scholarly journals Genus expansion of open free energy in 2d topological gravity

2021 ◽  
Vol 2021 (3) ◽  
Author(s):  
Kazumi Okuyama ◽  
Kazuhiro Sakai
Keyword(s):  
Free Energy ◽  
Riemann Surfaces ◽  
Two Dimensions ◽  
Intersection Theory ◽  
Genus Zero ◽  
Kdv Hierarchy ◽  
Kdv Equations ◽  
Higher Genus ◽  
Topological Gravity ◽  
Genus Expansion

Abstract We study open topological gravity in two dimensions, or, the intersection theory on the moduli space of open Riemann surfaces initiated by Pandharipande, Solomon and Tessler. The open free energy, the generating function for the open intersection numbers, obeys the open KdV equations and Buryak’s differential equation and is related by a formal Fourier transformation to the Baker-Akhiezer wave function of the KdV hierarchy. Using these properties we study the genus expansion of the free energy in detail. We construct explicitly the genus zero part of the free energy. We then formulate a method of computing higher genus corrections by solving Buryak’s equation and obtain them up to high order. This method is much more efficient than our previous approach based on the saddle point calculation. Along the way we show that the higher genus corrections are polynomials in variables that are expressed in terms of genus zero quantities only, generalizing the constitutive relation of closed topological gravity.

scholarly journals ON THE GENUS EXPANSION IN THE TOPOLOGICAL STRING THEORY

1995 ◽  
Vol 07 (03) ◽  
pp. 279-309 ◽  
Author(s):  
TOHRU EGUCHI ◽  
YASUHIKO YAMADA ◽  
SUNG-KIL YANG
Keyword(s):  
String Theory ◽  
Topological String ◽  
Minimal Models ◽  
Topological String Theory ◽  
Genus Zero ◽  
Zero Correlation ◽  
Higher Genus ◽  
Systematic Formulation ◽  
Genus One ◽  
Genus Expansion

A systematic formulation of the higher genus expansion in topological string theory is considered. We also develop a simple way of evaluating genus zero correlation functions. At higher genera we derive some interesting formulas for the free energy in the A1 and A2 models. We present some evidence that topological minimal models associated with Lie algebras other than the A-D-E type do not have a consistent higher genus expansion beyond genus one. We also present some new results on the CP1 model at higher genera.

scholarly journals Developments in topological gravity

2018 ◽  
Vol 33 (30) ◽  
pp. 1830029 ◽  
Author(s):  
Robbert Dijkgraaf ◽  
Edward Witten
Keyword(s):  
Matrix Model ◽  
Moduli Space ◽  
Riemann Surfaces ◽  
Degrees Of Freedom ◽  
Intersection Theory ◽  
Two Dimensional ◽  
The Matrix ◽  
Dimensional Gravity ◽  
Topological Gravity ◽  
The Relationship

This note aims to provide an entrée to two developments in two-dimensional topological gravity — that is, intersection theory on the moduli space of Riemann surfaces — that have not yet become well known among physicists. A little over a decade ago, Mirzakhani discovered[Formula: see text] an elegant new proof of the formulas that result from the relationship between topological gravity and matrix models of two-dimensional gravity. Here we will give a very partial introduction to that work, which hopefully will also serve as a modest tribute to the memory of a brilliant mathematical pioneer. More recently, Pandharipande, Solomon, and Tessler3 (with further developments in Refs. 4–6) generalized intersection theory on moduli space to the case of Riemann surfaces with boundary, leading to generalizations of the familiar KdV and Virasoro formulas. Though the existence of such a generalization appears natural from the matrix model viewpoint — it corresponds to adding vector degrees of freedom to the matrix model — constructing this generalization is not straightforward. We will give some idea of the unexpected way that the difficulties were resolved.

scholarly journals JT supergravity and Brezin-Gross-Witten tau-function

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Kazumi Okuyama ◽  
Kazuhiro Sakai
Keyword(s):  
Riemann Surfaces ◽  
Free Energies ◽  
Point Function ◽  
Tau Function ◽  
Kdv Hierarchy ◽  
The Matrix ◽  
Higher Genus ◽  
Super Riemann Surfaces ◽  
The One ◽  
Thermal Correlation

Abstract We study thermal correlation functions of Jackiw-Teitelboim (JT) supergravity. We focus on the case of JT supergravity on orientable surfaces without time-reversal symmetry. As shown by Stanford and Witten recently, the path integral amounts to the computation of the volume of the moduli space of super Riemann surfaces, which is characterized by the Brezin-Gross-Witten (BGW) tau-function of the KdV hierarchy. We find that the matrix model of JT supergravity is a special case of the BGW model with infinite number of couplings turned on in a specific way, by analogy with the relation between bosonic JT gravity and the Kontsevich-Witten (KW) model. We compute the genus expansion of the one-point function of JT supergravity and study its low-temperature behavior. In particular, we propose a non-perturbative completion of the one-point function in the Bessel case where all couplings in the BGW model are set to zero. We also investigate the free energy and correlators when the Ramond-Ramond flux is large. We find that by defining a suitable basis higher genus free energies are written exactly in the same form as those of the KW model, up to the constant terms coming from the volume of the unitary group. This implies that the constitutive relation of the KW model is universal to the tau-function of the KdV hierarchy.

THE ORIGIN OF SL(nC) CURRENT ALGEBRA IN GENERALIZED TWO-DIMENSIONAL GRAVITY

1992 ◽  
Vol 07 (18) ◽  
pp. 4353-4375 ◽  
Author(s):  
K. YOSHIDA
Keyword(s):  
Current Algebra ◽  
Riemann Surfaces ◽  
Mathematical Structure ◽  
Two Dimensions ◽  
Projective Structure ◽  
Two Dimensional ◽  
Induced Gravity ◽  
Higher Genus ◽  
Dimensional Gravity

The origin of local SL(nC) symmetry in induced gravity in two dimensions, enlarged with the so-called W fields, is considered on higher genus Riemann surfaces. In the simplest case of W2, the mathematical structure of Polyakov’s two-dimensional gravity precisely corresponds to the projective structure of Riemann surfaces. Some speculations on the generalization to higher Wn algebras are presented.

scholarly journals Padé approximants on Riemann surfaces and KP tau functions

2021 ◽  
Vol 11 (4) ◽  
Author(s):  
Marco Bertola
Keyword(s):  
Riemann Surface ◽  
Riemann Surfaces ◽  
Hilbert Problem ◽  
Line Bundles ◽  
Fixed Degree ◽  
Tau Function ◽  
Tau Functions ◽  
Deformation Parameters ◽  
Higher Genus ◽  
Geometric Solutions

AbstractThe paper has two relatively distinct but connected goals; the first is to define the notion of Padé approximation of Weyl–Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the Riemann surface and a measure on it, together with the additional datum of a local coordinate near a point and a divisor of degree g. The denominators of the resulting Padé-like approximation also satisfy an orthogonality relation and are sections of appropriate line bundles. A Riemann–Hilbert problem for a square matrix of rank two is shown to characterize these orthogonal sections, in a similar fashion to the ordinary orthogonal polynomial case. The second part extends this idea to explore its connection to integrable systems. The same data can be used to define a pairing between two sequences of line bundles. The locus in the deformation space where the pairing becomes degenerate for fixed degree coincides with the zeros of a “tau” function. We show how this tau function satisfies the Kadomtsev–Petviashvili hierarchy with respect to either deformation parameters, and a certain modification of the 2-Toda hierarchy when considering the whole sequence of tau functions. We also show how this construction is related to the Krichever construction of algebro-geometric solutions.

Level-one SU(3) Wess-Zumino model on higher-genus Riemann surfaces

1990 ◽  
Vol 41 (2) ◽  
pp. 478-483 ◽  
Author(s):  
R. K. Kaul ◽  
R. P. Malik ◽  
N. Behera
Keyword(s):  
Riemann Surfaces ◽  
Higher Genus ◽  
Zumino Model

A Two-Dimensional Free Energy Model for Single Crystalline Ferroelectrics

2005 ◽  
Vol 881 ◽  
Author(s):  
Sang-Joo Kim ◽  
Stefan Seelecke ◽  
Brian L. Ball ◽  
Ralph C. Smith ◽  
Chang-Hoan Lee
Keyword(s):  
Free Energy ◽  
Evolution Equations ◽  
Saddle Points ◽  
Ferroelectric Materials ◽  
Energy Model ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Energy Potential ◽  
Single Crystalline ◽  
Free Energy Model

AbstractThe one-dimensional free energy model for ferroelectric materials developed in [1-3] is general-ized to two dimensions. The proposed two-dimensional energy potential consists of four energy wells corresponding to four variants of the material, four saddle points representing the barriers for 900 switching processes, and a local energy maximum across which 1800-switching processes take place. The free energy potential is combined with the evolution equations based on the theory of thermally activated processes. The prediction of the model is compared with the recent measurements on a Ba- TiO3 single crystalline ferroelectric in [4]. The responses of the model at various loading frequencies are calculated and the kinetics of 900 and 1800 switching processes are discussed.

scholarly journals Laughlin States on Higher Genus Riemann Surfaces

2019 ◽  
Vol 367 (3) ◽  
pp. 837-871 ◽  
Author(s):  
Semyon Klevtsov
Keyword(s):  
Riemann Surfaces ◽  
Laughlin States ◽  
Higher Genus
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