Prog. Theor. Exp. Phys. 2020, 023B07 (9 pages)
DOI: 10.1093/ptep/ptz170
The null identities for boundary operators in the
(2, 2p + 1) minimal gravity
Goro Ishiki
1,2,∗
, Hisayoshi Muraki
3,∗
, and Chaiho Rim
4,∗
1
Tomonaga Center for the History of the Universe, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
2
Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan
3
Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 37673, Korea
4
Department of Physics, Sogang University, Seoul 04107, Korea
∗
E-mail: ishiki@het.ph.tsukuba.ac.jp, hmuraki@sogang.ac.kr, rimpine@sogang.ac.kr
Received November 8, 2019; Accepted December 18, 2019; Published February 22, 2020
...................................................................................................................
By using the matrix model representation, we show that correlation numbers of boundary-
changing operators (BCOs) in (2, 2p + 1) minimal Liouville gravity satisfy some identities,
which we call the null identities. These identities enable us to express the correlation numbers of
BCOs in terms of those of boundary-preserving operators. We also discuss a physical implication
of the null identities as the manifestation of the boundary interaction.
...................................................................................................................
Subject Index B83
1. Introduction
Two-dimensional gravity coupled with a minimal model of CFT has been studied as a good exam-
ple of well-defined quantum gravitational theories [1], and also allows a non-perturbative discrete
formulation given by matrix models [2–4] (for more references, see, for example, Ref. [5]).
In this paper we follow the one-matrix model description [6,7]ofthe(2, 2p + 1) minimal gravity
on Riemann surfaces, but focus on the description in the presence of boundaries [8,9]. The boundary
conditions of the minimal gravity, also referred to as FZZT branes [10], are specified by the value
of the boundary cosmological constant μ
B
and the Kac label (1, m) of the matter Cardy state. In
Ref. [9] it was shown that such boundary conditions are realized in the matrix model by introducing
a generalization of the resolvent operators. The disk partition n function for the (1, m) Cardy state is
given by
F
m
=−tr log f
m
(M ), (1)
where f
m
(M ) is a monic polynomial of the Hermitian matrix M with degree m, and ··· stands
for the expectation value of the one-matrix model. After some renormalizations, the coefficients
of f
m
(M ) are related to the sources of boundary operators, which preserves the (1, m) boundary
condition.
One can introduce some impurities on the boundary, which interpolate two different boundary
conditions. These are called boundary-changing operators (BCOs). Between two boundary segments
of (1, m
1
) and (1, m
2
) with different boundary cosmological constants, one can put a (1, k) primary
operator dressed by the Liouville factor e
β
k
φ
, where k =|m
1
−m
2
|+1, |m
1
−m
2
|+3, ..., m
1
+m
2
−1,
β
k
=
(k+1)b
2
, and b
2
= 2/(2p + 1). It was shown in Ref. [9] that these operators are described in the
© The Author(s) 2020. Published by Oxford University Press on behalf of the Physical Society of Japan.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/),
which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.
Funded by SCOAP
3