The kinematical AdS5 × S5 Neumann coefficient
Published for SISSA by
Springer
Received: January 7, 2016
Accepted: January 26, 2016
Published: February 22, 2016
Zoltan Bajnoka and Romuald A. Janikb
a
MTA Lendület Holographic QFT Group, Wigner Research Centre,
P.O.B. 49, Budapest 114, H-1525 Hungary
b
Institute of Physics, Jagiellonian University,
ul. Lojasiewicza 11, Kraków, 30-348 Poland
E-mail: ,
Abstract: For the case of two particles a solution of the string field theory vertex axioms
can be factorized into a standard form factor and a kinematical piece which includes the
dependence on the size of the third string. In this paper we construct an exact solution of
the kinematical axioms for AdS5 × S 5 which includes all order wrapping corrections w.r.t.
the size of the third string. This solution is expressed in terms of elliptic Gamma functions
and ordinary elliptic functions. The solution is valid at any coupling and we analyze its
weak coupling, pp-wave and large L limit.
Keywords: AdS-CFT Correspondence, Integrable Field Theories, String Field Theory
ArXiv ePrint: 1512.01471
Open Access, c The Authors.
Article funded by SCOAP3 .
doi:10.1007/JHEP02(2016)138
JHEP02(2016)138
The kinematical AdS5 × S5 Neumann coefficient
Contents
1
2 String field theory vertex axioms
3
3 The pp-wave Neumann coefficient
5
4 Interacting relativistic integrable QFT’s
7
5 The AdS5 × S 5 elliptic curve
8
6 Functional equations on the AdS torus
6.1 Elliptic Gamma function and the monodromy condition
10
13
7 The kinematical AdS5 × S 5 Neumann coefficient
7.1 Singularity structure
7.2 The pp-wave limit
14
15
16
8 Weak coupling limit
8.1 Decompactifed spin chain calculation
17
19
9 The large L limit
20
10 Conclusions
21
1
Introduction
Recently there has been significant progress in our understanding of string interactions
for string theories in curved backgrounds which exhibit integrability. In our previous
paper [1] we formulated a set of functional equations for the (light-cone) String Field
Theory (SFT) three-string vertex for the case when the worldsheet theory is integrable.
The axioms per-se apply to the case when two of the strings are large (more precisely they
are decompactified) while the third string can be of an arbitrary finite size L. The axioms
depend in a nontrivial way on the size L. The decompactification limit corresponds to
cutting the string pants diagram (see figure 1) along one edge. Since the third string has a
finite size, the decompactification limit includes arbitrary number of wrapping corrections
w.r.t. L. This can be explicitly seen in the case of the pp-wave background geometry where
we have at our disposal an exact explicit solution for any value of L. Unfortunately we
do not have, for the moment, a solution in the most interesting case of the AdS5 × S 5
geometry. This paper is a step in that direction.
–1–
JHEP02(2016)138
1 Introduction
Figure 1. The SFT vertex and its decmpactified version.
–2–
JHEP02(2016)138
In [2] a different approach was developed explicitly geared towards the computation
of OPE coefficients in N = 4 SYM. Here the string vertex was cut along three edges into
two hexagons. This corresponds to the decompactification limit of all three strings. In
this context, functional equations for the hexagon in AdS5 × S 5 have been solved exactly.
The passage to finite volume incorporating wrapping effects involves, however, an iterative
prescription for gluing the hexagons together through integrating over an arbitrary number
of particles on the edges being glued. Thus wrapping effects are build on iteratively.
Recently there appeared some further nontrivial checks of this proposal [3, 4] and it was
even related [5] in the HHL (L = 0) case to diagonal finite volume form factors. This is
the structure which was conjectured in [6] and checked at weak coupling in [7].
In contrast, the finite L solution of the SFT vertex axioms should at once resum an
infinite set of wrapping corrections and thus should provide some helpful information for
the hexagon gluing procedure.
In this paper we would like to find the simplest possible solutions of the SFT vertex
axioms concentrating on exactly treating the L dependence. Of course any solution is given
up to some analogs of CDD factors which a-priori can also be L dependent (although the
equations that they satisfy do not contain L). So what we are aiming at is providing a
‘minimal’ L dependent solution. It will then remain an important problem whether this
solution is physical or whether it has to be suplemented by some additional CDD-like factors. A similar question will arise for solutions for relativistic interacting integrable QFT’s
(e.g. sinh-Gordon or the O(N) model on the decompactified pants diagram), which we will
briefly also mention. It would be very interesting to cross-check these simplest relativistic solutions in some other way and to understand whether in that case any additional
CDD-like factors are in fact necessary. This would be important for our understanding
of the required analytical structure. Perhaps some integrable lattice realizations of these
integrable relativistic QFT’s might shed light on these issues.
The plan of this paper is as follows. In section 2 we will briefly review the String
Field Theory vertex axioms proposed in [1] and concentrate on the case of two particles
relevant for the present paper. Then we will review the structure of the pp-wave Neumann
coefficient in section 3 and consider the trivial relativistic solutions for sinh-Gordon and
O(N) in section 4. In the following section we will review the AdS5 × S 5 elliptic curve and
proceed to analyze and solve the relevant functional equations on the AdS5 × S 5 torus.
Finally we will describe the pp-wave, weak coupling and large L limits of the obtained
solutions. We close the paper with a discussion and outlook.
2
String field theory vertex axioms
The universal exponential part of the light cone string field theory vertex both in flat
spacetime and in the pp-wave geometry has the form
3 X
1 X
rs +(r) +(s)
|V i = exp
Nnm
an am
|0i .
(2.1)
2
n,m
r,s=1
+(r)
3|2;1
NL3 |L2 ;L1
θ1 , . . . , θ n
0
θ10 , . . . , θm
; θ100 , . . . , θl00
.
(2.2)
As argued in [1], we will consider the decompactified vertex with the strings #2 and #3
being infinite, and the string #1 being of size L (see figure 1).
3|2;1
N∞|∞;L
θ1 , . . . , θ n
0
θ10 , . . . , θm
; θ100 , . . . , θl00
.
(2.3)
In this case the functional equations will only depend explicitly on the particles in strings
#2 and #3, so we can use a shorthand notation
3|2
N•,L
0
θ10 , . . . , θm
θ1 , . . . , θ n
(2.4)
where the • stands for a specific state on string #1: • ≡ {θ100 , . . . , θl00 }.
In this paper we will restrict ourselves to amplitudes with just two particles. In analogy
to the Minkowski and pp-wave case we will use the term Neumann coefficients for them.
Without loss of generality we can take the two particles to be in the incoming (...truncated)
