Program for the Thematic Period
Schedule: Present Week
10h00



WYLLARD



11h30






12h30






14h00






15h30






16h30

WYLLARD





The lectures will take place in room P3.10, on the third floor of the
Department of Mathematics, IST.
Lectures
BOROT, A MATRIX MODEL FOR SIMPLE HURWITZ NUMBERS AND THE BOUCHARDMARIÑO CONJECTURE
Abstract: Following the work in 0906.1206[mathph] 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.
CHEKHOV, QUANTUM RIEMANN SURFACES AND MATRIX MODELS
Abstract: Lecture 1: We begin with the perturbative approach to solving βmodels 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 1differential; (ii) A and Bcycles; (iii) occupation numbers; (iv) holomorphic differentials and special geometry relations; (v) period matrix; (vi) recursion kernels and bidifferentials, relation to the twopoint 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.
CIRAFICI, CALABIYAU CRYSTALS AND TOPOLOGICAL STRINGS
Abstract: CalabiYau crystals are a dual description of the topological string on toric CalabiYau manifolds. In these lectures I will review this duality and its geometric
interpretation. I will also explain the role of Dbranes in the duality and the relation of the CalabiYau crystal with topological YangMills and the enumerative problem of DonaldsonThomas
invariants. Refs: hepth/0309208, hepth/0312022, hepth/0404246. Part 2:
I will continue my lectures with some recent developments on CalabiYau crystals and their relations with enumerative geometry.
I will study the crystal partition functions recently proposed by Szendroi for the conifold and OoguriYamazaki for generic toric CalabiYaus
and their relations with the socalled Noncommutative DonaldsonThomas invariants.
Refs: 0705.3419[math.AG], 0811.2801[hepth],
0902.3996[hepth].
EYNARD, MATRIX MODELS FOR PARTITIONS, PLANE PARTITIONS AND TOPOLOGICAL VERTEX FORMULAE
Abstract: GromovWitten 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[mathph]
(conifold), 0905.0535[mathph] (framed vertex), 0810.4944[mathph]
(SeibergWitten), and then for general toric CY 3folds. As a consequence, since matrix models satisfy the topological recursion, then, automatically,
the GromovWitten 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 Bmodel proposal" for all toric geometries.
We will also present some further developments of these methods.
IMBIMBO, THE COUPLING OF CHERNSIMONS THEORY TO TOPOLOGICAL GRAVITY AND TOPOLOGICAL STRINGS
Abstract: We describe the coupling of ChernSimons gauge theory —and of certain higherghost deformations of it— to 3dimensional topological gravity,
with the aim to determine its topological anomalies. The complete solution of this problem requires the full generality of the BatalinVilkovisky 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 higherghost
deformations of the ChernSimons theory describe, from the first quantized point of view, topological string amplitudes which involve vertex operators corresponding to
the extended moduli space of the Amodel.
IRIE, MACROSCOPIC LOOP AMPLITUDES IN THE MULTICUT MATRIX MODELS
Abstract: Study of multicut matrix models is a new direction of noncritical string theory which stems from the discovery of the correspondence
between twocut 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 onecut and twocut 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 multicut twomatrix 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[hepth], 0909.1197[hepth].
KLEMM, 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
hepth/0305132, which is based on the duality of the topological Astring with ChernSimons 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 Bstring
and matrix models hepth/0211098, 0709.1453[hepth].
We discuss the integrable structure underlying topological string theory hepth/0312085 and the application
of duality symmetries to higher genus calculations hepth/0612125,
0809.1674[hepth]. Refs: "Mirror Symmetry", C. Vafa and E. Zaslow Eds., Clay Mathematics 2003;
"ChernSimons Theory, Matrix Models and Topological Strings", Marcos Mariño, Oxford 2005.
KOSTOV, 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 twodimensional 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 twomatrix model, ADE matrix chains, the sixvertex and O(n) models. A few words about the generalized matrix integrals
a.k.a. βensembles. Refs: hepth/9907060, Nucl. Phys.
B285 (1987) 481503, 0912.2137[hepth], 0811.3531[mathph], hepth/9208053.
ORANTIN, 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[mathph],
mathph/0603003, mathph/0611087, hepth/0407261.
PASQUETTI, NONPERTURBATIVE EFFECTS IN MATRIX MODELS AND TOPOLOGICAL STRINGS
Abstract: Lecture 1: Asymptotics series, largeorder behavior, nonperturbative ambiguity. The transseries method.
Instantons configurations in matrix models. Lecture 2: Instantons and large order in c=1 matrix models and topological strings.
The SchwingerBorel completion and the Stokes phenomenon. Lecture 3: The matrix model nonperturbative partition function.
Large N duality beyond the genus expansion. Refs: 0805.3033[hepth],
0711.1954[hepth], 0809.2619[hepth],
0907.4082[hepth], 0911.4692[hepth].
VONK, AN INTRODUCTION TO CALABIYAU 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 socalled CalabiYau manifold. In these lectures I will give a basic introduction to CalabiYau manifolds and to topological field and string theories.
Refs: hepth/0504147, hepth/9702155.
WYLLARD, 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[hepth], the connection to CFTs was uncovered in 0906.3219[hepth] and the connection to matrix models in 0909.2453[hepth].
Schedule: Spring 2010
10h00




KOSTOV


11h30






12h30






14h00






15h30






16h30






10h00



KOSTOV



11h30






12h30






14h00






15h30






16h30

KOSTOV





10h00






11h30






12h30






14h00






15h30






16h30






10h00


CHEKHOV

PASQUETTI

CHEKHOV

PASQUETTI

11h30






12h30






14h00






15h30






16h30

PASQUETTI





10h00






11h30






12h30






14h00






15h30






16h30






10h00






11h30






12h30






14h00






15h30






16h30






10h00



WYLLARD



11h30






12h30






14h00






15h30






16h30

WYLLARD





10h00



WYLLARD



11h30






12h30






14h00






15h30






16h30

WYLLARD





10h00


WYLLARD




11h30






12h30






14h00






15h30






16h30






Schedule: Fall 2009
10h00






11h00




VONK

VONK

12h30






14h00


VONK




15h30






16h30






10h00






11h00




CIRAFICI

CIRAFICI

12h30






14h00






15h00


CIRAFICI




16h30






10h00






11h00


ORANTIN


ORANTIN


12h30






14h00





ORANTIN

15h00






16h30






10h00






11h00


ORANTIN


ORANTIN

ORANTIN

12h30






14h00






15h00






16h30






10h00






11h00






12h30






14h00






15h00






16h30






10h00






11h00


KLEMM

KLEMM



12h30






14h00






16h00




BOROT


16h30

KLEMM





10h00






11h00


EYNARD



EYNARD

12h30






14h00






15h00






16h00




EYNARD


10h00






11h00


EYNARD

IRIE



12h30






14h00






15h00






16h30

EYNARD





10h00






11h00






12h30






14h00






15h00






16h30






10h00






11h00



IMBIMBO



12h30






14h00






15h00






16h30






10h00






11h00



IMBIMBO



12h30






14h00






15h00






16h30






10h00






11h00


CIRAFICI

CIRAFICI


CIRAFICI

12h30






14h00






15h00






16h30






