Mathematical Physics Group

Seminars:

Geometric structures in Riemannian Geometry (Raffero)

Abstract: A Riemannian manifold may admit further geometric structures besides the Riemannian metric, and various meaningful examples are related to Riemannian holonomy groups. In this talk, I will give an introduction to this topic, with special emphasis on geometric structures that are relevant in supersymmetry, namely SU(3)- and G2-structures.

Why should geometers care about supersymmetry (and holography)? (Martelli)

Abstract: In this talk I will try to provide some answers to the question raised in the title. In particular, the talk will be an informal exposition directed mainly to the differential geometry audience, highlighting the emergence of different types of geometric structures rather directly from the study of supersymmetry (and holography). No previous knowledge of string theory is required, nor will be ever needed to work and contribute actively to this research field. Rather than on general theories the discussion will be driven by a number of concrete examples of such mathematical structures.

Program: The workshop revolves around the notion of equivariant volumes and their applications to mathematics and physics, especially Geometry, Quantum Field Theory, Supergravity and Strings. The workshop includes introductory lectures and seminars on the latest developments in different sub-fields.

Lecturers:

Zabzine

• Toric Kähler manifolds and equivariant volumes: JK integrals, counting of states and categorical approach;

• Analytical properties, compact vs non-compact geometry;

• (Quantum) deformations of equivariant volumes and semi-classical expansions.

Zaffaroni

•  A brief excursus on toric geometry;

•  The symplectic geometry of toric orbifolds;

•  Equivariant forms and localization;

• The equivariant volume of toric orbifolds;

• Examples and relation between different approaches.

Seminars:

Exploring Supersymmetric Quantum Field Theories on Orbifolds (Pittelli)

Abstract: Supergravity solutions exhibiting orbifold singularities represent non-trivial contributions to the path integral of quantum gravity. The holographic duals correspond to supersymmetric QFTs on orbifolds with conical singularities. Physics-wise, partition functions of systems living on these singular orbifolds encode dualities between distinct models and contain essential physical information, e.g. the entropy of accelerating black holes. Mathematics-wise, the path integral of such theories can be associated with topological invariants inherent to the underlying orbifold, generalizing results known for smooth manifolds. I will extensively delve into the exploration of supersymmetric QFTs on orbifolds with conical singularities, including general circle fibrations over spindles and some ongoing research in four and five dimensions.

Symplectic Cuts and Open Strings (Zabzine)

Abstract: I will discuss how to define genus zero open Gromov-Witten invariants of a Lagrangian toric A-brane on toric CY3-fold via the notion of symplectic cut of quantum equivariant volumes. I will explain the interplay between open and closed Gromov-Witten invariants in the context toric CY3 geometry and stress the role of the equivariance in this construction.

Equivariant Localization in Supergravity (Benetti Genolini)

Abstract: Equivariant localization may be applied to AdS/CFT to compute various BPS observables in gravity without solving the supergravity equations. The key ingredient is that supersymmetric solutions with an R-symmetry are equipped with a set of equivariantly closed forms. These may in turn be used to impose flux quantization and compute observables for supergravity solutions, using only topological information and the fixed point formula.

Equivariant Localization and Holography (Zaffaroni)

Abstract: The equivariant volume of the internal manifold can be used to characterise universally the geometry of a large class of supersymmetric solutions in holography. In particular, many extremization problems in quantum field theory can be formulated in terms of an extremization of the equivariant volume. In this talk, we provide the general picture and various explicit examples.

Schedule:

Day 1

• 10.00 - 11.00 Lecture (Zabzine)

• 11.00 - 11.30 Coffee Break

• 11.30 - 12.30 Lecture (Zabzine)

• 12.30 - 14.30 Lunch Break

• 14.30 - 15.15 Seminar (Pittelli)

• 15.15 - 15.45 Coffee Break

• 15.45 - 16.30 Seminar (Zabzine)

• 16.30 - 17.30 Informal Discussions

Day 2

• 10.00 - 11.00 Lecture (Zaffaroni)

• 11.00 - 11.30 Coffee Break

• 11.30 - 12.30 Lecture (Zaffaroni)

• 12.30 - 14.30 Lunch Break

• 14.30 - 15.15 Seminar (Benetti Genolini)

• 15.15 - 15.45 Coffee Break

• 15.45 - 16.30 Seminar (Zaffaroni)

• 16.30 - 17.30 Informal Discussions

Program: These are two introductory lectures on the Classical Geometry of symplectic and toric varieties and the far-reaching relationships with the Quantum Mathematical Physics motivated by String Theory. The covered topics include:

• Classical Geometry:

-      Symplectic manifolds and symplectic volumes;

-      Moment maps, symplectic quotients and symplectic cuts;

-      Basics of toric varieties and their torus-equivariant cohomology;

-      Equivariant volumes, localization formulas and Jeffrey-Kirwan residues.

• Quantum Geometry/Gromov-Witten invariants:

-      Genus zero invariants and quantum cohomology;

-      Batyrev ideal and Picard-Fuchs equations;

-      Givental I and J functions, mirror map;

-      Periods and mirror symmetry.

Ref.s:

-       M. Audin, “Torus Actions on Symplectic Manifolds”, Birkhäuser (2012);

-       D. Cox and S. Katz, “Mirror Symmetry and Algebraic Geometry”, AMS (1999);

-       K. Hori et al., “Mirror Symmetry”, AMS (2003).