# Institut de Recherche Mathématique Avancée, UMR 7501

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## 86e rencontre entre physiciens théoriciens et mathématiciens : Théorie des modules en mathématiques et en physique

IRMA, 2-4 septembre 2010

La 86e Rencontre entre mathématiciens et physiciens théoriciens aura lieu à l’IRMA, du 2 au 4 septembre 2010, sur le thème : "Théorie des modules en mathématiques et en physique".

The 86th Encounter between Mathematicians and Theoretical Physicists will take place at Institut de Recherche Mathématique Avancée (University of Strasbourg and CNRS) on September 2-4, 2010. The theme will be : "Moduli spaces in Mathematics and in Physics".

All talks will be in english.

List of registered participants

Contact : V. Fock or A. Papadopoulos

## 2 septembre 2010

### Alexey Sossinsky - Independent University and CNRS--IUM Poncelet Laboratory, Moscow.

Abstract : Planar mechanical linkages (or hinge mechanisms in another terminology) are a classical subject of mathematics and engineering going back to the end of the 19th century, with important work by James Watt, Maxwell, Cayley, Peaucellier, and Chebyshev. It was revived in the 1980ies by Thurston, who shifted the focus of interest to the study of configuration spaces (or moduli spaces) of linkages, and motivated many other researchers to work in this field, e.g. Connelly, King, Mnev, D.Zvonkine, Kapovich & Millson, Steiner, and the speaker. At present, the main area of application of the theory is robotics, in particular in the work of Farber (and many others). In the talk, I will concentrate on the study of specific planar hinge mechanisms with two degrees of freedom in order to demonstrate the type of results that can be obtained and the methods used. The mechanisms studied are the {\it pentagon}, which is a closed chain of five rectilinear links joined by hinges, one of the links being fixed on the plane, and the {\it $n$-legged spider}, which consists of a central hinge (the "body" of the spider) to which $n$ "legs" (of two links joined by a mobile hinge, the endpointof each leg being a hinge fixed in the plane) are attached. A complete classification of moduli spaces of pentagons (independently obtained by many authors, including D.Zvonkine, Kapovich & Millson, Steiner & Curtis) will be described, in a version due to the speaker's PhD student, A. Kondakova. In this version, the classification of the 19 different moduli spaces is obtained via a kind of Vassiliev finite-type invariant defined on the parameter space of all pentagons; this approach leans heavily on the work of Kapovich & Millson, and involves Morse surgery of surfaces arising when ones crosses the discriminant in the parameter space. The moduli spaces of $n$-legged spiders are much more complicated than that of pentagons (actually, pentagons can be regarded as two-legged spiders). No complete classification result can be expected even for $n=3$, however, a number of nontrivial results about $n$-legged spiders have been obtained by A.Kondakova, in particular, a finiteness theorem (for each $n$, there is only a finite number of different moduli spaces of generic $n$-legged spiders), and upper and lower bounds for the Euler characteristics, the number of connected components, the number and types of singularities of moduli spaces of $n$-legged spiders. It follows from these results that for any positive integer $N$ there exists a spider whose moduli space is an orientable surface of genus $M>N$, but such values of $M$ for large $N$ are very scarce; in particular, there is no spider whose moduli space is a surface of genus 10. In conclusion, a list of (hopefully accessible) open problems aboutlinkages (not limited to the mechanisms considered and to the case of two degrees of freedom) will be presented.. The talk is aimed for a general audience.

Coffee break

### Feng Luo - Rutgers U.

Thurston’s equation on triangulated 3-manifolds.

Abstract : In 1978, Thurston introduced an algebraic equation defined over each triangulated 3-manifold to find hyperbolic structures. Thurston's theory can be considered as a discrete SL(2,C Chern-Simons theory on the manifold. We propose a finite dimensional variational principle on triangulated 3-manifolds so that its critical points are solutions to Thurston's equation and to Haken's normal surface equation. The action functional is the volume. This is a generalization of an earlier program by Casson and Rivin for compact 3-manifolds with torus boundary. Combining the work of Futer-GueritaudSegerman Tillmann and Luo-Tillmann, we obtain a new finite dimensional variational formulation of the Poncaré conjecture. This provides a step toward a new proof the Poincaré rconjecture without using the Ricci flow.

### 11h30

Unsolved problems in moduli theory.

Abstract : Moduli theory has shown itself as a branch of mathematics containing the whole mathematics as a subset. The aim of this short talk is to present (certainly very subjective) evidence in favour of this thesis by showing relations (sometimes mysterious, and almost always partially conjectural) to hyperkähler geometry, categorification, integrable systems, affine Lie groups and other apparently unrelated subjects some of which closely related to the talks of the workshop.

Lunch break

### Kirill Krasnov - University of Nottingham

Moduli space of shapes of a tetrahedron and SU(2) intertwiners.

Abstract : I will show how the vector space of SU(2intertwiners is obtained via geometric quantization of n copies of S^2 modulo the diagonal action of SU(2). For n=4 the symplectic manifold in question is the moduli space of "shapes" of a flat tetrahedron with its face areas fixed. A decomposition of the identity formula as an integral over the moduli space is proved.

### Louis Funar - University of Grenoble

Mapping class group representations.

Abstract: Topological quantum field theories in dimension 3 are a rich source of interesting mapping class group representations. We discuss first several applications of this theory to finding geometric group properties of mapping class groups. We talk further about the first steps towards an hypothetical construction of moduli spaces of conformal field theories.

### 16h00

Departure for a boat trip around the old city offered to all participants. We shall depart from the Math Institute to the landing stage situated in front of the Palais des Rohan. The boat leaves at 16:30 and the trip lasts about 1h15'.

### 18h00

Apéritif-reception in the the City Hall (Mairie) Place Broglie, offered to all participants by the Mayor of Strasbourg. We shall go there directly after the boat trip.

## 3 septembre 2010

### Nikita Nekrasov - IHES

Quantum integrable systems and symplectic geometry.

Abstract: Four dimensional N=2 supersymmetric gauge theory has an equivariant generalization, the so-called Omega-deformation, which has a partition function, "the instanton partition function", similar to the generating function of Gromov-Witten invariants. As in the context of the mirror symmetry, this generating function of the equivariant enumerative invariants has a classical geometric interpretation. We argue, for a class of N=2 gaugetheories which correspond to the Riemann surfaces ofgenus g with n marked points and a Lie group G,that the instanton partition function captures the geometry of thevariety of ^{L}G-opers, and present an explicit construction for G=SL_2. More precisely, we construct a set of Darboux coordinate systems on the moduli space of SL(2, C)-connections on the Riemann surface with punctures, using the complexified hyperbolic geometry. We then consider the generating function of the variety of SL_2-opers which is a Lagrangian submanifold in the moduli space of flat connections. This generating function is identified with the instanton partition function of a four dimensional gauge theory. We also comment on the Bethe ansatz and separation of variables for the quantum Hitchin system. Based on the joint work with A.Rosly and S.Shatashvili

Coffee break

### Lizhen Ji - U. of Michigan

Analysis and geometry of moduli spaces of Riemann surfaces.

Abstract : The moduli space of Riemann surfaces has been extensively studied in algebraic geometry, geometry topology and geometric topology in connection with Teichmuller spaces and mapping class groups and others. On the other hand, it also admits many natural metrics. Therefore, a natural problem is to understand the differential geometry and spectral theory of the moduli spaces. In this talk, I will discuss some natural questions and results in this general direction. The talk is aimed for a general audience.

### Sergei Tabachnikov - Penn. State U.

Algebra, geometry and dynamics of the pentagram map.

Abstract: Introduced by R. Schwartz almost 20 years ago, the pentagram map acts on plane n-gons, considered up to projective equivalence, by drawing the diagonals that connect second-nearest vertices and taking the new n-gon formed by their intersections. I shall survey recent work on the pentagram map, in particular, I shall demonstrate that the dynamics of the pentagram map is very regular (the map is completely integrable). I shall also explain that the pentagram map is a discretization of the Boussinesq equation, a well known completely integrable partial differential equation. A surprising relation between the spaces of polygons and combinatorial objects called the 2-frieze patterns (generalizing the frieze patterns of Coxeter) will be revealed. Eight new configuration theorems of projective geometry will be demonstrated. The talk will be illustrated by computer animation. Based on joint work with S. Morier-Genoud, V. Ovsienko and R. Schwartz. The talk is accessible to a wide audience.

Lunch break

### Jean-Marc Schlenker - U. of Toulouse

An introduction to anti-de Sitter geometry.

Abstract: This is an introductory talk for non-specialists. I shall speak about Riemannian manifolds of dimensions 2 and 3 of constant curvature, and the analogues to the Lorentzian case. I shall also talk about hyperbolic space, quasi-Fuchsian manifolds, anti-de Sitter space and globally hyperbolic manifolds. If time permits, I shall talk about relations between these various notions

### Alexandru Oancea - U. of Strasbourg

Contact homology as an S^1-equivariant theory.

Abstract: I will prove the following theorem. Using rational coefficients, linearized contact homology of a contact manifold with a symplectic filling is isomorphic to S^1-equivariant symplectic homology of the filling. This statement allows to solve by geometric means some serious transversality issues arising in the study of moduli spaces of holomorphic curves for symplectic field theory. Joint work with F. Bourgeois (Brussels).

Coffee break

### Richard Wentworth - U. of Maryland

Bosonization formulas on Riemann surfaces.

Abstract : The bosonization formulas on Riemann surfaces relate zeta-regularized determinants of Laplace operators acting on sections of line bundles to determinants of scalar laplacians. They are tantamount to a relationship between the metrics defined by Quillen and Faltings on the determinant of cohomology. In this talk I will explain how to compute a precise expression for all of the previously undetermined constants appearing in these formulas. In the process, we derive a new factorization formula for determinants of Dolbeault operators on Riemann surfaces.

### 19h30

Conference dinner, offered to the participants, at the restaurant "Le petit bois vert", 2 Quai de la Bruche.

## 4 septembre 2010

### Andrei Losev - ITEP, Moscow

Geometry, QFT, their deformations and their obstructions.

Abstract : We start with explaining that theoretical physics today is like geometry in 19 century. In this analogy the main problem is proper definition of QFT that is analogous to proper definition of a manifold. Then we recall Segal's definition, its power and weakness, and speculate about various generalizations. We explain that functional integral (when it makes sense) may be considered just as a representation of the Segal construction. We explain (in the Segal picture) the origin of the observables in QFT associated to submanifolds of the space-time. In particular, we show how these observables may be considered as deformations of QFT ( we restrict ourself with point observables - "deformations of the Lagrangian", line-observables - "Feymnan diagrams" and surface-observables - "string theory") and illustrate it by mirror-symmetry motivated examples. Finally, we discuss obstructions to deformations, configuration space approach to renormalization, associated homotopical Maurer-Cartan equation and evidence for existence of homotopy differential geometry (coming from the extension of Polyakov conjecture to 2d homotopically conformal theories).

Coffee break

### 10h30

The braid semigroup and cluster varieties.

Abstract : We shall explain a (relatively old) construction associating a quiver (graph with oriented edges) to a word in letters being simple roots of a simple Lie algebra and such that application of the braid group relations to the word corresponds to cluster mutation of quivers. Thus for any element of the braid semigroup one can associate a cluster manifold. The main aim of this talk will be the particular example of this construction giving some known manifolds, in particular the ones appearing in the talk of S.Tabchnikov.

### Francesco Costantino - Strasbourg

On spin networks and their generating series.

Abstract: After recalling the basic facts and definitions on classical and quantum spin networks, I will sketch a new proof (joint with Julien Marché) of a result of B. Westbury computing the generating series of a classical spin network. Then I will discuss some possible extentions and open problems.

Dernière mise à jour le 2-09-2010