Let be a compact Lie group. I will review the construction of Mabuchi geodesic families of –invariant Kähler structures on , via Hamiltonian flows in imaginary time generated by a strictly convex invariant function on , and the corresponding geometric quantization. At infinite geodesic time, one obtains a rich mixed polarization of , the Kirwin-Wu polarization, which is then continuously connected to the vertical polarization of . The geometric quantization of along this family of polarizations is described by a generalized coherent state transform that, as geodesic time goes to infinity, describes the convergence of holomorphic sections to distributional sections supported on Bohr-Sommerfeld cycles. These are in correspondence with coadjoint orbits . One then obtains a concrete (quantum) geometric interpretation of the Peter-Weyl theorem, where terms in the non-abelian Fourier series are directly related to geometric cycles in . The role of a singular torus action in this construction will also be emphasized. This is joint work with T. Baier, J. Hilgert, O. Kaya and J. Mourão.
A wide class of two-dimensional gravitational models, including (2,p) minimal strings and Jackiw-Teitelboim gravity, are described by certain double-scaled matrix models. This allows for the study of non-perturbative effects in such theories, via instanton calculus in the corresponding matrix model dual. In matrix models, non-perturbative contributions to observables organize themselves in two distinct classes, associated to eigenvalue tunneling (corresponding to ZZ branes) and to determinant insertions (corresponding to FZZT). We provide a recipe for systematically computing ZZ brane contributions in such models, using a non-perturbative formulation of the topological recursion. Also, we show how the competing ZZ and FZZT brane effects show up in the Borel plane and in the large genus asymptotics of several observables.
In this talk we will describe the connection formulae constructed from the non-linear Stokes data associated to the resurgent two-parameter transseries of the Painlevé I and II equations. We will discuss the monodromy orbits and how these transition functions agree with those inherited from the isomonodromy method. We will further discuss preliminary results on the case of the inhomogeneous and deformed Painlevé II equations.
I will start out by discussing a recent conjecture for the complete resurgent Stokes data
of the Painlevé I/II equations, which is based upon their resurgent structure [1].
This provides access to an infinite amount of transcedental numbers which completely
characterize the global properties of the transseries solutions of Painlevé I/II and
the hermitian quartic matrix model. I will then outline how to obtain the exact same data from
(off-critical and double-scaled) matrix integral calculations [2]. The full resonant
resurgent transseries may be obtained by generalizing physical-sheet eigenvalue
tunneling to all (non-physical) sheets of the spectral geometry.
This provides the complete semiclassical decoding of resurgent matrix models and minimal strings.
I will end by presenting explicit examples.
[1] S. Baldino, R. Schiappa, M. S., R. Vega
Resurgent Stokes Data for Painlevé Equations and Two-Dimensional Quantum (Super) Gravity,
(2022), arXiv:2203.13726 [hep-th]
[2] M. Mariño, R. Schiappa, M. S.
To appear (2022).
Dissipative relativistic hydrodynamics is expected to describe the late times, thermalised behaviour of strongly coupled fluids such as a strongly coupled super Yang-Mills plasma. These systems are then accurately described by a hydrodynamic series expansion in small gradients. Surprisingly, this hydrodynamic expansion is accurate even when the systems are still quite anisotropic: the non-hydrodynamic modes governing the non-equilibrium behaviour at very early-times become exponentially close to the hydrodynamic solution in an early process called hydrodynamization. This early success is intimately related with the fact that the hydrodynamic expansion is asymptotic. The theory of transseries and resurgence explicitly shows how the non hydrodynamic modes are in fact encoded in this late-time expansion. In this talk we will focus on a MIS-type model and use exponentially accurate summations of the the late-time resurgent transseries to recover the behaviour of the fluid before hydrodynamisation, and effectively match it to any given initial non-equilibrium condition. We will further show that such summations can provide analytic predictions beyond the late time regime.
The spectrum of conformal field theories displays Regge trajectories: families of operators whose dimensions are smooth functions of the spin. We discuss the structure of Regge trajectories of the six-dimensional N=(2,0) SCFTs, in particular of operators coupling to the stress tensors super-multiplet. We show how these operators organize themselves in Regge trajectories starting at spin -3, and how different Regge trajectories are intertwined by supersymmetry. We then describe an iterative procedure to "bootstrap" the stress tensor multiplet four-point function using the Lorentzian inversion formula. Taking as input protected data we obtain an approximate low twist spectrum and compare it rigorous numerical bootstrap bounds.
The spectrum of conformal field theories displays Regge trajectories: families of operators whose dimensions are smooth functions of the spin. We discuss the structure of Regge trajectories of the six-dimensional N=(2,0) SCFTs, in particular of operators coupling to the stress tensors super-multiplet. We show how these operators organize themselves in Regge trajectories starting at spin -3, and how different Regge trajectories are intertwined by supersymmetry. We then describe an iterative procedure to “bootstrap” the stress tensor multiplet four-point function using the Lorentzian inversion formula. Taking as input protected data we obtain an approximate low twist spectrum and compare it rigorous numerical bootstrap bounds.
In general interacting theories — quantum mechanical, field, gauge, or string theories — perturbation theory is divergent: perturbative expansions have zero radius of convergence and seemingly cannot be summed. Nonperturbatively well-defined results can still be constructed out of perturbation theory by the uses of resurgence and transseries.
Asymptotic series require the use of resurgence and transseries in order for their associated observables to become nonperturbatively well-defined. Resurgent transseries encode the complete large-order asymptotic behaviour of the coefficients from a perturbative expansion, generically in terms of (multi) instanton sectors and for each problem in terms of its Stokes coefficients. By means of two very explicit examples, we plan to introduce the aforementioned resurgent, large- order asymptotics of general perturbative expansions, including discussions of transseries, Stokes phenomena, generalized steepest-descent methods, Borel transforms, nonlinear resonance, and alien calculus.
Time permitting, we will discuss advanced examples in matrix models, minimal and topological strings, and Jackiw-Teitelboim gravity. The discussion will focus both on the construction of the resurgent transseries, their resonance, and the computation of associated Stokes data.
The program of the lectures will include:
Dissipative relativistic hydrodynamics is expected to describe the late times, thermalised behaviour of strongly coupled fluids such as a strongly coupled super Yang-Mills plasma. These systems are then accurately described by a hydrodynamic series expansion in small gradients. Surprisingly, this hydrodynamic expansion is accurate even when the systems are still quite anisotropic: the non-hydrodynamic modes governing the non-equilibrium behaviour at very early-times become exponentially close to the hydrodynamic solution in an early process called hydrodynamization. This early success is intimately related with the fact that the hydrodynamic expansion is asymptotic. The theory of transseries and resurgence explicitly shows how the non-hydrodynamic modes are in fact encoded in this late-time expansion. In this talk we will focus on a MIS-type model and use exponentially accurate summations of the the late-time resurgent transseries to recover the behaviour of the fluid before hydrodynamisation, and effectively match it to any given initial non-equilibrium condition. We will further show that such summations can provide analytic predictions beyond the late time regime.