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.

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.

TOPOLOGICAL σ MODELS AND LARGE N MATRIX INTEGRAL

1995 ◽  
Vol 10 (29) ◽  
pp. 4203-4224 ◽  
Author(s):  
TOHRU EGUCHI ◽  
KENTARO HORI ◽  
SUNG-KIL YANG
Keyword(s):  
Matrix Model ◽  
Riemann Surfaces ◽  
Holomorphic Maps ◽  
Integrable Structure ◽  
Toda Hierarchy ◽  
Matrix Integral ◽  
Large N ◽  
The Matrix ◽  
Lax Operator ◽  
The One

In this paper we describe in some detail the representation of the topological CP1 model in terms of a matrix integral which we have introduced in a previous article. We first discuss the integrable structure of the CP1 model and show that it is governed by an extension of the one-dimensional Toda hierarchy. We then introduce a matrix model which reproduces the sum over holomorphic maps from arbitrary Riemann surfaces onto CP1. We compute intersection numbers on the moduli space of curves using a geometrical method and show that the results agree with those predicted by the matrix model. We also develop a Landau-Ginzburg (LG) description of the CP1 model using a superpotential eX + et0,Q e-X given by the Lax operator of the Toda hierarchy (X is the LG field and t0,Q is the coupling constant of the Kähler class). The form of the superpotential indicates the close connection between CP1 and N=2 supersymmetric sine-Gordon theory which was noted sometime ago by several authors. We also discuss possible generalizations of our construction to other manifolds and present an LG formulation of the topological CP2 model.

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.

FUNCTIONAL INTEGRAL METHODS FOR FERMIONIC STRING AND SUPER RIEMANN SURFACES

1989 ◽  
Vol 04 (09) ◽  
pp. 2283-2315 ◽  
Author(s):  
KEN-JI HAMADA ◽  
MASARU TAKAO
Keyword(s):  
Light Cone ◽  
Riemann Surfaces ◽  
Functional Integral ◽  
Light Cone Gauge ◽  
Spin Structures ◽  
Integral Methods ◽  
Integral Measure ◽  
Fermionic String ◽  
Super Riemann Surfaces ◽  
The One

We investigate the light-cone gauge formulation of fermionic string of Mandelstam in a superspace. To formulate the fermionic string in a superspace, we use the theory of super Riemann surfaces (SRS). We define the Neumann functions and the Mandelstam mappings in a superspace by means of the so-called Abelian differentials of the first and the third kinds on SRS. In the super Schottky parametrization of SRS these superdifferentials are constructed at the multiloop level. The functional integral measure of a super light-cone diagram, which consists of 6g−6+2N even moduli parameters and 4g−4+2N odd ones, are specified in our formulation. In one-loop case with even spin structures we explicitly evaluate the superdeterminants on the super light-cone diagrams and calculate the one-loop amplitudes with N massless vector states. It is shown that the result is written in the form of a correlation function of the vertex operators. Furthermore, we evaluate the 3- and 4-particle amplitudes explicitly, which agree exactly with those calculated in the Green-Schwarz formulation.

scholarly journals Intersection numbers on $$ {\overline{M}}_{g,n} $$ and BKP hierarchy

2021 ◽  
Vol 2021 (9) ◽  
Author(s):  
Alexander Alexandrov
Keyword(s):  
Moduli Spaces ◽  
Riemann Surfaces ◽  
Intersection Numbers ◽  
Expansion Formula ◽  
Tau Function ◽  
Kdv Hierarchy ◽  
Bkp Hierarchy ◽  
Simple Expansion ◽  
Similar Description ◽  
Hypergeometric Solutions

Abstract In their recent inspiring paper, Mironov and Morozov claim a surprisingly simple expansion formula for the Kontsevich-Witten tau-function in terms of the Schur Q-functions. Here we provide a similar description for the Brézin-Gross-Witten tau-function. Moreover, we identify both tau-functions of the KdV hierarchy, which describe intersection numbers on the moduli spaces of punctured Riemann surfaces, with the hypergeometric solutions of the BKP hierarchy.

scholarly journals N = 1 and 2 Superstrings as Supertopological Models on Higher-Genus Super Riemann Surfaces

1994 ◽  
Vol 92 (1) ◽  
pp. 249-264
Author(s):  
G. Konisi ◽  
R. Kubo ◽  
T. Saito
Keyword(s):  
Riemann Surfaces ◽  
Higher Genus ◽  
Super Riemann Surfaces

scholarly journals Path-Integral Quantization of the (2,2) String

1997 ◽  
Vol 12 (27) ◽  
pp. 4933-4971 ◽  
Author(s):  
Jan Bischoff ◽  
Olaf Lechtenfeld
Keyword(s):  
Path Integral ◽  
Riemann Surfaces ◽  
Dimensional Space ◽  
Space Time ◽  
Gauge Fixing ◽  
Complete Treatment ◽  
Dimensional Space Time ◽  
Higher Genus ◽  
Instanton Number ◽  
Super Riemann Surfaces

A complete treatment of the (2,2) NSR string in flat (2 + 2)-dimensional space–time is given, from the formal path integral over N = 2 super Riemann surfaces to the computational recipe for amplitudes at any loop or gauge instanton number. We perform in detail the superconformal gauge fixing, discuss the spectral flow, and analyze the supermoduli space with emphasis on the gauge moduli. Background gauge field configurations in all instanton sectors are constructed. We develop chiral bosonization on punctured higher-genus surfaces in the presence of gauge moduli and instantons. The BRST cohomology is recapitulated, with a new space–time interpretation for picture-changing. We point out two ways of combining left- and right-movers, which lead to different three-point functions.

Preparation of Electron Transparent Metal-Matrix Composites

1968 ◽  
Vol 26 ◽  
pp. 334-335 ◽  
Author(s):  
M. R. Pinnel ◽  
A. Lawley
Keyword(s):  
Stainless Steel ◽  
Load Transfer ◽  
Volume Fraction ◽  
Steel Wire ◽  
Initial Step ◽  
Interfacial Region ◽  
Matrix Composites ◽  
Specific Test ◽  
The Matrix ◽  
The One

Numerous phenomenological descriptions of the mechanical behavior of composite materials have been developed. There is now an urgent need to study and interpret deformation behavior, load transfer, and strain distribution, in terms of micromechanisms at the atomic level. One approach is to characterize dislocation substructure resulting from specific test conditions by the various techniques of transmission electron microscopy. The present paper describes a technique for the preparation of electron transparent composites of aluminum-stainless steel, such that examination of the matrix-fiber (wire), or interfacial region is possible. Dislocation substructures are currently under examination following tensile, compressive, and creep loading. The technique complements and extends the one other study in this area by Hancock.The composite examined was hot-pressed (argon atmosphere) 99.99% aluminum reinforced with 15% volume fraction stainless steel wire (0.006″ dia.).Foils were prepared so that the stainless steel wires run longitudinally in the plane of the specimen i.e. the electron beam is perpendicular to the axes of the wires. The initial step involves cutting slices ∼0.040″ in thickness on a diamond slitting wheel.

scholarly journals Divergent ⇒ complex amplitudes in two dimensional string theory

2021 ◽  
Vol 2021 (2) ◽  
Author(s):  
Ashoke Sen
Keyword(s):  
Scattering Amplitudes ◽  
String Theory ◽  
Matrix Model ◽  
Two Dimensional ◽  
Full Matrix ◽  
Instanton Contribution ◽  
The Matrix ◽  
Theory Result ◽  
The One ◽  
Remarkable Agreement

Abstract In a recent paper, Balthazar, Rodriguez and Yin found remarkable agreement between the one instanton contribution to the scattering amplitudes of two dimensional string theory and those in the matrix model to the first subleading order. The comparison was carried out numerically by analytically continuing the external energies to imaginary values, since for real energies the string theory result diverges. We use insights from string field theory to give finite expressions for the string theory amplitudes for real energies. We also show analytically that the imaginary parts of the string theory amplitudes computed this way reproduce the full matrix model results for general scattering amplitudes involving multiple closed strings.

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