Séminaire Laurent Schwartz — EDP et applications

I will start by an introduction to Donaldson Thomas theory and some of the statements about its modularity properties, as well as its connection to S-duality conjecture in superstring theory, made formerly by physicists Gaiotto, Strominger, Yin. I will then provide an algebraic geometric approach to prove this conjecture for DT invariants of sheaves supported on hyperplane sections of the quintic Calabi-Yau threefold.

Séminaire Laurent Schwartz — EDP et applications

We will consider abelian coverings of smooth projective varieties over finite fields which are wildly ramified along a divisor D with normal crossings, and will describe the corresponding abelianized fundamental group via modified logarithmic de Rham-Witt sheaves.

According to Hochster's direct summand conjecture (1973), a regular ring R is a direct summand, as an R-module, of every finite extension ring. We shall outline our recent proof which relies on perfectoid techniques. Similar arguments also establish the existence of big Cohen-Macaulay algebras for complete local domains of mixed characteristics.

We shall address the duality between supersymmetric gauge theories in various dimensions and elliptic integrable systems such as Ruijsenaars-Schneider model and periodic intermediate long wave hydrodynamics. These models arise in instanton counting problems and are described by certain elliptic algebras. We discuss the correspondence between the two types of models by employing the large-n limit of the dual gauge theory.

For many years researchers have tried to glean hints about quantum gravity from black hole thermodynamics. We try a different approach using a minimal statistical mechanical model for the horizon fluid based on Damour-Navier-Stokes (DNS) equation. For asymptotically flat black hole spacetimes in General Relativity, we show explicitly that at equilibrium the entropy of the horizon fluid is the Bekenstein-Hawking entropy. Further we show that, for the bulk viscosity of the fluctuations of the horizon fluid to be identical to Damour, a confinement scale exists for these fluctuations, implying quantization of the horizon area.

Classical Teichmüller space describes the space of conformal structures on a given topological surface S. It plays an important role in several areas of mathematics as well as in theoretical physics.

Higher Teichmüller theory generalizes several aspects of classical Teichmüller theory to the context of Lie groups of higher rank, such as the symplectic group PSp(2n; R) or the special linear group PSL(n; R). So far, two families of higher Teichmüller spaces are known. The Hitchin component, which is defined when the Lie group is a split real forms, and the space of maximal representations, which is defined for Lie groups of Hermitian type. Interestingly, both families are linked with various notions of positivity in Lie groups.

In this talk I will give an introduction to higher Teichmüller theory, introduce new positive structures on Lie groups and discuss the (partly conjectural) relation between the two.

Ten dimensional supergravities are well-known to arise as low-energy limits of string theory. I will show that in fact, these pure supergravities are complete string theories in their own right. The strings in question describe holomorphic maps to the space of null geodesics in complexified space-time, known as ambitwistor space, and have no massive states in their spectrum. Worldsheet correlators describe scattering amplitudes in pure supergravity, in a form found recently by Cachazo, He & Yuan, in which the integral over the moduli space of marked curves is localized to solutions of the Gross-Mende equations.