ReNewQuantum

In this talk, I will review the problem of quantizing an algebraic curve $P(y,x)=0$ into a differential equation $Q(\hbar \partial_x,x) \psi=0$ known in the literature as a "quantum curve". In particular, using recent works with N. Orantin, B. Eynard and E. Garcia-Failde, I will show how the topological recursion may be used to construct a matrix wave function $\Psi$, the quantum curve and an associated Lax pair that share the same pole structure as the initial spectral curve. I will also discuss the connections with integrable systems and isomonodromic deformations and some open problems regarding them.

Asymptotic series abounds in physics. And the resurgence theory provides a powerful tool to deal with them. In particular the resurgence theory reveals that different asymptotic series at different saddle points are related to each other by Stokes automorphisms characterised by Stokes constants. We argue that the Stokes constants can be treated as new invariants of the theories in question, and in some non-trivial examples they are integers and they can be interpreted as counting of BPS states. We support this statement with examples in Seiberg-Witten theory, complex Chern-Simons theory, and topological string theory.

After recalling the definition of resurgent series as those formal series whose formal Borel transforms define endlessly continuable germs at the origin of the Borel plane (germs with locally discrete singular locus), I will review the proof of the fact that they form a subalgebra. The proof requires to follow the analytic continuation of the convolution product of endlessly continuable germs. A similar proof shows that endless continuability is also stable under Hadamard product. Then I will report on a recent result about the Moyal star product in deformation quantization: "algebro-resurgent" series (a subspace of formal series with coefficients in C{q_1,...,q_N,p_1,...p_N}, with algebraic singular locus after Borel transform) are stable under Moyal star product.

Abstract - tba

In this talk, I will discuss the quantization of hyper-Kahler manifolds using a class of two-dimensional quantum field theories known as topological A-models. In particular, I will illustrate how studying "branes" living on these manifolds can lead to new insights into the representation theory of non-commutative algebras such as double affine Hecke algebras.

We will give a panoramic of the interaction between certain well behaved cohomological classes on the moduli spaces of curves, the origin of their integrals from enumerative problems, their integrable hierarchies counterpart, and their relation with topological and geometric recursion. The red thread of this interaction will pass through some worked out problems as well as some work in progress. If time allows, some possible links with resurgence will be discussed.

The BV-BFV formalism introduced by Cattaneo, Mnev and Reshetikhin is a tool that allows for the perturbative quantization of degenerate theories on manifolds with boundary in a way that is compatible with cutting and gluing. I'll review this formalism and discuss some first results of its application to Chern-Simons theory.

Airy structures - a notion introduced by Maxim Kontsevich and Yan Soibelman - are certain systems of linear ODEs admitting a unique solution (the so-called partition function) which is computed by a topological recursion. Airy structure therefore serve as initial data for the recursion. I will review the strategy to obtain Airy structures from the W(gl_r) vertex operator algebra (from joint work with Bouchard, Chidambaram, Creutzig, Noshchenko). This strategy depends on the choice of an element of the Weyl group of gl_r, used for "twisting". For a Coxeter element this is equivalent to Bouchard-Eynard topological recursion on spectral curves with higher order ramification. For other twisting we show with Aghaei, Kramer and Schueler that this defines a topological recursion on singular spectral curves. Schueler has classified which twisting do lead to Airy structures: surprisingly there are some restrictions, posing non-trivial conditions on the type of spectral curves allowed in this definition, and whose interpretation is still mysterious.

The quantum curve conjecture claims that one can associate a differential equation to a spectral curve, whose solution can be reconstructed by topological recursion applied to the original spectral curve. I will explain how starting just from loop equations, one can construct a system of PDEs which will annihilate the wave function built from topological recursion, solving the conjecture affirmatively for all hyperelliptic curves. The families of spectral curves can be parametrized with the so-called times, which can be defined as periods on second type cycles on the curves. These deformation parameters giving rise to families of spectral curves will play a key role when producing our system of PDEs. Our system of PDEs can be used to prove that the WKB solution of many isomonodromic systems coincide with the topological recursion wave function. If time permits, I will comment on how to generalize this procedure to spectral curves of arbitrary rank, but with simple ramifications, which is work in progress.