arXiv subjects

T. Eguchi

Publications and source records attributed to T. Eguchi.

7 recordsLinked to original sources

arXiv1995

Topological $σ$-Models and Large-$N$ Matrix Integral

In this paper we describe in some detail the representation of the topological $CP^1$ model in terms of a matrix integral which we have introduced in a previous article. We first discuss the integrable structure of the $CP^1$ model and show that it is governed by an extension of the 1-dimensional Toda hierarchy. We then introduce a matrix model which reproduces the sum over holomorphic maps from arbitrary Riemann surfaces onto $CP^1$. We compute intersection numbers on the moduli space of curves using 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 $CP^1$ model using a superpotential $e^X+e^{t_{0,Q}}e^{-X}$ given by the Lax operator of the Toda hierarchy ($X$ is the LG field and $t_{0,Q}$ is the coupling constant of the Kähler class). The form of the superpotential indicates the close connection between $CP^1$ and $N=2$ supersymmetric sine-Gordon theory which was noted some time ago by several authors. We also discuss possible generalizations of our construction to other manifolds and present a LG formulation of the topological $CP^2$ model.

hep-th↗

arXiv1994

The Topological CP^1 Model and the Large-N Matrix Integral

We discuss the topological $CP^1$ model which consists of the holomorphic maps from Riemann surfaces onto $CP^1$. We construct a large-$N$ matrix model which reproduces precisely the partition function of the $CP^1$ model at all genera of Riemann surfaces. The action of our matrix model has the form ${\rm Tr}\, V(M)=-2{\rm Tr}\, M(\log M -1) +2\sum t_{n,P}{\rm Tr}\, M^n(\log M-c_n) +\sum 1/n\cdot t_{n-1,Q}{\rm Tr}\, M^n~(c_n=\sum_1^n 1/j )$ where $M$ is an $N\times N$ hermitian matrix and $t_{n,P}\, (t_{n,Q}),~(n=0,1,2,\cdots)$ are the coupling constants of the $n$-th descendant of the puncture (Kähler) operator.

hep-th↗

arXiv1994

On the Genus Expansion in the Topological String Theory

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 $A_1$ and $A_2$ 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 $CP^1$ model at higher genera.

hep-th↗

arXiv1994

Toda Lattice Hierarchy and the Topological Description of the c=1 String Theory

The Toda lattice hierarchy is discussed in connection with the topological description of the $c=1$ string theory compactified at the self-dual radius. It is shown that when special constraints are imposed on the Toda hierarchy, it reproduces known results of the $c=1$ string theory, in particular the $W_{1+\infty}$ relations among tachyon correlation functions. These constraints are the analogues of string equations in the topological minimal theories. We also point out that at $c=1$ the Landau-Ginzburg superpotential becomes simply a $U(1)$ current operator.

hep-th↗

arXiv1993

Topological Field Theories and the Period Integrals

We discuss topological Landau-Ginzburg theories coupled to the 2-dimensional topological gravity. We point out that the basic recursion relations for correlation functions of the 2-dimesional gravity have exactly the same form as the Gauss-Manin differential equations for the period integrals of superpotentials. Thus the one-point functions on the sphere of the Landau-Ginzburg theories are given exactly by the period integrals. We discuss various examples, A-D-E minimal models and the $c=3$ topological theories.

hep-th↗

arXiv1993

Topological Strings, Flat Coordinates and Gravitational Descendants

We discuss physical spectra and correlation functions of topological minimal models coupled to topological gravity. We first study the BRST formalism of these theories and show that their BRST operator $Q=Q_s+Q_v$ can be brought to $Q_s$ by a certain homotopy operator $U$, $UQU^{-1}=Q_s$ ($Q_s$ and $Q_v$ are the $N=2$ and diffeomorphism BRST operators, respectively). The reparametrization (anti)-ghost $b$ mixes with the supercharge operator $G$ under this transformation. Existence of this transformation enables us to use matter fields to represent cohomology classes of the operator $Q$. We explicitly construct gravitational descendants and show that they generate the higher-order KdV flows. We also evaluate genus-zero correlation functions and rederive basic recursion relations of two-dimensional topological gravity.

hep-th↗

arXiv1992

$W_{\infty} Algebra in Two-Dimensional Black Hole

We study the $SL(2;R)/U(1)$ coset model of two-dimensional black hole and its relation to the Liouville theory coupled to c=1 matter. We uncover a basic isomorphism in the algebraic structures of these theories and show that the black hole model has the same physical spectrum as the c=1 model, i.e. tachyons, $W_\infty$ currents and the ground ring elements. we also identify the operator responsible for the creation of the mass of the black hole.

hep-th↗