aainstitutetext: Landau Institute for Theoretical Physics, 142432
Chernogolovka, Russiabbinstitutetext: Skolkovo Institute of Science and
Technology, 121205, Moscow, Russiaccinstitutetext: Physics Department, Ariel
University, Ariel 40700, Israel

# Torus one-point correlation numbers in minimal Liouville gravity

A. Artemev c V. Belavin<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We present a method for the first principles calculation of tachyon one-point
amplitudes in $(2,2p+1)$ minimal Liouville gravity defined on a torus. The
method is based on the higher equations of motion in the Liouville CFT. These
equations were earlier successfully applied for analytic calculations of the
amplitudes in the spherical topology. We show that this approach allows to
reduce the moduli integrals entering the definition of the torus amplitudes to
certain boundary contributions, which can be calculated explicitly. The
results agree with the calculations performed in the matrix models approach.

###### Keywords:

CFT, Matrix Models, Liouville gravity

##  1 Introduction

Minimal Liouville gravity (MLG) is a model of non-critical string theory
Polyakov1 . In a series of papers Belavin:2013nba ; Belavin:2014cua ;
Belavin:2014hsa ; Belavin:2014xya ; Belavin:2015ffa  for some correlators in
genus zero it was shown that MLG results can also be obtained from Matrix
Models Douglas:1989dd  (in the double-scaling limit) provided appropriate
identifications between coupling constants MSS ; Belavin:2008kv . In the
continuous approach an essential progress has been achieved thanks to
discovery of the so-called higher equations of motion (HEM) in the Liouville
conformal field theory HEM . However it was not clear whether it is possible
to develop a method based on HEM for considering higher genus topology. Even
though some numerical checks were performed Belavin:2010pj ; Belavin:2010sr ,
which gave evidence that the correspondence between MLG and Matrix Models
persists at higher genus, the proof was not possible because the analytic
calculation of the moduli integral in the torus topology was not available.
The purpose of this paper is to describe a method for computing the amplitudes
in MLG for a basic class of physical observables, which are correlation
functions of tachyon vertex operators, in the case of torus topology.

The plan of the paper is as follows. In Section 2 we recall some facts about
continuous formulation of MLG, which are relevant for our consideration. In
Section 3 we consider the particular examples of the torus one-point
correlation numbers. Section 4 is a conclusion in which we discuss in
particular some open questions.

##  2 Preliminaries

###  2.1 Generalities of minimal models and Lioville field theory

##### Liouville CFT.

Liouville conformal field theory is a 2-dimensional CFT, which has the action
of the form

$A_{L}=\int
d^{2}x\,\sqrt{\hat{g}}\left(\frac{1}{4\pi}\hat{g}^{ab}\partial_{a}\phi\partial_{b}\phi+\mu
e^{2b\phi}+\frac{Q}{4\pi}\hat{R}\phi\right)\;.$ (1)

Here $\hat{g}$ is a reference metric with scalar curvature $\hat{R}$,
introduced to have the theory in a covariant form. The parameter $Q=b+b^{-1}$;
the Virasoro central charge of the theory is then given by $c_{L}=1+6Q^{2}$.
$\mu$ is the cosmological constant parameter; the dependence of the
correlation functions on $\mu$ is fixed, e.g., by noticing that $\mu$ can be
set to one by shifting $\phi\to\phi-\frac{1}{2b}\log\mu$.

The normal ordered exponential operators $V_{a}=:e^{2a\phi}:$ are primary
fields of the model of conformal dimension $\Delta^{\text{L}}_{a}=a(Q-a)$.
Operators $V_{a}$ and $V_{Q-a}$ have the same conformal dimension; they are
identified up to a factor $R_{L}(a)$, which is called the “reflection
coefficient”.

The so-called degenerate fields $V_{m,n}\equiv V_{a_{m,n}}$ in the Liouville
theory arise for the special values of the dimension parameter

$a_{m,n}=-b^{-1}\frac{(m-1)}{2}-b\frac{(n-1)}{2}\;.$ (2)

These are the primary fields (Virasoro highest vectors) with a descendant at
level $mn$ in the corresponding highest weight representation, which itself is
a highest vector. In other words, the corresponding Verma module has a
submodule, in CFT such submodules are usually decoupled by putting its highest
vector to zero. Decoupling conditions can be written as
$D^{(L)}_{m,n}V_{m,n}=0$, where $D^{(L)}_{m,n}$ is a polynomial of Virasoro
generators $L_{-k}$ of degree $mn$: $D^{(L)}_{m,n}=L_{-1}^{mn}+\dots\,$. This
condition leads to the fact that only finite number of conformal families, the
so called fusion channels, contribute to the OPE with a degenerate field.

An expression for the sphere three-point functions (or structure constants) of
Liouville theory $C_{L}(a_{1},a_{2},a_{3})$ was first proposed by two groups
of authors dornotto1994  and LFT . It can be motivated e.g. by deriving
recursion relations following from conformal bootstrap/crossing symmetry
equations for four-point correlator with degenerate field $V_{1,2}$. A
relevant solution of these relations is called Dorn-Otto-Zamolodchikov-
Zamolodchikov (DOZZ) three-point function

$C_{L}(a_{1},a_{2},a_{3})=(\pi\mu\gamma(b^{2})b^{2-2b^{2}})^{(Q-a)/b}\frac{\Upsilon_{b}(b)\Upsilon_{b}(2a_{1})\Upsilon_{b}(2a_{2})\Upsilon_{b}(2a_{3})}{\Upsilon_{b}(a-Q)\Upsilon_{b}(a-2a_{1})\Upsilon_{b}(a-2a_{2})\Upsilon_{b}(a-2a_{3})}\;,$
(3)

where $\Upsilon_{b}(x)$ is a certain special function, which obey two
important properties: first, the shift relations

$\displaystyle\Upsilon_{b}(x+b)=\gamma(bx)b^{1-2bx}\Upsilon_{b}(x)\;,$ (4)
$\displaystyle\Upsilon_{b}(x+b^{-1})=\gamma(x/b)b^{2b/x-1}\Upsilon_{b}(x)\;,$
(5)

and second is that this function has zeroes for $x=-mb-\frac{n}{b}$ and
$x=Q+\frac{m}{b}+nb$ for non-negative integers $m,n$.

The OPE $V_{a_{1}}(x)V_{a_{2}}(0)$ in Liouville CFT is most simply written
when $a_{1},a_{2}$ lie in the so-called “basic domain” defined by

$\left|\frac{Q}{2}-\text{Re }a_{1}\right|+\left|\frac{Q}{2}-\text{Re
}a_{2}\right|<\frac{Q}{2}\;.$ (6)

In this case one can write

$V_{a_{1}}(x)V_{a_{2}}(0)=\int\limits_{-\infty}^{\infty}\frac{dP}{4\pi}C_{L}(a_{1},a_{2},\frac{Q}{2}-iP)(x{\overline{x}})^{\Delta^{\text{L}}_{Q/2+iP}-\Delta^{\text{L}}_{a_{1}}-\Delta^{\text{L}}_{a_{2}}}[V_{Q/2+iP}(0)]\;,$
(7)

where $[\dots]$ stands for the contribution of the Virasoro representation
associated with the corresponding primary field. The operators in the
Liouville theory are parametrized by complex $a$, but OPE is given in terms of
contour integral and fields that appear here are parametrized by one real
number $P$. This is because only operators with
$a=\frac{Q}{2}+iP,\,P\in\mathbb{R}$, correspond to normalizable, or
“physical”, states of the theory. This is why in the OPE the sum goes over
these states only.

When parameters of the correlators are not in this domain, some poles of the
structure constants may cross the integration contour. In order to keep the
analyticity in the parameters in such a case one should either deform the
contour of integration over $P$ or (equivalently) keep the contour unchanged
but explicitly add the contributions from these poles which are referred to as
“discrete terms”. In fact, when one of the fused fields is degenerate, only a
finite number of these terms contribute to the OPE (the “continuous” part
becomes zero), as one could expect.

##### Minimal model.

Minimal models (MM) are CFTs in which only finite number of conformal families
of operators exist. All primary fields have to be degenerate for consistency,
so that after the fusion of two of them we would also have only a finite
number of contributions. For the special values of the central charge defined
by

$c_{M}=1-6(\beta^{-1}-\beta)^{2},\quad\beta=\sqrt{\frac{r}{r^{\prime}}},\quad
r<r^{\prime}-\text{ coprime numbers }$ (8)

in the lattice of degenerate fields $\Phi_{m,n}$ there is a subset called the
Kac table that consists of fields $\Phi_{m,n}$ with $0<m<r,\,0<n<r^{\prime}$,
on which the OPE is closed. There is also Kac symmetry identification
$\Phi_{m,n}=\Phi_{r-m,r^{\prime}-n}$, which is similar to the reflection
property in the Liouville CFT. A theory that contains only conformal families
from such a Kac table is called a minimal model $\mathcal{M}_{r,r^{\prime}}$.
The conformal dimensions of the primary fields are parametrized as
$\Delta_{\alpha}^{\text{M}}=\alpha(\alpha-\beta^{-1}+\beta)$ with

$\alpha_{m,n}=\frac{\beta^{-1}-\beta}{2}+\frac{n\beta-m\beta^{-1}}{2}\;.$ (9)

The “kinematical” properties of the minimal model strongly remind of those of
the Liouville theory for imaginary $b$. For example, the decoupling conditions
defining the fusion rules have the form $D^{(M)}_{m,n}\Phi_{m,n}=0$, where
operator $D^{(M)}$ can be obtained from $D^{(L)}$ by replacing
$b^{2}\to-\beta^{2}$; the recursion relations allowing to obtain the structure
constants are also given by analytic continuation $b\to-i\beta,a_{i}\to
i\alpha_{i}$.

Relevant for the minimal model solutions to the recursion relations above are
not analytic continuations of DOZZ structure constants. In fact, they are
given by Zamolodchikov:2005fy

$C_{M}(\alpha_{1},\alpha_{2},\alpha_{3})=\frac{\mathcal{A}\Upsilon_{\beta}(\alpha-2\alpha_{3}+\beta)\Upsilon_{\beta}(\alpha-2\alpha_{1}+\beta)\Upsilon_{\beta}(\alpha-2\alpha_{2}+\beta)\Upsilon_{\beta}(\alpha+2\beta-\beta^{-1})}{\prod\limits_{i=1}^{3}(\Upsilon_{\beta}(2\alpha_{i}+\beta)\Upsilon_{\beta}(2\alpha_{i}+2\beta-\beta^{-1}))^{1/2}}\;,$
(10)

where

$\alpha\equiv\alpha_{1}+\alpha_{2}+\alpha_{3}\;,\quad\mathcal{A}\equiv\frac{\beta^{\beta^{-2}-\beta^{2}-1}[\gamma(\beta^{2})\gamma(\beta^{-2}-1)]^{1/2}}{\Upsilon_{\beta}(\beta)}\;.$
(11)

When specialized to the degenerate values of $b$ and $\alpha$, (10) coincides
with the minimal model structure constants whenever they are non-zero, but it
does not necessarily conform with the fusion rules (i.e., it can give a non-
zero answer even if the sphere three-point function
$\langle\Phi_{\alpha_{1}}\Phi_{\alpha_{2}}\Phi_{\alpha_{3}}\rangle$ should be
zero; most explicit example is that $C_{M}(\alpha_{1},\alpha_{2},0)$ can be
non-zero even for
$\Delta^{\text{M}}_{\alpha_{1}}\neq\Delta^{\text{M}}_{\alpha_{2}}$).

###  2.2 Minimal Liouville gravity

The MLG partition function is combined from the matter (M), Liouville (L) and
ghosts (G) sectors

$Z_{\text{MLG}}=Z_{\text{M}}\cdot Z_{\text{L}}\cdot Z_{\text{G}}\;,$ (12)

Ghosts are just a free fermionic system with fields $B,C$ of conformal
dimensions $(2,-1)$ respectively, with the corresponding antiholomorphic
counterparts ${\overline{B}},{\overline{C}}$ (its central charge is equal to
$-26$). All three sectors obey conformal symmetry and the zero total central
charge condition, $c_{\text{M}}+c_{\text{L}}=26$, which follows from the Weyl
invariance of the string action and ensures the nilpotency of the BRST-
symmetry charge

$\mathcal{Q}\equiv\oint dz\,\left(C(T_{L}+T_{M})+:C\partial CB:\right)(z)$
(13)

in MLG (and, similarly, for its antiholomorphic part
${\overline{\mathcal{Q}}}$). In this paper we are interested in Yang-Lee
series of minimal models in the matter sector, $r=2,r^{\prime}=2p+1$, where
$p$ is positive integer, see (8). In this case the central charge balance
condition constraints the Liouville coupling constant to be
$b=\sqrt{2/(2p+1)}$.

We are interested in the physical operators, or BRST cohomologies $T_{m,n}$,
constructed by dressing the minimal model primaries $\Phi_{m,n}(z)$111 For
Yang-Lee series $m=1$ and Kac symmetry restricts $1\leq n\leq p$. with the
Liouville (non-degenerate) primary exponential fields $V_{a}(z)$:

$T_{m,n}=C\bar{C}\,U_{m,n}\;,\quad U_{m,n}\equiv\Phi_{m,n}V_{a_{m,-n}}\;,$
(14)

where $a_{m,-n}$ is the solution of the dimensional constraint

$\Delta^{\text{M}}_{m,n}+\Delta^{\text{L}}_{a}=1\;.$ (15)

Note that these cohomology classes can be represented either by inserting
local operators $T_{m,n}$ at some point $z$, or by integrating the local
density $U_{m,n}$ over the surface. Since the
$\mathcal{Q},{\overline{\mathcal{Q}}}$-variation of $U_{m,n}$ is a total
derivative, such an integral is BRST-invariant up to boundary terms. However,
the second option becomes relevant only for higher multipoint correlators,
which are not considered in this paper.222For constructing an invariant
correlation number, it may be necessary to additionally dress $T_{m,n}$ with
$B$-ghosts to provide a covariant correlator transformation, see the beginning
of Section 3.

###  2.3 Ground ring in MLG and higher equations of motion

Apart from the tachyon physical operators (14), there exist a class of ground
ring operators wit1992 , constructed from degenerate Liouville primary fields,

$O_{m,n}=H_{m,n}{\overline{H}}_{m,n}\underbrace{\Phi_{m,n}V_{a_{m,n}}}_{\equiv\Theta_{m,n}}\;.$
(16)

Here $H_{m,n}$ are polynomials of degree $mn-1$ of Virasoro generators and
ghosts $B$ and $C$. Unlike the tachyons, the ground ring operators are built
of the Virasoro descendants. The reason for their BRST invariance is that the
result of the $\mathcal{Q}$ action is proportional to the singular descendants
in the Liouville theory $D^{(L)}_{m,n}V_{a_{m,n}}$ and in the minimal model
$D^{(M)}_{m,n}\Phi_{m,n}$ (which we put to zero):

$\mathcal{Q}O_{m,n}={\overline{H}}_{m,n}C\mathcal{D}_{m,n}\Theta_{m,n}\;,\quad\mathcal{D}_{m,n}=D^{(M)}_{mn}-(-1)^{mn}D^{(L)}_{mn}\;.$
(17)

No general form of $H_{m,n}$ is known, but it is easily found case by case
when demanding the requirement above. We cite expressions for $H_{m,n}$ for
the particular cases we will need in the next section (see e.g. imbimbo1992 )

$\displaystyle H_{1,2}=M_{-1}-L_{-1}+b^{2}CB\;,$ (18) $\displaystyle
H_{1,4}=\text{(ghostless part)}+9b^{4}BC(L_{-2}-M_{-2})+\partial
B\,C\left((12b^{4}-\frac{15b^{2}}{2})L_{-1}-(12b^{4}+\frac{15b^{2}}{2})M_{-1}\right)$
$\displaystyle\qquad\quad+\frac{\partial^{2}\,BC}{2}9b^{2}(4b^{4}-1)\;,$ (19)

where by $M$ we denote Virasoro modes of the minimal model and by $L$ those of
the Liouville theory. The ground ring operators have two important properties:
independence of the correlators on their positions in a BRST-invariant
environment and simple OPE with tachyons. These properties turn out to be
useful in calculating MLG correlation numbers on the sphere Belavin:2005jy .

There is an important connection between the MLG gound ring operators and
Zamolodchikov’s higher equations of motion (HEM) highereoms2004 , which are a
set of operator relations in Liouville QFT, that we briefly describe below.
HEM refers to “logarithmic” operators of the form $\phi e^{2a\phi}$; such
operators can be expressed as

$V^{\prime}_{a}=\frac{1}{2}\frac{\partial}{\partial a}V_{a}\;.$ (20)

We denote by $V^{\prime}_{m,n}$ such a logarithmic operator evaluated at the
value of the parameter corresponding to the degenerate dimension $a=a_{m,n}$.
The simplest example is a Liouville field $\phi$ itself $V^{\prime}_{1,1}$,
which is subject to the ordinary Liouville equation of motion

$\partial{\overline{\partial}}\phi\equiv\partial{\overline{\partial}}V^{\prime}_{1,1}=\pi
b\mu e^{2b\phi}\equiv\pi b\mu V_{1,-1}\;.$ (21)

In general, HEM is the relation between the field, which is obtained by
applying the singular vector creating operator to the corresponding
logarithmic field, and the primary field $V_{m,-n}$

$D^{(L)}_{m,n}{\overline{D}}^{(L)}_{m,n}V^{\prime}_{m,n}=B_{m,n}V_{m,-n}\;,$
(22)

where the constant

$B_{m,n}=(\pi\mu\gamma(b^{2})b^{2-2b^{2}})^{n}\frac{\Upsilon^{\prime}_{b}(2a_{m,n})}{\Upsilon_{b}(2a_{m,-n})}\;.$
(23)

In the minimal Liouville gravity, these equations, together with some
additional observations, lead to relations that are sometimes called the
cohomological version of the HEM Belavin:2005jy . Since $V_{m,-n}$ in the RHS
of HEM are precisely the dressing operators for the minimal model primaries
$\Phi_{m,n}$ that appear in the MLG tachyons and these primaries are
annihilated by operators $D^{(M)}_{m,n}$ and ${\overline{D}}^{(M)}_{m,n}$, one
gets

$U_{m,n}=V_{m,-n}\Phi_{m,n}=B_{m,n}^{-1}D^{(L)}_{m,n}{\overline{D}}^{(L)}_{m,n}\underbrace{V^{\prime}_{m,n}\Phi_{m,n}}_{\equiv\Theta^{\prime}_{m,n}}=B_{m,n}^{-1}\mathcal{D}_{m,n}{\overline{\mathcal{D}}}_{m,n}\Theta^{\prime}_{m,n}\;.$
(24)

The RHS can be further rewritten using that

$\mathcal{D}_{m,n}{\overline{\mathcal{D}}}_{m,n}\Theta^{\prime}_{m,n}=(\partial
H_{m,n}-\mathcal{Q}R_{m,n})({\overline{\partial}}\,{\overline{H}}_{m,n}-{\overline{\mathcal{Q}}}\,{\overline{R}}_{m,n})\Theta^{\prime}_{m,n}\;,\quad
R_{m,n}\equiv B_{-1}H_{m,n}\;.$ (25)

Thus, up to $\mathcal{Q}$-exact terms $U_{m,n}$ can be represented as a
derivative of the operator $O^{\prime}_{m,n}\equiv
H_{m,n}{\overline{H}}_{m,n}\Theta^{\prime}_{m,n}$ which is a logarithmic
counterpart of the ground ring operator $O_{m,n}$, eq. (16).

Another form of higher equations of motion, which was first found in the
context of super-MLG Belavin:2008vc , but is also valid in the bosonic case,
reads

$T_{m,n}\equiv
C{\overline{C}}V_{m,-n}\Phi_{m,n}=B_{m,n}^{-1}\mathcal{Q}{\overline{\mathcal{Q}}}\left(O^{\prime}_{m,n}\right)\;.$
(26)

It is easy to derive (25) from this equation by applying $B_{-1}$ and
commuting it with the BRST-charge.

###  2.4 Some properties of CFT correlators on a torus

In this section we collect some additional facts about CFTs on the torus
(i.e., about ghosts, MM and Liouville CFT) relevant to our problem. In what
follows $\tau$ is the modular parameter of the torus and $q\equiv\exp(2\pi
i\tau)$.

First, we list the properties that are valid for all of the mentioned above
CFTs; most important are conformal Ward identities, which in particular
express the correlators of the primary field $\Phi_{\Delta}$ with additional
insertions of the stress-energy tensor $T$ in terms of the correlator without
these insertions. For single and double stress-tensor insertions they read

$\langle
T(z)\Phi_{\Delta}(x)\rangle=\left[\Delta\left(\mathcal{P}(z-x)+2\eta_{1}\right)+(\zeta(z-x)+2\eta_{1}x)\partial_{x}+2\pi
i\frac{\partial}{\partial\tau}\right]\langle\Phi_{\Delta}(x)\rangle\;,$ (27)
$\displaystyle\langle
T(z)T(w)\Phi_{\Delta}(x)\rangle=\frac{c}{12}\mathcal{P}^{\prime\prime}(z-w)\langle\Phi_{\Delta}\rangle+$
$\displaystyle\quad+\left[2(P(z-w)+2\eta_{1})+(\zeta(z-w)+2\eta_{1}w)\partial_{w}\right]\langle
T(w)\Phi_{\Delta}(x)\rangle+$
$\displaystyle\quad+\left[\Delta(P(z-x)+2\eta_{1})+(\zeta(z-x)+2\eta_{1}x)\partial_{x}\right]\langle
T(w)\Phi_{\Delta}(x)\rangle+2\pi i\frac{\partial}{\partial\tau}\langle
T(w)\Phi_{\Delta}(x)\rangle\;.$ (28)

Here elliptic $\zeta$-function and Weierstrass $\mathcal{P}$-function behave
as $\zeta(z)\sim 1/z$ and $\mathcal{P}(z)\sim 1/z^{2}$ at $z=0$ (in this sense
they are doubly periodic analogues of the corresponding terms arising in the
conformal Ward identities on the sphere) and $\eta_{1}$ is a function of
$\tau$ which is given by the following series expansion

$\eta_{1}=(2\pi)^{2}\left[\frac{1}{24}+\sum\limits_{n=1}^{\infty}\frac{nq^{n}}{1-q^{n}}\right]\;.$
(29)

The derivation of the identities (27) and (28), as well as the precise
definitions of $\zeta(z)$ and $\mathcal{P}(z)$, which we will not need in this
paper, can be found in Eguchi:1986sb .333Note that our normalization of $T$
differs from the one of Eguchi:1986sb  by a factor of $2\pi$.

Another common feature of the considered CFTs is that one-point functions of
primary fields $f_{\Delta}(\tau)\equiv\langle\Phi_{\Delta}\rangle_{\tau}$ are
modular forms of weight $\Delta$, which means that under transformation
$\tau\to\tau+1$ they are invariant and under $\tau\to-\frac{1}{\tau}$ they
transform as

$f_{\Delta}(-\frac{1}{\tau})=(\tau{\overline{\tau}})^{\Delta}f_{\Delta}(\tau)\;.$
(30)

This condition is important for a conformal theory to be consistent on higher
genus Riemann surfaces Sonoda:1988fq  (for the proof in case of minimal models
see Felder:1989vx , where it is also shown that similar transformation rules
take place for Virasoro descendant fields).

Let us now turn to the expressions for specific correlators in the QFTs we are
interested in. In the Hamiltonian formulation torus correlators are
represented as traces over the full set of states representing the spectrum of
the theory.

We start with the ghost CFT. As pointed out for instance in polchinski_1998 ,
$BC$-system on the torus is to be quantized with the same boundary conditions
as fields in the other sectors (i.e. periodic rather than antiperiodic in both
directions), because it comes from the Faddeev-Popov determinant. This means
that the fields decompose in integer Fourier modes: $B(z)=\sum e^{2\pi
inz}B_{-n},\,C(z)=\sum e^{2\pi inz}C_{-n}$ and the ghost correlator is to be
understood as a trace with the insertion of the fermion parity operator
$(-1)^{F}$. Operator coefficients $B_{l},C_{k}$ satisfy
$\\{B_{l},C_{k}\\}=\delta_{l+k,0}$.

There are two degenerate vacua
$\left|\uparrow\right\rangle,\left|\downarrow\right\rangle$ that form the two-
dimensional representation of the subalgebra of zero modes
$\\{B_{0},C_{0}\\}=1$ (e.g. we have
$C_{0}\left|\uparrow\right\rangle=\left|\downarrow\right\rangle$,
$B_{0}\left|\uparrow\right\rangle=0$) and are annihilated by $B_{k},C_{k}$
with positive $k$. The basis states in the theory are built by applying to
these two vacua modes $B_{l},C_{l}$ with $l<0$; they can be labeled by two
strictly increasing Young diagrams $\lambda_{1},\lambda_{2}$.

For every basis state
$B_{-\lambda_{1}}C_{-\lambda_{2}}\left|\uparrow\right\rangle$ there is a pair
$B_{-\lambda_{1}}C_{-\lambda_{2}}\left|\downarrow\right\rangle$ built from
another vacuum with the same $L_{0}$ eigenvalue, but different fermion number.
In particular, this implies that $\langle\mathbb{I}\rangle_{gh}$ is zero. The
same is valid for the antiholomorphic ghosts ${\overline{B}},{\overline{C}}$.
The simplest nonzero correlator is $\langle
B(z)C(w){\overline{B}}({\overline{z}}){\overline{C}}({\overline{w}})\rangle$;
it is given by

$\langle
B(z)C(w){\overline{B}}({\overline{z}}){\overline{C}}({\overline{w}})\rangle=|\eta(q)|^{4},\quad\eta(q)\equiv
q^{1/24}\prod\limits_{n=1}^{\infty}(1-q^{n})\;.$ (31)

In what follows, we will often use its independence from the positions of the
ghosts. In the Lagrangian formulation this is equivalent to the fact that the
functional integral over the ghosts vanishes unless there are insertions to
saturate the zero-modes, and there is one-zero mode for both $B$ and $C$ (they
are just constant functions of $z$).

Minimal model one-point correlators (we will not need the higher multipoint
ones) are just given by traces over Hilbert space, as mentioned before.
Schematically,

$\langle\Phi\rangle_{\tau}=\sum\limits_{\Delta,\lambda,{\overline{\lambda}}}q^{\Delta(\lambda)-\frac{c_{L}}{24}}{\overline{q}}^{\Delta({\overline{\lambda}})-\frac{c_{L}}{24}}\left\langle\Delta,\lambda,{\overline{\lambda}}\right|\Phi(z)\left|\Delta,\lambda,{\overline{\lambda}}\right\rangle\;,$
(32)

where $\Delta$ are conformal dimensions and $\lambda$, ${\overline{\lambda}}$
— Young diagrams, labeling Virasoro descendants. The ratio
$\left\langle\Delta,\lambda,{\overline{\lambda}}\right|\Phi(z)\left|\Delta,\lambda,{\overline{\lambda}}\right\rangle/\left\langle\Delta\right|\Phi(z)\left|\Delta\right\rangle$
is completely fixed by conformal symmetry and the series over
$\lambda,{\overline{\lambda}}$ for any $\Delta$ in the spectrum sums up to the
toric conformal blocks (see e.g. Fateev:2009aw ). Explicitly one gets

$\langle\Phi_{1,k}\rangle=\sum\limits_{m=1}^{p}C_{m,(1,k)}^{(M)m}|q|^{2\Delta_{1,m}^{\text{M}}-\frac{1}{12}+\frac{(b^{-1}-b)^{2}}{2}}|F_{M}(\Delta^{\text{M}}_{1,k},\Delta^{\text{M}}_{1,m},q)|^{2}\;.$
(33)

A similar formula is valid in the Liouville theory:

$\langle
V_{a}\rangle_{\tau}=\int\limits_{\gamma}\frac{dP}{4\pi}C^{(L)Q/2+iP}_{a,Q/2+iP}(q{\overline{q}})^{-1/24+P^{2}}\times|F_{L}(\Delta^{\text{L}}_{a},\Delta^{\text{L}}_{Q/2+iP},q)|^{2}\;,$
(34)

where the “diagonal” structure constant is (it follows from (3))

$C^{(L)Q/2+iP}_{a,Q/2+iP}=(\pi\mu\gamma(b^{2})b^{2-2b^{2}})^{-a/b}\frac{\Upsilon(b)\Upsilon(2a)\Upsilon(2iP)\Upsilon(-2iP)}{\Upsilon^{2}(a)\Upsilon(a+2iP)\Upsilon(a-2iP)}\;.$
(35)

It has four series of poles in $P$ at $\pm 2iP=-mb-nb^{-1}-a$ and $\pm
2iP=Q+mb+nb^{-1}-a$. For $0<a<Q$ (as for instance for the Liouville fields
with $a=a_{1,-n}$ which dress the minimal model primary fields) the
integration contour $\gamma$ is just the real line; it “separates” the four
series of poles in this case. For other $a$ (for example, degenerate ones),
from the requirement of analyticity, as in the discussion of OPE, it is
necessary to take into account the residues of the poles that cross the
contour under analytic continuation.

##  3 Calculation of torus one-point numbers

Because of the presence of ghost zero modes, in order to get a nonzero answer
for a tachyon one-point correlator on the torus one needs to insert additional
fields $B$ and ${\overline{B}}$ together with $T_{m,n}$. Due to the properties
described in the previous section, the correlator $\langle
T_{m,n}B{\overline{B}}\rangle_{\tau}$ is a modular form of weight $(2,2)$. It
follows that we get a well-defined correlation number if we integrate this
correlator over one-punctured torus moduli space, which is a fundamental
domain of the $PSL(2,\mathbb{Z})$ action on upper half-plane.444Because of
this property of the correlator, the correlation number is independent of the
choice of the fundamental domain. Hence our main object is

$\int\limits_{F}d^{2}\tau\,\langle
B{\overline{B}}T_{1,n}\rangle_{\tau}=\int\limits_{F}d^{2}\tau\,\langle
B{\overline{B}}C{\overline{C}}V_{1,-n}\Phi_{1,n}\rangle_{\tau}\;.$ (36)

In this section we illustrate the principal steps of the calculation on the
simplest in the technical sense case $(m,n)=(1,2)$ and then describe how it
generalizes to a more difficult case $(m,n)=(1,4)$, where we need to address
additional questions.

###  3.1 Case of $T_{1,2}$

####  3.1.1 Reduction to boundary terms

First, note that the correlator is independent of ghost positions due to (31).
Correlators with the number of ghosts $B$ or $C$ less than one are zero, as
well as correlators of the form $\langle\partial^{k}B\,\partial^{l}C\rangle$,
$k\text{ or }l>0$. Let’s move $C$ to the same point where $V\Phi$ stands, and
move $B$ to some other point $z$. Then we can rewrite the tachyon operator
dressed by $C$ ghosts using HEM in the form (26). The explicit expression for
the operator $H$ in the considered case is (18).

One can commute the BRST operator with the remaining $B$-ghosts using
$\\{Q,B(z)\\}=T(z)$, where $T=T_{L}+T_{M}+T_{gh}$ is the stress-energy tensor
of the full theory, and discard $Q$-exact terms. This gives

$\int d^{2}\tau\,\langle
B(z){\overline{B}}({\overline{z}})\,\mathcal{Q}{\overline{\mathcal{Q}}}\left(O^{\prime}_{m,n}\right)\rangle_{\tau}=\int
d^{2}\tau\,\langle
T(z){\overline{T}}({\overline{z}})O^{\prime}_{m,n}\rangle_{\tau}\;.$ (37)

The fact that $Q$-exact terms can be ignored can be understood as follows. In
the Hamiltonian formulation, the torus average of BRST-exact operator
$\mathcal{Q}(\dots)$ becomes the (super)trace over the Hilbert space of the
form
$\text{Tr}\,\left[(-1)^{F}q^{L_{0}}{\overline{q}}^{{\overline{L}}_{0}}\\{\mathcal{Q},\dots\\}\right]$.
Using the cyclic property of the trace and the fact that $\mathcal{Q}$
anticommutes with $(-1)^{F}$ and commutes with Virasoro modes, it formally
evaluates to zero. However, using this argument in our case can be problematic
since the action of the logarithmic operators is defined only on an extension
of the MLG Hilbert space. As is known from the consideration of correlators in
genus zero, $\mathcal{Q}$-descendants of the logarithmic operators do not
necessarily give zero in a BRST-invariant environment. Noticing this subtle
point, we nevertheless discard $\mathcal{Q}$-exact terms in (37) in order to
move on.

Taking the derivative $\partial/\partial a$ out of the integral over $\tau$,
we are left with

$\frac{1}{2}\frac{\partial}{\partial a}\int d^{2}\tau\,\langle
T(z){\overline{T}}({\overline{z}})\,(L_{-1}-M_{-1}+b^{2}BC){\overline{(\dots)}}V_{a}\Phi_{1,2}\rangle=\frac{1}{2}\frac{\partial}{\partial
a}\int d^{2}\tau\,\langle
T(z){\overline{T}}({\overline{z}})\,b^{4}BC{\overline{B}}{\overline{C}}V_{a}\Phi_{1,2}\rangle\;.$
(38)

The omitted terms are zero due to the absence of ghost operators either in the
holomorphic or antiholomorphic sector (there are ghosts in $T_{gh}$, but at
least one ghost in each term has a derivative with respect to $z$). Now we can
use Ward identities (27) to get rid of $T,{\overline{T}}$ insertions, since
other fields in the correlator are primary. This is one of the simplifications
that occurs in the case $(1,2)$. From the Ward identities, we will also
discard terms with derivatives over $w$ and ${\overline{w}}$ on the basis of
translational invariance of the one-point function.555For more discussion on
this point see Section 4.2.

At $a=a_{1,2}$ the total dimension of the field $BCV_{a}\Phi_{1,2}$ is $0$.
When differentiating over $a$, we either do not differentiate the prefactors
coming from the Ward identities, which gives

$(2\pi)^{2}\frac{\partial^{2}}{\partial\tau\partial{\overline{\tau}}}\langle
BC{\overline{BC}}V^{\prime}_{1,2}\Phi_{1,2}\rangle_{\tau}\;,$ (39)

or we differentiate the conformal dimension $\Delta_{\Phi}$ in one of these
prefactors, then we get 2 terms

$2\pi\Delta^{\text{L}^{\prime}}_{a}\left(i\frac{\partial}{\partial\tau}\left[{\overline{\mathcal{P}}}+2{\overline{\eta}}_{1}\right]-i\frac{\partial}{\partial{\overline{\tau}}}\left[\mathcal{P}+2\eta_{1}\right]\right)\langle
BC{\overline{BC}}V_{1,2}\Phi_{1,2}\rangle_{\tau}\;.$ (40)

Here $\partial/\partial\tau,\partial/\partial{\overline{\tau}}$ act on
everything that stands on the right. We note that (40) has some expicit
$z$-dependence, although this dependence was absent in the initial expression.

The listed above terms are given by derivatives on the moduli space, so they
can be reduced to the boundary contributions. The standard representation of
the one-punctured torus moduli space is a region in $\tau$-plane
$F_{0}=\\{\text{Im }\tau>0,\,|\text{Re }\tau|<\frac{1}{2},\,|\tau|>1\\}$ (see
Figure 1a). The boundary components at $\tau_{1}\equiv\text{Re }\tau=\pm 1/2$
are identified, as well as the arcs of the circle $|\tau|=1$, which are
symmetric with respect to $\text{Re }\tau=0$.666Note that gluing two arcs on
the boundary component $|\tau|=1$ can be done only if the integrand is modular
invariant, which is not the case for $O^{\prime}_{1,2}$. This causes an
additional contribution, as described below. Therefore, it is natural to think
of the moduli space boundary as a horizontal segment of length 1 at
$\tau_{2}\equiv\text{Im }\tau=+\infty$. In this limit, most of the terms
appearing in Liouville/MM/ghost correlators die out, being proportional to
positive powers of $q=\exp(2\pi i\tau)$. Thus, we expect that in order to
obtain an exact answer, we need to calculate only a finite number of terms in
the correlator of order $q^{0}$. This is the main motivation for the upcoming
calculation.

Let’s address first the $z$ dependency problem in (40). The solution is
somewhat surprising: (40) is actually zero, because the average $\langle
BC{\overline{BC}}V_{1,2}\Phi_{1,2}\rangle$ is independent of
$\tau,{\overline{\tau}}$. There are two ways to illustrate this. The first one
is algebraic, and consists of the following chain of relations (and analogous
one for ${\overline{\tau}}$)

$\displaystyle 2\pi i\frac{\partial}{\partial\tau}\langle
BC{\overline{BC}}V_{1,2}\Phi_{1,2}\rangle=\text{(Ward identities)}=\langle
T(z)BC{\overline{BC}}V_{1,2}\Phi_{1,2}\rangle=\text{(as discussed before)}$
$\displaystyle=\langle
T(z)O_{1,2}^{full}\rangle=\langle(\mathcal{Q}B(z))O_{1,2}^{full}\rangle=\langle\mathcal{Q}(B(z)O_{1,2}^{full})\rangle-\langle
B(z)\cancel{(\mathcal{Q}O_{1,2}^{full})}\rangle=0\;.$

This is an inversion of the argument presented earlier, analogous to the fact
that correlators on the sphere do not depend on the positions of
$\mathcal{Q}$-closed operators, if all other operators are BRST-closed.
Another way is to calculate $\langle O_{1,2}\rangle$ brute force using
explicit expressions for single point functions (33), (34) and toric conformal
blocks Fateev:2009aw ; since we have a degenerate field in the Liouville
sector, only discrete terms contribute to the Liouville one-point function
(continuous contribution vanishes) and the expression for $\langle
O_{1,2}\rangle_{\tau}$ is simple. We checked the independence of $q$ of the
correlation function $\langle
BC{\overline{BC}}V_{1,2}\Phi_{1,2}\rangle_{\tau}$ up to the second order. In
particular, this implies an interesting identity for toric conformal blocks
with degenerate dimensions

$\frac{\eta(q)^{2}}{q^{1/12}}\times
F_{L}(\Delta^{\text{L}}_{1,2},\Delta^{\text{L}}_{Q/2+ib/4},q)\times
F_{M}(\Delta^{\text{M}}_{1,2},\Delta^{\text{M}}_{1,p},q)=1\;.$ (41)

The brute force method is more difficult to adapt to the case of general
$O_{1,k}$, but it can also follow from such non-trivial identities.

Now let’s address the issue of modular non-covariance. As discussed in the
previous section, one-point functions of primaries are modular forms (30). For
$\Delta=0$, as, for example, for the ground ring operator $O_{1,2}$, this
means that the correlator is modular invariant. For the logarithmic fields
like $O_{1,2}^{\prime}$, however, they transform non-covariantly. Taking
$\Phi_{\Delta}=BC{\overline{BC}}V_{a}\Phi_{1,2}$ and differentiating both
sides of (30) with respect to $a$ and then setting $a$ such that
$\Delta(a)=0$, we get:

$\frac{1}{2}\frac{\partial}{\partial
a}f_{\Delta}(-\frac{1}{\tau})=\frac{1}{2}\frac{\partial}{\partial
a}f_{\Delta}(\tau)+\frac{1}{2}\frac{\partial}{\partial
a}\Delta^{\text{L}}_{a}\cdot\log(\tau{\overline{\tau}})f_{\Delta}(\tau)\;.$
(42)

We will denote the average $\langle
O^{\prime}\rangle_{\tau}\equiv\frac{1}{2}\frac{\partial}{\partial
a}f_{\Delta}(\tau)$ as $F(r,\varphi)$, where $\tau=ire^{i\varphi}$. Then the
equation above can be rewritten as

$F(1/r,-\varphi)=F(r,\varphi)+\frac{1}{2}\frac{\partial}{\partial
a}\Delta^{\text{L}}_{a}\cdot\log(r^{2})f_{\Delta}(\tau)\;.$ (43)

With this knowledge, let us rewrite (39) as the boundary integral using Gauss
theorem:

$\int\limits_{F}dS\,\left(\vec{\nabla},\vec{A}\right)=\int\limits_{\partial
F}dl\,(\vec{n},\vec{A})\;.$ (44)

The contribution from the boundary at infinity is

$(2\pi)^{2}\int\limits_{F}dS\,\underbrace{\frac{1}{4}\left(\vec{\nabla},\vec{\nabla}\right)}_{\frac{\partial^{2}}{\partial\tau\partial{\overline{\tau}}}}\langle
O^{\prime}\rangle=\frac{1}{4}(2\pi)^{2}\int\limits_{-1/2}^{1/2}d\tau_{1}\,\frac{\partial}{\partial\tau_{2}}\langle
O^{\prime}\rangle_{\tau_{2}\to\infty}+\dots$ (45)

and the integral over the arc segment of the boundary can be rewritten as

$-\frac{1}{4}(2\pi)^{2}\int\limits_{-\pi/6}^{\pi/6}d\varphi\,\frac{\partial}{\partial
r}F(r,\varphi)\mid_{r=1}=-\frac{1}{4}(2\pi)^{2}\int\limits_{0}^{\pi/6}d\varphi\,\left[\frac{\partial}{\partial
r}F(r,\varphi)+\frac{\partial}{\partial r}F(r,-\varphi)\right]\mid_{r=1}\;.$
(46)

There is no contribution from the vertical parts of the boundary with
$\text{Re }\tau=\pm 1/2$ because the integrand is invariant under
$\tau\to\tau+1$. Differentiating (43) with respect to $r$ and then setting
$r=1$, we get

$-\frac{\partial}{\partial r}F(r,-\varphi)\mid_{r=1}=\frac{\partial}{\partial
r}F(r,\varphi)\mid_{r=1}+\frac{1}{2}\frac{\partial}{\partial
a}\Delta^{\text{L}}_{a}\cdot 2\cdot f_{\Delta}(\tau)\;.$ (47)

Therefore, the integral over the arc is defined by the expression

$+\frac{1}{4}(2\pi)^{2}\frac{1}{2}\frac{\partial}{\partial
a}\Delta^{\text{L}}_{a}\cdot 2\int\limits_{0}^{\pi/6}d\varphi\,\langle
O\rangle_{\tau=i\exp(i\varphi)}\;.$ (48)

Using the fact that $\langle O_{1,2}\rangle_{\tau}$ does not depend on $\tau$,
which we stated earlier, eq. (39) can be finally rewritten as

$\frac{1}{4}(2\pi)^{2}\int\limits_{-1/2}^{1/2}d\tau_{1}\,\frac{\partial}{\partial\tau_{2}}\langle
O^{\prime}\rangle_{\tau_{2}\to\infty}+\frac{1}{24}(2\pi)^{3}\cdot\frac{1}{2}\frac{\partial}{\partial
a}\Delta^{\text{L}}_{a}\cdot\langle O\rangle\;.$ (49)

####  3.1.2 Calculation of $\langle O_{1,2}\rangle_{\tau}$ and $\langle
O^{\prime}_{1,2}\rangle_{\tau},\,\tau_{2}\to\infty$

In this section we will discuss how to calculate $\langle O\rangle$ and
$\langle O^{\prime}\rangle$. First, the average in the minimal model is given
as a sum over $p$ conformal families (33)

$\langle\Phi_{1,2}\rangle=\sum\limits_{m=1}^{p}C_{m,(1,2)}^{(M)m}|q|^{2\Delta^{\text{M}}_{1,m}-\frac{1}{12}+\frac{(b^{-1}-b)^{2}}{2}}|F_{M}(\Delta^{\text{M}}_{1,2},\Delta^{\text{M}}_{1,m},q)|^{2}\;.$
(50)

In the case of $\Phi_{1,2}$, the diagonal structure constant
$C_{m,(1,2)}^{(M)m}$ is only nonzero for $m=p$, because $\Phi_{1,p}$ and
$\Phi_{1,p+1}$ are identified. In this particular case it is equal to

$C_{p,(1,2)}^{(M)p+1}=\left(\frac{\gamma(2-2b^{2})\gamma(1-pb^{2})}{\gamma(1-b^{2})\gamma(2-b^{2}(p+1))}\right)^{1/2}\;.$
(51)

The correlator of the ghosts is given by (31), it can be written as a series
expansion in $q$, which starts with $|q|^{1/6}$. The Liouville part is given
by (34).

The normalization of the conformal blocks $F_{L}$ and $F_{M}$ is such that
they start with $1$ when expanded into series in $q$. Let us look at leading
terms in the series expansion in $q$ of the total correlator. Substituting
$\Delta^{\text{M}}_{1,m}=\Delta^{\text{M}}_{1,p}$, we get that it is of order
$|q|^{(b^{2}/8+2P^{2})}$. If $P$ is real, even this leading term tends to zero
on the boundary of moduli space. Therefore, the continuous contribution to the
Liouville correlator vanishes. The Liouville structure constant (35) also
linearly tends to zero when $a\to a_{1,2}=-\frac{b}{2}$. There are, however,
discrete contributions from the poles of structure constant (which have
imaginary $P$) that cross the integration contour when $a\to a_{1,2}$. If two
of $\Upsilon$-functions in the denominator of the structure constant become
zero when $a\to a_{1,2}$, the residues do not vanish in the limit. There are
two such poles $P=\pm\frac{ia}{2}\to\mp\frac{ib}{4}$. Note that for such
Liouville momenta both contributions to correlator from these poles are of
order $|q|^{0}$ in the limit. Moreover, they are equal, because they are
connected via reflection $P\to-P$. We now have an expression for the
correlator

$\langle O_{1,2}\rangle_{\tau_{2}\to\infty}=b^{4}\times
C_{p,(1,2)}^{(M)p+1}\times\frac{2\pi i\times
2}{4\pi}\text{Res}_{P=ia/2}C_{a,Q/2+iP}^{(L)Q/2+iP}\mid_{a=a_{1,2}}\;.$ (52)

Let’s see how this consideration is modified for the logarithmic field
$\langle O^{\prime}\rangle$. We are interested in the contributions
proportional to $\tau_{2}=\text{Im }\tau$, because in the boundary integral we
have to differentiate with respect to $\tau_{2}$. This can only appear when we
keep the $a$-dependence in the prefactor $|q|^{2P^{2}(a)}$ in discrete terms
and differentiate it with respect to $a$ (differentiating the structure
constant would instead give zero when we take the limit, so we leave it
untouched). In our case $P(a)=\pm ia/2$ and the differentiation gives

$\displaystyle\frac{\partial}{\partial\tau_{2}}\langle
O^{\prime}_{1,2}\rangle_{\tau_{2}\to\infty}=\lim\limits_{a\to
a_{1,2}}\left[\frac{\partial}{\partial\tau_{2}}\frac{1}{2}\frac{\partial}{\partial
a}\exp\left(2\pi\tau_{2}a^{2}/2\right)\langle
O_{1,2}\rangle+O(a-a_{1,2})\right]=$ $\displaystyle=\pi a_{1,2}\langle
O_{1,2}\rangle=-\frac{\pi b}{4}\cdot 2\langle O_{1,2}\rangle\;.$ (53)

Now, using

$\frac{1}{2}\frac{\partial}{\partial
a}\Delta^{\text{L}}_{a}=\frac{Q}{2}-a=(a=a_{1,k}=-\frac{(k-1)b}{2})=\frac{b^{-1}}{2}+\frac{kb}{2}=\frac{b}{4}(2p+1+2k)\;,$
(54)

the two terms in (49) together finally yield

$\frac{(2\pi)^{3}}{24}\frac{b}{4}(2p+1+4)\langle
O_{1,k}\rangle-\frac{(2\pi)^{2}}{4}\frac{4\pi b}{8}\langle
O_{1,2}\rangle=\frac{(2\pi)^{3}b}{96}(2p-1)\langle O_{1,2}\rangle\;.$ (55)

###  3.2 Case of $T_{1,4}$

We now want to consider the case when the descendants of the fields $V$ and
$\Phi$ appear in the operator $H_{m,n}$ and become relevant, in particular,
the case $(1,4)$, see eq. (19). We will use the Ward identities on the torus
(27) and (28), discarding terms with derivatives over $z$, as before. To make
sure everything goes according to the same scenario as for the $(1,2)$ case,
we need to check the following statements

  1. 1.

The average of the ground ring operator $\langle O_{1,4}\rangle$ is $\tau$ and
${\overline{\tau}}$ independent. As before, taking into account that $O_{1,4}$
is BRST-closed, it is enough to prove that the derivative with respect to
$\tau$ (${\overline{\tau}}$) can be obtained by inserting $\mathcal{Q}$-exact
stress energy tensor: $2\pi i\frac{\partial}{\partial\tau}\langle
O_{1,4}\rangle=\langle T(z)O_{1,4}\rangle$.

  2. 2.

Given the first point, we need to prove that for the logarithmic operator

$\frac{\partial}{\partial a}\langle
T(z){\overline{T}}({\overline{z}})H_{1,4}{\overline{H}}_{1,4}V_{a}\Phi_{1,4}\rangle\mid_{a=a_{1,4}}=(2\pi)^{2}\frac{\partial^{2}}{\partial\tau\partial{\overline{\tau}}}\frac{\partial}{\partial
a}\langle
H_{1,4}{\overline{H}}_{1,4}V_{a}\Phi_{1,4}\rangle\mid_{a=a_{1,4}}\;.$ (56)

To this end we first consider the average of the form (focusing on the
holomorphic sector)

$\displaystyle\langle T(z)H_{1,4}V_{a}\Phi_{1,4}\rangle=9b^{4}\langle
T(z)BC(L_{-2}-M_{-2})V_{a}\Phi_{1,4}\rangle+$ $\displaystyle+\langle
T(z)\partial
B\,C(\\#L_{-1}-\\#M_{-1})V_{a}\Phi_{1,4}\rangle+\frac{9b^{2}(4b^{4}-1)}{2}\langle
T(z)\partial^{2}B\,CV_{a}\Phi_{1,4}\rangle\;.$ (57)

Here we dropped the ghostless part in $H_{1,4}$, because, as before, the
corresponding expectation value vanishes, since the ghost zero modes are not
saturated.

####  3.2.1 Reduction to boundary terms

##### Terms with ghost derivatives in $H$.

Let us first consider the second and third terms in (57). Since the two-point
ghost function $\langle\partial^{k}B\,C\rangle=0$, for $k={1,2}$, the
contribution from $(T_{L}+T_{M})(z)$ in these terms vanishes because of the
ghost sector. However, the four-point function involving the derivatives of
the ghosts (and thus the ghost expectation values $\langle
T_{gh}(z)\partial^{k}B\,C\rangle$ is non-zero. In fact, from the Ward
identity, using the coordinate invariance of the two-point function without
derivatives, we get

$\displaystyle\partial_{x}^{k}\langle
T_{gh}(z)B(x)C(0)\rangle=\partial_{x}^{k}\left[2(\mathcal{P}(z-x)+2\eta_{1})-(\mathcal{P}(z)+2\eta_{1})+2\pi
i\frac{\partial}{\partial\tau}\right]\langle B(x)C(0)\rangle=$
$\displaystyle=2(-1)^{k}\mathcal{P}^{(k)}(z-x)\langle BC\rangle\;.$ (58)

Taking into account these facts, we conclude that the second term in (57)
vanishes because of translational invariance (now in MM and Liouville sector)
and the third one contributes

$9b^{2}(4b^{4}-1)\mathcal{P}^{\prime\prime}(z)\langle
BCV_{a}\Phi_{1,4}\rangle\;.$ (59)

##### The first term.

For the term without ghost derivatives, we represent Virasoro modes as contour
integrals of the stress-energy tensor: $L_{-2}=\frac{1}{2\pi
i}\oint\frac{dw}{w}T_{L}(w)$ and rewrite it as

$\frac{1}{2\pi
i}\oint\frac{dw}{w}\langle(T_{L}+T_{M}+T_{gh})(z)BC(T_{L}(w)-T_{M}(w))V_{a}\Phi_{1,4}(0)\rangle\;.$
(60)

Let us first list the terms that do not need Ward identities for double $T$
insertion:

$\displaystyle
T_{gh}(T_{L}-T_{M}):~{}~{}\left[1(\mathcal{P}(z)+2\eta_{1})+2\pi
i\frac{\partial}{\partial\tau_{gh}}\right]\langle
BC(L_{-2}-M_{-2})V_{a}\Phi_{1,4}(0)\rangle\;,$ (61) $\displaystyle
T_{L}(-T_{M}):~{}~{}\left[\Delta^{\text{L}}_{a}(\mathcal{P}(z)+2\eta_{1})+2\pi
i\frac{\partial}{\partial\tau_{L}}\right]\langle
BC(-M_{-2})V_{a}\Phi_{1,4}(0)\rangle\;,$ (62) $\displaystyle
T_{M}T_{L}:~{}~{}\left[\Delta^{\text{M}}_{1,4}(\mathcal{P}(z)+2\eta_{1})+2\pi
i\frac{\partial}{\partial\tau_{M}}\right]\langle
BCL_{-2}V_{a}\Phi_{1,4}(0)\rangle\;.$ (63)

We denote by partial $\tau$-derivative with a subscript the derivative acting
only on the corresponding factor in the (factorized) one-point correlator. The
total derivative is a sum
$\frac{\partial}{\partial\tau}=\frac{\partial}{\partial\tau_{gh}}+\frac{\partial}{\partial\tau_{L}}+\frac{\partial}{\partial\tau_{M}}$.
Now we write one of the remaining 2 terms, the one with double $T_{L}$
insertion, using (28)

$\displaystyle\frac{c_{L}}{12}\mathcal{P}^{\prime\prime}(z)\langle
BCV_{a}\Phi_{1,4}\rangle+\oint\frac{dw}{(2\pi
i)w}\left[2(\mathcal{P}(z-w)+2\eta_{1})+(\zeta(z-w)+2\eta_{1}w)\partial_{w}\right]\times$
$\displaystyle\times\left[\Delta^{\text{L}}_{a}(\mathcal{P}(w)+2\eta_{1})+2\pi
i\frac{\partial}{\partial\tau_{L}}\right]\langle
BCV_{a}\Phi_{1,4}\rangle+\left[\Delta^{\text{L}}_{a}(\mathcal{P}(z)+2\eta_{1})+2\pi
i\frac{\partial}{\partial\tau_{L}}\right]\langle
BCL_{-2}V_{a}\Phi_{1,4}\rangle\;.$ (64)

Let us carefully extract the terms of order $1/\omega$ which contribute to the
integral in the second term of the equation above. Taking into account that
$\mathcal{P}(w)\sim\frac{1}{\omega^{2}},\,\omega\to 0$, we obtain

$\displaystyle\left[2(\mathcal{P}(z-w)+2\eta_{1})+(\zeta(z-w)+2\eta_{1}w)\partial_{w}\right]\left[\Delta^{\text{L}}_{a}(\mathcal{P}(w)+2\eta_{1})+2\pi
i\frac{\partial}{\partial\tau_{L}}\right]=$
$\displaystyle=2\left(\mathcal{P}(z)+2\eta_{1}-w\mathcal{P}^{\prime}(z)+\frac{w^{2}}{2}\mathcal{P}^{\prime\prime}(z)\right)\left(2\eta_{1}\Delta_{a}+2\pi
i\frac{\partial}{\partial\tau_{L}}+\frac{\Delta^{\text{L}}_{a}}{w^{2}}+\dots\right)+$
$\displaystyle+\left(\zeta(z)+(2\eta_{1}-\zeta^{\prime}(z))w+\frac{w^{2}}{2}\zeta^{\prime\prime}(z)-\frac{w^{3}}{6}\zeta^{\prime\prime\prime}(z)\right)\left(-\frac{2\Delta^{\text{L}}_{a}}{w^{3}}+\dots\right)=$
$\displaystyle=2\left(\mathcal{P}(z)+2\eta_{1}\right)\left(2\eta_{1}\Delta^{\text{L}}_{a}+2\pi
i\frac{\partial}{\partial\tau_{L}}\right)+\frac{2\Delta^{\text{L}}_{a}}{3}\mathcal{P}^{\prime\prime}(z)\;.$
(65)

We used that $\zeta^{\prime\prime\prime}(z)=-\mathcal{P}^{\prime\prime}(z)$.
With single-$T$ insertion Ward identities

$\langle
BCL_{-2}V_{a}\Phi_{1,4}\rangle=\left(2\eta_{1}\Delta^{\text{L}}_{a}+2\pi
i\frac{\partial}{\partial\tau_{L}}\right)\langle BCV_{a}\Phi_{1,4}\rangle\;.$
(66)

the expression (64) can be rewritten as follows:

$\left(\frac{c_{L}}{12}+\frac{2\Delta^{\text{L}}_{a}}{3}\right)\mathcal{P}^{\prime\prime}(z)\langle
BCV_{a}\Phi_{1,4}\rangle+\left[(\Delta^{\text{L}}_{a}+2)(\mathcal{P}(z)+2\eta_{1})+2\pi
i\frac{\partial}{\partial\tau_{L}}\right]\langle
BCL_{-2}V_{a}\Phi_{1,4}\rangle\;.$ (67)

Similarly, the term coming from $T_{M}(z)T_{M}(w)$ gives

$\left(-\frac{c_{M}}{12}-\frac{2\Delta^{\text{M}}_{1,4}}{3}\right)\mathcal{P}^{\prime\prime}(z)\langle
BCV_{a}\Phi_{1,4}\rangle+\left[(\Delta^{\text{M}}_{1,4}+2)(\mathcal{P}(z)+2\eta_{1})+2\pi
i\frac{\partial}{\partial\tau_{M}}\right]\langle
BC(-M_{-2})V_{a}\Phi_{1,4}\rangle\;.$ (68)

Summing all the contributions, we obtain

$\displaystyle\left[9b^{4}\left(\frac{c_{L}-c_{M}}{12}+\frac{2(\Delta^{\text{L}}_{a}-\Delta^{\text{M}}_{1,4})}{3}\right)+9b^{2}(4b^{4}-1)\right]\mathcal{P}^{\prime\prime}(z)\langle
BCV_{a}\Phi_{1,4}\rangle+$
$\displaystyle+(\Delta^{\text{L}}_{a}+\Delta^{\text{M}}_{1,4}+3)(\mathcal{P}(z)+2\eta_{1})\langle
BC(L_{-2}-M_{-2})V_{a}\Phi_{1,4}\rangle+$ $\displaystyle+2\pi
i\frac{\partial}{\partial\tau}\langle
BC(L_{-2}-M_{-2})V_{a}\Phi_{1,4}\rangle\;.$ (69)

As expected, the prefactors in the first and second lines vanish at
$a=a_{1,4}$ (for the expectation value $\langle T(z)O_{1,4}\rangle$), leaving
only the derivative with respect to the modular parameter. This proves the
$\tau$ and ${\overline{\tau}}$-independence of $\langle O_{1,4}\rangle$. To
prove it for the logarithmic operator, we use the following brief notation:

$\displaystyle\left[9b^{4}\left(\frac{c_{L}-c_{M}}{12}+\frac{2(\Delta^{\text{L}}_{a}-\Delta^{\text{M}}_{1,4})}{3}\right)+9b^{2}(4b^{4}-1)\right]\mathcal{P}^{\prime\prime}(z)=\beta(a-a_{1,4})\;,$
$\displaystyle(\Delta^{\text{L}}_{a}+\Delta^{\text{M}}_{1,4}+3)(\mathcal{P}(z)+2\eta_{1})=\gamma(a-a_{1,4})\;.$
(70)

We also use analogous notation to describe the antiholomorphic dependence.
Then we have

$\displaystyle\langle
T(z){\overline{T}}({\overline{z}})H_{1,4}{\overline{H}}_{1,4}V_{a}\Phi_{1,4}\rangle=\beta(a-a_{1,4})\langle{\overline{T}}BC{\overline{H}}_{1,4}V_{a}\Phi_{1,4}\rangle+$
$\displaystyle+\left(2\pi
i\frac{\partial}{\partial\tau}+\gamma(a-a_{1,4})\right)\langle{\overline{T}}H_{1,4}{\overline{H}}_{1,4}V_{a}\Phi_{1,4}\rangle=$
$\displaystyle=\beta(a-a_{1.4})\left[{\overline{\beta}}(a-a_{1,4})\langle
BC{\overline{BC}}V_{a}\Phi_{1,4}\rangle+\left(-2\pi
i\frac{\partial}{\partial{\overline{\tau}}}+{\overline{\gamma}}(a-a_{1,4})\right)\langle
BC{\overline{H}}_{1,4}V_{a}\Phi_{1,4}\rangle\right]+$
$\displaystyle+\left(-2\pi
i\frac{\partial}{\partial{\overline{\tau}}}+{\overline{\gamma}}(a-a_{1,4})\right)\left(2\pi
i\frac{\partial}{\partial\tau}+\gamma(a-a_{1,4})\right)\langle
H_{1,4}{\overline{H}}_{1,4}V_{a}\Phi_{1,4}\rangle+$
$\displaystyle+{\overline{\beta}}(a-a_{1,4})\left(2\pi
i\frac{\partial}{\partial\tau}+\gamma(a-a_{1,4})\right)\langle
H_{1,4}{\overline{BC}}V_{a}\Phi_{1,4}\rangle\;.$ (71)

We see that after differentiating with respect to $a$ and substituting
$a=a_{1,4}$ the only nonzero term777 All other terms either contain higher
powers of $(a-a_{1,4})$, or are zero because the expectation values of the
ground ring elements do not depend on $\tau$. is

$\frac{1}{2}\frac{\partial}{\partial a}\langle
T(z){\overline{T}}({\overline{z}})H_{1,4}{\overline{H}}_{1,4}V_{a}\Phi_{1,4}\rangle=(4\pi^{2})\frac{\partial^{2}}{\partial\tau\partial{\overline{\tau}}}\frac{1}{2}\frac{\partial}{\partial
a}\langle
H_{1,4}{\overline{H}}_{1,4}V_{a}\Phi_{1,4}\rangle=(4\pi^{2})\frac{\partial^{2}}{\partial\tau\partial{\overline{\tau}}}\langle
O^{\prime}_{1.4}\rangle\;.$ (72)

####  3.2.2 Calculation of $\langle O_{1,4}\rangle_{\tau}$ and $\langle
O^{\prime}_{1,4}\rangle_{\tau},\,\tau_{2}\to\infty$

Similarly to the $(1,2)$ case, here we have 2 contributions to the sum in the
matter correlator (corresponding to $m=p\sim p+1$ and $m=p-1\sim p+2$) or 4
Liouville discrete terms (two pairs connected by reflection) with Liouville
momenta $P=\pm\frac{ib}{4}$ and $P=\pm\frac{3ib}{4}$. When one multiplies the
OPE coming from both sectors, there appear 4 bilinear combinations of
conformal blocks, such that the leading power of $q$ is not necessarily zero.
Namely, we have

$\displaystyle m=p-1:\quad
P=\pm\frac{ib}{4}-\langle\dots\rangle\sim|q|^{b^{2}}\;,\quad
P=\pm\frac{3ib}{4}-\langle\dots\rangle\sim|q|^{0}$
$\displaystyle\,\,\,m=p:\quad
P=\pm\frac{ib}{4}-\langle\dots\rangle\sim|q|^{0}\;,\quad
P=\pm\frac{3ib}{4}-\langle\dots\rangle\sim|q|^{-b^{2}}$

The two contributions which are not of the order $q^{0}$ actually vanish,
because there is an additional coefficient coming from $L,M$ Virasoro modes in
(19). From the conformal Ward identity (27) it follows that

$\langle(L_{-2}-M_{-2})V_{1,4}\Phi_{1,4}\rangle=\left[2\eta_{1}\left(\Delta^{\text{L}}_{1,4}-\Delta^{\text{M}}_{1,4}\right)+2\pi
i\left(\frac{\partial}{\partial\tau_{L}}-\frac{\partial}{\partial\tau_{M}}\right)\right]\langle
V_{1,4}\Phi_{1,4}\rangle\;.$ (73)

Differentiating the leading terms of the four considered contributions with
respect to $\tau$ reduces to an additional numeric factor, which depends on
$m$ and $P$. We thus replace the result of the action of the derivative
operators with this numeric factor as follows:

$\frac{\partial}{\partial\tau_{L}}-\frac{\partial}{\partial\tau_{M}}\to
2\pi\left(\Delta^{\text{M}}_{m}-\Delta_{P}^{L}+\frac{c_{L}-c_{M}}{24}\right)\;.$
(74)

Moreover (again, up to leading order in $q$) we have
$\eta_{1}\approx\frac{1}{24}(2\pi)^{2}$, then in total RHS of (73) can be
rewritten in the following way

$(2\pi)^{2}\left[\frac{1}{12}\left(\Delta^{\text{L}}_{1,4}-\Delta^{\text{M}}_{1,4}\right)+\left(\frac{(b^{-1}-b)^{2}}{4}+\Delta^{\text{M}}_{m}-P^{2}\right)\right](1+O(q))\langle
V_{1,4}\Phi_{1,4}\rangle\;.$ (75)

It is easy to verify that for the cases $(m=p-1,P=\pm\frac{ib}{4})$ and
$(m=p,P=\pm\frac{3ib}{4})$ the prefactors are zero, and for the other two
combinations it simplifies to $\pm(2\pi)^{2}\frac{b^{2}}{2}$. Since such a
factor comes from both holomorphic and antiholomorphic Virasoro modes, the
sign is irrelevant and in the terms of interest we can replace Virasoro modes
$(L_{-2}-M_{-2})$ with a numeric factor $(2\pi)^{4}\frac{b^{4}}{4}$. To
calculate the expectation value $\langle O_{1,4}\rangle$ it remains to
calculate the product of the degenerate structure constants corresponding to
two remaining choices of $m$ and $p$.

An interesting observation is that all 4 nonzero terms (if the Liouville
contributions associated with reflection are taken into account separately)
turn out to be equal. The contributions to the correlator from $m=p$ and
$m=p-1$ with the corresponding Liouville discrete terms each give half of
$\langle O_{1,4}\rangle$. This is because we have an identity

$C_{p,(1,4)}^{(M)p+1}\text{Res}_{P=ib/4}C_{a_{1,4},Q/2+iP}^{(L)Q/2+iP}=C_{p-1,(1,4)}^{(M)p+2}\text{Res}_{P=3ib/4}C_{a_{1,4},Q/2+iP}^{(L)Q/2+iP}\;.$
(76)

When considering the logarithmic operator $O^{\prime}_{1,4}$ instead of
$O_{1,4}$, two pairs of reflection-related discrete terms after
differentiation with respect to $a$ are additionally multiplied by
corresponding Liouville momenta (up to a factor of $i/2$ they are equal to
$a_{1,4}=\frac{-3b}{2}$ and $a_{1,4}+b=\frac{-b}{2}$), so we would have

$\frac{\partial}{\partial\tau_{2}}\langle
O^{\prime}_{1,4}\rangle_{\tau_{2}\to\infty}=-\pi\cdot\frac{1}{2}\left(\frac{3b}{2}\langle
O_{1,4}\rangle+\frac{b}{2}\langle O_{1,4}\rangle\right)=-\frac{\pi b}{4}\cdot
4\langle O_{1,4}\rangle\;.$ (77)

We proved above that the expression in the form (49) is also valid in case
$(1,4)$. We thus have the final answer for tachyon one-point function

$\langle T_{1,4}\rangle=\frac{(2\pi)^{3}}{24}\frac{b}{4}(2p+1+8)\langle
O_{1,4}\rangle-\frac{(2\pi)^{2}}{4}\frac{8\pi b}{8}\langle
O_{1,4}\rangle=\frac{(2\pi)^{3}b}{96}(2p-3)\langle O_{1,4}\rangle\;.$ (78)

##  4 Discussion

###  4.1 General case $T_{1,2k}$ and comparison with matrix model

Building on the observations for the example of $O_{1,4}$ about the fact that
all 4 terms in the sum for $\langle O\rangle$ turn out equal, as well as
extrapolating formulas (77) and (53) we conclude by conjecturing the following
generalized formulas

$\langle O_{1,2k}\rangle=2k\;,\quad\langle O^{\prime}_{1,2k}\rangle=-\frac{\pi
b}{4}\cdot 2k\langle O_{1,2k}\rangle=-\pi bk^{2}\;.$ (79)

Then, if all the steps leading to (49) are valid, we would get for even $k$

$\langle T_{1,k}\rangle=\frac{(2\pi)^{3}}{24}\frac{b}{4}(2p+1+2k)\langle
O_{1,k}\rangle-\frac{(2\pi)^{2}}{4}\frac{2\pi b}{8}k\langle
O_{1,k}\rangle=\frac{(2\pi)^{3}b}{96}(2p+1-k)\langle O_{1,k}\rangle\sim
k(2p+1-k)\;.$ (80)

This is precisely the matrix model result obtained in Belavin:2010pj .

###  4.2 Comments on the case $T_{1,k}$ with odd $k$

In our examples, we ignored terms with derivatives over $w$ and
${\overline{w}}$ on the basis of translational invariance of the one-point
function. It is known from considering ordinary Liouville EOM that this can be
a subtle issue for logarithmic operators — we have a non-zero average of
operator $\partial{\overline{\partial}}\phi$ given by derivatives of the
Liouville field. However, in that case, zero arising from the translational
invariance cancels with the singularity of the diagonal structure constant
$C_{0,Q/2+iP,Q/2-iP}$. There exists a regularisation in which this statement
can be made precise; we describe it in Appendix A. We do not have such a
singularity in the expectation value of $V_{1,2}$, as well as of any
$V_{1,2k}$, so we do not expect such problems. However, for $\langle
T_{1,k}\rangle$ with odd $k$ we have to treat carefully the terms containing
derivatives with respect to the position of the tachyon.

For odd $k$ tachyons, there is also another problem (or another manifestation
of the problem described above). The simplest demonstration can be done for
$k=1$; in this case we have $H_{1,1}=1$, containing no Virasoro modes nor
ghosts. After using HEM and commuting the BRST-charge with the field $B(z)$,
we obtain the following correlator:

$\langle T(z){\overline{T}}(z)V^{\prime}_{1,1}\Phi_{1,1}\rangle=\langle
T(z){\overline{T}}(z)\phi\,\mathbb{I}_{M}\rangle\;.$ (81)

It is easy to see that the average in the ghost sector vanishes for this
correlator. The expression we started with is nonzero (in fact, this
particular tachyon one-point function was computed analytically long before
with another methods in Bershadsky:1990xb ), which is an obvious
contradiction. However, the Liouville part of this correlator
$\langle\phi\rangle$ is in fact infinite (even more so than the torus
partition function $\langle\mathbb{I}_{L}\rangle$, divergent because of
continuity of the spectrum of the theory). It is likely that the zero from the
ghost sector may somehow cancel with this infinity, but we have not yet found
a regularization that would allow to demonstrate this cancellation explicitly.

Despite the caveats above, the correlation numbers $T_{1,k}$ with odd $k$ can
still be calculated with the method described in this paper, if before
applying HEM we use that tachyons related by reflection in MLG are
idenfitifed: $T_{1,k}=T_{1,2p+1-k}$. This reflection interchanges even and odd
$k$, so after using this relation, the problems described above become
irrelevant. Note that the expression obtained with our methods (80) evidently
respects the reflection symmetry.

###  4.3 Other choices of fundamental domain

In this subsection we want to check that our construction is consistent with
the requirement of modular symmetry for the tachyon amplitudes (36). The
partition function is modular invariant, which can be checked explicitly using
the transformation properties of the ghost correlation function $\langle
B\bar{B}C\bar{C}\rangle$ and modular properties of the characters. The
correlation functions with insertions of the vertex operators are modular
covariant and transform according to the transformation law for the vertex
operators (see eq.(30)). This condition combined with the dimensional
constraint (15) leads to the modular invariance of the amplitude (36). The
question is whether our regularization procedure used in Sections 2 and 3 is
consistent with this requirement.

As an example we consider the region $F_{1}$ which is obtained from the
fundamental region $F_{0}$ by applying $S$ transformation, see Fig. 1. In
fact, one could think that this choice even more preferable, because the upper
boundary element of $F_{0}$ maps to point $\tau=0$, while
$(\tilde{d},\tilde{a})$ was considered previously and other two elements
$(\tilde{a},0)$, $(\tilde{d},0)$ are mapped to each other by certain modular
transformation (see below), so that similar consideration as in Section 3,
based on the transformation property of the logarithmic field, can be used to
evaluate corresponding contributions explicitly. However, the expression for
one-point amplitude has a singularity at $\tau=0$ which requires some care.

-1.0-0.500.51$1$dacb (a) Region $F_{0}$

-1.0-0.500.51$1$$\tilde{a}$$\tilde{b}$$\tilde{c}$$\tilde{d}$ (b) Region $F_{1}$

Figure 1: Fundamental regions.

Let us illustrate how to come to the formula (49) for this choice of
fundamental domain, starting with (39). First of all, the contribution of an
integral over the arc $(\tilde{a},\tilde{d})$ is given by the same formula
(48), but with a different sign:

$-\frac{1}{4}(2\pi)^{2}\frac{1}{2}\frac{\partial}{\partial
a}\Delta^{\text{L}}_{a}\cdot 2\int\limits_{0}^{\pi/6}d\varphi\,\langle
O\rangle_{\tau=i\exp(i\varphi)}$ (82)

since for a new fundamental domain outward normal direction is different. The
two arcs $(0,\tilde{a})$ and $(0,\tilde{d})$ can be parametrized respectively
as

$\begin{cases}\tau_{1}=r_{1}e^{i\varphi_{1}}-1,\quad
r_{1}=1,\;\;\varphi_{1}\in(0,\pi/3)\\\
\tau_{2}=1-r_{2}e^{-i\varphi_{2}},\;\;r_{2}=1,\;\;\varphi_{2}\in(0,\pi/3)\\\
\end{cases}$ (83)

These two arcs are glued together using a modular transformation

$\tau_{2}=\frac{\tau_{1}}{\tau_{1}+1}\;,\quad\text{ or
}\quad\varphi_{1}=\varphi_{2}\;,\quad r_{2}=\frac{1}{r_{1}}$ (84)

under which the correlation function of primaries transforms as

$f_{\Delta}\left(\frac{\tau}{\tau+1}\right)=|\tau+1|^{2\Delta}f_{\Delta}(\tau)\;.$
(85)

Then for the logarithmic operator $O^{\prime}$ we have

$\langle O^{\prime}\rangle_{\tau/(\tau+1)}=\langle
O^{\prime}\rangle_{\tau}+\Delta^{\text{L}^{\prime}}_{a}\log|\tau+1|\times\langle
O^{\prime}\rangle_{\tau_{1}}\;.$ (86)

We integrate over these two arcs the normal derivatives of this correlator
(i.e. derivatives with respect to $r_{1}=|\tau_{1}+1|$ and $r_{2}$, with
additional minus sign). Differentiating the transformation rule above with
respect to $r_{1}$, as before, we get that only the term with
$\Delta^{\prime}$ survives when we sum the two contributions; it integrates to

$+\frac{1}{4}(2\pi)^{2}\frac{\pi}{3}\Delta^{\text{L}^{\prime}}_{a}\langle
O\rangle\;.$ (87)

The two terms (82) and (87) add to the same answer as in (48).

Now let us carefully deal with the singularity at $\tau=0$. Let us regularize
the boundary adding the arc $\gamma_{0}\equiv(\tilde{b},\tilde{c})$ of small
circle of radius $r$ with the center at $\tau=0$. It is easy to calculate that
the length of this arc is $l\approx r^{2}$ for $r\to 0$. From modular
properties we know that

$\langle O^{\prime}\rangle_{ire^{i\varphi}}=\langle
O^{\prime}\rangle_{ie^{i\varphi}/r}-\Delta^{\text{L}^{\prime}}_{a}\log
r\langle O\rangle_{\tau}\;.$ (88)

For $\tau\to i\infty$, $\tau_{2}\approx(-i\tau)=\frac{1}{r}$ and
$\frac{\partial}{\partial\tau_{2}}\langle
O^{\prime}\rangle\approx\text{const},\,\tau_{2}\to+\infty$, as we discussed in
the main part of the work. Because of that, the considered integral over the
arc $\gamma_{0}$ can be rewritten as

$\frac{1}{4}(2\pi)^{2}\int\limits_{\gamma_{0}}dl\,\left(-\frac{\partial}{\partial
r}\langle
O^{\prime}\rangle_{ire^{i\varphi}}\right)=\frac{1}{4}(2\pi)^{2}\int\limits_{0}^{r^{2}}dl\,\frac{1}{r^{2}}\frac{\partial}{\partial\tau_{2}}\langle
O^{\prime}\rangle_{\tau_{2}\to\infty}+\frac{1}{4}(2\pi)^{2}\Delta^{\text{L}^{\prime}}_{a}\int\limits_{0}^{r^{2}}dl\,\frac{1}{r}\langle
O\rangle\;.$ (89)

The second term tends to zero when $r\to 0$; thus, we are left with only the
first one, which reproduces contribution from infinity (45) for the original
fundamental domain $F_{0}$.

###  4.4 Comments on generalized minimal gravity

In some contexts it might be interesting to generalize these results to a
gravity theory, in which the matter sector has generic central charge $c<1$
(equivalently, non-rational and negative $b^{2}$) and the structure constants
are analogous to (10). There are several possible definitions of such
“generalized minimal models” (GMM). In the original work by Zamolodchikov
Zamol3pt  this term is used to describe a theory with continuous set of
primary fields $\Phi_{\alpha},\,\alpha\in\mathbb{C}$ and the structure
constants given by (10) with additional identification of fields
$\Phi_{\alpha}$ and $\Phi_{b^{-1}-b-\alpha}$. In a more recent work
clessthan12015  this notion, GMM, is used for a theory with an irrational
central charge, but a discrete set of fields organised as an infinite Kac
table, while the model with the continuous spectrum introduced in Zamol3pt  is
referred to as “$c<1$ Liouville theory”.888We note that the normalization of
the primary fields is different in the references Zamol3pt  and clessthan12015
.

In clessthan12015  it is argued that the GMM with the discrete spectrum is
ill-defined on higher genus Riemann surfaces: there is a discrete set of
possible correlators and all of them are either zero, if they do not satisfy
the fusion rules, or infinite, similar to the partition function. On the other
hand, in $c<1$ Liouville theory the OPE is always continuous and has no
discrete terms, even in the presence of degenerate fields (the analytic
structure constant (10) does not obey the fusion rules), which makes the
matter sector correlator difficult to calculate, even if it turns out to be
finite. It would be interesting to see if the answer that we found, being
analytic in $p$, could be continued to the generalized case where $p$ is non-
integer.

## Acknowledgments

We thank Alexander Belavin, Alexey Litvinov, Mikhail Lashkevich and Sergey
Parkhomenko for useful discussions.

##  Appendix A Equations of motion on the torus in Liouville CFT

##### Regularisation.

EOM in Liouville theory on the torus read

$\partial{\overline{\partial}}\phi=\pi be^{2b\phi}$ (90)

(we put $\mu=1$). Average on the torus of the RHS is given by (see e.g.
Belavin:2010sr )

$\pi b\langle
e^{2b\phi}\rangle_{\tau}=\frac{1}{16\pi^{2}}\frac{1}{|\eta(q)|^{2}\tau_{2}^{3/2}}=\frac{1}{16\pi^{2}\tau_{2}^{3/2}(q{\overline{q}})^{1/24}}\left(1+q+2q^{2}+3q^{3}+\dots\right)\left(1+{\overline{q}}+2{\overline{q}}^{2}+3{\overline{q}}^{3}+\dots\right)$
(91)

which is obtained using eq. (34) and

$C^{Q/2+iP}_{b,Q/2+iP}=\frac{4P^{2}}{\pi b}\;.$ (92)

In this appendix we show how to obtain the same result from the LHS of (90).
Defining $\varphi$ as $\frac{1}{2}\frac{\partial}{\partial
a}\underbrace{V_{a}}_{e^{2a\varphi}}\mid_{a=0}$, we have two problems: 1) the
correlator $\langle\partial{\overline{\partial}}\phi\rangle$ seems to vanish
due to translation invariance of one-point functions; 2) the structure
constant, entering the expression for the one-point function

$C^{Q/2+iP}_{a,Q/2+iP}=(\pi\mu\gamma(b^{2})b^{2-2b^{2}})^{-a/b}\frac{\Upsilon(b)\Upsilon(2a)\Upsilon(2iP)\Upsilon(-2iP)}{\Upsilon^{2}(a)\Upsilon(a+2iP)\Upsilon(a-2iP)}\approx\frac{2}{a}\;,\quad
a\to 0$ (93)

is singular in the limit $a\to 0$ (even more so if we differentiate with
respect to $a$). This divergence appears due to continuous spectrum of
physical states.999E.g., for free bosonic string genus 1 partition function we
also have this divergence, which is related to the target space volume. To
solve these problems, we need to introduce some regularization. The explicit
expression of the correlator

$\langle
V_{a}\rangle_{\tau}=\sum\limits_{\Delta,\lambda,{\overline{\lambda}}}q^{\Delta(\lambda)-\frac{c_{L}}{24}}{\overline{q}}^{\Delta({\overline{\lambda}})-\frac{c_{L}}{24}}\left\langle\Delta,\lambda,{\overline{\lambda}}\right|V_{a}(z)\left|\Delta,\lambda,{\overline{\lambda}}\right\rangle\;.$
(94)

Restricting first the sum to the primary field level, one can rewrite the
matrix elements in terms of the correlators on the sphere with complex
coordinate $\zeta=e^{z}$ in the following way:

$\left\langle\Delta_{3}\right|V_{1}(\zeta_{1})\left|\Delta_{2}\right\rangle=\lim_{\zeta_{3}\to\infty,\zeta_{2}\to
0}|\zeta|_{3}^{4\Delta_{3}}\langle
V_{3}(\zeta_{3})V_{1}(\zeta_{1})V_{2}(\zeta_{2})\rangle=\frac{C(a_{1},a_{2},a_{3})}{|\zeta_{1}|^{2(\Delta_{1}+\Delta_{2}-\Delta_{3})}}\;.$
(95)

We need to transform the coordinates from sphere to cylinder/torus, which
leads to transformation $V_{1}(z)=V_{1}(\zeta_{1})|\zeta_{1}|^{2\Delta_{1}}$.
Then the matrix element on a cylinder

$\left\langle\Delta_{3}\right|V_{1}(z)\left|\Delta_{2}\right\rangle=\frac{C(a_{1},a_{2},a_{3})}{|\zeta_{1}|^{2(\Delta_{2}-\Delta_{3})}}$
(96)

is translation invariant (independent of $z$) if $\Delta_{2}=\Delta_{3}$. Only
such elements appear in the trace for the torus correlator.

With these observations, we note that introducing the position dependence as
well as regularizing the diagonal structure constant can both be done by
replacing the trace with the following expression

$\langle V_{a}\rangle_{\tau}=\lim\limits_{\epsilon\to
0}\sum\limits_{\Delta,\lambda,{\overline{\lambda}}}q^{\Delta(\lambda)-\frac{c_{L}}{24}}{\overline{q}}^{\Delta({\overline{\lambda}})-\frac{c_{L}}{24}}\left\langle\Delta(P),\lambda,{\overline{\lambda}}\right|V_{a}(z)\left|\Delta(P+\epsilon),\lambda,{\overline{\lambda}}\right\rangle\;.$
(97)

##### Level zero.

First, let us check what happens at level zero
$\lambda={\overline{\lambda}}=0$. The matrix element and its derivative are
given by

$\displaystyle\left\langle\Delta(P)\right|V_{1}(z)\left|\Delta(P+\epsilon)\right\rangle=\frac{C_{a_{1},Q/2+i(P+\epsilon)}^{Q/2+iP}}{|\zeta_{1}|^{4P\epsilon+2\epsilon^{2}}}\;,$
$\displaystyle\partial{\overline{\partial}}\left\langle\Delta(P)\right|V_{1}(z)\left|\Delta(P+\epsilon)\right\rangle=4P^{2}\epsilon^{2}(1+O(\epsilon))\frac{C_{a_{1},Q/2+i(P+\epsilon)}^{Q/2+iP}}{|\zeta_{1}|^{4P\epsilon+2\epsilon^{2}}}\;,$
(98)

with the structure constant

$C_{a_{1},Q/2+i(P+\epsilon)}^{Q/2+iP}=(\pi\mu\gamma(b^{2})b^{2-2b^{2}})^{-(a+i\epsilon)/b}\frac{\Upsilon(b)\Upsilon(2a)\Upsilon(2iP)\Upsilon(-2i(P+\epsilon))}{\Upsilon(a+i\epsilon)\Upsilon(a-i\epsilon)\Upsilon(a+i\epsilon+2iP)\Upsilon(a-i\epsilon-2iP)}\;.$
(99)

We differentiate it w.r.t. $a$ and take the limit $a\to 0$, keeping in mind
that only terms coming from differentiation of $\Upsilon(2a)$ are relevant,
otherwise for nonzero $\epsilon$ we get zero in the limit. Taking into account
$\Upsilon^{\prime}(0)=\Upsilon(b)$, we get

$\frac{1}{2}\frac{\partial}{\partial
a}C_{a_{1},Q/2+i(P+\epsilon)}^{Q/2+iP}\mid_{a=0}=(\pi\mu\gamma(b^{2})b^{2-2b^{2}})^{-i\epsilon/b}\frac{\Upsilon(b)^{2}\Upsilon(2iP)\Upsilon(-2i(P+\epsilon))}{\Upsilon(i\epsilon)\Upsilon(-i\epsilon)\Upsilon(i\epsilon+2iP)\Upsilon(-i\epsilon-2iP)}\approx\frac{1}{\epsilon^{2}}+\dots$
(100)

Now we multiply these results and keep only the finite contribution for
$\epsilon\to 0$. We obtain the leading contribution in $q$ for the LHS of (90)

$\langle\partial{\overline{\partial}}\phi\rangle=\int\frac{dP}{4\pi}\times
4P^{2}\times(q{\overline{q}})^{-1/24+P^{2}}\times(1+\dots)\;,$ (101)

which coincides with the integral for RHS of EOM, taking into account (92).

##### Level one.

The matrix elements with descendants can be expressed using the following
commutation relations

$[L_{n},V_{a}(z)]=\zeta^{n}\left(\partial V_{a}+n\Delta_{a}V_{a}\right)\;.$
(102)

For example, on the first level we get

$\displaystyle\left\langle\Delta\right|L_{1}V_{a}(z)L_{-1}\left|\Delta+\delta\right\rangle=2\left\langle\Delta\right|V_{a}L_{0}\left|\Delta+\delta\right\rangle+\zeta(\partial+\Delta_{a})\left\langle\Delta\right|V_{a}L_{-1}\left|\Delta+\delta\right\rangle=$
$\displaystyle=2(\Delta+\delta)\left\langle\Delta\right|V_{a}\left|\Delta+\delta\right\rangle+\zeta^{-1}(\partial-\Delta_{a})\zeta(\partial+\Delta_{a})\left\langle\Delta\right|V_{a}\left|\Delta+\delta\right\rangle=$
$\displaystyle=\frac{C_{a_{1},a_{\Delta+\delta}}^{a_{\Delta}}}{|\zeta|^{2\delta}}\left[2(\Delta+\delta)+(1-\delta-\Delta_{a})(\Delta_{a}-\delta)\right]\;.$

Note the same $\zeta$-dependence as we had for level zero. Again, this
dependence disappears for $\delta=0$, as expected from translation invariance
property of the torus correlator. This means that additional differentiation
$\partial{\overline{\partial}}$ brings out the same factor $\epsilon^{2}$
which cancels $1/\epsilon^{2}$ singularity of structure constant’s derivative
at $a=0$. The kinematic factor in the brackets is regular and non-zero as
$a,\epsilon\to 0$; that means that in our limiting procedure we don’t need to
differentiate it with respect to $a$ nor leave $\epsilon$-dependence. With
appropriate normalization
$\left\langle\Delta\right|L_{1}L_{-1}\left|\Delta\right\rangle=2\Delta$, it
gives coefficient $1$ for $O(q)$ contribution to the toric conformal block, as
expected from (91).

In general, at higher levels we do not get any new subtleties and reasoning
analogous to the original proof of HEM on the sphere HEM  (based on the fact
that the conformal properties of $\partial{\overline{\partial}}\varphi$ and
$e^{2b\varphi}$ are the same) should work for descendant contributions to the
torus correlators.

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