Section de Mathématiques et Département de Physique Théorique

2013

Abstract: We discuss various aspects of multi–instanton configurations in generic multi–cut matrix models. Explicit formulae are presented in the two–cut case and, in particular, we obtain general formulae for multi–instanton amplitudes in the one–cut matrix model case as a degeneration of the two–cut case. These formulae show that the instanton gas is ultra–dilute, due to the repulsion among the matrix model eigenvalues. We exemplify and test our general results in the cubic matrix model, where multi–instanton amplitudes can be also computed with orthogonal polynomials. As an application, we derive general expressions for multi–instanton contributions in two–dimensional quantum gravity, verifying them by computing the instanton corrections to the string equation. The resulting amplitudes can be interpreted as regularized partition functions for multiple ZZ–branes, which take into full account their back–reaction on the target geometry. Finally, we also derive structural properties ...

2008

We discuss various aspects of multi-instanton configurations in generic multi-cut matrix models. Explicit formulae are presented in the two-cut case and, in particular, we obtain general formulae for multi-instanton amplitudes in the one-cut matrix model case as a degeneration of the two-cut case. These formulae show that the instanton gas is ultra-dilute, due to the repulsion among the matrix model eigenvalues. We exemplify and test our general results in the cubic matrix model, where multi-instanton amplitudes can be also computed with orthogonal polynomials. As an application, we derive general expressions for multi-instanton contributions in two-dimensional quantum gravity, verifying them by computing the instanton corrections to the string equation. The resulting amplitudes can be interpreted as regularized partition functions for multiple ZZ-branes, which take into full account their back-reaction on the target geometry. Finally, we also derive structural properties of the trans-series solution to the Painlevé I equation.

Circumnavigating Theoretical Physics - Ian Kogan Memorial Collection (In 3 Volumes), 2005

We derive the non-perturbative corrections to the free energy of the two-matrix model in terms of its algebraic curve. The eigenvalue instantons are associated with the vanishing cycles of the curve. For the (p, q) critical points our results agree with the geometrical interpretation of the instanton effects recently discovered in the CFT approach. The form of the instanton corrections implies that the linear relation between the FZZT and ZZ disc amplitudes is a general property of the 2D string theory and holds for any classical background. We find that the agreement with the CFT results holds in the presence of infinitesimal perturbations by order operators and observe that the ambiguity in the interpretation of the eigenvalue instantons as ZZ-branes (four different choices for the matter and Liouville boundary conditions lead to the same result) is not lifted by the perturbations. We find similar results to the c = 1 string theory in the presence of tachyon perturbations.

Physica D: Nonlinear Phenomena, 2007

Annals of Physics, 2004

We consider specific quantum mechanical model problems for which perturbation theory fails to explain physical properties like the eigenvalue spectrum even qualitatively, even if the asymptotic perturbation series is augmented by resummation prescriptions to "cure" the divergence in large orders of perturbation theory. Generalizations of perturbation theory are necessary which include instanton configurations, characterized by nonanalytic factors exp(−a/g) where a is a constant and g is the coupling. In the case of one-dimensional quantum mechanical potentials with two or more degenerate minima, the energy levels may be represented as an infinite sum of terms each of which involves a certain power of a nonanalytic factor and represents itself an infinite divergent series. We attempt to provide a unified representation of related derivations previously found scattered in the literature. For the considered quantum mechanical problems, we discuss the derivation of the instanton contributions from a semi-classical calculation of the corresponding partition function in the path integral formalism. We also explain the relation with the corresponding WKB expansion of the solutions of the Schrödinger equation, or alternatively of the Fredholm determinant det(H − E) (and some explicit calculations that verify this correspondence). We finally recall how these conjectures naturally emerge from a leading-order summation of multi-instanton contributions to the path integral representation of the partition function. The same strategy could result in new conjectures for problems where our present understanding is more limited.

Arxiv preprint hep-th/9211085, 1992

We discuss the basic features of the double scaling limit of the one dimensional matrix model and its interpretation as a two dimensional string theory. Using the collective field theory formulation of the model we show how the fluctuations of the collective field can be interpreted as the massless "tachyon" of the two dimensional string in a linear dilaton background. We outline the basic physical properties of the theory and discuss the nature of the Smatrix. Finally we show that the theory admits of another interpretation in which a certain integral transform of the collective field behaves as the massless "tachyon" in the two dimensional string with a blackhole background. We show that both the classical background and the fluctuations are non-singular at the black hole singularity.

Fortschritte der Physik, 2004

In this contribution we describe how to obtain instanton effects in four dimensional gauge theories by computing string scattering amplitudes in D3/D(-1) brane systems. In particular we show that the disks with mixed boundary conditions, which are typical of the D3/D(-1) system, are the sources for the classical instanton solution.

Journal of High Energy Physics, 2003

We study the D3/D(−1) brane system and show how to compute instanton corrections to correlation functions of gauge theories in four dimensions using open string techniques. In particular we show that the disks with mixed boundary conditions that are typical of the D3/D(−1) system are the sources for the classical instanton solution. This can then be recovered from simple calculations of open string scattering amplitudes in the presence of D-instantons. Exploiting this fact we also relate this stringy description to the standard instanton calculus of field theory.

Nuclear Physics B, 2005

The FZZT and ZZ branes in (p, p + 1) minimal string theory are studied in terms of continuum loop equations. We show that systems in the presence of ZZ branes (D-instantons) can be easily investigated within the framework of the continuum string field theory developed by Yahikozawa and one of the present authors [1]. We explicitly calculate the partition function of a single ZZ brane for arbitrary p. We also show that the annulus amplitudes of ZZ branes are correctly reproduced. *

2016

It is generally accepted that the double-scaled 1D matrix model is equivalent to the c = 1 string theory with tachyon condensation. There remain however puzzles that are to be clarified in order to utilize this connection for our quest towards possible non-perturbative formulation of string theory. We discuss some of the issues that are related to the space-time interpretation of matrix models, in particular, the questions of leg poles, causality, and black hole background. Finally, a speculation about a possible connection of a deformed matrix model with the idea of Dirichret brane is presented.