String field theory vertex from integrability

Published for SISSA by Springer Received: February 17, 2015 Accepted: March 10, 2015 Published: April 9, 2015 Zoltan Bajnoka and Romuald A. Janikb a MTA Lendület Holographic QFT Group, Wigner Research Centre, H-1525 Budapest 114, P.O.B. 49, Hungary b Institute of Physics, Jagiellonian University, ul. Lojasiewicza 11, 30-348 Kraków, Poland E-mail: , Abstract: We propose a framework for computing the (light cone) string field theory vertex in the case when the string worldsheet QFT is a generic integrable theory. The prime example and ultimate goal would be the AdS5 × S 5 superstring theory cubic string vertex and the chief application will be to use this framework as a formulation for N = 4 SYM theory OPE coefficients valid at any coupling up to wrapping corrections. In this paper we propose integrability axioms for the vertex, illustrate them on the example of the pp-wave string field theory and also uncover similar structures in weak coupling computations of OPE coefficients. Keywords: AdS-CFT Correspondence, Integrable Field Theories, String Field Theory ArXiv ePrint: 1501.04533 Open Access, c The Authors. Article funded by SCOAP3 . doi:10.1007/JHEP04(2015)042 JHEP04(2015)042 String field theory vertex from integrability Contents 1 2 Insight from the spectral problem and form factors 4 3 The pp-wave light cone string field theory vertex 8 4 The decompactified string vertex and the SFT axioms 4.1 The decompactified SFT axioms 13 13 5 The free massive boson example (or the pp-wave SFT vertex) 5.1 A review of LSNS formulas 5.2 The decompactification limit of the LSNS formulas and their analyticity properties 5.3 Asymptotic limit 5.4 Reconstruction of Γ̃µ (θ) from the SFT axioms 18 18 6 Axioms for the nondiagonal case 25 7 The program for the finite volume string vertex 7.1 The program for the simplest plane wave SFT vertex 29 31 8 Weak coupling cross-checks with OPE coefficients 8.1 The su(1|1) sector 8.2 The su(2) sector 32 33 34 9 Conclusions 35 A The decompactified vertex formulation and solution 37 B Properties of the Γ̃µ (θ) functions 41 C Details on the su(1|1) and su(2) OPE coefficients C.1 The su(1|1) sector C.2 The su(2) sector 42 42 44 1 20 22 24 Introduction The integrability properties of string theory in AdS5 × S 5 background [1] together with the AdS/CFT correspondence [2] allows for obtaining exact results for various observables in N = 4 Super-Yang-Mills (SYM) theory for any value of the gauge theory coupling in the planar, large Nc limit. Currently this program is very well developed for the spectral –1– JHEP04(2015)042 1 Introduction –2– JHEP04(2015)042 problem, namely for the determination of the scaling dimensions of all local operators [3]– [8]. For other observables we have currently only partial results like various strong and weak coupling expansions or exact answers but restricted to some particular concrete observables like generalized cusp Wilson loops, circular loops or for some ingredients of scattering amplitudes. A class of observables for which it would be crucial to obtain a similar level of understanding as for the scaling dimensions are the OPE coefficients or, equivalently, the 3-point correlation functions of local operators. Namely, these quantities provide the remaining fundamental data for any conformal field theory (CFT). Indeed, higher point functions do not carry any independent dynamical content and can be reduced to scaling dimensions, OPE coefficients and conformal blocks determined by conformal symmetry alone. On the string side of the AdS/CFT correspondence these quantities are also interesting for their own sake, namely the AdS/CFT string diagram corresponding to a 3-point function can be interpreted as a three string interaction. In fact, the first wave of interest in OPE coefficients of (unprotected) operators in N = 4 SYM theory [9]–[12] came from the proposed link with the 3-string string field theory vertex in the pp-wave [13] string field theory (SFT) [14]–[19]. The SFT vertex is also interesting as it is related to the first 1/Nc corrections to the string hamiltonian/scaling dimensions, too. Unfortunately, there is practically no information on generalizing the pp-wave SFT to the full AdS5 × S 5 case. This is not an issue of technical or calculational complexity but rather a more fundamental one. A unique feature of the pp-wave geometry is that, although it is curved, the worldsheet quantum field theory of the string in an appropriate light cone gauge reduces to free massive bosons and fermions [20], thus allowing for the use of mode expansions in implementing continuity conditions for the SFT (light cone) vertex [14] similarly as for the flat space SFT vertex [21]. For an interacting worldsheet QFT, as is the case for the full AdS5 × S 5 geometry, we do not have any techniques so far for finding the SFT vertex. Thus the main goal of the present paper is to provide a new formulation for the problem of determining the (light cone) SFT vertex in the case when the worldsheet theory is a generic integrable QFT, which includes as a key special case the AdS5 × S 5 background. We propose an integrable bootstrap formulation of the SFT vertex, namely a set of coupled functional equations for the SFT amplitudes understood as the value of the vertex with specific string excited states on each of the three legs. The dependence on the concrete background/worldsheet QFT enters through the appearance of the S-matrix in the SFT vertex axioms. This formulation should be valid up to exponential ‘wrapping corrections’. The bootstrap approach for obtaining various physical quantities in two dimensional integrable quantum field theories has already a long and successful history. Basically it amounts to implementing very general functional and analyticity properties of the various observables and using in addition key properties of integrability like factorized scattering etc. Initially, the bootstrap program was developed for determining the scattering amplitudes (and at the same time the particle content, hence the name bootstrap) for a theory on a two-dimensional plane [22]–[24]. The result is the explicit knowledge of the 2-particle scattering S-matrix and the mass spectrum of the theory, e.g. the masses of bound states 2 1 3 in terms of the masses of the fundamental particles. Subsequently this information was used to obtain the spectrum of such a theory on a cylinder of finite size [25, 26]. Since then, the bootstrap program was extended to cover theories with integrable boundary conditions [27], providing exact formulas for reflection factors; as well as for theories with integrable defects [28]. A whole new field of research started when bootstrap was applied to more fine-grained, and in a certain sense off-shell observables such as form factors [29]–[31]. Here, in contrast to ordinary scattering amplitudes the number of incoming and outgoing particles does not need to be bala (...truncated)