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arXiv:2101.10211 · Journal of High Energy Physics

Virasoro algebras, kinematic space and the spectrum of modular Hamiltonians in CFT2

Suchetan Das, Bobby Ezhuthachan, Somnath Porey, Baishali Roy

Published Aug 23, 2021• 1 citation
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Abstract

Abstract We construct an infinite class of eigenmodes with integer eigenvalues for the Vacuum Modular Hamiltonian of a single interval N in 2d CFT and study some of its interesting properties, which includes its action on OPE blocks as well as its bulk duals. Our analysis suggests that these eigenmodes, like the OPE blocks have a natural description on the so called kinematic space of CFT2 and in particular realize the Virasoro algebra of the theory on this kinematic space. Taken together, our results hints at the possibility of an effective description of the CFT2 in the kinematic space language.

DOI
10.1007/jhep08(2021)123

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Abstract We construct an infinite class of eigenmodes with integer eigenvalues for the Vacuum Modular Hamiltonian of a single interval N in 2d CFT and study some of its interesting properties, which includes its action on OPE blocks as well as its bulk duals. Our analysis suggests that these eigenmodes, like the OPE blocks have a natural description on the so called kinematic space of CFT2 and in particular realize the Virasoro algebra of the theory on this kinematic space. Taken together, our results hints at the possibility of an effective description of the CFT2 in the kinematic space language.
We construct an infinite class of eigen modes with integer eigenvalues for the Vacuum Modular Hamiltonian of a single interval in 2d CFT. An interesting feature of this construction is that these eigen modes together with the Modular Hamiltonian form a Virasoro algebra. The construction is geometric in the sense that it is intimately related to the partition of the interval (NNN) into subintervals (N′N'N′) and (N′′N''N′′). In particular, we show that given a set consisting of a causal diamond associated to an interval (NNN) and its sub-intervals (N′N'N′, N′′N''N′′) on a constant time slice in a 2d CFT, one can construct modular eigen modes, satisfying the Virasoro algebra, which we call the 'modular Virasoro algebra'. For (n=±1,0n=\pm 1, 0n=±1,0), the corresponding generators are related to the modular Hamiltonians of the interval and its sub intervals. Another interesting feature of our construction is that OPE blocks associated with the interval NNN transform as the modes of the primary highest weight field under transformations generated by these 'modular Virasoro generators'. We end with some comments on the bulk dual of the modular Virasoro algebra.
Abstract We study the parallel transport of modular Hamiltonians encoding entanglement properties of a state. In the case of 2d CFT, we consider a change of state through action with a suitable diffeomorphism on the circle: one that diagonalizes the adjoint action of the modular Hamiltonian. These vector fields exhibit kinks at the interval boundary, thus together with their central extension they differ from usual elements of the Virasoro algebra. The Berry curvature associated to state-changing parallel transport is the Kirillov-Kostant symplectic form on an associated coadjoint orbit, one which differs appreciably from known Virasoro orbits. We find that the boundary parallel transport process computes a bulk symplectic form for a Euclidean geometry obtained from the backreaction of a cosmic brane, with Dirichlet boundary conditions at the location of the brane. We propose that this gives a reasonable definition for the symplectic form on an entanglement wedge.
We generalize the Quantum Geometric Tensor by replacing a Hamiltonian with a modular Hamiltonian. The symmetric part of the Quantum Geometric Tensor provides a Fubini-Study metric, and its anti-symmetric sector gives a Berry curvature. Our generalization dubbed Quantum Modular Geometric Tensor gives the metric and curvature of a Kinematic Space. We also use the result of the identity Virasoro block to relate the connected correlator of two Wilson lines to the two-point function of a modular Hamiltonian. This relation realizes a novel holographic entanglement formula for two intervals of a general separation. This result also provides a new interpretation to the connected correlators as the mutual information. Our study remains valid at the quantum level. Hence it provides an opportunity to explore the Kinematic Space in a generic case without conformal symmetry.
Abstract Virasoro conformal blocks are universal ingredients of correlation functions of two-dimensional conformal field theories (2d CFTs) with Virasoro symmetry. It is acknowledged that in the (classical) limit of large central charge of the Virasoro algebra and large external, and intermediate conformal weights with fixed ratios of these parameters Virasoro blocks exponentiate to functions known as Zamolodchikovs’ classical blocks. The latter are special functions which have awesome mathematical and physical applications. Uniformization, monodromy problems, black holes physics, quantum gravity, entanglement, quantum chaos, holography, N \mathcal{N} N N = 2 gauge theory and quantum integrable systems (QIS) are just some of contexts, where classical Virasoro blocks are in use. In this paper, exploiting known connections between power series and integral representations of (quantum) Virasoro blocks, we propose new finite closed formulae for certain multi-point classical Virasoro blocks on the sphere. Indeed, combining classical limit of Virasoro blocks expansions with a saddle point asymptotics of Dotsenko-Fateev (DF) integrals one can relate classical Virasoro blocks with a critical value of the “Dotsenko-Fateev matrix model action”. The latter is the “DF action” evaluated on a solution of saddle point equations which take the form of Bethe equations for certain QIS (Gaudin spin models). A link with integrable models is our main motivation for this research line. For instance, analogous quantities as the “DF on-shell action” appear in 2d CFT realization of the so-called Richardson’s solution of the reduced BCS model describing physics of ultra-small superconducting grains. Precisely, the Richardson solution and its particular limit characterizing the rational Gaudin model have known implementation in 2d CFT in terms of a free field representation of certain (perturbed and unperturbed) WZW blocks. The WZW analogue of the “DF on-shell action” appears here and plays a crucial role, it is a generating function for eigenvalues of quantum integrals of motion. An exploration of relationships between power expansions and Coulomb gas representations of the aforementioned WZW blocks could pave a way for new analytical tools useful in the study of models of the Richardson-Gaudin type.
Abstract We derive a universal asymptotic formula for generic boundary conditions for the average value of the bulk-to-boundary and boundary Operator Product Expansion coefficients of any unitary, compact two-dimensional Boundary CFT (BCFT) with c > 1. The asymptotic limit consists of taking one or more boundary primary operators — which transform under a single copy of the Virasoro algebra — to have parametrically large conformal dimension for fixed central charge. In particular, we find a single universal expression that interpolates between distinct heavy regimes, exactly as in the case of bulk OPE asymptotics [1]. The expression depends universally on the boundary entropy and the central charge, and not on any other details of the theory. We derive these asymptotics by studying crossing symmetry of various correlation functions on higher genus Riemann surfaces with open boundaries. Essential in the derivation is the use of the irrational versions of the crossing kernels that relate holomorphic Virasoro blocks in different channels. Our results strongly suggest an extended version of the Eigenstate Thermalization Hypothesis for boundary OPE coefficients, where the hierarchy between the diagonal and non-diagonal term in the ansatz is further controlled by the boundary entropy. We finally comment on the applications of our results in the context of AdS3/BCFT2, as well as on the recent relation of BCFTs with lower dimensional models of evaporating black holes.