We study the connection between minimal Liouville string theory and generalized
open KdV hierarchies. We are interested in generalizing Douglas string equation
formalism to the open topology case. We show that combining the results of the
closed topology, based on the Frobenius manifold structure and resonance
transformations, with the appropriate open case modification, which requires the
insertion of macroscopic loop operators, we reproduce the well-known result for the
expectation value of a bulk operator for the FZZT brane coupled to the general

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q\geq2$ being relatively prime integers), with ghosts and Liouville field theory~\cite{ LFT}, described by scalar boson field $\phi(z)$, arising due to the conformal anomaly effect. The MLG partition function splits into three parts \be Z_{\text{MLG}} = Z_{\text{MM}} \cdot Z_{\text{Liouville}} \cdot Z_{\text{gh}}\;, \ee where all three sectors obey conformal symmetry and the zero total central charge condition, $c_{\text{MM}}+c_{\text{Liouville}} -26=0$, which follows from the Weyl invariance of the string action and ensures BRST invariance of MLG theory. This condition constraints the Liouville coupling constant to be $b=\sqrt{q/p}$. In this paper we are interested in the physical operators, or BRST cohomologies $\OO_{m,n}$, constructed by dressing the minimal model primaries $\Phi_{m,n}(z)$ (where $1\leq m\leq q-1$ and $1\leq n\leq p-1$, modulo Kac symmetry, implemented below by restricting $p m-q n> 0$) with the Liouville \emph{non-degenerate} primary exponential fields $V_a(z)\equiv :e^{2a \phi(z)}:$ \be\label{tachyon} \OO_{m,n} = \int \dd^2 z \Phi_{m,n}(z) V_{a_{m,-n}} (z)\;, \ee where $a_{m,-n}$ is the solution of the dimensional constraint $\Delta^{\text{MM}}_{m,n}+\Delta^{\text{Liouville}}(a)=1$, in the standard Liouville parametrization~\cite{LFT}. We note that apart from these \emph{tachyon} physical operators~\eqref{tachyon}, there exist an important class of \emph{ground ring} operators, constructed from \emph{degenerate} Liouville primaries, which is relevant in particular for constructing multi-point tachyon correlation numbers in the worldsheet approach~\cite{Belavin-ml-2006ex,Belavin-ml-2009cb}, and which will not be considered in this paper. \paragraph{Generating function.} One can pack multi-point bulk correlators $\langle \OO_{m_1,n_1}\ldots \OO_{m_N,n_N}\rangle_{g}$ in a generating function $Z_{g}(\lambda)$, where $g$ is a genus and $\lambda=\{\lambda_{m,n}\}$ is a set of (MLG) Liouville coupling constants,\footnote{We do not discuss here subtleties of the gauge fixing and defining spherical 1- and 2-point functions.} \be\label{gen-fun} Z_{g}(\lambda) = \langle \exp\left(\sum \lambda_{m,n} \OO_{m,n} \right) \rangle_g\;. \ee Our discussion here is limited to a sphere, $g=0$, or a disk worldsheet topology, relevant for the planar limit, so that in what follows we omit the subscript $g$, meaning either spherical or disk topology. The coupling $\lambda_{1,-1}=\mu$ is the Liouville bulk cosmological constant, which corresponds to the operator $e^{2b\phi(x)}$, measuring the area of the worldsheet Riemann surface. The generating function~\eqref{gen-fun} has simple scaling properties, being quasi-homogeneous function of the cosmological constant $\mu$, which allows to assign~\cite{KPZ} the gravitational dimensions $\delta_{m,n}$ to the physical fields $\OO_{m,n}$, and to $Z(\lambda)$. This quasi-homogeneity property together with the conformal selection rules for admissible correlation functions are important constraints for constructing the dual representation of the generating function via the resonance transformation~\cite{Belavin-ml-2014hsa}, which will be described in section~\ref{section:general}. ]]>

0} n t_n x^{n-1}/2 - g\sum_{n\ge 0} \frac{\partial}{\partial t_n} x^{-n-1}\;, \ee which up to normalization is the current of the chiral free boson $\phi(x)$, \be\label{J-phi} J(x) =\frac{g}{\sqrt{2}}\pd \phi(x)\;. \ee Hence, one can consider an energy momentum tensor of this boson, $T(x) = :\pd \phi(x)^2:$, where dots stand for the normal ordering, and the equation~\eqref{ward1} takes the form \be \langle T(x)\rangle = reg\;, \; \text{ or } T(x)Z(t) = 0\;, \; x \to 0\;, \ee where in the second equality we consider $T(x)$ as a differential operator acting on functions of $t$. If we expand $T(x) = \sum_n L_n x^{-n-2},$ then the latter equation can be written in terms of Virasoro constraints: \be L_n Z(t) = 0, \; n \ge -1\;, \ee where $L_n$ are found from the definition of $T(x)$: \ba L_{-1}& = \sum_{k\ge 1} k t_k \frac{\pd}{\pd t_{k-1}}, \\ L_n &= \sum_k k t_k \frac{\pd}{\pd t_{k+n}} + g^2 \sum_{i+j=n} \frac{\pd^2} {\pd t_i \pd t_j}, \quad n \ge 0 \ea and by construction satisfy Virasoro algebra with $c=1$. In the double scaling limit the Virasoro structure also persists with the deference that the boson becomes twisted, that is it acquires half-integer modes in the expansion \be \label{twistedboson} \frac12 g \pd \phi(x) = x^{1/2} - \sum_{n\ge0}(n+1/2)t_n x^{n-1/2}- \frac14 g^2\sum_{n\ge0}\frac{\pd}{\pd t_n} x^{-n-3/2}\;. \ee Here now $t_n$ stand for new (re-scaled) KdV couplings, which are some functions of the ``bare'' couplings of the underlying matrix model~\eqref{pf}. The derivatives $\pd/\pd_{t_k}$ are interpreted as insertions of operators $\OO_k$ in the correlation function. In order to motivate this change we note that in the semiclassical limit the eq.~\eqref{ward1} becomes \be \label{speccurve} y^2 = P(x)\;, \ee where $y:= \lim_{N\to\infty}\langle J(x) \rangle$, and $P(x)$ is a polynomial, which arises from the r.h.s.\ of~\eqref{ward1}. This can be interpreted as an equation for a so-called spectral curve. The boson $\phi(x)$ is then defined on this curve rather then on the $x$-plane (for more details, see e.g.~\cite{Dijkgraaf-ml-2018vnm}). In the double scaling limit the Virasoro constraints, which arise for the twisted free boson~\eqref{twistedboson}, become: \ba \label{vir1} L_{-1} &= \sum_{k\ge 1} (k+1/2) t_k \frac{\pd}{\pd t_{k-1}}+ \frac{1}{2g^2}t_0^2\;, \\ L_{0} &= \sum_{k\ge 0} (k+1/2) t_k \frac{\pd}{\pd t_{k}}+ \frac{1}{16}\;, \\ L_{n}& = \sum_{k\ge 0} (k+1/2) t_k \frac{\pd}{\pd t_{k+n}}+ \frac{1}{8}g^2 \sum_{i+j=n-1} \frac{\pd^2}{\pd t_i \pd t_j}\;,\qquad n>0\;. \\ \ea Finally we note, that any function of KdV parameters, which is annihilated by all these operators, is uniquely defined and represents in fact a (square root of a) tau-function of a KdV hierarchy~\cite{Makeenko-ml-1990in}. ]]>

0} n t_n x^{n-1}/2 - g^2\sum_{n\ge 0} \pd_n x^{-n-1} + \frac {g N_b}{2(z-x)}\;, \ee which obeys the Virasoro symmetry, with the stress tensor $T(x) = J(x)^2/g^2$. It is convenient to introduce an extra factor $e^{-N_b W(z)/(2g)}$ in the matrix integral, which is trivial because it does not depend on the matrix variables, \be\label{Z-open-2} Z(t,z) := \frac{1}{\text{vol}(\U(N))} \int \dd M \, \det(z-M)^{N_b} \, e^{-\frac{N_b}{2g} W(z)}\, e^{-\frac{1}{g}\Tr W(M)}\;. \ee In this setting $J(x)$ as a differential operator is given by the same formula~\eqref{bos1} as in the closed string case due to the factor $e^{-N_b W(z)/(2g)}$. The new point is that the refined boundary partition function~\eqref{Z-open-2} is equivalent to \be \langle \det(z-M)^{N_b} e^{-N_b W(z)/2g} \rangle = \langle V(z) \rangle\;, \ee \sloppy{where $V(z) = :e^{-N_b\phi(z)/\sqrt{2}}:$ is the primary vertex operator, constructed form the \hbox{new~boson.}} Repeating computation of the closed case we get: \be \label{wardopen} \langle T(x) \cdot V(z) \rangle = \left(\frac{N_b^2}{4(x-z)^2} + \frac{1}{x-z}\pd_z +P(x,z)\right) \langle V(z)\rangle\;, \ee where $P(x,z)$ is regular at $x=0$. We can now expand all the quantities in series in powers of $x$ and consider the terms at negative powers: \be (L_n^c+L_n^o) \langle V(z) \rangle = 0, \; \quad n\ge -1\;. \ee Here $L_n^c$ come from the expansion of $T(x)$ and are the same as in the closed case~\eqref{vir1} and $L_n^o$ come from the right hand side of~\eqref{wardopen}: \be L_n^o = -z^{n+1} \frac{\pd}{\pd z} - \frac{N_b^2}4(n+1) z^n\;. \ee Now we turn to the loop operator, which corresponds to a surface with one boundary component. To this end we take the first coefficient in the series expansion in $N_b$, \ba \label{loopFormula2} w(z) =& \frac{\pd}{\pd N_b} \langle \det(z-M)^{N_b} e^{-\frac{N_b}{2g} W(z)} \rangle|_{N_b=0} \\ =& \langle \left(-\frac1{2g} W(z) + \tr \frac{1}{z-M}\right) e^{-\frac{N_b}{2g} W(z)} \rangle \sim \langle \phi(z) \rangle\;. \ea In order to compare with the result of the direct minimal Liouville gravity approach we introduce the inverse Laplace transform $w(l)$ of $w(z)$, according to $w(z)= l^{-1}\int_{0}^{\infty} \dd z \, e^{-lz} \, w(l)$. The singular part of this operator is given by~\eqref{twistedboson}, and the loop~\eqref{loopFormula2} takes the form of the following differential operator \be \label{loop2} w(l) = \sum_{k=0}^{\infty} \frac{l^{k+1/2}}{\Gamma(k+1/2+1)} \OO_{k}\;. \ee ]]>

q+1$), which was due to inappropriate solution of the string equation in $\dd x$ integration. ]]>