Integrable spin chain for stringy Wess-Zumino-Witten models

Abstract

Building on arXiv:1804.01998 we investigate the integrable structure of the Wess-Zumino-Witten (WZW) model describing closed strings on AdS 3 × S 3 × T 4 . Using the recently-proposed integrable S matrix we show analytically that all wrapping corrections cancel and that the theory has a natural spin-chain interpretation. We construct the integrable spin chain and discuss its relation with the WZW description. Finally we compute the spin-chain spectrum in closed form and show that it matches the WZW prediction on the nose. arXiv:1806.00422v2 [hep-th] The correspondence between gravity on AdS 3 and conformal field theory in two dimensions (CFT 2 ) is a key example of holographic duality [1,2]. From the seminal work of Brown and Henneaux [3] to the remarkable developments in string theory [4], see e.g. ref.[5] for a review, the AdS 3 /CFT 2 correspondence stands out for the possibility of performing exact computations by conformal field theory techniques. In string theory such techniques arise in two distinct ways. Firstly, the dual conformal field theory is two-dimensional and, at weak string tension, is described by an almost-free CFT-more specifically, by the symmetric-product orbifold of a free CFT. Secondly, there exist superstring backgrounds that are supported by Neveu-Schwarz-Neveu-Schwarz (NS-NS) fluxes only. These can be described by a CFT 2 on the worldsheet, without the usual complications due to Ramond-Ramond (R-R) fluxes [6]. Such a CFT 2 involves an sl(2, R) Wess-Zumino-Witten model describing (the chiral part of) AdS 3 , which can be studied in detail following Maldacena and Ooguri [7], see also refs. [8][9][10][11][12].A more recent development is that strings on maximally supersymmetric AdS 3 backgrounds are classically integrable [13][14][15], and indeed this integrability seems to carry over to the quantum theory when this is constructed in light-cone gauge, see ref.[16] for a review (see also refs. [17, 18] for broader reviews of AdS/CFT integrability). More specifically, an exact worldsheet scattering matrix was constructed for strings on R-R backgrounds [19][20][21] and the relevant dressing factors were proposed [22][23][24]. Quite remarkably, even backgrounds supported by a mixture of R-R and NS-NS fluxes are classically integrable [15] and their S matrix can similarly be determined [25][26][27], though no proposal exists for the dressing factors yet. Curiously, until recently no proposal for the worldsheet S matrix at the pure NS-NS point existed-despite the fact that, at least in the RNS formalism, the resulting theory is substantially simpler than a generic mixed-flux one. Technically this is due to the fact that the light-cone symmetry algebra contracts at the NS-NS point. 1 Recently, in ref.[31], an exact integrable worldsheet S matrix for strings on AdS 3 × S 3 × T 4 with pure-NS-NS fluxes was proposed by a different approach, based on TT deformations [32][33][34][35]. This S matrix, including its dressing factor, is much simpler than its mixed-flux [27] (or even pure-R-R [36]) count...

Non-linear supersymmetry and $$ T\overline{T} $$-like flows

“…10 The model analyzed in [21] is the analogue of the D = 4 model of [64,65] and related on-shell to the Goldstino model of [66]. 11 This section is devoted to showing that these D = 4 N = 1 Goldstino models satisfy a supercurrent-squared flow driven by the operator O T 2 of the supersymmetric BI, in agreement with the arguments given above.…”

“…The alternative representation of the Goldstino action that interests us was originally introduced by Casalbuoni et al in [64], and later rediscovered and made fashionable by 10 We refer to [63] for a discussion of various models possessing non-linearly realized (2, 2) supersymmetry. 11 Note that the Goldstino models of [64][65][66] were shown in [59,62] to be identical to the fermionic truncation of the supersymmetric BI action up to a field redefinition of the Goldstino.…”

“…This is a beautiful connection between TT deformations and a field theory which classically possesses a non-linearly realized D = 3 Lorentz symmetry; for other connections between TT and classical string theory, see, for example, [8][9][10][11][12][13][14][15].…”

The plane-wave limit of AdS3×S3×S3×S1

“…This wealth of parameters will allow for a precision-spectroscopy comparison of new proposals for computing the spectrum of strings on AdS 3 backgrounds, see e.g. [45,46]. This paper is organised as follows.…”

$$ T\overline{T} $$ type deformation in the presence of a boundary

“…This is a very important result in the study of the renormalization group flow and this is the reason why it attracted much attention recently [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. In particular, this modification was studied in the context of holography in [19], where it was proposed that the UV deformation is geometrically realized by a cutoff that removes the asymptotic region of AdS 3 space and replaces it by wall at finite distance from the boundary, where a QFT with Dirichlet boundary conditions is defined.…”