Modular properties of characters of the W3 algebra

Iles, Nicholas J.; Watts, Gérard

doi:10.1007/jhep01(2016)089

Cited by 19 publications

(19 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The expectation values of these higher spin charges were studied in[40,42,43]. The modular transformations were studied in[34], which obtains results in line with the expectations of[32], as noted in the next subsection.…”

mentioning

confidence: 73%

Thermal correlation functions of KdV charges in 2D CFT

Maloney

Ross

et al. 2019

J. High Energ. Phys.

View full text Add to dashboard Cite

Two dimensional CFTs have an infinite set of commuting conserved charges, known as the quantum KdV charges, built out of the stress tensor. We compute the thermal correlation functions of the these KdV charges on a circle. We show that these correlation functions are given by quasi-modular differential operators acting on the torus partition function. We determine their modular transformation properties, give explicit expressions in a number of cases, and give a general form which determines an arbitrary correlation function up to a finite number of functions of the central charge. We show that these modular differential operators annihilate the characters of the (2m + 1, 2) family of non-unitary minimal models. We also show that the distribution of KdV charges becomes sharply peaked at large level. Introduction and summary of resultsTwo dimensional conformal field theories (CFTs) have an infinite-dimensional symmetry algebra -the Virasoro algebra of local conformal transformations -which powerfully constrains the structure of the theory. Our goal is to explore the consequences of this symmetry structure for the finite-temperature behaviour of the theory. We will study two dimensional CFTs at finite temperature on the circle. Our interest in this paper is obtaining exact results: we will keep the radius of the circle finite, and will not study the high temperature or large c limits, though these will be discussed in a companion paper [1].Our starting point is the observation that, from the Virasoro algebra, one can define an infinite set of mutually commuting conserved charges [2-4]. In the large c limit these charges generate the KdV hierarchy of differential equations, so they are often referred to as the quantum KdV charges. We will follow the notation of [4] where the charges are denoted I 2m−1 with m = 1, 2, . . . , and the subscript labels the spin of the charge. The first KdV charge I 1 is just L 0 = T (z)dz, the zero mode of the holomorphic part of the stress tensor. The KdV charge I 2m−1 is an m th order polynomial in the Virasoro generators L n .Given this set of mutually commuting charges, one can define a Generalized Gibbs Ensemble (GGE) for two-dimensional CFTs where we introduce a chemical potential for each KdV charge: 1 Z[β, µ 3 , µ 5 , . . .] = Tr [e −βE+µ 3 I 3 +µ 5 I 5 +... ] = Tr [e µ 3 I 3 +µ 5 I 5 +... q L 0 −c/24 ], q ≡ e −β , (1.1)In this equation we have introduced chemical potentials for the left-moving (holomorphic) generators L 0 , I 3 , I 5 , . . . . In general, of course, one should also introduce potentials for the right-moving (anti-holomorphic) generatorsL 0 ,Ī 3 ,Ī 5 , . . . . However, all of our results involve the use just of the Virasoro algebra so the left-and right-moving sectors completely factorize.1 See [5-9] for some recent work on GGEs and [10,11] for recent experimental realisations. This ensemble was studied from a holographic perspective in [12]; see also [13].These KdV charges act within each Virasoro module, so can be simultaneously diagonalized level by level with...

show abstract

mentioning

confidence: 73%

Thermal correlation functions of KdV charges in 2D CFT

Maloney

Ross

et al. 2019

J. High Energ. Phys.

View full text Add to dashboard Cite

show abstract

“…(24) and (25) agree with, and generalize, the results of ref. [39]. We conclude that, up to quadratic order in µ, the modular transformation of the partition function is given by…”

Section: Partition Function and Modular Transformationsmentioning

confidence: 70%

“…Once the chemical potential is turned on, the partition function is no longer modular invariant. While a closed-form expression for the transformed partition function is not known, it is possible to determine this transformation perturbatively around µ = 0 [38,39]. In particular, note that the constraints given in eq.…”

Section: Partition Function and Modular Transformationsmentioning

confidence: 99%

“…Following refs. [39,42], 6 The transformation (20) is also featured in ref. [41] in the context of generalized Jacobi Theta functions.…”

Section: Partition Function and Modular Transformationsmentioning

confidence: 99%

“…Following ref. [39], we first find a composite primary field of dimension 6 whose zero mode contains W 2 0 τ,τ . This field is given by…”

Section: Acknowledgmentsmentioning

confidence: 99%

See 2 more Smart Citations

Bounds on CFTs with W3 algebras and AdS3 higher spin theories

Apolo

2017

Phys. Rev. D

View full text Add to dashboard Cite

The scaling dimension of the first excited state in two-dimensional conformal field theories (CFTs) satisfies a universal upper bound. Using the modular bootstrap, we extend this result to CFTs with W3 algebras which are generically dual to higher spin theories in AdS3. Assuming unitarity and modular invariance, we show that the conformal weights h,h of the lightest charged state satisfy h < c/12 + O(1) andh 4|h − c/24|/ √ 10πc + O(1). We discuss hints on the existence of stronger bounds and comment on the interpretation of our results in the dual higher spin theory.

show abstract

General results for higher spin Wilson lines and entanglement in Vasiliev theory

Hegde

Kraus

Perlmutter

2016

J. High Energ. Phys.

View full text Add to dashboard Cite

We develop tools for the efficient evaluation of Wilson lines in 3D higher spin gravity, and use these to compute entanglement entropy in the hs [λ] Vasiliev theory that governs the bulk side of the duality proposal of Gaberdiel and Gopakumar. Our main technical advance is the determination of SL(N) Wilson lines for arbitrary N , which, in suitable cases, enables us to analytically continue to hs [λ] via N → −λ. We apply this result to compute various quantities of interest, including entanglement entropy expanded perturbatively in the background higher spin charge, chemical potential, and interval size. This includes a computation of entanglement entropy in the higher spin black hole of the Vasiliev theory. These results are consistent with conformal field theory calculations. We also provide an alternative derivation of the Wilson line, by showing how it arises naturally from earlier work on scalar correlators in higher spin theory. The general picture that emerges is consistent with the statement that the SL(N) Wilson line computes the semiclassical W N vacuum block, and our results provide an explicit result for this object.

show abstract

Modular properties of characters of the W3 algebra

Cited by 19 publications

References 15 publications

Thermal correlation functions of KdV charges in 2D CFT

Thermal correlation functions of KdV charges in 2D CFT

Bounds on CFTs with W3 algebras and AdS3 higher spin theories

General results for higher spin Wilson lines and entanglement in Vasiliev theory

Contact Info

Product

Resources

About