January 11, 2023
TITLE: Modularity of BPS indices on Calabi-Yau threefolds

ABSTRACT: Unlike in cases with maximal or half-maximal supersymmetry, the spectrum of BPS states in type II string theory compactified on a Calabi-Yau threefold with generic SU(3) holonomy remains partially understood. Mathematically, the BPS indices coincide with the generalized Donaldson-Thomas invariants associated to the derived category of coherent sheaves, but they are rarely known explicitly. String dualities indicate that suitable generating series of rank 0 Donaldson-Thomas invariants counting D4-D2-D0 bound states should transform as vector-valued mock modular forms, in a precise sense. I will spell out and test these predictions in the case of one-modulus compact Calabi-Yau threefolds such as the quintic hypersurface in P⁴, where rank 0 DT invariants can (at least in principle) be computed from Gopakumar-Vafa invariants, using recent mathematical results by S. Feyzbakhsh and R. Thomas. Work in progress with S. Alexandrov, S. Feyzbakhsh, A. Klemm and T. Schimannek.

Slides of Lecture

January 11, 2023
TITLE: Virasoro constraints: vertex algebras and wall-crossing

ABSTRACT: This talk will be the second one concerning the Virasoro constraints in moduli spaces of sheaves (see Woonam’s abstract), based on joint work with A. Bojko and W. Lim. In this talk, I will focus on the connection between Virasoro constraints and the vertex algebra that D. Joyce recently introduced to study wall-crossing. It turns out that this vertex algebra can be endowed with a conformal element that induces the Virasoro operators that had appeared previously in the literature. In this language, our conjectures/results say that moduli of sheaves define physical/primary states in this vertex operator algebra. From this point of view and Joyce’s theory, we can prove that the Virasoro constraints are compatible with wall-crossing. This is the main new technical tool that allows to prove the constraints for torsion-free sheaves on curves and surfaces by reducing everything to rank 1.

January 11, 2023
TITLE: Virasoro constraints; history and moduli of sheaves

ABSTRACT: Virasoro constraints were first conjectured for the moduli of stable curves (the Witten conjecture) and stable maps. These conjectures provide a set of universal relations among descendent invariants described by a representation of half of the Virasoro algebra. Recently, the analogous constraints were conjectured in several sheaf theoretic contexts. In joint work with A. Bojko and M. Moreira, we provide a unifying viewpoint to Virasoro constraints for general moduli of sheaves and prove the conjecture for torsion-free sheaves on curves and surfaces.

January 12, 2023
TITLE: An introduction to some aspects of the swampland distance conjectures

ABSTRACT: The swampland program aims at distinguishing the quantum field theories which can be consistently coupled to quantum gravity at high energies from those which cannot. It leads to the formulation of many interesting problems at the intersection of geometry and physics. The first part of this talk will be an introduction to the distance conjectures, concerning the moduli spaces of vacua of the theories which admit a consistent quantum gravity completion. In the second part I will present ongoing work on twisted connected sum G2-manifolds related to the distance conjectures.
Slides of Lecture

January 12, 2023
TITLE: Quantum gravity conjectures and asymptotic Hodge theory

ABSTRACT: In this talk I will explain how asymptotic Hodge theory can be used to provide general evidence for some of the quantum gravity conjectures. In particular, I will describe how the orbit theorems of Hodge theory can be used in addressing the so-called Distance Conjecture. For Calabi-Yau manifolds, I will sketch the implied classification of asymptotic regions of the moduli space and comment on the special properties of infinite distance boundaries. I will highlight that the quantum gravity conjectures can actually lead to new mathematical theorems by presenting a finiteness theorem generalizing a famous result of Cattani, Deligne, and Kaplan on Hodge classes. This new result is based on work with B. Bakker, C. Schnell, and J. Tsimerman.

Slides of Lecture

January 11, 2023
TITLE: Wall-crossing for Calabi-Yau fourfolds and applications

ABSTRACT: To count SU(4)-instantons on Calabi-Yau fourfolds, Borisov-Joyce and later on Oh-Thomas formulated a theory which counts Gieseker stable sheaves. An interesting question is what happens when one changes the stability condition and crosses walls where sheaves get destabilized. Relying on the ideas introduced by D. Joyce, I will explain a formulation of wall-crossing that is particularly useful in solving existing conjectures by discussing some of its applications.

Slides of Lecture

January 10, 2023
TITLE: Embedding superconformal vertex algebras from Killing spinors and (0,2) mirror symmetry

ABSTRACT: I will construct embeddings of the N=2 superconformal vertex algebra under appropriate conditions inspired by the Killing spinor equations in supergravity. Firstly, when these equations are formulated on a quadratic Lie algebra, they become purely algebraic conditions that induce an embedding in the corresponding superaffine vertex algebra. Secondly, when they are formulated on a suitable class of Courant algebroids, they induce an embedding in the vertex algebra of global sections of the corresponding chiral de Rham complex. As an application, I will present an example of (0,2) mirror symmetry given by pairs of homogeneous Hopf surfaces equipped with a Bismut-flat pluriclosed metric. Joint work with Andoni De Arriba De La Hera and Mario Garcia-Fernandez (arxiv:2012.01851, to appear in IMRN, and further work in progress).

Slides of Lecture

September 8, 2022
TITLE: Equivariant Birational Geometry

Slides of Lecture

• 09/08/2023: Area-minimizing integral currents: singularities and structure

September 8, 2023
TITLE: Area-minimizing integral currents: singularities and structure

ABSTRACT: Let T be an area-minimizing integral current of dimension m in a smooth closed Riemannian manifold of dimension m + n. It is known since the work of De Giorgi, Fleming, Almgren, Simons, and Federer in the sixties and seventies that, when n = 1, the (interior) singular set of T has dimension at most m − 7. In higher codimension Almgren’s big regularity paper proved in 1980 that the singular set has dimension at most m − 2, laying the grounds for a theory which has been simplified and extended in the last 15 years. Both theorems are optimal, but at the qualitative level there is a quite important mismatch between the singular sets of the known examples and a general closed set of the same dimension. In a celebrated work in the nineties Simon proved, for n = 1, that the singular set is m − 7- rectifiable and that the tangent cone is unique Hm−7-a.e.. The counterpart of Simon’s theorem in higher codimension has been reached very recently by Paul Minter, Anna Skorobogatova and myself and, independently, by Krummel and Wickramasekera. Even though it would be natural to expect much stronger structural results, our theorem is indeed close to optimal, as a recent result of Liu shows that the singular set can in fact be a fractal of any Hausdorff dimension α ≤ m − 2.

September 14, 2022
TITLE: 5D SCFTs from C³ Orbifolds

ABSTRACT: In this talk, we will consider the orbifold singularities X = C³/Γ where Γ is a finite subgroup of SU(3). M-theory on X gives rise to a rich class of 5D SCFTs. We will study these SCFTs via 3D McKay correspondence which relates the group theoretical data of \Gamma to the physical properties of the 5D SCFT. We find some theories in this class can be obtained by gauging certain discrete symmetries in other 5D SCFTs. I will report on recent progress on relating 5D SCFTs to 4D electric-magnetic duality in a class of G2 compactification.

Slides of Lecture