Applications of the bootstrap to superconformal field theories require the construction of superconformal blocks for four-point functions of arbitrary supermultiplets. Up until recently, only sporadic results had been obtained. In my talk I explain the key ingredients of a new systematic construction that apply to a large class of superconformal field theories, including 4-dimensional models with any number N of supersymmetries. It hinges on a universal construction of the relevant Casimir differential equations. In order to find these equations, we model superconformal blocks as functions on the supergroup and pick a distinguished set of coordinates. The latter are chosen so that the superconformal Casimir operator can be written as a perturbation of the Casimir operator for spinning bosonic blocks by a fermionic (nilpotent) term. Solutions to the associated eigenvalue problem can be obtained through a quantum mechanical perturbation theory that truncates at some finite order so that all results are exact.

I will review a method to evaluate correlation functions of certain statistical systems via chord diagrams and apply it to compute correlators in the double-scaled version of SYK model, in particular in its large N limit. The results are exact at all energies and allow to extract corrections to the maximal Lyapunov exponent. Time permitting, I will comment on the suggested relation of this model to a Hamiltonian reduction of quantum particle moving on the non-compact quantum group SU_q(1,1).

We apply recently constructed functional bases to the numerical conformal bootstrap for 1D CFTs. We argue and show that numerical results in this basis converge much faster than the traditional derivative basis. In particular, truncations of the crossing equation with even a handful of components can lead to extremely accurate results, in opposition to hundreds of components in the usual approach. We explain how this is a consequence of the functional basis correctly capturing the asymptotics of bound-saturating extremal solutions to crossing. We discuss how these methods can and should be implemented in higher dimensional applications.

Séminaire Laurent Schwartz — EDP et applications

Séminaire Laurent Schwartz — EDP et applications

We apply the endoscopic classification of automorphic representations for inner forms of unitary groups to bound the growth of cohomology in congruence towers of locally symmetric spaces associated with U(n-1,1). Our bound is sharper than the bound predicted by Sarnak-Xue for general locally symmetric spaces. This is joint work with Simon Marshall.

Séminaire Laurent Schwartz — EDP et applications

The Teichmüller space parametrizes Riemann surfaces of
fixed topological type and is fundamental in various contexts of mathematics
and physics.  It can be defined as a component of the moduli space of flat G=PSL(2,R)
connections on the surface.  Higher Teichmüller space extends these notions
to appropriate higher rank classical Lie groups G, and N=1 super Teichmüller
space likewise studies the extension to the super Lie group G=OSp(1|2).
In this talk, which reports on joint work with Yi Huang, Ivan Ip and Robert Penner,
I will discuss our solution to the long-standing problem of giving Penner-type
coordinates on super-Teichmüller space and its higher analogues and will
also talk about several applications of this theory including our recent
generalization of the McShane identity to the super case.

Let G be a real reductive Lie group and H a closed subgroup of G. According to the Benoist-Kobayashi criterion, the properness of an action on G/H is controlled by the Cartan projection of H. In this talk, we give some examples of closed subgroups that are not reductive in G but whose Cartan projection is computable (one example is an abelian horospherical subgroup of G). Applying these nonreductive subgroups, we show that some homogeneous spaces of reductive type do not have compact Clifford-Klein forms.

I will discuss results of recent numerical bootstrap work performed for systems with cubic symmetry. Under certain assumptions, we find an isolated region in parameter space, which given prior intuition with the numerical bootstrap, indicates the existence of a CFT in this region. We find critical exponents for the conjectured CFT which are in discrepancy with the epsilon expansion (but in good agreement with experiments for structural phase transitions). The disagreement of critical exponents for structural phase transitions calculated in the epsilon expansion with those measured in experiments is something that was noticed since the 70s. I will briefly discuss some resolutions proposed at the time.

We explore in detail the properties of two melonic quantum mechanical theories which can be formulated either as fermionic matrix quantum mechanics in the new large D limit, or as disordered models. Both models have a mass parameter m and the transition from the perturbative large m region to the strongly coupled « black-hole » small m region is associated with several interesting phenomena. One model, with U(n)^2 symmetry and equivalent to complex SYK, has a line of first-order phase transitions terminating, for a strictly positive temperature, at a critical point having non-trivial, non-mean-field critical exponents for standard thermodynamical quantities. Quasi-normal frequencies, as well as Lyapunov exponents associated with out-of-time-ordered four-point functions, are also singular at the critical point, leading to interesting new critical exponents. The other model, with reduced U(n) symmetry, has a quantum critical point at strictly zero temperature and positive critical mass m*. For 0<m<m*, it flows to a new gapless IR fixed point, for which the standard scale invariance is spontaneously broken by the appearance of distinct scaling dimensions Δ+ and Δ- for the Euclidean two-point function when t→ +∞ and t→ -∞