All (Poincaré + gauge)-symmetry-preserving lattice theories can be constructed using a discrete derivative and its inverse (lattice integral) that seem to satisfy the same calculi of the continuum. In this sense the discrete operations are exact for which there are certain preliminary evidences. Noting that the mathematical tool needed to construct a classical field theory in the continuum is that of the Schwartz class functions and tempered distributions, we embark on attacking the full-fledged problem at hand, i.e. to develop the tempered distribution theory on the lattice. Some of the key results our construction leads to are as follows: (1) Exact operations satisfy exactly the same calculi of the continuum. Kronecker delta satisfies the same continuum distributional identities of the Dirac delta on the lattice with respect to the exact derivative. This allows one to prove Noether's theorems and derive symmetry algebras for the lattice theories. (2) Lattice configuration space (Schwartz space) corresponds to a subspace of the continuum configuration space that carries lattice cutoff as a parameter such that: (A) Lattice path integral is same as the continuum path integral over this subspace. By construction, the latter is UV divergence free. This must be true non-perturbatively. (B) The subspace is closed under Poincaré and gauge transformations. This is the root cause for the symmetries being preserved on the lattice. (C) Preliminary observations indicate that this may be a dense subspace.

We generalize recent results in two-dimensional Jackiw-Teitelboim gravity  to study factorization of the Hilbert space of eternal black holes in quantum gravity with a negative cosmological constant in any dimension. We approach the problem by computing the  trace  of two-sided observables as a sum over a recently constructed family of semiclassically well-controlled black hole microstates. These microstates, which contain heavy matter shells behind the horizon and form an overcomplete basis of the  Hilbert space, exist in any theory of gravity with general relativity as its low energy limit. Using this representation of the microstates, we show that the  trace of operators dual to functions of the Hamiltonians of the  left and right holographic CFTs factorizes into a product over left and right factors to leading order in the semiclassical limit.  Under certain conditions this implies factorization of the Hilbert space.

I describe a new exact correspondence between correlation functions in the double scaled SYK model and Schur indices in 4D N=2 Seiberg-Witten theory. The operator algebra on both sides of the duality is isomorphic to the algebra of Wilson lines in SL(2,C) Chern-Simons theory on the punctured sphere. I comment on the relation with the complex Liouville string and on the potential application of these results to low dimensional de Sitter holography.

(1) Shreya Vardhan: Estimating time in quantum chaotic systems and black holes

Abstract: A time-evolved state in a unitary chaotic quantum many-body system macroscopically resembles a thermal density matrix. However, while a thermal density matrix does not evolve with time, a pure state undergoing unitary evolution must continue to evolve with time unless it is an energy eigenstate. We sharpen this difference by considering an information-theoretic task where we attempt to estimate the time for which the state has been evolved by making measurements on the state. We quantify the effectiveness of the time estimate using a quantity from quantum metrology called the quantum Fisher information (QFI). This quantity shows an interesting interplay between expectations from thermalization and constraints from unitarity. In the context of evaporating black holes, these results imply that the ability to make local time estimates suddenly improves after the Page time. 

(2) Herman Verlinde: Double Scaled SYK Model and de Sitter Holography

(3) Philip Argyres: Light ray operators and lightlike conformal defects