Program

The conference takes place in aula Vitali (ground floor)

Thursday, December 18, 2014
10:00
Opening
11:00
Fabrizio Nieri (University of Surrey, United Kingdom)
Holomorphic blocks for supersymmetric gauge theories in various dimensions. download abstract - view abstract Fabrizio Nieri
Holomorphic blocks for supersymmetric gauge theories in various dimensions.
In recent years, due to the method of supersymmetric localization, many exact results have been achieved in the study of supersymmetric gauge theories on compact spaces of various dimension and topology, discovering surprising structures. For example, it is becoming clear that partition functions of a large variety of such theories can be described in terms of fundamental ``holomorphic blocks" and gluing rules. In this talk, I will introduce the subject by reviewing one of the best understood example, namely 3d N=2 theories defined on manifolds admitting a Heegaard decomposition in solid tori, where partition functions can be built by fusing 3d holomorphic blocks by elements in SL(2,Z). This was originally checked for S2xS1 and S3, and recently for any lens space L(r,1). Moreover, this picture can be lifted to 4d N=1 theories defined on L(r,1)xS1. Holomorphic blocks have been also proposed for 5d N=1 theories, and I will discuss how partition functions on S4xS1 and any toric Sasaki-Einstein manifold can be built out of them.
12:30
Lunch
14:00
Enrico Manfredi (Alma Mater Studiorum Università di Bologna, Italy)
Representations of lens spaces. download abstract - view abstract Enrico Manfredi
Representations of lens spaces.
Lens spaces are the simplest family of closed 3-manifolds; they may be defined in diffent ways. In this talk the focus will be on the genus one Heegaard splitting definition: a lens space is the glueing of the two solid tori through an homeomorphism of their boundaries that can be represented by an element of S L(2,Z). Lens spaces may be defined also by a quotient of the 3-sphere S^3 under the action of Z_p, by a glueing of the boundary of the ball B^3 and by integral or rational Dehn surgery on knots/links. The equivalence of all these definitions can be easily described through some pictures. At last, the classification results of lens spaces will be outlined, both up to diffeomorphism and up to homotopy type. If time will permit, it will be shown that every closed 3-manifold may be described by an Heegaard splitting, as well as Dehn surgery on knots/links.
15:00
Luca Migliorini (Alma Mater Studiorum Università di Bologna, Italy)
The P=W conjecture: statement, motivation, known cases. download abstract - view abstract Luca Migliorini
The P=W conjecture: statement, motivation, known cases.
This is a survey talk explaining the statement of the so called P=W conjecture (due to M. A. de Cataldo, T. Hausel and myself), its motivation, stemming from the determination, due to T. Hausel and F. Rodriguez-Villegas, of the Mixed Hodge polynomial of some character varieties, the few cases in which this conjecture was proved and the problems one finds to extend these proofs to the general case.
16:00
Break
16:30
Michele Schiavina (Universität Zürich, Switzerland)
Classical and quantum gauge theories on manifolds with boundaries. download abstract - view abstract Michele Schiavina
Classical and quantum gauge theories on manifolds with boundaries.
One of the long term goals of modern mathematical physics is to understand the quantisation of physical theories under a general and mathematically sound framework. Among physical theories, those presenting local symmetries of the action (gauge theories) are of fundamental relevance, and the related issues that appear when ensuing their quantisation are addressed under a variety of points of view. One of the most celebrated mechanism to quantise gauge symmetric theories is the BRST formalism, but there are many important examples where it does not apply or it is not optimal. The extension of BRST to more general symmetries that do not come from the action of a Lie group (or that do, up to the equations of motion) goes under the name of BV formalism. More recently it turned out that this formalism provides a natural way to treat gauge theories on manifolds with boundary, even when the theory can be cast in the BRST formalism. In this talk I will outline the generalities of the CMR (Cattaneo, Mnev, Reshetikhin) formalism for gauge theories on manifolds with boundary, focusing on examples both worked out and in progress.
Friday, December 19, 2014
09:30
Andrea Cattaneo (Università di Parma, Italy)
Elliptic 3-folds in P^2-bundles over surfaces download abstract - view abstract Andrea Cattaneo
Elliptic 3-folds in P^2-bundles over surfaces
In this talk we want to study the elliptic fibrations whose total space is a Calabi-Yau variety embedded as an anticanonical divisor in a suitable class of $\mathbb{P}^2$-bundles over a surface. Once we fix the base surface, we will show that the number of such elliptic 3-folds is finite, using only intersection theoretic properties of the base. When the base surface is $\mathbb{P}^2$, we will show that some of these fibrations admit in a very natural way some non-Kodaira fibres.
10:30
Break
11:00
Marco Fazzi (Université libre de Bruxelles, Belgium)
F-Theory on singular spaces. download abstract - view abstract Marco Fazzi
F-Theory on singular spaces.
I will review the new approach to F-theory compactifications pioneered in Arxiv:1410.4867. According to it, instead of resolving or deforming the singularities of the internal space, one can extract physically relevant information by adding more structure to the singular space itself. This new perspective also allows to treat more exotic brane configurations, such as T-branes. The mathematical framework in which the approach is formulated is known as Eisenbud '92s matrix factorizations. I will present an affine example of how this works, as well as a global, compact one.
12:00
Nicolò Piazzalunga (SISSA Trieste, Italy)
(Unoriented) Gopakumar-Vafa invariants. download abstract - view abstract Nicolò Piazzalunga
(Unoriented) Gopakumar-Vafa invariants.
After a review of topological string theory and BPS invariants, I will discuss the type IIA physical realisation of the unoriented topological string introduced by Walcher, its M-theory lift, and I will show how it allows to compute open and unoriented topological amplitudes in terms of one-loop diagrams of BPS\M2-brane states.
13:00
Lunch
14:30
Danilo Lewanski (Universiteit van Amsterdam, Netherlands)
Gromov-Witten invariants. download abstract - view abstract Danilo Lewanski
Gromov-Witten invariants.
What are they? In algebraic geometry they represent the intersection theory on moduli spaces of pseudo- holomorphic curves in almost K\'e4hler manifolds. In mathematical physics they are symplectic invariants that can be read as coefficients in the multiplication table of the quantum cohomology ring of homogeneous varieties. In A-model string theory they represent well defined path integrals of the theory, and they play a fundamental role in the Mirror Symmetry for Calabi-Yau manifolds. Yes, but how can I compute them? They are usually hard to compute, we will go through some examples: ( 1) GW({pt}) and the Witten conjecture ( 2) GW(P ^1) and the recently found quantum curve (3) GW(P ^2) and the amazing Kontsevich-Manin Formula (4) Other intriguing examples, like the comparison between the quantum cohomologies of P^3 and a smooth quadric 3-fold that have very similar classical cohomology rings.
15:30
Break
16:00
Simone Marzioni (Aarhus Universitet, Denmark)
The Andersen-Kashaev TQFT. download abstract - view abstract Simone Marzioni
The Andersen-Kashaev TQFT.
In the 1989 work "Quantum Field Theories and Jones Polynomial" E. Witten showed how the Quantum Field Theories formalism with the Chern-Simons action and compact gauge group can produce important invariants in low dimensional topology like the Jones polynomial. A combinatorial, formally correct, model for such theories was constructed right afterwards by Reshetikhin and Turaev. In order to be able to define quantum Chern-Simons theory with non compact gauge group, we will introduce a combinatorial model for a TQFT coming from the quantisation of the Teichmuller Space, as described by J.E. Andersen and R.M. Kashaev. This will permit us to define a new quantum invariant for triangulated 3-manifolds, conjecturally related to the hyperbolic volume. On the way we will need to introduce the Penner coordinates for the decorated Teichmuller space, and angle structures on triangulated pseudo 3-manifolds. If the time permits I will explain how this quantisation program can be extended to the case when these coordinates are complex.