Schedule of the Workshop: Algebra, Geometry and Physics of BPS States

Chris Brav: Shifted symplectic structures and derived critical loci

We show that a derived scheme with -1-shifted symplectic structure, in the sense of Pantev-Toen-Vezzosi-Vaquie, can be Zariski locally presented as the critical locus of a polynomial. In particular, this shows that moduli schemes of simple coherent sheaves on Calabi-Yau threefolds are Zariski locally critical loci. This is joint work with Vittoria Bussi, Delphine Dupont, and Dominic Joyce.

Tom Bridgeland: Quadratic differentials and stability conditions

I will give a precise description of stability conditions and BPS invariants for a certain class of CY3 quiver algebras in terms of quadratic differentials on Riemann surfaces and their finite-length trajectories. This is joint work with Ivan Smith, and takes its inspiration from a paper of physicists Gaiotto, Moore and Neitzke.

Vittoria Bussi: Categorification of Donaldson-Thomas invariants and of Lagrangian intersections

We study the behaviour of perverse sheaves of vanishing cycles under action of symmetries and stabilization, and we investigate to what extent they depend on the function which defines them. We investigate the relation between perverse sheaves of vanishing cycles associated to isomorphic critical loci with their symmetric obstruction theories, pointing out the necessity for an extra "derived data". Similar results are proved for mixed Hodge modules and motivic Milnor fibres.
These results will be used to construct perverse sheaves and mixed Hodge modules on moduli schemes of stable coherent sheaves on Calabi-Yau 3-folds equipped with ‘orientation data’, giving a categorification of Donaldson-Thomas invariants. This will be a consequence of the fact that a quasi-smooth derived scheme with a (-1)-shifted symplectic structure and orientation data has a "categorification". Finally we categorify intersections of Lagrangians in a complex symplectic manifold, describing the relation with Fukaya categories and deformation-quantization.

Lothar Goettsche: Refined curve counting on surfaces

An old conjecture of mine gives a generating function for the numbers of δ-nodal curves in linear systems on surfaces. In this talk we want to propose a refinement of the conjecture, where the numbers of curves are replaced my polynomials in a variable y, which for y=1 specialize to the numbers of curves. This refinement is defined in terms of χ_y-genera of relative Hilbert schemes of points for the family of curves in the linear system. For rational surfaces these refined invariants are related to Welschinger invariants and have an interpretation in tropical geometry.

Martijn Kool: Reduced classes and curve counting on surfaces.

Counting nodal curves in linear systems |L| on smooth projective surfaces S is a problem with a long history. The Göttsche conjecture, now proved by several people, states that these counts are universal and only depend on c₁(L)², c₁(L)⋅c₁(S), c₁(S)² and c₂(S). We present a quite general definition of reduced Gromov-Witten and stable pair invariants on S. The reduced stable pair theory is entirely computable. Moreover, we prove that certain reduced Gromov-Witten and stable pair invariants with many point insertions coincide and are both equal to the nodal curve counts appearing in the Göttsche conjecture. This can be see as version of the MNOP conjecture for the canonical bundle K_S. This is joint work with R. P. Thomas.

Andrew Morrison: Values of Behrend's microlocal function

By a theorem of Behrend Donaldson-Thomas invariants can be defined in terms of a certain constructible function. We will compute this function at all points in the Hilbert scheme of points in three dimensions and see that it is constant. We use equivariant zeta functions and a certain commuting matrix calculation. As a corollary we see that this Hilbert scheme of points is generically reduced and its components have the same dimension mod 2. This gives an application of the techniques of BPS state counting to a problem in Algebraic Geometry.

Daniel Persson: Generalised Moonshine in the Elliptic Genus of K3

I will discuss the analogue of Norton's Generalized Monstrous Moonshine for the recently discovered connection between the  largest Mathieu group M24, K3-surfaces and weak Jacobi forms.  I the second part of the talk I will also explain the connection with  automorphic Borcherds lifts and wall-crossing in N=4 string theory.

Markus Reineke: Motivic DT invariants of quivers with stability and abelian analogues

We will review the definition of motivic DT invariants for quivers with stability and state the main conjectures as well as known results and formulas for these invariants. Against the background of MPS degeneration, we will consider abelian DT invariants and discuss positivity properties and explicit formulas.

Roberto Volpato: Mathieu Moonshine and sigma models on K3

The Mathieu Moonshine conjecture is a relationship between the largest Mathieu group M24 and the elliptic genus of the K3, which may be viewed as the supersymmetric analog of the more famous Monstrous Moonshine conjecture. The Mathieu Moonshine suggests the existence of a web of surprising and not yet understood connections among seemingly unrelated topics in mathematics and physics, from modular functions to conformal field theories, from generalised Kac-Moody algebras to BPS degeneracy formulae and wall-crossing phenomena in supergravity theories. We review the evidence in support of the Mathieu Moonshine, describe its relationship with non-linear sigma models on K3 and discuss the open problems in its interpretation.