Models of discretized moduli spaces, cohomological field theories, and Gaussian means

We prove combinatorially the explicit relation between genus filtrated $s$-loop means of the Gaussian matrix model and terms of the genus expansion of the Kontsevich--Penner matrix model (KPMM). The latter is the generating function for volumes of discretized (open) moduli spaces $M_{g,s}^{\mathrm{disc}}$ given by $N_{g,s}(P_1,\dots,P_s)$ for $(P_1,\dots,P_s)\in{\mathbb Z}_+^s$. This generating function therefore enjoys the topological recursion, and we prove that it is simultaneously the generating function for ancestor invariants of a cohomological field theory thus enjoying the Givental decomposition. We use another Givental-type decomposition obtained for this model by the second authors in 1995 in terms of special times related to the discretisation of moduli spaces thus representing its asymptotic expansion terms (and therefore those of the Gaussian means) as finite sums over graphs weighted by lower-order monomials in times thus giving another proof of (quasi)polynomiality of the discrete volumes. As an application, we find the coefficients in the first subleading order for ${\mathcal M}_{g,1}$ in two ways: using the refined Harer--Zagier recursion and by exploiting the above Givental-type transformation. We put forward the conjecture that the above graph expansions can be used for probing the reduction structure of the Delgne--Mumford compactification $\overline{\mathcal M}_{g,s}$ of moduli spaces of punctured Riemann surfaces.Comment: 36 pages in LaTex, 6 LaTex figure

A Numerical Approach to Virasoro Blocks and the Information Paradox

We chart the breakdown of semiclassical gravity by analyzing the Virasoro conformal blocks to high numerical precision, focusing on the heavy-light limit corresponding to a light probe propagating in a BTZ black hole background. In the Lorentzian regime, we find empirically that the initial exponential time-dependence of the blocks transitions to a universal $t^{-\frac{3}{2}}$ power-law decay. For the vacuum block the transition occurs at $t \approx \frac{\pi c}{6 h_L}$, confirming analytic predictions. In the Euclidean regime, due to Stokes phenomena the naive semiclassical approximation fails completely in a finite region enclosing the `forbidden singularities'. We emphasize that limitations on the reconstruction of a local bulk should ultimately stem from distinctions between semiclassical and exact correlators.Comment: 45 pages, 23 figure