scaling limit
of certain matrix models (see e.g. franc1995  for the review). So far, there
is a lot of evidence for this conjecture, although it has not been proven in
full since it is difficult to perform analytic calculations on the worldsheet.
The dual description is convenient because calculation of perturbative string
correlators is significantly more streamlined (up to some subtleties
concerning the precise matching of correlators Moore:1991ir , belzam2009 ). It
also allows to address questions about non-perturbative physics of the theory
(see e.g. Gregori:2021tvs , Eniceicu:2022dru ).

In some sense the minimal string can be regarded as a deformation of Witten-
Kontsevich topological gravity Witten:1990hr , for which there are at least
three formulations: using field theory (with a certain $c=-2$ worldsheet
matter CFT), matrix models or intersection theory of tautological classes on
the moduli space of stable curves. One can not, however, immediately
generalize the methods used in establishing these equivalent formulations to
different non-critical string theories, including minimal string, for which a
matrix model description is lacking direct worldsheet derivation. Despite
that, more recent studies (in particular, studies of the semiclassical limit
of large matter central charge) reveal that in minimal string correlators also
have some interpretation in terms of moduli space geometry (Artemev_2022 ,
eberhardt20232d ).

Another nontrivial theory was discovered recently; it is the so-called
Virasoro minimal string (VMS, Collier:2023cyw ). The worldsheet CFT for this
theory consists of Liouville CFT of central charge $c>25$; “timelike”, or
“$c<1$ Liouville CFT” clessthan12015  as “matter” and BRST ghosts. Exploring
previously proposed links with quantization of Teichmuller space, the authors
of Collier:2023cyw  managed to find an intersection-theoretic formula for
correlators in this theory which serves as a bridge between worldsheet and
double-scaled matrix model descriptions. Attempt on generalizing this work to
better understand the correspondence between worldsheet CFT and matrix models
for minimal string is the main motivation for our study.

This paper is structured as follows. In section 2.1, we introduce notations
and give some background information on the objects of our interest —
“correlation numbers” in minimal string theory, focusing on $(2,2p+1)$ case.
In section 2.2, we describe the previously proposed matrix model analogues of
them. Then we come to the main results of this paper: in section 3.1, we
relate several definitions of “correlation numbers” in the “topological
recursion” language, one of which, although applicable only for certain values
of parameters, is particularly convenient to use for calculations. In sections
3.2 and 3.3 we describe how one can obtain an intersection-theoretic formula
for these correlation numbers. Although a general formula for topological
recursion data exists for any double-scaled matrix model, we propose and prove
a different one, which is, interestingly, very close to the VMS answer. We
discuss some possible interpretations of this formula in section 3.4. In
section 3.5, we study what form the simplest “string” and “dilaton” recurrent
equations in the topological recursion framework obtain for correlation
numbers and comment on the possible CFT interpretation of these equations. We
finish with some concluding thoughts on further directions to study in section
4.

##  2 Preliminaries

###  2.1 Minimal string worldsheet theory

#### Definition and notations.

Worldsheet CFT for minimal string theory (also known as minimal Liouville
gravity) is a CFT of total central charge zero consisting of a CFT minimal
model as “matter” (we will discuss only $(2,2p+1)$ series), a Liouville CFT
and anticommuting $BC$-ghosts of $c_{gh}=-26$

$\displaystyle A_{MLG}=A_{L}+A_{M_{2,2p+1}}+\underbrace{\frac{1}{\pi}\int
d^{2}x\,\left(C{\overline{\partial}}B+{\overline{C}}\partial{\overline{B}}\right)}_{A_{ghost}};\,$
(1) $\displaystyle A_{L}=\int
d^{2}x\,\sqrt{\hat{g}}\left(\frac{1}{4\pi}\hat{g}^{ab}\partial_{a}\phi\partial_{b}\phi+\mu
e^{2b\phi}+\frac{Q}{4\pi}\hat{R}\phi\right),\,Q\equiv b+b^{-1}$ (2)

The condition of zero central charge leads to the relation between $p$ and
Liouville parameter $b$: $b=\sqrt{2/(2p+1)}$.

To set up notations, we will describe primary operators in Liouville and
matter sectors. In $(2,2p+1)$ minimal model, the spectrum consists of $p$
primary fields from the Kac table $\Phi_{1,k},\,k=1\dots 2p$ with the
additional identification $\Phi_{1,k}=\Phi_{1,2p+1-k}$. Dimension of
$\Phi_{1,k}$ is $\Delta^{M}_{1,k}=\alpha_{1,k}(\alpha_{1,k}-b^{-1}+b)$, where
$\alpha_{1,k}=\frac{b}{2}(k-1)$. Virasoro representation, corresponding to
each primary field in minimal model, is doubly degenerate.

In Liouville sector, primary operators are exponentials $V_{a}=\exp(2a\phi)$
with an additional identification $V_{a}=R(a)V_{Q-a}$. Their (bulk) conformal
dimension is $\Delta_{a}^{L}=a(Q-a)$111If we consider the boundary operator
represented by the same exponential, its dimension is
$\Delta_{a}^{L,b}=2a(Q-2a)$. Dependent on circumstances, it may be convenient
to parametrize them instead of $a$ via “Liouville momenta”
$P:a=\frac{Q}{2}+iP$, or, for special values of $a$, with a pair of integers
$(m,n):a_{m,n}=\frac{-(m-1)b^{-1}-(n-1)b}{2}$ (positive $m,n$ correspond to
degenerate Virasoro representations).

#### BRST cohomology.

From the requirement of zero total central charge it also follows that the
theory has a nilpotent ($Q_{\text{BRST}}^{2}=0$) BRST-symmetry with generator

$Q_{\text{BRST}}\equiv\mathcal{Q}+{\overline{\mathcal{Q}}};\,\mathcal{Q}\equiv\oint
dz\,\left(C(T^{L}+T^{M})+:C\partial CB:\right)$ (3)

and an analogous expression for antiholomorphic charge $\mathcal{Q}$.
Equivalently one can write a mode expansion

$\mathcal{Q}=\sum\limits_{n}c_{n}(L_{-n}-\delta_{n,0})-\frac{1}{2}\sum\limits_{n,m}(m-n):c_{-m}c_{-n}b_{n+m}:$
(4)

String theoretic observables are given by cohomology classes of this BRST
operator; these are additionally graded by their ghost number. In minimal
string theory the classes represented by (bulk) local operators were fully
classified by Lian and Zuckerman in lianzuck . It turns out that (in
$(2,2p+1)$ case) there is one such cohomology class with a definite ghost
number for each Liouville parameter $a=a_{1,-n};n\in\mathbb{Z},n\text{ mod
}(2p+1)\neq 0$ (these correspond to Liouville momenta
$P=\frac{ib}{4}(2p+1-2n)$). Lian and Zuckerman also describe a general
procedure of constructing the corresponding representatives.

#### Tachyon correlators.

The simplest cohomology classes that will be of interest to us are “tachyon”
operators $\mathcal{T}_{1,n}$ of ghost number $1$, corresponding to
$n=1,\dots,p$; they are obtained simply by dressing $\Phi_{1,n}$ with
Liouville operators $V_{1-n}$ and ghosts $C{\overline{C}}$ so that we have a
primary field of total conformal dimension $0$. Due to identifications in both
sectors, similarly constructed representatives for $n=p+1,\dots,2p$ are
idenified with the ones described above up to a constant factor:
$\mathcal{T}_{1,n}\sim\mathcal{T}_{1,2p+1-n}$.

To obtain a non-vanishing correlator, one should saturate the ghost zero modes
in the functional integral, i.e. have a specific number of ghost insertions;
to this end, when considering correlators of tachyon operators, one may
equivalently “strip away” the $C$-ghosts and integrate the obtained operator
of dimension $1$ over its position. Thus, on general surface, tachyon
correlation numbers can be defined as follows:

$Z^{g}_{k_{1}\dots
k_{n}}:=\int\limits_{{\overline{\mathcal{M}}}_{g,n}}Z_{BC}\times\left\langle\prod\limits_{i=1}^{n}\underbrace{V_{1,-k_{i}-1}(z_{i})\Phi_{1,k_{i}+1}}_{\Delta=\Delta^{L}_{1,-k-1}+\Delta^{M}_{1,k+1}=1}(z_{i})\right\rangle_{L+MM}$
(5)

with $Z_{BC}$ being a “ghost partition function” (in fact, a certain
correlator that saturates the zero modes) and the integration is over the
moduli space of genus $g$ curves with $n$ marked points.

Tachyon correlators are difficult to calculate analytically from worldsheet;
integration over the moduli space usually can not be done explicitly. Some
reliable analytic results are available only for complex 1-dimensional moduli
spaces, i.e. sphere 4-point Belavin:2005jy  and torus 1-point Artemev:2022sfi
correlators. Other than that difficulty, for Liouville correlator calculation
involves integrating a product of generic conformal blocks over intermediate
Liouville momenta $P$. The integration contour is a real line for small enough
external momenta, but is deformed (equivalently, one should account for
“discrete terms” — residues from the poles of Liouville structure constants
that cross the integration contour zamzam1996 ) if for some $i,j$

$|P_{i}|+|P_{j}|>\frac{Q}{2},\text{ or }k_{i}+k_{j}<p-\frac{5}{2}.$ (6)

#### Boundary states.

Another type of correlators that will appear in the discussion below are
string partition functions with boundaries. In worldsheet approach, they are
constructed as follows: first, for a given bordered Riemann surface at each
boundary component we impose a conformal boundary condition in all three
sectors of worldsheet theory. These conditions can be represented as a
“boundary state”

$\left|\sigma;1,l\right\rangle=\left|\sigma\right\rangle_{L}\otimes\left|1,l\right\rangle_{MM}\otimes\left|\text{Gh}\right\rangle$
(7)

In Liouville, we consider an FZZT state Fateev:2000ik , parametrized by a
continuous parameter $\sigma$, or boundary cosmological constant
$\mu_{b}\sim\cosh 2\pi b\sigma$; in minimal model, we take the Cardy state
Cardy:1989ir  and $\left|\text{Gh}\right\rangle$ in the ghost sector can be
described as a “coherent state” (see the explicit formula in polchinski_1998
), solving

$(c_{n}+{\overline{c}}_{-n})\left|\text{Gh}\right\rangle=0,\,(b_{n}-{\overline{b}}_{-n})\left|\text{Gh}\right\rangle=0.\,$
(8)

We will not need explicit formulas for these states in this work. The main
“kinematic” property of Liouville/minimal model physical states is that they
satisfy conformal boundary conditions
$(L^{L/MM}_{n}-{\overline{L}}^{L/MM}_{-n})\left|\sigma/1,l\right\rangle\rangle=0$.
These conditions lead to
$\mathcal{Q}\left|\sigma;k,l\right\rangle=-{\overline{\mathcal{Q}}}\left|\sigma;k,l\right\rangle$
(see (4)), or $Q_{\text{BRST}}\left|\sigma;k,l\right\rangle=0$ — boundary
states are formally representatives of BRST-cohomology. They, however, include
Virasoro descendants at all levels (being constructed from “Ishibashi states”
Ishibashi:1988kg ) and thus can not be represented by local operators.

To obtain the string partition function, the CFT correlator described above
should be integrated over the moduli space of bordered Riemann surfaces. The
result is referred to FZZT-brane amplitude. This calculation is even more
technically involved than for the correlators on the closed surfaces; the only
examples of explicit worldsheet calculations in minimal string theory the
authors are aware of are disk and cylinder partition functions.

In the seminal work of Seiberg:2003nm , it is the FZZT-amplitudes, where a
state of the form $\left|\sigma_{i};1,1\right\rangle$ is associated to each
boundary component, that are argued to have the simplest dual “matrix model”
interpretation (to be discussed in the next section). Moreover, it is argued
that more general boundary states of the form $\left|\sigma;k,l\right\rangle$
are BRST-equivalent to a certain linear combination of the ones with matter
Cardy labels $(1,1)$, so restricting to this case is enough to study all
boundary amplitudes.

###  2.2 Minimal string matrix model

#### Minimal string and topological recursion.

$(2,2p+1)$ minimal string is usually described using a certain hermitian “one-
matrix model” after double scaling limit. In the following, we will describe
how to obtain perturbative data (correlation numbers and boundary partition
functions) in this approach. We will mostly use terminology from a more
general framework of topological recursion Eynard:2007kz , of which double-
scaled matrix models are a particular case. Along the way we will comment on
what exactly are the objects that we consider in the “matrix model” framework.

The object that we start with is the “spectral curve” — a curve in
$\mathbb{C}^{2}$ defined parametrically by equations

$\displaystyle\begin{cases}x(z)=z^{2}\\\
y(z)=\frac{1}{4\pi}\sin\left(\frac{2}{b^{2}}\arcsin\pi
b^{2}z\right)\end{cases}$ (9) $\displaystyle
z\in\mathbb{CP}^{1},\,b^{2}\equiv\frac{2}{2p+1}$ (10)

For $b^{2}$ as above, $y(z)$ can be equivalently written as a Chebyshev
polynomial of the first kind $\frac{(-1)^{p}}{4\pi}T_{2p+1}(\pi b^{2}z)$.

These equations (more precisely, the 1-form $\omega_{0,1}=y\,dx$) and the
bidifferential

$B(z_{1},z_{2})=\frac{dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}$ (11)

define the recursion kernel

$K(z_{n+1},z)=\frac{\int\limits_{-z}^{z}B(z_{n+1},\cdot)}{2(y(z)-y(-z))dx(z)}=\frac{\pi}{(z_{n+1}^{2}-z^{2})\sin\left(\frac{2}{b^{2}}\arcsin\pi
b^{2}z\right)}\frac{dz_{n+1}}{dz}$ (12)

and “symplectic invariants” — $n$-differentials $\omega_{g,n}(z_{1}\dots
z_{n})\equiv W_{g,n}(z_{1},...,z_{n})dz_{1}\dots dz_{n}$. In matrix model,
$W_{g,n}$ differ from double-scaled expansion coefficients for correlators of
resolvent operators $R_{g,n}(-z_{1}^{2}\dots-z_{n}^{2})$ by a factor of
$\prod\limits_{k=1}^{n}(-2z_{k})$. Topological recursion is a relation
allowing to calculate $W_{g,n}$ starting from $\omega_{0,1}$ and
$\,\omega_{0,2}=B$; it looks as follows

$\displaystyle\omega_{g,n+1}(z_{1},\dots,z_{n},z_{n+1})=\,$
$\displaystyle\underset{z=0}{\text{Res}}\;K(z_{n+1},z)\left(\omega_{g-1,n+2}(z_{1},\dots
z_{n},z,-z)+\right.$
$\displaystyle\left.+\sum\limits_{g_{1}=0}^{g}\sum\limits_{J_{1}+J_{2}=\\{z_{1}\dots
z_{n}\\}}\omega_{g_{1},|J|+1}(J_{1},z)\omega_{g-g_{1},|J_{2}|+1}(J_{2},-z)\right)$
(13)

The first few symplectic invariants are

$W_{0,3}=\frac{1}{z_{1}^{2}z_{2}^{2}z_{3}^{2}};\,W_{0,4}=\left(2\pi^{2}-\frac{\pi^{2}}{2}b^{4}+3\sum\limits_{i=1}^{4}\frac{1}{z_{i}^{2}}\right)\frac{1}{z_{1}^{2}z_{2}^{2}z_{3}^{2}z_{4}^{2}};\,W_{1,1}=\left(\frac{\pi^{2}}{12}-\frac{\pi^{2}}{48}b^{4}+\frac{1}{8}\frac{1}{z_{1}^{2}}\right)\frac{1}{z_{1}^{2}}.$

In different contexts to conform with the other references it might be
convenient to switch to different parametrization, rescaling $z$, or to shift
or rescale $x$ and $y$ by a constant; all these transformations do not change
the spectral curve in a significant way. For example, one might put equal to
$1$ the highest order coefficient for the polynomial $y$: $y=z^{2p+1}+\dots$.

$W_{g,n}(z)$ (or, more precisely, resolvent correlators) are supposed to
compute minimal string multi-boundary amplitudes with FZZT boundary conditions
and matter lables $(1,1)$ mertens2021 , where $1-2z^{2}$ for each boundary has
a meaning of boundary cosmological constant Seiberg:2003nm . “Free energies”
$\mathcal{F}_{g}$ (double-scaled expansion coefficients for logarithm of the
partition function) can also be calculated using topological recursion data.
For genus $g\geq 2$, they can be defined using the dilaton equation (see (49)
below), identifying $\mathcal{F}_{g}\equiv W_{g,0}$. For genus $0$ and $1$,
definitions are more involved (see e.g Eynard:2007kz ).

#### Correlation numbers.

One can deform the spectral curve and calculate how free energy in different
genus changes. A class of interesting deformations is when $x(z)$ and $y(z)$
are still polynomials of the same parity and the same degree, but their
coefficients are varied: $x=z^{2}+x_{0},y=z^{2p+1}+y_{0}z^{2p-1}+\dots$. Then,
it is convenient to use the so-called “KdV coordinates” $t_{k}$ in the space
of parameters. They are defined as follows (see e.g. Marshakov:2009mn ,
although we use a different normalization):

$t_{k}=-\frac{1}{2}\frac{g_{k}}{g_{-2}}\underset{z=\infty}{\text{Res}}\;\left(x^{1/2+k-p}y\,dx\right),\,g_{k}=\frac{(p-k-1)!}{(2p-2k-3)!!}$
(14)

To compare with the other approaches, an additional condition $t_{-1}=0$ can
be imposed on the polynomials $x$ and $y$.

There are several ways to calculate free energies in this approach. A
convenient method for small genus makes use of “Douglas string equation”
(belzam2009 , tarn2011 , beltar2010 ); we will not review it here, since we
will not use it. Derivatives $\frac{\partial^{n}\mathcal{F}_{g}}{\partial
t_{k_{1}}\dots\partial t_{k_{n}}}$ are referred to as $n$-point “correlation
numbers in KdV frame”.

This is not precisely what we are interested in, however. To compare the
results with worldsheet CFT approach, we need to take derivatives with respect
to different couplings $\tau_{k}$, which are related to $t$ via the so-called
“resonance transformations”. Let us parametrize
$t_{0}=-\frac{1}{2}\frac{g_{0}}{g_{-2}}u_{0}^{2}$. Then, other KdV times are
expressed in terms of $\tau_{k}$ as follows (tarn2011 ):

$\frac{g_{k}}{g_{-2}}t_{k}=(2p+1)u_{0}^{k+2}\sum\limits_{n=1}^{\lfloor\frac{k+2}{2}\rfloor}\sum\limits_{\begin{subarray}{c}m_{1}\dots
m_{n}=0\\\ \sum
m_{l}=k+2-2n\end{subarray}}\frac{\tau_{m_{1}}\dots\tau_{m_{n}}}{n!}\frac{(2p-2k+2n-5)!!}{(2p-2k-3)!!}$
(15)

That means that $\tau_{0}=-1/2$. As a check of this formula, one can put
$\tau_{1}\dots\tau_{p-1}=0$ and $u_{0}=2^{-p/2}$. Then, the couplings
$t_{k}^{(0)}$ are such that the spectral curve becomes

$\begin{cases}2x=T_{2}(z)\\\ 2^{2p}y=T_{2p+1}(z)\\\ \end{cases}$ (16)

i.e. coincides with the undeformed one (9) up to rescaling and shifts in $x,y$
and $z$.

Parameter $u_{0}^{2}$ is proportional to the coupling constant $\mu$ for the
exponential interaction term $\mu e^{2b\phi}$ in the worldsheet Liouville CFT.
It is known that worldsheet correlation functions have a power-like dependence
on $\mu$; this is why we have this simple dependence on $u_{0}$ above.

Because of resonance transformations, $n$-th derivatives of free energy with
respect to $\tau$ generally can not be expressed only in terms of $n$-point
KdV correlation numbers. Lower-point contributions appear when changing
$k_{i}$ if nonlinear terms in resonance transformations become relevant. In
the semiclassical limit $b\to 0$, or $p\to\infty$, when $k_{i}$ is of order
$p$, rescaled correlation numbers have a continuous, but non-analytic limit as
functions of $\kappa_{i}\equiv k_{i}/p$ due to these additional terms. There
is, however, analyticity in certain domains in parameter space. Consider the
case when (other than $\tau_{0}$) for all non-zero couplings
$\tau_{k_{i}},\tau_{k_{j}}$ $k_{i}+k_{j}\geq p-3$. Then it is easy to see that
$\forall i,j,l$ $\frac{\partial^{2}t_{l}}{\partial\tau_{i}\partial\tau_{j}}=0$
and the same is valid for higher derivatives. This means that $n$-point
correlation number in CFT is just a linear combination of $n$-point
correlation numbers in KdV frame and no non-analytic contributions appear. One
can note that this condition coincides with the negation of (6), i.e. in
worldsheet formulation it means that no discrete terms appear in the Liouville
OPE.

In the domain we described, semiclassical limit of correlation numbers is
known to coincide with Weil-Petersson volumes, analytically continued to
imaginary lengths $l_{n}=2\pi i(1-\frac{2k_{n}}{2p+1})$ (turiaci2021 ,
Artemev_2022 ). In such a normalization, corrections to the semiclassical
answer are polynomials in $b^{4}$.

#### p-deformed volumes.

An observable that is more straightforward to study systematically than
correlation numbers described above was introduced in mertens2021  by the name
of “$p$-deformed volumes”. They are given by certain integral transforms of
“symplectic invariants” $W_{g,n}(z_{1},\dots z_{n})$

$\frac{W_{g,n}(z_{i})}{\prod\limits_{k=1}^{n}2z_{k}}=\\#\prod\limits_{k=1}^{n}\left(\int\limits_{0}^{\infty}d\beta_{k}\,e^{(1-2\pi^{2}b^{4}z_{i}^{2})\beta_{k}}\int\limits_{0}^{\infty}d\lambda_{k}\,\lambda_{k}\,\tanh\pi\lambda_{k}K_{i\lambda_{k}}(\beta_{k})\right)V_{g,n}(\lambda_{1}\dots\lambda_{n})$
(17)

This integral transform is complicated to perform in general, but one can
easily apply it for our purposes, when transforming polynomials in
$1/z_{i}^{2}$, according to

$\frac{1}{z^{2(i+1)}}\rightarrow(2\pi^{2}b^{4})^{i+1/2}\frac{\sqrt{\frac{1}{2\pi}}}{2^{i}\Gamma(i+1)\Gamma(i+3/2)}\prod\limits_{j=1}^{i}\left(\lambda^{2}+\frac{(2j-1)^{2}}{4}\right)$
(18)

##  3 Main results

###  3.1 Correlation numbers and p-deformed volumes

An a priori nontrivial observation that can be noted when studying known
correlation numbers in low genus is that, in fact, $p$-deformed volumes and
correlation numbers in the domain $k_{i}+k_{j}>p-3$ are related via analytic
continuation:

$\left.\frac{\partial^{n}\mathcal{F}_{g}}{\partial\tau_{k_{1}}\dots\partial\tau_{k_{n}}}\right|_{\tau_{0}=-\frac{1}{2},\tau_{1}\dots\tau_{p-1}=0}\sim
V_{g,n}\left(\lambda_{j}=\frac{i}{2}(2(p-k_{j})-1)\right)$ (19)

Thus, $\lambda$, up to a factor, can be thought of as Liouville momentum for
dressing fields: $\lambda=\frac{2P}{b}$. To our knowledge, this relation has
not been noted before in the literature. This formula will allow us to
systematically examine finite $p$ corrections to tachyon correlators (at least
in some domain) in “matrix model” using topological recursion.

Let us explain why this particular integral transform of symplectic invariants
is related to correlation numbers in the CFT frame. We would like to find a
deformation of the spectral curve such that e.g. $V_{g,1}$ calculate the
corresponding variations of free energies and compare this deformation with
what happens when we change the couplings $\tau_{k}$. Deformations of the
spectral curve data in terms of small changes of functions
$(x(z),y(z))\in\mathbb{C}((z))\times\mathbb{C}((z))$ can be parametrized by a
$(1,1)$-form

$\Omega=\delta y\,dx-\delta x\,dy$ (20)

Here $dx,dy\in\mathbb{C}((z))dz$ are $(1,0)$ forms ($1$-forms on the spectral
curve) and variations $\delta x$ and $\delta y$ are $(0,1)$ forms. In terms of
coordinates $\\{(x_{k},y_{k})\\}_{k}$ such that

$\displaystyle
x(z)=\sum\limits_{k}x_{k}z^{k},\;y(z)=\sum\limits_{k}y_{k}z^{k},$ (21)

$\delta x$ and $\delta y$ are given by

$\displaystyle\delta x(z)=\sum\limits_{k}\delta x_{k}\,z^{k},\;\delta
y(z)=\sum\limits_{k}\delta y_{k}\,z^{k}.$ (22)

In the following we will use the same notation $\Omega$ for $1$-form on the
curve obtained by substitution of a specific deformation into $\Omega$.

The form $\Omega$ does not depend on whether we vary $x,y$ or both of them and
is an invariant characteristic of the deformation; in particular, it is zero
if $\delta$ corresponds to reparametrization $z\to f(z)$. According to the
general theory Eynard:2007kz , if $\Omega$ can be expressed as

$\Omega(p)=\oint\limits_{(\infty)}\Lambda(q)B(p,q)$ (23)

free energy is deformed as follows

$\delta\mathcal{F}_{g}=\oint\limits_{\infty}\omega_{g,1}(z)\Lambda(z)$ (24)

Similar formula is valid for symplectic invariants themselves and, thus, can
be used for higher derivatives: $n$-th derivative of free energy is expressed
via $n$-ple contour integral of $\omega_{g,n}$. Equating
$\delta\mathcal{F}_{g}(\Lambda_{k})=V_{g,1}(\frac{i}{2}(2p-2k-1))$, from the
explicit transformation formulas (18) one can find $\Lambda$ as a polynomial
(in fact it is a Chebyshev polynomial of the first kind, see (30))

$\Lambda(z)=\sum\limits_{i=0}^{p-k}\frac{z^{2i+1}}{\Gamma(i+1)\Gamma(i+3/2)}\prod\limits_{j=1}^{i}\left(\frac{(2j-1)^{2}-(2p-2k-1)^{2}}{4}\right)$
(25)

$\Omega$ then can be calculated, using (11). It turns out to be proportional
to a Chebyshev polynomial of the second kind

$\Omega(z)\sim U_{2(p-k-1)}(z)\,dz$ (26)

If we keep $x(z)$ unchanged, to realize this deformation we need to consider
non-polynomial deformation of $y$ of the form $\delta
y\sim\frac{U_{2(p-k-1)}(z)}{z}$. The same deformation of spectral curve
equation appears in Seiberg:2003nm , where it is argued to describe an
insertion of tachyon vertex operator $\mathcal{T}_{1,k+1}$ in worldsheet CFT.
This is precisely what we wanted to show. In Seiberg:2003nm , only one-point
function on the disk was considered, so only linear in
$\tau_{1}\dots\tau_{p-1}$ terms in resonance transformations could be probed.
However, they are still nontrivial and in general necessary to account for to
obtain this $(1,0)$-form $\Omega$. In Appendix A, we rederive this expression
for $\Omega$ using explicit formulas for KdV times (14) and resonance
transformations (15), illustrating it on the simplest nontrivial case $p=4$,
or $(2,9)$ model.

If we know the connection between correlation numbers and $p$-deformed
volumes, integral transform (17) can be motivated by the following idea: LHS,
as mentioned before, is supposed to compute the FZZT brane amplitude for
minimal string with boundary cosmological constants parametrized by $z_{i}$,
and the first Laplace transform brings us to the partition function of the
theory on surfaces with boundaries with “fixed-length” boundary conditions.
These partition functions are decomposed as the correlator of local operators
on the punctured surface (which is $V$), to which once-punctured disks are
glued (the Bessel function is a one-point function on a disk with these b.c.
see e.g. mertens2021 ), summed over a certain complete basis of operators that
can be inserted (see some comments on this after (35)). Inverse formula ((28),
see below) heuristically describes the opposite operation of gluing punctured
disks with bulk operator insertion to surface with boundaries.

###  3.2 Comparing spectral curve data for minimal and Virasoro minimal string

For the following sections we will use a slightly different parameterization
for $(2,2p+1)$ minimal string spectral curve:

$\displaystyle\begin{cases}x(z)=T_{2}(\pi b^{2}z)=2\pi^{2}b^{4}z^{2}-1\\\
y(z)=(-1)^{p}\,T_{2p+1}(\pi b^{2}z)=\sin\left(\frac{2}{b^{2}}\arcsin\left(\pi
b^{2}z\right)\right)\end{cases}$ (27)

Bidifferential $B(z_{1},z_{2})$, needed to define other symplectic invariants,
is the same as before:
$B(z_{1},z_{2})=\frac{dz_{1}dz_{2}}{(z_{1}-z_{2})^{2}}$. Note that in this
normalization $W_{g,n}$ differ from the ones defined using $(\ref{MertensSC})$
by a factor $(8\pi^{3}b^{4})^{2g-2+n}$. The $p-$deformed volumes are expressed
via $W_{g,n}$ using transformation

$\frac{V_{g,n}\left(\lambda_{j}\right)}{(8\pi^{3}b^{4})^{2g-2+n}}=\prod\limits_{k=1}^{n}\left(\frac{\cosh\pi\lambda_{k}}{\pi^{2}b^{2}}\int\limits_{0}^{\infty}\frac{d\beta_{k}}{\beta_{k}}K_{i\lambda_{k}}(\beta_{k})\int\limits_{-i\infty+\epsilon}^{i\infty+\epsilon}\frac{dz_{k}}{2\pi
i}\,e^{(2\pi^{2}b^{4}z_{k}^{2}-1)\beta_{k}}\right)W_{g,n}(z_{1},\dots,z_{n})$
(28)

which is inverse to $(\ref{VtoW})$.

One can integrate over $\beta_{k}$ (Appendix B) and reduce this to the
following formula:

$V_{g,n}(\lambda_{j})=(8\pi^{3}b^{4})^{2g-2+n}\underset{z_{1}\dots
z_{n}=0}{\text{Res}}\;W_{g,n}(z_{1},...,z_{n})\prod\limits_{k=1}^{n}\frac{dz_{k}}{2\pi
b^{2}\lambda_{k}}e^{2\lambda_{k}\arcsin\pi b^{2}z_{k}}$ (29)

For $\lambda_{k}=\frac{i}{2}(2n_{k}+1)$ it can be simplified more and we have

$\frac{V_{g,n}\left(\frac{i}{2}(2n_{j}+1)\right)}{(8\pi^{3}b^{4})^{2g-2+n}}=\underset{z_{1}\dots
z_{n}=0}{\text{Res}}\;W_{g,n}(z_{1},...,z_{n})\prod\limits_{k=1}^{n}\frac{dz_{k}}{2\pi}\frac{2p+1}{2n_{k}+1}(-1)^{n_{k}}T_{2n_{k}+1}\left(\frac{2\pi
z}{2p+1}\right)$ (30)

The form of transformation (29) suggests local change of variable:

$z(s)=\frac{1}{\pi b^{2}}\sin\pi bs$ (31)

1-form $\omega_{0,1}$ and bidifferential $\omega_{0,2}$ in terms of $s$ become

$\displaystyle\tilde{\omega}_{0,1}(s)=2\pi b\sin(2\pi bs)\sin(2\pi
b^{-1}s)ds,\,$ $\displaystyle B(z(s_{1}),z(s_{2}))=(\pi b)^{2}\frac{\cos\pi
bs_{1}\cos\pi bs_{2}}{(\sin\pi bs_{1}-\sin\pi bs_{2})^{2}}ds_{1}ds_{2}$ (32)

Other symplectic invariants $\tilde{\omega}_{g,n}(s_{1},..,s_{n})$ are also
given by pullback of $\omega_{g,n}(z_{1},...,z_{n})$ under (31). It follows
from the formula (2.2), which defines $\omega_{g,n}$ only in terms of
$\omega_{0,1}$ and $B$. To calculate their coefficients $\tilde{W}_{g,n}$ from
topological recursion, there is some freedom in choosing an embedding of the
spectral curve in $\mathbb{C}^{2}\supset(\tilde{x}(s),\tilde{y}(s))$, such
that $\tilde{\omega}_{0,1}=\tilde{y}\,d\tilde{x}$. A convenient choice is

$\displaystyle\begin{cases}\tilde{x}(s)=s^{2}\\\
\tilde{y}(s)=y(z(s))\frac{dx(z(s))}{d\tilde{x}(s)}\\\ \end{cases}$ (33)

The change of variable (31) allows one to compare topological recursion data
for MS and VMS Collier:2023cyw . In fact the initial $1$-forms for both
theories coincide and (as one can notice from (29)) $p$-deformed volumes for
MS can be obtained from $\tilde{W}_{g,n}$ by simple residue formula

$\displaystyle
V_{g,n}(\lambda_{j})=(8\pi^{3}b^{4})^{2g-2+n}\underset{z_{k}=0}{\text{Res}}\;\tilde{W}_{g,n}(s_{1},...,s_{n})\prod\limits_{k=1}^{n}\frac{ds_{k}}{2\pi
b^{2}\lambda_{k}}e^{2\pi b\lambda_{k}s},$ (34)

which has exactly the same form as for VMS. The only difference between the
recursion data is the nontrivial bidifferential $\tilde{W}_{0,2}(s)$ for MS in
contrast to the standard one for VMS. This form of topological recursion gives
us an alternative (to the “matrix model”) way to calculate perturbative data.
We find this relation between two theories very interesting and suggestive.

Unlike $W(z)$, $\tilde{W}(s)$ — coefficients for $\tilde{\omega}_{g,n}$ — do
not have an immediate worldsheet interpretation for the minimal string. The
same is valid for VMS: its matrix model resolvents are obtained from FZZT
amplitudes by a double Laplace transform, first with respect to $s^{2}$ and
then the “boundary length” $\beta$.

One can obtain an intriguing alternative formula for the inversion of (34);
for $\Im s<0$

$\tilde{W}_{g,n}(s_{1},...,s_{n})=\frac{1}{(8\pi^{3}b^{4})^{2g-2-n}}\sum\limits_{n_{k}=0}^{\infty}V_{g,n}\left(\frac{i}{2}(2n_{k}+1)\right)\prod\limits_{k=1}^{n}(2\pi
b)^{2}i\frac{ib}{2}\left(2n_{k}+1\right)e^{-2\pi s_{k}\frac{ib}{2}(2n_{k}+1)}$
(35)

Unlike (17), this formula represents $\tilde{W}$ as a discrete sum over
physical operators (almost all Liouville momenta in the sum correspond to
BRST-cohomology classes of lianzuck , including not only tachyons, but
operators of other ghost numbers) rather than an integral. One can not exactly
take this interpretation at face value: even for tachyons, $V$ coincide with
correlation numbers only in a certain region of parameter space. For operators
other than tachyons, the relation between $V$ and “genuine” correlation
numbers is even less clear: it is not known how to introduce deformation
parameters so that the corresponding free energy derivatives would compute a
correlator with e.g. insertion of several ground ring operators. It would be
interesting to better understand this question.

###  3.3 Intersection theory formula for p-deformed volumes

In this section, we will use the topological recursion described above to find
an intersection-theoretic formula for $V_{g,n}$ which is equivalent, but
appears to be more suggestive than the one that can be derived from spectral
curve (9) and the standard Bergmann kernel. It might be surprising that the
same quantities $V_{g,n}$ can be expressed by two different formulas that use
tautological classes. However, it can be shown that equivalence of these two
formulas is nothing more than a consequence of Virasoro constraints.

We apply the technique developed in Eynard:2011sc  to represent $p$-deformed
volumes in terms of intersection of tautological classes on
$\mathcal{\overline{M}}_{g,n}$. We recall relevant definitions and properties
in Appendix C; for a more thorough introduction to intersection theory on
$\mathcal{\overline{M}}_{g,n}$, see e.g. Zvonkine2012AnIT .

According to Eynard:2011sc , symplectic invariants $\tilde{W}_{g,n}$ for our
topological recursion have the following representation:

$\tilde{W}_{g,n}=2^{3g-3+n}\sum\limits_{j_{1},...,j_{n}\geq
0}\frac{d\xi_{j_{1}}}{ds_{1}}...\frac{d\xi_{j_{n}}}{ds_{n}}\int\limits_{\mathcal{\overline{M}}_{g,n}}\psi_{1}^{j_{1}}...\psi_{n}^{j_{n}}e^{\;\sum\limits_{k\geq
0}\tilde{t}_{k}\kappa_{k}+\sum\limits_{k,l\geq 0}\tilde{B}_{k,l}\delta_{k,l}}$
(36)

Integrand in (36) is understood as the formal power series in cohomological
classes, and integration means pairing of its top degree term with fundamental
homology class of ${\overline{\mathcal{M}}}_{g,n}$. $1$-forms $d\xi_{j}$ are
defined by

$d\xi_{j}(s)=-\underset{s^{\prime}=0}{\text{Res}}\;\tilde{B}(s,s^{\prime})\frac{(2j-1)!!}{2^{j}s^{\prime
2j+1}}$ (37)

and the times $\tilde{t}_{k}$ are computed from the Laplace transform of the
$1$-form $\tilde{\omega}_{0,1}=\tilde{y}\,d\tilde{x}$:

$e^{-\sum_{k}\tilde{t}_{k}u^{-k}}=\frac{2u^{3/2}}{\sqrt{\pi}}\int\limits_{0}^{\infty}e^{-u\tilde{x}(s)}\tilde{\omega}_{0,1}(s)=4b\pi^{3}e^{\frac{b^{2}+b^{-2}}{4}(2\pi)^{2}u^{-1}}\frac{(2\pi)^{2}u^{-1}/2}{\sinh\left((2\pi)^{2}u^{-1}/2\right)}$
(38)

Finally, parameters $\tilde{B}_{k,l}$ are defined by the double Laplace
transform of the regularized bidifferential (Appendix D):

$\displaystyle\frac{e^{-(u_{1}+u_{2})\sum\limits_{k,l\geq
0}\tilde{B}_{k,l}(u_{1}+u_{2})^{k}(u_{1}u_{2})^{l}}-1}{u_{1}+u_{2}}=\frac{1}{2\pi\sqrt{u_{1}u_{2}}}\int\limits_{s_{1}\in\mathbb{R}+i\epsilon}\int\limits_{s_{2}\in\mathbb{R}}e^{-\frac{s_{1}^{2}}{u_{1}}-\frac{s_{2}^{2}}{u_{2}}}\left(\tilde{B}-\overset{\circ}{\tilde{B}}\right)(s_{1},s_{2})$
(39)

We see that $\tilde{B}_{k,l}=0$ for $l\neq 0$ and
$\tilde{B}_{k,0}=(2\pi)^{2k+2}\tilde{b}_{k}$, where the first few
$\tilde{b}_{k}$’s are given by

$\displaystyle\tilde{b}_{0}=\frac{(ib/2)^{2}}{12}$
$\displaystyle\tilde{b}_{3}=-\frac{59(ib/2)^{8}}{56700}$
$\displaystyle\tilde{b}_{1}=-\frac{(ib/2)^{4}}{90}$
$\displaystyle\tilde{b}_{4}=\frac{232(ib/2)^{10}}{467775}$
$\displaystyle\tilde{b}_{2}=\frac{8(ib/2)^{6}}{2835}$

Using (29) we find (in terms of $P_{k}=\frac{b}{2}\lambda_{k}$):

$\displaystyle
V_{g,n}=\\#(8\pi^{3}b^{4})^{2g-2+n}\sum\limits_{j_{1},...,j_{n}\geq
0}\prod\limits_{k=1}^{n}\underset{s_{k}=0}{\text{Res}}\;\frac{e^{2\pi
b\lambda_{k}s_{k}}d\xi_{j_{k}}}{2\pi
b^{2}\lambda_{k}}\int\limits_{\mathcal{\overline{M}}_{g,n}}\psi_{1}^{j_{1}}...\psi_{n}^{j_{n}}e^{\sum\limits_{k\geq
0}\tilde{t}_{k}\kappa_{k}+\sum\limits_{k\geq 0}\tilde{B}_{k,0}\delta_{k,0}}=$
$\displaystyle=\\#(8\pi^{3}b^{4})^{2g-2+n}\sum\limits_{j_{1},...,j_{n}\geq
0}\prod\limits_{k=1}^{n}\frac{-1}{b}\frac{(\pi
b\lambda_{k})^{2j_{k}}}{j_{k}!}\int\limits_{\mathcal{\overline{M}}_{g,n}}\psi_{1}^{j_{1}}...\psi_{n}^{j_{n}}e^{\sum\limits_{k\geq
0}\tilde{t}_{k}\kappa_{k}+\sum\limits_{k\geq 0}\tilde{B}_{k,0}\delta_{k,0}}=$
$\displaystyle=\\#(8\pi^{3}b^{4})^{2g-2+n}\int\limits_{\mathcal{\overline{M}}_{g,n}}e^{\frac{c-13}{24}(2\pi)^{2}\kappa_{1}-\sum\limits_{m\geq
1}\frac{B_{2m}(2\pi)^{4m}}{(2m)(2m)!}\kappa_{2m}+\sum\limits_{k=1}^{n}(2\pi)^{2}P_{k}^{2}\psi_{k}+\sum\limits_{k\geq
0}\tilde{b}_{k}(2\pi)^{2k+2}\delta_{k,0}}=$
$\displaystyle=(8\pi^{2}b^{2})^{3g-3+n}\int\limits_{\mathcal{\overline{M}}_{g,n}}e^{\frac{c-13}{24}\kappa_{1}-\sum\limits_{m\geq
1}\frac{B_{2m}}{(2m)(2m)!}\kappa_{2m}+\sum\limits_{k=1}^{n}P_{k}^{2}\psi_{k}+\sum\limits_{k\geq
0}\tilde{b}_{k}\delta_{k,0}}$ (40)

Laplace transform of $d\xi_{k}$, used in the second line, is calculated in
Appendix D. A few examples of the volumes computed using this formula as well
as comments on how to compare them with the other approaches can be found in
Appendix E.

In the formula above, by $c$ we mean the Liouville central charge
$c=13+6b^{2}+6b^{-2}$, and $P^{2}$ parametrize dimensions for external
Liouville insertions as $\Delta^{L}=\frac{c-1}{24}+P^{2}$. However, the same
answers for $p$-deformed volumes are obtained after replacing
$b^{2}\to-b^{2}$, $P_{k}^{2}\to-P_{k}^{2}$; then the formula is written in
terms of parameters of the matter theory. Coefficients $-P_{k}^{2}$ before
$\psi$-classes are then related to matter insertions’ conformal dimensions in
the same way: $\Delta^{M}=\frac{c_{M}-1}{24}-P^{2}$.

###  3.4 Discussion on the obtained answer

Similarity of this formula with the intersection-theoretic answer for VMS (the
only difference being the additional factor with $\delta_{k,0}$-classes)
suggests a hypothetical relation to worldsheet approach in the spirit of
Collier:2023cyw . We think that it can be helpful in establishing a more
direct understanding (or derivation) of minimal string — matrix model duality.

What would be a possible explanation for the emergence of these
$\delta$-classes? Recall that in VMS, the integrand over moduli space is
interpreted as a product of Chern character of “line bundle of Liouville
conformal blocks” and Todd class of $T\mathcal{M}_{g,n}$. The integrand then
is reminiscent of the one in Hirzebruch-Riemann-Roch theorem, which computes
dimensions of the space of holomorphic sections of this bundle (more
precisely, its Euler characteristic)222We note in passing that for the minimal
string, albeit in the region of parameter space not studied in this paper,
correlation numbers on the sphere are also known to be proportional to
dimensions of the space of conformal blocks in the _matter_ sector
Artemev_2022 ..

Line bundles of conformal blocks are well-studied objects in the case of
affine Lie algebras Beauville_1994 . Such bundles are defined over the moduli
space of semistable bundles for a given complex curve; over moduli space of
curves $\mathcal{M}_{g,n}$ (or its compactification) they then give rise to
vector bundles of finite rank via pushforward. Analogous construction for
Liouville blocks is less clear: if we think of them as a vector bundle over
${\overline{\mathcal{M}}}_{g,n}$, it would be of infinite rank and would not
have a well-defined Chern character. Rigorous construction of line bundles
over Teichmuller space $\mathcal{T}_{g,n}$ whose holomorphic sections are
Liouville conformal blocks has started to develop recently in
guillarmou2024review ; however, even if this construction can be used to
formalize the derivation for VMS, there are still confusing issues with
applying HRR theorem to line bundles on non-compact Teichmuller space. We
believe that subtleties with treatment of boundary classes and switching
between $\mathcal{T}_{g,n}$, $\mathcal{M}_{g,n}$ and
${\overline{\mathcal{M}}}_{g,n}$ in Collier:2023cyw  need to be understood
better.

For rational CFTs, rigorous definitions of vector bundles of conformal blocks
over moduli space of stable curves exist when their rank is finite; in
particular, for affine blocks tsuchiya1989  and in cases like minimal models
Damiolini_2021 . For these examples, on the open part of moduli space the
Chern character is given by the same expression as what appears in the VMS
formula (up to multiplication by the rank of the bundle). On
${\overline{\mathcal{M}}}_{g,n}$, however, boundary $\delta$-classes are
added, with coefficients depending on allowed conformal dimensions in all
possible intermediate channels. The general formula for the affine case is
given in marian2016chern2 , but it is complicated; instead of giving that, we
will formulate a property from which the coefficients for the boundary classes
can be obtained in this context. It characterizes the bundle of conformal
blocks $E(\vec{\mu})$ on ${\overline{\mathcal{M}}}_{g,n}$ ($\vec{\mu}$ is a
set of irreps of vertex operator algebra associated to marked points) in
degeneration limit; namely, it states that pullback under the attaching map

$j:\mathcal{\overline{M}}_{g-1,n+2}\sqcup\bigsqcup\limits_{h,I}\mathcal{\overline{M}}_{h,I}\times\mathcal{\overline{M}}_{g-h,\bar{I}}\longrightarrow\mathcal{\overline{M}}_{g,n}$

decomposes as

$j^{*}E(\vec{\mu})=\mathop{{\bigoplus}}\limits_{\lambda}E(\vec{\mu},\lambda,{\overline{\lambda}})$
(41)

i.e. as a direct sum of conformal block bundles on the corresponding moduli
spaces, where we associate to two additional marked points all possible irreps
$\lambda$ and its dual ${\overline{\lambda}}$. Pullback of the Chern character
of conformal block bundle is also given by the sum of characters for bundles
with $n+2$ marked points. Although it is best understood for the affine case,
this property is very natural to have for a vector bundle of conformal blocks
associated to any vertex operator algebra.

Let us try to analyze MS and VMS answers from this perspective. Tautological
classes that we work with are of the form $\exp(\dots)$ and start with zero
degree cohomology class $1$. Chern characters of vector bundles, on the other
hand, start with $\text{rk }E$; thus the best we can assume is that we
actually calculate the (properly regularized333We should expect this bundle to
be of infinite rank. ) ratio of Chern character of this bundle to its rank. In
what follows we denote it as “ch $E$”. Property (41) for conformal block
bundles would lead to $j^{*}\text{``ch $E$''}$ decomposing as the sum of such
classes for $n+2$ marked points, weighted by ratios of ranks.

First, for VMS: if Chern character does not contain boundary classes, as it is
defined in Collier:2023cyw , the pullback does not lead to anything nontrivial
(since $j^{*}\kappa_{k}-\kappa_{k}=j^{*}\psi-\psi=0$). If we interpret the
integrand in terms of HRR formula for compact moduli space
${\overline{\mathcal{M}}}_{g,n}$, we should have as a second factor a Todd
class $\text{td}(T{\overline{\mathcal{M}}}_{g,n})$ with added $\delta$-classes
Eberhardt:2022wlc :

$\text{td}({\overline{\mathcal{M}}}_{g,n})=\exp\left(-\frac{13}{24}\kappa_{1}+\frac{1}{24}\sum\psi_{k}+\frac{11}{24}\delta_{0,0}-\sum\limits_{m=1}^{\infty}\frac{B_{2m}}{2m(2m)!}(\kappa_{2m}-\delta_{2m-1,0})\right)$
(42)

Thus, we should add boundary classes to the Chern character as well with the
opposite sign, so that the full formula remains the same; what we want to
interpret as Chern character of some bundle is

$\text{``ch
$E$''}_{VMS}=\exp\left(\frac{c}{24}\kappa_{1}+\sum\limits_{k=1}^{n}(P_{k}^{2}-\frac{1}{24})\psi_{k}-\frac{11}{24}\delta_{0,0}-\sum\limits_{m=1}^{\infty}\frac{B_{2m}}{2m(2m)!}\delta_{2m-1,0}\right)$
(43)

If there is a bundle of conformal blocks (for a certain VOA) associated to
these expressions, we expect that when calculating the pullback, we will
obtain an expression of the same form, multiplied by a sum over some spectrum
$\sum\limits_{\lambda}\exp\left((P^{2}_{\lambda}-\frac{1}{24})(\psi_{n+1}+\psi_{n+2})\right)$,
where $\Delta_{\lambda}\equiv P^{2}_{\lambda}+\frac{c-1}{24}$ is a Virasoro
highest weight of VOA irrep $\lambda$ (we used that its dual
${\overline{\lambda}}$ has the same highest weight). Since

$\delta_{k,0}-j^{*}\delta_{k,0}=(\psi_{n+1}+\psi_{n+2})^{k+1}$ (44)

pullback of “ch $E$” is given by

$\displaystyle j^{*}\text{``ch
$E$''}_{VMS}=\exp\left(\frac{c}{24}\kappa_{1}+\sum\limits_{k=1}^{n}(P_{k}^{2}-\frac{1}{24})\psi_{k}-\frac{11}{24}\delta_{0,0}-\sum\limits_{m=1}^{\infty}\frac{B_{2m}}{2m(2m)!}\delta_{2m-1,0}\right)\times$
$\displaystyle\times\exp\left(-\frac{1}{24}u+\left[\frac{u}{2}+\sum\limits_{m=1}^{\infty}\frac{B_{2m}}{2m(2m)!}u^{2m}\right]\right)=\exp(\dots)\times\exp\left(-\frac{u}{24}\right)\frac{\exp(u)-1}{u}$
(45)

where $u\equiv\psi_{n+1}+\psi_{n+2}$. The last expression can be rewritten in
several ways as an integral $\int d\Delta\,\exp((\Delta-\frac{c}{24})u)$,
similarly to the expected sum over some spectrum.

In the minimal string case there appears an additional factor. It is expressed
via the generating function found in Appendix D and looks like

$\exp(-\sum\limits_{k\geq
0}\tilde{b}_{k}u^{k+1})=\frac{b^{2}}{2}u\sum\limits_{k=0}^{\infty}\frac{2k+1}{2}\exp\left(-u\frac{b^{2}}{4}(k+\frac{1}{2})^{2}\right)$
(46)

When we multiply by this the answer that was calculated for VMS, additional
factor that appears after taking the pullback becomes

$\exp\left(-\frac{u}{24}\right)\sum\limits_{k=0}^{\infty}\frac{2k+1}{2(2p+1)}\left[\exp\left(u(1-\frac{b^{2}}{16}(2k+1)^{2})\right)-\exp\left(-u\frac{b^{2}}{16}(2k+1)^{2})\right)\right]$
(47)

This sum is reminiscent of what we expect — note that
$P_{k}^{2}=-\frac{b^{2}}{16}(2k+1)^{2}$ are Liouville momenta for physical
operators in minimal string. In accordance with earlier discussion, rational
factors can perhaps be interpreted as ratios of ranks. However, we do not know
if there is an interpretation for most notable elements of this formula —
terms with different sign and conformal dimension shifted by $1$.

There are other plausible cohomological interpretations of the expression for
MS “$p$-deformed volumes”; in particular, there is a resemblance to Chiodo
classes Lewanski_2016  that appear when we try to raise the canonical line
bundle to rational power. We hope that it will be possible to formulate a more
precise explanation for our formula in the future.

###  3.5 Recurrence equations: string and dilaton

Symplectic invariants, calculated using topological recursion, also satisfy
simpler linear recurrence relations, commonly referred to as “string” and
“dilaton” equations. They are the consequence of first two Virasoro
constraints. Adapted to our case, they are as follows chaudhuri2019local

$\underset{z=0}{\text{Res}}\;y(z)\frac{\omega_{g.n+1}(z,\vec{z})}{\prod\limits_{i=1}^{n}2z_{i}}=-\sum\limits_{k=1}^{n}dz_{k}\,\frac{\partial}{\partial(z_{k}^{2})}\left(\frac{\omega_{g.n}(\vec{z})}{dz_{k}}\frac{1}{\prod\limits_{i=1}^{n}2z_{i}}\right);\,\vec{z}=(z_{1},\dots,z_{n})\leavevmode\nobreak\
\text{(string)}$ (48)
$\underset{z=0}{\text{Res}}\;\Phi(z)\omega_{g.n+1}(z,\vec{z})=(2g-2+n)\omega_{g.n}(\vec{z}),\,d\Phi=y\,dx\leavevmode\nobreak\
\text{(dilaton)}$ (49)

They can be rewritten in a quite simple way for our objects of interest —
$p$-deformed volumes in minimal string theory. Analogous equations were
previously derived for usual WP volumes do2006weilpetersson  and for “quantum
volumes” in Virasoro minimal string.

We will derive the string equation for $V_{g,n}$ in detail. $W_{g,n}$ is a
polynomial in $\frac{1}{z_{i}^{2}}$ of the form

$W_{g,n}=\sum\limits_{i_{1}\dots i_{n}=0}\frac{c_{i_{1}\dots
i_{n}}}{z_{1}^{2(i_{1}+1)}\dots z_{n}^{2(i_{n}+1)}}$ (50)

$k$th term in the sum in the RHS of the string equation then reads

$\mathcal{W}^{(k)}_{g,n}=\sum\limits_{i_{1}\dots
i_{n}=0}(i_{k}+3/2)\frac{c_{i_{1}\dots i_{n}}}{z_{1}^{2(i_{1}+1)}\dots
z_{n}^{2(i_{n}+1)}}\frac{1}{z_{k}^{2}}$ (51)

After the integral transform we obtain $p$-deformed volumes

$V_{g,n}=\sum\limits_{i_{1}\dots i_{n}=0}\tilde{c}_{i_{1}\dots
i_{n}}\prod\limits_{k=1}^{n}\prod\limits_{j_{k}=1}^{i_{k}}\left(\lambda_{k}^{2}+\frac{(2j_{k}-1)^{2}}{4}\right),\,\tilde{c}:=c\cdot\underbrace{\prod\limits_{m=1}^{n}\frac{(2\pi^{2}b^{4})^{i_{m}+1/2}\sqrt{\frac{1}{2\pi}}}{2^{i_{m}}\Gamma(i_{m}+1)\Gamma(i_{m}+3/2)}}_{\equiv
F_{1\dots n}}$ (52)

and on the RHS of the string equation

$\mathcal{V}^{(k)}_{g,n}=\sum\limits_{i_{1}\dots i_{n}=0}\tilde{c}_{i_{1}\dots
i_{n}}\prod\limits_{m=1}^{n}\prod\limits_{j_{m}=1}^{i_{m}}\left(\lambda_{m}^{2}+\frac{(2j_{m}-1)^{2}}{4}\right)\cdot\frac{2\pi^{2}b^{4}\cancel{(i_{k}+3/2)}}{2(i_{k}+1)\cancel{(i_{k}+3/2)}}\left(\lambda_{k}^{2}+\frac{(2i_{k}+1)^{2}}{4}\right)$
(53)

$\mathcal{V}^{(k)}_{g,n}$ is a polynomial with the following properties:

  * •

$\mathcal{V}^{(k)}_{g,n}(\lambda_{k}=\pm\frac{i}{2})=0$, because every summand
necessarily has a factor $(\lambda_{k}^{2}+\frac{1}{4})$;

  * •

It solves a difference equation

$i\frac{\sin\frac{\partial}{\partial\lambda}}{2\pi^{2}b^{4}}\mathcal{V}_{g,n}^{(k)}(\lambda)\equiv\frac{1}{4\pi^{2}b^{4}}\left(\mathcal{V}^{(k)}_{g,n}(\lambda_{k}+i)-\mathcal{V}^{(k)}_{g,n}(\lambda_{k}-i)\right)=i\lambda_{k}V_{g,n}(\vec{\lambda})$
(54)

The solution of this equation with the initial condition
$\mathcal{V}^{(k)}_{g,n}(\lambda_{k}=\frac{i}{2})=0$ in the class of
polynomials can be written using the integro-differential operator
$\frac{\frac{\partial}{\partial x}}{\sin\frac{\partial}{\partial x}}$ (it is
understood as its Maclaurin series and its action on polynomials eventually
truncates, so the expression is well-defined)

$\mathcal{V}^{(k)}_{g,n}(\tilde{\lambda})=2\pi^{2}b^{4}\int\limits_{i/2}^{\lambda_{k}}dx\frac{\frac{\partial}{\partial
x}}{\sin\frac{\partial}{\partial
x}}\left(xV_{g,n}(\vec{\lambda}\,\backslash\,\lambda_{k},x)\right)$ (55)

Since $V$ is an even function of $\lambda_{j}$, $\mathcal{V}$ is as well and
we also have $\mathcal{V}(-i/2)=0$. Because of that, as a simple consequence
of the difference equation, for $\lambda_{k}=\frac{i}{2}(2n+1)$ we can rewrite
the answer as a finite sum:

$\mathcal{V}_{g,n}\left(\lambda_{k}=\frac{i}{2}(2n+1)\right)=4\pi^{2}b^{4}\left.\sum\limits_{k=0}^{2n-1-4k>-1}xV_{g,n}(\lambda_{k}=x)\right|_{x=\frac{i}{2}(2n-1-4k)}$
(56)

In the worldsheet CFT, these $\lambda$ are precisely what corresponds to
Liouville momenta of Lian-Zuckerman physical states.

Now, in the LHS we calculate $\text{Res}\;y(z)\omega_{g,n+1}(z,\vec{z})$, with
$y\sim T_{2p+1}(\pi b^{2}z)$. From (30) we know that calculating residue of
$\omega\cdot T_{2n+1}(z)$ is equivalent to transformation (18) with
substituting $\lambda=\frac{i}{2}(2n+1)$. Thus, in the LHS of the string
equation after integral transform with respect to $z_{1}\dots z_{n}$ we get
$V_{g,n+1}\left(\frac{i(2p+1)}{2},\vec{\lambda}\right)$.

Using parametrization $l=2\pi b^{2}\lambda$ and “Weil-Petersson” normalization
(when at leading order in $b$
$V_{g,n}=\int\limits_{{\overline{\mathcal{M}}}_{g,n}}\exp(2\pi^{2}\kappa_{1}+\frac{1}{2}\sum
l_{i}^{2}\psi_{i})$), string equation assumes the form

$V_{g,n+1}(2\pi i,\vec{l})=\sum\limits_{k=1}^{n}\int\limits_{\pi
ib^{2}}^{l_{k}}dx\,\frac{2\pi b^{2}\frac{\partial}{\partial x}}{\sin 2\pi
b^{2}\frac{\partial}{\partial
x}}\left(xV_{g,n}(x,\vec{l}\,\backslash\,l_{k})\right)$ (57)

The dilaton equation in the same normalization reads

$V_{g,n+1}(2\pi i(1-b^{2}),\vec{l})-V_{g,n+1}(2\pi
i(1+b^{2}),\vec{l})=8\pi^{2}b^{2}(2g-2+n)V_{g,n}(\vec{l})$ (58)

It is the same as for Virasoro minimal string.

The reason we are interested in these equations is that they are supposed to
have a known interpretations in CFT terms, corresponding to the insertion of
the simplest kind of operators in the bulk (area operator, for dilaton
equation) and on the boundary (length operator, for string equation). Let us
make a few comments on that.

Liouville momenta $P_{\pm}=\frac{l_{\pm}}{4\pi b}=\frac{i}{2}(b^{-1}\pm b)$,
appearing in the LHS of dilaton equation, correspond (respectively) to unit
operator $V_{1,1}$ and bulk operator of dimension $1$ $V_{1,-1}$. Insertion of
the latter is equivalent to differentiating over Liouville bulk cosmological
constant, dependence on which is fixed and power-like (KPZ scaling); thus, for
minimal string correlation numbers we have

$\langle\mathcal{T}_{1,1}\prod\limits_{i}\mathcal{T}_{1.k_{i}}\rangle_{g}\sim\frac{1}{b}\left(\sum\limits_{i}\frac{b(k_{i}+1)}{2}-\frac{b^{-1}+b}{2}(2-2g)\right)\langle\prod\limits_{i}\mathcal{T}_{1.k_{i}}\rangle_{g}$
(59)

This simpler equation, however, is not valid for $p$-deformed volumes. The
correlation number on the LHS is necessarily in the other region of
analyticity and one can not analytically continue formulas for $V_{g,n}$ to
calculate it. The dilaton equation that is valid for $p$-deformed volumes can
be obtained if one imagines the analogous relation with the insertion of
matter “screening operator” of dimension 1.

$\langle\mathcal{T}_{1,-1}\prod\limits_{i}\mathcal{T}_{1.k_{i}}\rangle_{g}\sim\frac{1}{b}\left(\sum\limits_{i}\frac{b(k_{i}-1)}{2}-\frac{b^{-1}-b}{2}(2-2g)\right)\langle\prod\limits_{i}\mathcal{T}_{1.k_{i}}\rangle_{g}$
(60)

and subtracts it from the equation above. In MS such operator does not
“exist”, being outside of the Kac table, and can only be considered formally.

The result of substraction reads

$\langle(\mathcal{T}_{1.1}-\mathcal{T}_{1,-1})\prod\limits_{i}\mathcal{T}_{1.k_{i}}\rangle_{g}\sim\left(\sum\limits_{i}1-2+2g\right)\langle\prod\limits_{i}\mathcal{T}_{1.k_{i}}\rangle_{g}=\left(2g-2+n\right)\langle\prod\limits_{i}\mathcal{T}_{1.k_{i}}\rangle_{g}$
(61)

which is precisely the dilaton equation.

As for the string equation, Liouville momentum of additional insertion in the
LHS $P=\frac{ib^{-1}}{2}$ corresponds to Liouville operator
$V_{1,0}=\exp(b\varphi)$. It is easy to check that one cannot construct a
meaningful bulk operator of integer dimension combining it with any minimal
model operators from the Kac table. However, it has boundary dimension $1$, as
expected of the length operator. Interpretation of the string equation in this
terms can also be seen from the form we started with (48). In the RHS, we have
a sum of derivatives of FZZT boundary amplitude $R=\frac{W}{\prod 2z_{i}}$
over $z_{k}^{2}$. Since $z_{k}^{2}$ has a meaning of boundary cosmological
constant for each boundary component, differentiating over it is precisely
what brings down the length operator.

In “matrix model”/KdV framework it is known that introduction of the boundary
operator can be reduced to a modification of the couplings, corresponding to
bulk operators. This modification (as presented e.g. in Martinec:1991ht )
should be consistent with what we have in the RHS of (57), if we account for
(linearized) resonance transformations. It would be interesting to understand
how this RHS could appear in the worldsheet approach. This would require to
understand formula (30) in this language and how introducing the length
operator interferes with it.

##  4 Conclusions

To conclude, we list some speculations and ideas on possible directions for
further study:

  * •

Our main aim was to clarify connection between worldsheet and “matrix model”
formulations for the minimal string like the one of Collier:2023cyw ; we have
not reached that goal yet. Difference between our formula and the one for VMS
is an additional factor that includes only $\delta$-classes; we have already
speculated on a possible explanation for this modification in the main text.
Other than that, it might be possible to interpret this additional factor as a
character of some other bundle, tensored with the one of Collier:2023cyw .
Since $\delta$-classes in a sense “have support” on the boundary of moduli
space $\partial{\overline{\mathcal{M}}}_{g,n}$, it would be interesting to
describe the sections of this bundle as having modified asymptotic behaviour
on the boundary.

  * •

The fact that VMS and MS spectral curve data are related in such a simple way
needs to be better understood. Perhaps it is connected to the fact that in
both cases, the worldsheet CFTs do not have any extra structure other than the
Virasoro symmetry (although in the matter sector one deals with either generic
or degenerate, respectively, representations of it). The 1-form (3.2) has
several known interpretations: its coefficient gives the universal Cardy
density of states (which is the same as modular kernel for the vacuum
character) and the form itself coincides with the Plancherel measure, related
to representations of the modular double
$\mathcal{U}_{q}(\text{sl}(2,\mathbb{R}))\times\mathcal{U}_{\tilde{q}}(\text{sl}(2,\mathbb{R}))$
Fan:2021bwt . It would be interesting to investigate whether this quantum
group perspective is useful to understand these two worldsheet theories and
their matrix model duals.

  * •

Formula (30) suggests that open string (boundary) states can be expanded in
local operators, which allows for an LSZ-type formula connecting open and
closed string amplitudes. An analogous idea was already considered for matrix
model operators e.g. in Moore:1991ir , Gaiotto:2003yb . Is there an associated
relation in BRST cohomology on the worldsheet between boundary and Lian-
Zuckerman states?

  * •

An interesting question is whether worldsheet correlators including cohomology
classes other than tachyons (e.g. ground ring, when at least the simplest
correlators in genus zero can be directly calculated) can be obtained from the
“matrix model” perspective. KPZ scaling arguments suggest that they might be
associated with differentiating over higher KdV couplings; agreement with
fusion rules for these correlators, however, would demand them to be
incorporated into resonance transformations. It is problematic, since these
times have non-positive KPZ dimension. Perhaps a useful way to address this is
to consider $(2p+1)$-reduced KP hierarchy instead of $2$-reduced. In a related
description of MS using Frobenius manifolds, it is known that an analogous
reformulation Belavin:2015ffa  takes care of resonance transformations
automatically.

##  5 Acknowledgements

The authors are grateful to B. Feigin, M. Kazarian, A. Litvinov and A.
Marshakov for numerous stimulating and insightful discussions. The work of
A.A. is supported by the Russian Science Foundation grant (project no.
23-12-00333).

##  Appendix A Spectral curve variations corresponding to CFT couplings:
example (2,9)

Spectral curve for $(2,9)$ minimal string and general KdV times looks as
follows

$\begin{cases}x=z^{2}+X_{0}\\\
y=z^{9}+Y_{0}z^{7}+Y_{1}z^{5}+Y_{2}z^{3}+Y_{3}z\end{cases}$ (62)

KdV times are calculated via (14). Condition $t_{-1}=0$ yields
$Y_{0}=\frac{9X_{0}}{2}$ (this is implied in the following). Other times are
expressed as functions of the polynomial’s coefficients as follows

$\cdot\begin{cases}\frac{g_{-2}t_{0}}{g_{0}}=\frac{1}{8}\left(8Y_{1}-63X_{0}^{2}\right)\\\
\frac{g_{-2}t_{1}}{g_{1}}=\frac{1}{8}\left(105X_{0}^{3}-20X_{0}Y_{1}+8Y_{2}\right)\\\
\frac{g_{-2}t_{2}}{g_{2}}=\frac{1}{128}\left(-945X_{0}^{4}+240X_{0}^{2}Y_{1}-192X_{0}Y_{2}+128Y_{3}\right)\\\
\frac{g_{-2}t_{3}}{g_{3}}=\frac{1}{64}\left(63X_{0}^{5}-20X_{0}^{3}Y_{1}+24X_{0}^{2}Y_{2}-32X_{0}Y_{3}\right)\end{cases}$
(63)

Resonance transformations (15) for $u_{0}=2^{-p/2}=1/4$, on the other hand,
read

$\begin{cases}\frac{g_{-2}t_{0}}{9g_{0}}=2^{-4}\tau_{0}\\\
\frac{g_{-2}t_{1}}{9g_{1}}=2^{-6}\tau_{1}\\\
\frac{g_{-2}t_{2}}{9g_{2}}=2^{-8}(\tau_{2}+\frac{3\tau_{0}^{2}}{2})\\\
\frac{g_{-2}t_{3}}{9g_{3}}=2^{-10}(\tau_{3}+\tau_{0}\tau_{1})\end{cases}$ (64)

Background couplings, when $x$ and $y$ are proportional to Chebyshev
polynomials, correspond to
$X_{0}=\frac{-1}{2},Y_{1}=\frac{27}{16},Y_{2}=\frac{-15}{32},Y_{3}=\frac{9}{256}$.
For CFT couplings it means $\tau_{0}=-\frac{1}{2},\,\tau_{1\dots 3}=0$, as
expected. Differentiating the equations above, one can now find how variations
$\delta\tau_{k}$ of the couplings near this point are related to variations of
polynomial’s coefficients $\delta X_{0},\delta Y_{i}$ and calculate the
$(1,0)$-form $\Omega$. The answer is

$\Omega=dz\cdot\frac{9}{512}\left(\delta\tau_{0}\,U_{6}(z)+\delta\tau_{1}\,U_{4}(z)+\delta\tau_{2}\,U_{2}(z)+\delta\tau_{3}\,U_{0}(z)\right)$
(65)

with $U$ being Chebyshev polynomials of the second kind, as expected.

##  Appendix B Reducing integral transform (18) to residue

We want to simplify the integral transform (28) for $V_{g,n}$; with respect to
each variable $z_{i}$ we need to calculate the integral of the form

$I(\lambda)\equiv\frac{\cosh\pi\lambda}{\pi^{2}b^{2}}\int\limits_{0}^{\infty}\frac{dl}{l}K_{i\lambda}(l)\frac{1}{2\pi
i}\int\limits_{-i\infty+\epsilon}^{i\infty+\epsilon}dz\,e^{(2\pi^{2}b^{4}z^{2}-1)l}W(z),$
(66)

where $W(z)$ is a polynomial in $\frac{1}{z^{2}}$

$W(z)=\sum\limits_{i=0}\frac{c_{i}}{z^{2(i+1)}}.$ (67)

One has

$\displaystyle
I(\lambda)=\frac{\cosh\pi\lambda}{\pi^{2}b^{2}}\int\limits_{0}^{\infty}\frac{dl}{l}K_{i\lambda}(l)\frac{1}{2\pi
i}\int\limits_{-i\infty+\epsilon}^{i\infty+\epsilon}dz\,e^{(2\pi^{2}b^{4}z^{2}-1)l}\frac{\partial}{\partial
z}\sum\limits_{i=0}\frac{-c_{i}/(2i+1)}{z^{2i+1}}=$
$\displaystyle=4b^{2}\cosh\pi\lambda\int\limits_{0}^{\infty}dl\,K_{i\lambda}(l)\frac{1}{2\pi
i}\int\limits_{-i\infty+\epsilon}^{i\infty+\epsilon}dz\,e^{(2\pi^{2}b^{4}z^{2}-1)l}\sum\limits_{i=0}\frac{c_{i}/(2i+1)}{z^{2i}}=$
$\displaystyle=4b^{2}\cosh\pi\lambda\frac{1}{2\pi
i}\int\limits_{-i\infty+\epsilon}^{i\infty+\epsilon}dz\sum\limits_{i=0}\frac{c_{i}/(2i+1)}{z^{2i}}\int\limits_{0}^{\infty}dl\,K_{i\lambda}(l)e^{-l(1-2\pi^{2}b^{4}z^{2})}$
(68)

Integral over $l$ is known (see gradshteyn2007  6.611, 3); then,

$\displaystyle I(\lambda)=4b^{2}\cosh\pi\lambda\frac{1}{2\pi
i}\int\limits_{-i\infty+\epsilon}^{i\infty+\epsilon}dz\sum\limits_{i=0}\frac{c_{i}/(2i+1)}{z^{2i}}\frac{\pi
i\sin\left(\frac{2\lambda}{i}\arcsin\pi
b^{2}z\right)}{\sinh\pi\lambda\sin\left(2\arcsin\pi b^{2}z\right)}=(\pi
b^{2}z=\sin\pi bs)$ $\displaystyle 2\pi b\coth\pi\lambda\frac{1}{2\pi
i}\int\limits_{-i\infty+\epsilon}^{i\infty+\epsilon}ds\frac{i\sin\left(2\lambda\pi
b\frac{s}{i}\right)}{\sin\left(\pi
bs\right)}\sum\limits_{i=0}\frac{c_{i}/(2i+1)}{(\frac{1}{\pi b^{2}}\sin\pi
bs)^{2i}}$ (69)

Now, we can replace $i\sin\frac{2\pi\lambda bs}{i}=\sinh(2\pi\lambda
bs)=\exp(2\pi\lambda bs)-\cosh(2\pi\lambda bs)$ and rewrite

$\int\limits_{-i\infty+\epsilon}^{i\infty+\epsilon}\cosh(2\pi\lambda
bs)\dots=\frac{1}{2}\left(\int\limits_{-i\infty+\epsilon}^{i\infty+\epsilon}-\int\limits_{-i\infty-\epsilon}^{i\infty-\epsilon}\right)\exp(2\pi\lambda
bs)\dots,$

since the other factor in the integrand is an odd function. This expression is
equal to half-residue of the integrand at $s=0$. Integral of the first term
then can also be reduced to summing residues at the poles in the left half-
plane; overall, we have

$\displaystyle I(\lambda)=\frac{2}{b}\coth\pi\lambda\left(\frac{1}{2\pi
i}\int\limits_{-i\infty+\epsilon}^{i\infty+\epsilon}ds\;e^{2\lambda\pi
bs}\sum\limits_{i=0}\frac{\frac{c_{i}}{(2i+1)}}{(\frac{\sin\pi bs}{\pi
b^{2}})^{2i+1}}-\frac{1}{2}\underset{s=0}{\text{Res}}\;e^{2\lambda\pi
bs}\sum\limits_{i=0}\frac{\frac{c_{i}}{(2i+1)}}{(\frac{\sin\pi bs}{\pi
b^{2}})^{2i+1}}ds\right)=$ $\displaystyle=\frac{\coth\pi\lambda}{\pi\lambda
b^{3}}\left(\frac{1}{2\pi
i}\int\limits_{-i\infty+\epsilon}^{i\infty+\epsilon}ds\cos\left(\pi
bs\right)e^{2\lambda\pi bs}W\left(\frac{\sin\pi bs}{\pi
b^{2}}\right)-\frac{1}{2}\underset{s=0}{\text{Res}}\cos\left(\pi
bs\right)e^{2\lambda\pi bs}W\left(\frac{\sin\pi bs}{\pi
b^{2}}\right)ds\right)=$ $\displaystyle=\frac{\coth\pi\lambda}{\pi\lambda
b^{3}}\left(\sum\limits_{n=0}^{\infty}\underset{s=-\frac{n}{b}}{\text{Res}}\cos\left(\pi
bs\right)e^{2\lambda\pi bs}W\left(\frac{\sin\pi bs}{\pi
b^{2}}\right)ds-\frac{1}{2}\underset{s=0}{\text{Res}}\cos\left(\pi
bs\right)e^{2\lambda\pi bs}W\left(\frac{\sin\pi bs}{\pi
b^{2}}\right)ds\right)=$ $\displaystyle=\frac{1}{\pi\lambda
b^{3}}\coth\pi\lambda\left(\sum\limits_{n=0}^{\infty}(-1)^{n}e^{-2\lambda\pi
n}-\frac{1}{2}\right)\underset{s=0}{\text{Res}}\cos\left(\pi
bs\right)e^{2\lambda\pi bs}W\left(\frac{\sin\pi bs}{\pi b^{2}}\right)ds=$
$\displaystyle=\frac{1}{2\pi\lambda
b^{2}}\underset{s=0}{\text{Res}}\;e^{2\lambda\pi bs}W\left(\frac{\sin\pi
bs}{\pi b^{2}}\right)d\frac{\sin\pi bs}{\pi b^{2}}=\frac{1}{2\pi
b^{2}\lambda}\underset{z=0}{\text{Res}}\;e^{2\lambda\arcsin\pi b^{2}z}W(z)dz$
(70)

##  Appendix C Tautological classes on the moduli space of stable curves

Here we shortly introduce the $\psi$-, $\kappa$\- and $\delta$\- cohomological
classes. One can find more details in a nice exposition Zvonkine2012AnIT .

Let ${\overline{\mathcal{M}}}_{g,n}$ be the moduli space of stable $n$-pointed
curves of genus $g$. Then there is the universal curve

$\displaystyle
p:\overline{\mathcal{C}}_{g,n}\longrightarrow\overline{\mathcal{M}}_{g,n}$
(71)

and marked points define the structure sections

$\displaystyle
s_{i}:\overline{\mathcal{M}}_{g,n}\longrightarrow\overline{\mathcal{C}}_{g,n}.$
(72)

Let $\Delta\overset{i}{\hookrightarrow}\overline{\mathcal{C}}_{g,n}$ be the
set of nodes in the singular fibers. There is a holomorphic line bundle
$\mathcal{L}$ over $\overline{\mathcal{C}}_{g,n}$ whose restriction on
$\overline{\mathcal{C}}_{g,n}\setminus\Delta$ is cotangent to the universal
curve. It defines $n$ line bundles $\mathcal{L}_{i}$ over
$\overline{\mathcal{M}}_{g,n}$:

$\displaystyle\mathcal{L}_{i}=s_{i}^{*}\mathcal{L}$ (73)

The $\psi$-classes are their first Chern classes:

$\displaystyle\psi_{i}=c_{1}(\mathcal{L}_{i})\in
H^{2}(\overline{\mathcal{M}}_{g,n})$ (74)

Let $D_{i}$ be the divisor in $\overline{\mathcal{C}}_{g,n}$ given by $i$th
section $s_{i}$. One can twist $\mathcal{L}$ by these divisors and consider
the corresponding first Chern class:

$\displaystyle K=c_{1}(\mathcal{L}(\sum\limits_{i=1}^{n}D_{i}))\in
H^{2}(\overline{\mathcal{C}}_{g,n})$ (75)

The $\kappa$-classes are defined to be pushforwards of its powers:

$\displaystyle\kappa_{m}=p_{*}(K^{m+1})\in
H^{2m}(\overline{\mathcal{M}}_{g,n})$ (76)

To introduce the $\delta$-classes one needs to consider conormal bundle
$N^{\vee}$ on $\Delta$ and define the following classes on
$\overline{\mathcal{C}}_{g,n}$:

$\displaystyle\Delta_{k,l}=i_{*}(c_{1}(N^{\vee})^{k})\cdot\Delta^{l}\in
H^{2k+4l+4}(\overline{\mathcal{C}}_{g,n}),$ (77)

where by $\Delta$ we denote the Poincare dual element in
$H^{4}(\overline{\mathcal{C}}_{g,n})$ for the homology class of $\Delta$. Then
the $\delta$-classes are defined to be pushforwards of $\Delta_{k,l}$:

$\displaystyle\delta_{k,l}=p_{*}(\Delta_{k,l})\in
H^{2k+4l+2}(\overline{\mathcal{M}}_{g,n})$ (78)

In fact $\delta$-classes can be expressed via $\psi$-classes:

$\displaystyle\delta_{k,l}=\frac{1}{2}j_{*}((\psi_{n+1}+\psi_{n+2})^{k}(\psi_{n+1}\psi_{n+2})^{l}),$
(79)

where $j$ is the attaching map:

$j:\mathcal{\overline{M}}_{g-1,n+2}\sqcup\bigsqcup\limits_{h,I}\mathcal{\overline{M}}_{h,I}\times\mathcal{\overline{M}}_{g-h,\bar{I}}\longrightarrow\mathcal{\overline{M}}_{g,n}$
(80)

In the union $I$ runs over subsets of $\\{1,...,n+2\\}$; $\psi_{n+1}$ and
$\psi_{n+2}$ denote the $\psi$-classes corresponding to additional 2
punctures.

##  Appendix D Integrals and Laplace transforms

In this appendix, we present some technical calculations needed for the
purpose of section 3.3. First, Laplace transform of the 1-forms $d\xi_{j}$ is

$\displaystyle\underset{s=0}{\text{Res}}\;\frac{e^{2\pi b\lambda
s}d\xi_{j}}{2\pi
b^{2}\lambda}=-\underset{s=0}{\text{Res}}\;\underset{s^{\prime}=0}{\text{Res}}\;\frac{e^{2\pi
b\lambda s}}{2\pi
b^{2}\lambda}\cdot\frac{(2j-1)!!}{2^{j}s^{{}^{\prime}2j+1}}\tilde{B}(s,s^{\prime})=$
$\displaystyle=-\frac{1}{(2\pi
i)^{2}}\oint\limits_{|s|=2\epsilon}\oint\limits_{|s^{\prime}|=\epsilon}\frac{e^{2\pi
b\lambda s}}{2\pi
b^{2}\lambda}\cdot\frac{(2j-1)!!}{2^{j}s^{{}^{\prime}2j+1}}\tilde{B}(s,s^{\prime})=$
$\displaystyle=-\frac{1}{(2\pi
i)^{2}}\oint\limits_{|s^{\prime}|=\epsilon}\oint\limits_{|s|=\epsilon/2}\frac{e^{2\pi
b\lambda s}}{2\pi
b^{2}\lambda}\cdot\frac{(2j-1)!!}{2^{j}s^{{}^{\prime}2j+1}}\tilde{B}(s,s^{\prime})-\frac{1}{2\pi
i}\oint\limits_{|s^{\prime}|=\epsilon}\underset{s=s^{\prime}}{\text{Res}}\;\frac{e^{2\pi
b\lambda s}}{2\pi
b^{2}\lambda}\cdot\frac{(2j-1)!!}{2^{j}s^{{}^{\prime}2j+1}}\tilde{B}(s,s^{\prime})=\dots$
(81)

Here we deformed the integration contour for $s^{\prime}$-variable and took
into account the contribution from the pole at $s^{\prime}=s$. Since integrand
is regular at $s=0$, the first term vanishes and we proceed

$\displaystyle\dots$
$\displaystyle=-\underset{s^{\prime}=0}{\text{Res}}\;\frac{(2j-1)!!}{2^{j}s^{{}^{\prime}2j-1}}\underset{z=z(s^{\prime})}{\text{Res}}\;\frac{e^{2\lambda\arcsin\pi
b^{2}z}}{2\pi b^{2}\lambda}\frac{dzdz(s^{\prime})}{(z-z(s^{\prime}))^{2}}=$
$\displaystyle=-\underset{s^{\prime}=0}{\text{Res}}\;\frac{(2j-1)!!}{2^{j}s^{{}^{\prime}2j-1}}dz(s^{\prime})\left.\frac{\partial}{\partial
z}\right|_{z=z(s^{\prime})}\frac{e^{2\lambda\arcsin\pi b^{2}z}}{2\pi
b^{2}\lambda}=-\underset{s^{\prime}=0}{\text{Res}}\;\frac{(2j-1)!!}{2^{j}s^{{}^{\prime}2j-1}}\frac{\partial}{\partial
s^{\prime}}\frac{e^{2\pi b\lambda s^{\prime}}}{2\pi b^{2}\lambda}ds^{\prime}=$
$\displaystyle=-\frac{(2j-1)!!}{2^{j}b}\underset{s^{\prime}=0}{\text{Res}}\;\frac{e^{2\pi
b\lambda s^{\prime}}}{s^{{}^{\prime}2j-1}}ds^{\prime}=-\frac{1}{b}\frac{(\pi
b\lambda)^{2j}}{j!}$ (82)

Now, we compute Laplace transform of the bidifferential

$\displaystyle\int\limits_{s_{1}\in\mathbb{R}+i\epsilon}\int\limits_{s_{2}\in\mathbb{R}}e^{-\tilde{x}(s_{1})/u_{1}-\tilde{x}(s_{2})/u_{2}}\tilde{B}(s_{1},s_{2})=\int\limits_{\mathbb{R}+i\epsilon}ds_{1}\,e^{-s_{1}^{2}/u_{1}}\int\limits_{\mathbb{R}}ds_{2}\,e^{-s_{2}^{2}/u_{2}}(\pi
b)^{2}\frac{\cos\pi bs_{1}\cos\pi bs_{2}}{(\sin\pi bs_{1}-\sin\pi
bs_{2})^{2}}=$
$\displaystyle=\int\limits_{\mathbb{R}+i\epsilon}ds_{1}\,e^{-s_{1}^{2}/u_{1}}\int\limits_{\mathbb{R}}ds_{2}\,e^{-s_{2}^{2}/u_{2}}\sum\limits_{k\in\mathbb{Z}}\left(\frac{1}{(s_{1}-s_{2}+\frac{2k}{b})^{2}}-\frac{1}{(s_{1}+s_{2}+\frac{2k-1}{b})^{2}}\right)=$
$\displaystyle=\int\limits_{0}^{\infty}dl\,l\sum\limits_{k\in\mathbb{Z}}\;\int\limits_{\mathbb{R}+i\epsilon}ds_{1}\,e^{-s_{1}^{2}/u_{1}}\int\limits_{\mathbb{R}}ds_{2}\,e^{-s_{2}^{2}/u_{2}}\left(e^{il(s_{1}-s_{2}+\frac{2k}{b})}-e^{il(s_{1}+s_{2}+\frac{2k-1}{b})}\right)=$
$\displaystyle=\sqrt{\pi
u_{2}}\int\limits_{0}^{\infty}dl\,l\sum\limits_{k\in\mathbb{Z}}\;\int\limits_{\mathbb{R}+i\epsilon}ds_{1}\,e^{-s_{1}^{2}/u_{1}-\frac{u_{2}l^{2}}{4}+il(s_{1}+\frac{k}{b})+\pi
ik}=$
$\displaystyle=\pi\sqrt{u_{1}u_{2}}\int\limits_{0}^{\infty}dl\,le^{-(u_{1}+u_{2})l^{2}/4}\sum\limits_{k\in\mathbb{Z}}e^{i\frac{k}{b}(l-\pi
b)}=$
$\displaystyle=2\pi^{2}b\sqrt{u_{1}u_{2}}\sum\limits_{k\in\mathbb{Z}}\int\limits_{0}^{\infty}dl\,le^{-(u_{1}+u_{2})l^{2}/4}\delta(l-2\pi
b(k+1/2))=$ $\displaystyle=2\pi\sqrt{u_{1}u_{2}}\cdot\pi
b\sum\limits_{k\in\mathbb{Z}}\pi b|k+1/2|e^{-(u_{1}+u_{2})(\pi
b(k+1/2))^{2}}\sim$ (83)
$\displaystyle\sim\frac{2\pi\sqrt{u_{1}u_{2}}}{u_{1}+u_{2}}\left(1+\sum\limits_{m\geq
1}(-1)^{m}\,\frac{(2^{1-2m}-1)B_{2m}}{m!}((\pi
b)^{2}(u_{1}+u_{2}))^{m}\right)$ (84)

The last formula gives an asymptotic expansion for (83). It can be obtained
using methods similar to Mac2018 , or simply by zeta-regularization of
divergent series that appear when expanding the summand in series in
$(u_{1}+u_{2})$. To calculate coefficients $\tilde{B}_{k,l}$, one needs to
subtract

$\displaystyle\int\limits_{s_{1}\in\mathbb{R}+i\epsilon}\int\limits_{s_{2}\in\mathbb{R}}e^{-\tilde{x}(s_{1})/u_{1}-\tilde{x}(s_{2})/u_{2}}\overset{\circ}{\tilde{B}}(s_{1},s_{2})=\int\limits_{\mathbb{R}+i\epsilon}ds_{1}\,e^{-s_{1}^{2}/u_{1}}\int\limits_{\mathbb{R}}ds_{2}\,e^{-s_{2}^{2}/u_{2}}\frac{1}{(s_{1}-s_{2})^{2}}=$
$\displaystyle=\pi\sqrt{u_{1}u_{2}}\int\limits_{0}^{\infty}dl\,le^{-(u_{1}+u_{2})l^{2}/4}=\frac{2\pi\sqrt{u_{1}u_{2}}}{u_{1}+u_{2}}$
(85)

##  Appendix E Table of p-deformed volumes

Let us present some answers for $p$-deformed volumes obtained with our
intersection-theoretic formula. Below they are parametrized by “geodesic
lengths” $l_{k}=4\pi bP_{k}$ and are given in Weil-Petersson normalization. We
use a condensed notation $m_{\vec{\Lambda}}$ with
$\Lambda_{1}\geq\Lambda_{2}\geq\dots$ to denote a symmetric polynomial in
$l_{i}^{2}$; e.g. $m_{3,2,1}=\sum\limits_{i\neq j\neq
k}l_{i}^{6}l_{j}^{4}l_{k}^{2}$.

The simplest way to check the answers for genus zero is to use the explicit
formula in mertens2021  involving the heat capacity for minimal string. Genus
1 answers can be easily reproduced using “Douglas string equation” method of
beltar2010 , if we only use linearised resonance transformations.

We also carried out a few checks for $p$-deformed volumes for $g>1$ and zero
marked points (equivalently, free energy at genus $g$). They can be
alternatively calculated from topological recursion data using the dilaton
equation; this was done for the minimal string in Gregori:2021tvs . Their
formulas for $(g,n)=(2,0),(3,0),(4,0)$ also agree with our intersection-
theoretic answer.

$n$ | $V_{0,n}(\vec{l})$
---|---
4 | $\begin{aligned} 2\pi^{2}+\frac{3}{2}\pi^{2}b^{4}+\frac{1}{2}m_{1}\end{aligned}$
5 | $\begin{aligned} 10\pi^{4}+14\pi^{4}b^{4}+\frac{13}{2}\pi^{4}b^{8}+\left(3\pi^{2}+\frac{5}{2}\pi^{2}b^{4}\right)m_{1}+\frac{1}{2}m_{1,1}+\frac{1}{8}m_{2}\end{aligned}$
6 | $\begin{aligned} \frac{244}{3}\pi^{6}+\frac{493}{3}\pi^{6}b^{4}+\frac{1651}{12}\pi^{6}b^{8}+\frac{765}{16}\pi^{6}b^{12}+\left(26\pi^{4}+40\pi^{4}b^{4}+\frac{229}{12}\pi^{4}b^{8}\right)m_{1}+\\\ +\left(6\pi^{2}+\frac{21}{4}\pi^{2}b^{4}\right)m_{1,1}+\left(\frac{3}{2}\pi^{2}+\frac{31}{24}\pi^{2}b^{4}\right)m_{2}+\frac{3}{4}m_{1,1,1}+\frac{3}{16}m_{2,1}+\frac{1}{48}m_{3}\end{aligned}$
7 | $\begin{aligned} \frac{2758}{3}\pi^{8}+\frac{7252}{3}\pi^{8}b^{4}+\frac{34123}{12}\pi^{8}b^{8}+\frac{7109}{4}\pi^{8}b^{12}+\frac{31851}{64}\pi^{8}b^{16}+\\\ +\left(\frac{910}{3}\pi^{6}+\frac{3995}{6}\pi^{6}b^{4}+\frac{4645}{8}\pi^{6}b^{8}+\frac{805}{4}\pi^{6}b^{12}\right)m_{1}+\\\ +\left(80\pi^{4}+130\pi^{4}b^{4}+\frac{377}{6}\pi^{4}b^{8}\right)m_{1,1}+\left(20\pi^{4}+\frac{385}{12}\pi^{4}b^{4}+\frac{2939}{192}\pi^{4}b^{8}\right)m_{2}+\\\ +\left(15\pi^{2}+\frac{27}{2}\pi^{2}b^{4}\right)m_{1,1,1}+\left(\frac{15}{4}\pi^{2}+\frac{10}{3}\pi^{2}b^{4}\right)m_{2,1}+\left(\frac{5}{12}\pi^{2}+\frac{35}{96}\pi^{2}b^{4}\right)m_{3}+\\\ +\frac{3}{2}m_{1,1,1,1}+\frac{3}{8}m_{2,1,1}+\frac{3}{32}m_{2,2}+\frac{1}{24}m_{3,1}+\frac{1}{384}m_{4}\end{aligned}$
$n$ | $V_{1,n}(\vec{l})$
---|---
1 | $\begin{aligned} \frac{1}{12}\pi^{2}+\frac{1}{48}m_{1}\end{aligned}$
2 | $\begin{aligned} \frac{1}{4}\pi^{4}+\frac{5}{24}\pi^{4}b^{4}+\frac{7}{192}\pi^{4}b^{8}+\left(\frac{1}{12}\pi^{2}+\frac{1}{24}\pi^{2}b^{4}\right)m_{1}+\frac{1}{96}m_{1,1}+\frac{1}{192}m_{2}\end{aligned}$
3 | $\begin{aligned} \frac{14}{9}\pi^{6}+\frac{169}{72}\pi^{6}b^{4}+\frac{209}{144}\pi^{6}b^{8}+\frac{11}{32}\pi^{6}b^{12}+\left(\frac{13}{24}\pi^{4}+\frac{31}{48}\pi^{4}b^{4}+\frac{137}{576}\pi^{4}b^{8}\right)m_{1}+\\\ +\left(\frac{1}{8}\pi^{2}+\frac{1}{12}\pi^{2}b^{4}\right)m_{1,1}+\left(\frac{1}{24}\pi^{2}+\frac{35}{1152}\pi^{2}b^{4}\right)m_{2}\\\ +\frac{1}{96}m_{1,1,1}+\frac{1}{192}m_{2,1}+\frac{1}{1152}m_{3}\\\ \end{aligned}$
4 | $\begin{aligned} \frac{529\pi^{8}}{36}+\frac{1135\pi^{8}b^{4}}{36}+\frac{8879\pi^{8}b^{8}}{288}+\frac{3125\pi^{8}b^{12}}{192}+\frac{3973\pi^{8}b^{16}}{1024}+\\\ +\left(\frac{187}{36}\pi^{6}+\frac{689}{72}\pi^{6}b^{4}+\frac{517}{72}\pi^{6}b^{8}+\frac{421}{192}\pi^{6}b^{12}\right)m_{1}+\\\ +\left(\frac{17}{12}\pi^{4}+\frac{47}{24}\pi^{4}b^{4}+\frac{317}{384}\pi^{4}b^{8}\right)m_{1,1}+\left(\frac{41}{96}\pi^{4}+\frac{175}{288}\pi^{4}b^{4}+\frac{613\pi^{4}b^{8}}{2304}\right)m_{2}\\\ +\left(\frac{1}{4}\pi^{2}+\frac{3}{16}\pi^{2}b^{4}\right)m_{1,1,1}+\left(\frac{1}{12}\pi^{2}+\frac{17}{256}\pi^{2}b^{4}\right)m_{2,1}+\left(\frac{7}{576}\pi^{2}+\frac{23}{2304}\pi^{2}b^{4}\right)m_{3}\\\ +\frac{1}{64}m_{1,1,1,1}+\frac{1}{128}\sum\limits_{j<k}m_{2,1,1}+\frac{1}{384}m_{2,2}+\frac{1}{768}m_{3,1}+\frac{1}{9216}m_{4}\end{aligned}$
$g$ | $V_{g,0}(\vec{l})$
---|---
2 | $\begin{aligned} \frac{43\pi^{6}}{2160}+\frac{139\pi^{6}b^{4}}{8640}+\frac{73\pi^{6}b^{8}}{34560}-\frac{17\pi^{6}b^{12}}{9216}\end{aligned}$
3 | $\begin{aligned} \frac{176557\pi^{12}}{1209600}+\frac{2313247\pi^{12}b^{4}}{7257600}+\frac{19246753\pi^{12}b^{8}}{58060800}+\frac{11651873\pi^{12}b^{12}}{58060800}+\\\ +\frac{8154143\pi^{12}b^{16}}{132710400}-\frac{2202331\pi^{12}b^{20}}{619315200}-\frac{656431\pi^{12}b^{24}}{84934656}\end{aligned}$
4 | $\begin{aligned} \frac{1959225867017\pi^{18}}{493807104000}+\frac{1828160015713\pi^{18}b^{4}}{131681894400}+\frac{15954711860347\pi^{18}b^{8}}{658409472000}+\\\ +\frac{218467010151779\pi^{18}b^{12}}{7900913664000}+\frac{94910534310169\pi^{18}b^{16}}{4213820620800}+\frac{1113449946972163\pi^{18}b^{20}}{84276412416000}+\\\ +\frac{519201864600889\pi^{18}b^{24}}{101131694899200}+\frac{191976434526551\pi^{18}b^{28}}{224737099776000}-\frac{327865175088253\pi^{18}b^{32}}{1198597865472000}-\\\ -\frac{192144632177\pi^{18}b^{36}}{1304596316160}\end{aligned}$

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