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Virasoro algebra from modular Hamiltonians

Suchetan Das, Bobby Ezhuthachan, Somnath Porey, Baishali Roy

Published Jan 25, 2021• 0 citations
OpenAlex

Abstract

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 ($N$) into subintervals ($N'$) and ($N''$). In particular, we show that given a set consisting of a causal diamond associated to an interval ($N$) and its sub-intervals ($N'$, $N''$) 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 ($n=\pm 1, 0$), 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 $N$ 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.

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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 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.
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.
In this article, we explore the connection between the heating phase of periodically driven CFTs and the Modular Hamiltonian of a subregion in the vacuum state. We show that the heating phase Hamiltonian corresponds to the Modular Hamiltonian, with the fixed points mapping to the endpoints of the subregion. In the bulk dual, we find that these fixed points correspond to the Ryu-Takayanagi surface of the AdS-Rindler wedge. Consequently, the entanglement entropy associated to the boundary interval within two fixed points exactly matches with the Rindler entropy of AdS-Rindler. We observe the emergent Virasoro algebra in the boundary quantization of the Modular Hamiltonian has a striking similarity with the emergent near Horizon Virasoro algebra. This is a consequence of the fact that while obtaining the boundary Virasoro algebra, a cut-off with conformal boundary condition around the fixed point is introduced, which in the bulk is related to a stretched horizon, with an emergent two-dimensional conformal symmetry. We also argue that as one tunes the parameter space of Floquet Hamiltonians to transition from the non-heating to the heating phase the operator algebra type changes from Von Neumann type III to III1III_1III1​ factor, providing a non-equilibrium analogue of the Hawking-Page transition.
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.