MATRIX MODELS AND GEOMETRY --- Thematic Period @ IST 2009/2010
Matrix Models and GeometryCAMGSD Thematic Period @ IST | Fall 2009 — Spring 2010INSTITUTO SUPERIOR TÉCNICO, LISBON, PORTUGAL Speakers:Gaetan BOROT (CEA-Saclay, Paris) — A MATRIX MODEL FOR SIMPLE HURWITZ NUMBERS AND THE BOUCHARD-MARIÑO CONJECTURE
Abstract: Following the work in 0906.1206[math-ph] with B. Eynard, M. Mulase and B. Safnuk, I will present a matrix model computing simple Hurwitz numbers, defined as the number of "simple" coverings of CP^{1}. This was motivated by a conjecture of Bouchard and Mariño (0709.1458[math.AG]) which was itself an application of the BKMP conjecture: "The topological recursion of matrix models, introduced by B. Eynard and N. Orantin, with the Lambert curve y = x e^{-x} as a spectral curve, computes
generating functions of genus g simple Hurwitz numbers". We have obtained a proof of this proposal, that I will sketch.
Leonid CHEKHOV (Steklov, Moscow) — QUANTUM RIEMANN SURFACES AND MATRIX MODELS
Abstract: Lecture 1: We begin with the perturbative approach to solving β-models of the type ∫ ... ∫ dλ_{1} ... dλ_{N} |Δ(λ)|^{2β} e^{-N √β V(λ)}. We find the loop equation for correlation functions, solve it perturbatively, and find the symplectic invariants (terms in the double expansion for the free energy). We mention the relevance to calculating conformal blocks (the AGT conjecture). Lecture 2: Nonperturbative approach: Quantum Riemann surfaces as solutions to the loop equation. (i) resolvents: definition and correspondence to the special 1-differential; (ii) A- and B-cycles; (iii) occupation numbers; (iv) holomorphic differentials and special geometry relations; (v) period matrix; (vi) recursion kernels and bi-differentials, relation to the two-point correlation function; (vii) solving the loop equation and proving the symmetricity of the period matrix. Lecture 3: Recurrent solution for the quantum Riemann surfaces: correlation functions and symplectic invariants.
Michele CIRAFICI (IST, Lisbon) — CALABI-YAU CRYSTALS AND TOPOLOGICAL STRINGS
Abstract: Calabi-Yau crystals are a dual description of the topological string on toric Calabi-Yau manifolds. In these lectures I will review this duality and its geometric interpretation. I will also explain the role of D-branes in the duality and the relation of the Calabi-Yau crystal with topological Yang-Mills and the enumerative problem of Donaldson-Thomas invariants. Refs: hep-th/0309208, hep-th/0312022, hep-th/0404246. Part 2: I will continue my lectures with some recent developments on Calabi-Yau crystals and their relations with enumerative geometry. I will study the crystal partition functions recently proposed by Szendroi for the conifold and Ooguri-Yamazaki for generic toric Calabi-Yaus and their relations with the so-called Noncommutative Donaldson-Thomas invariants. Refs: 0705.3419[math.AG], 0811.2801[hep-th], 0902.3996[hep-th].
Bertrand EYNARD (CEA-Saclay, Paris) — MATRIX MODELS FOR PARTITIONS, PLANE PARTITIONS AND TOPOLOGICAL VERTEX FORMULAE
Abstract: Gromov-Witten invariants can be computed by topological vertex formulae, which are written as sums over partitions or plane partitions. We will show how to rewrite sums over partitions as matrix integrals, along the lines of 0804.0381[math-ph] (conifold), 0905.0535[math-ph] (framed vertex), 0810.4944[math-ph] (Seiberg-Witten), and then for general toric CY 3-folds. As a consequence, since matrix models satisfy the topological recursion, then, automatically, the Gromov-Witten invariants also satisfy the topological recursion attached to the matrix model's spectral curve. Then, we compute the matrix model's spectral curve, and we will show that it coincides (modulo symplectic transformations) with the mirror spectral curve. This proves the "remodeling the B-model proposal" for all toric geometries. We will also present some further developments of these methods.
Camillo IMBIMBO (Genoa U. & INFN, Genoa) — THE COUPLING OF CHERN-SIMONS THEORY TO TOPOLOGICAL GRAVITY AND TOPOLOGICAL STRINGS
We describe the coupling of Chern-Simons gauge theory —and of certain higher-ghost deformations of it— to 3-dimensional topological gravity, with the aim to determine its topological anomalies. The complete solution of this problem requires the full generality of the Batalin-Vilkovisky formalism. In the context of topological strings the topological anomalies we compute, which generalize the familiar framing anomaly, are canceled by couplings of the closed string sector. We determine such couplings and show that they are obtained by dressing the closed string field with topological gravity observables. We also show that the higher-ghost deformations of the Chern-Simons theory describe, from the first quantized point of view, topological string amplitudes which involve vertex operators corresponding to the extended moduli space of the A-model.
Hirotaka IRIE (NTU, Taiwan) — MACROSCOPIC LOOP AMPLITUDES IN THE MULTI-CUT MATRIX MODELS
Abstract: Study of multi-cut matrix models is a new direction of non-critical string theory which stems from the discovery of the correspondence between two-cut matrix models and type 0 superstring theories. These models still seem to have some correspondences with other kind of string theories and seem to have distinct structures which have not been observed in the traditional one-cut and two-cut system. In this talk, we first discuss a conjecture of correspondence with fractional superstring theory, and summarize current evidences and also issues we need to check in this correspondence. We then move on to macroscopic loop amplitudes in the multi-cut two-matrix models. We propose a proper large N ansatz for the Lax pair of the matrix models in Z_{k} symmetric background and discuss possible geometries appearing in the weak string coupling region. In particular, we show that solutions in the "unitary" models is given by the Jacobi polynomials. If possible, we also mention the cases of Z_{k} symmetry breaking backgrounds which should correspond to minimal fractional superstring theory. Refs: 0902.1676[hep-th], 0909.1197[hep-th].
Albrecht KLEMM (Universität Bonn, Bonn) — LARGE N METHODS IN TOPOLOGICAL STRING THEORY
Abstract: In these lectures we will explain recent developments in the solution of topological string theory. We focus on methods which use the
duality between string theory and large N gauge theory. We start with the description of the topological vertex hep-th/0305132, which is based on the duality of the topological A-string with Chern-Simons gauge theory in the large N expansion, and localization on toric manifolds. The mirror dual to this gauge/string duality is the duality between the topological B-string and matrix models hep-th/0211098, 0709.1453[hep-th]. We discuss the integrable structure underlying topological string theory hep-th/0312085 and the application of duality symmetries to higher genus calculations hep-th/0612125, 0809.1674[hep-th]. Refs: "Mirror Symmetry", C. Vafa and E. Zaslow Eds., Clay Mathematics 2003; "Chern-Simons Theory, Matrix Models and Topological Strings", Marcos Mariño, Oxford 2005.
Ivan KOSTOV (CEA-Saclay, Paris) — MATRIX MODELS AS CONFORMAL FIELD THEORIES
Abstract: In these lectures I will try to explain how the asymptotic properties of correlation functions of U(N) invariant
matrix integrals can be derived by means of conformal field theory. In the large N limit such
a CFT describes gaussian fields on a Riemann surface. Lecture 1: The hermitian matrix model is reformulated as a two-dimensional chiral
CFT of a free Dirac fermion. The loop equations are derived from the conformal Ward identities.
The collective field theory is obtained by bosonization. An unusual property of this CFT is that the bosonic field develops a large expectation value. The
classical solution for the bosonic field defines a hyperelliptic Riemann surface.
The quasiclassical expressions for the spectral kernel and the joint eigenvalue probabilities are
obtained as correlation functions of current, fermionic and twist operators. Lecture 2:
The 1/N expansion, which is the quasiclassical expansion of the bosonic field, is formulated in terms of a CFT on the Riemann surface defined by the classical solution.
An operator solution for the 1/N expansion is formulated in terms of twisted bosonic
fields on this Riemann surface, with special operators inserted at the branch points.
The operator solution yields a set of Feynman rules for the 1/N expansion, which represent a partial resummation of the diagrams that appear in the recursion procedure invented by
Eynard and collaborators. Lecture 3: Generalization of the CFT description
to multimatrix models. Examples: The two-matrix model, ADE matrix chains, the six-vertex and O(n) models. A few words about the generalized matrix integrals
a.k.a. β-ensembles. Refs: hep-th/9907060, Nucl. Phys.
B285 (1987) 481-503, 0912.2137[hep-th], 0811.3531[math-ph], hep-th/9208053.
Nicolas ORANTIN (CERN, Geneva) — FROM DISCRETE SURFACES AND MATRIX MODELS TO TOPOLOGICAL RECURSIONS
Abstract: Random matrix models represent a wonderful tool in the enumeration of random discrete surfaces of given topology. These lectures will address three issues. I will first properly define the concept of formal matrix integral used to build generating functions of discrete surfaces. I will then show that the enumeration of all possible ways of removing one edge from such a surface gives a set of loop equations which can be solved by induction in terms of an algebraic curve characterizing the considered matrix model: the spectral curve. Finally, I will present a generalization of this method allowing to associate by induction a set of correlation functions to a spectral curve, whether it comes from a matrix model or not. I will review some of the main properties shared by these correlation functions such as symplectic and modular invariance, holomorphic anomaly equations and deformations. Refs: 0811.3531[math-ph], math-ph/0603003, math-ph/0611087, hep-th/0407261.
Sara PASQUETTI (CERN, Geneva) — NONPERTURBATIVE EFFECTS IN MATRIX MODELS AND TOPOLOGICAL STRINGS
Abstract: Lecture 1: Asymptotics series, large-order behavior, nonperturbative ambiguity. The trans-series method. Instantons configurations in matrix models. Lecture 2: Instantons and large order in c=1 matrix models and topological strings. The Schwinger-Borel completion and the Stokes phenomenon. Lecture 3: The matrix model nonperturbative partition function. Large N duality beyond the genus expansion. Refs: 0805.3033[hep-th], 0711.1954[hep-th], 0809.2619[hep-th], 0907.4082[hep-th], 0911.4692[hep-th].
Marcel VONK (IST, Lisbon) — AN INTRODUCTION TO CALABI-YAU GEOMETRY AND TOPOLOGICAL STRING THEORY
Abstract: Topological string theory is a simplified version of string theory which is calculationally much more accessible than the usual superstring theories. It can be used as a toy model for superstrings, but also turns out to exactly describe some of its subsectors. Moreover, the topological string has many surprising and interesting mathematical applications in algebraic and differential geometry and topology and even number theory. The underlying geometric object which gives the theory its elegant features is a so-called Calabi-Yau manifold. In these lectures I will give a basic introduction to Calabi-Yau manifolds and to topological field and string theories. Refs: hep-th/0504147, hep-th/9702155.
Niclas WYLLARD (@ IST, Lisbon) — MATRIX MODELS, 2d CFTs AND 4d N=2 GAUGE THEORIES
Abstract: I will review the recent remarkable results relating quiver matrix models to 4d N=2 quiver gauge theories, and 2d conformal Toda field theories.
Refs: The relevant class of gauge theories was discussed in 0904.2715[hep-th], the connection to CFTs was uncovered in 0906.3219[hep-th] and the connection to matrix models in 0909.2453[hep-th].
Organizing Committee: Gabriel Lopes Cardoso Ricardo Schiappa Marcel Vonk Sponsors: FCT — Fundação para a Ciência e Tecnologia IST — Instituto Superior Técnico CAMGSD — Centro de Análise Matemática, Geometria e Sistemas Dinâmicos |