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Beta function of k deformed AdS5 × S 5 string theory
HJE
AdS5
Calan Appadu 0 1
Timothy J. Hollowood 0 1
Swansea 0 1
0 mal Field Models in String Theory, Renormalization Group
1 Department of Physics, Swansea University
We calculate the one loop beta function for the would-be marginal coupling on the world sheet of the k deformed sigma models associated to a quantum group with q = ei =k. This includes the bosonic principal chiral models and symmetric space sigma models but also the k deformed semi-symmetric space sigma model describing strings in a deformation of AdS5 S5. The world sheet sigma model is a current-current deformation of the gauged WZW model for the supergroup PSU(2; 2j4) with level k. In the string theory context the beta function is shown to vanish because of the vanishing of the Killing form of PSU(2; 2j4) which is another piece of evidence that the k deformed theories de ne consistent string theories.
Integrable Equations in Physics; Superstrings and Heterotic Strings; Confor-
-
5
1 Introduction
2 Integrable sigma models
3 k-deformed sigma models
Principal chiral models Symmetric space sigma models k deformed PCM k deformed SSSM
The k deformation Beta function
4
5
2.1
2.2
3.1
3.2
4.1
4.2
Discussion
A Conventions
1
Introduction
The semi-symmetric space sigma models
It is an important question to ask how the gauge-gravity correspondence can be deformed
whilst maintaining some of its special features. One such special feature is its integrability
(reviewed in [1, 2]) and one can ask what kind of deformations of the classic AdS5
background preserve the integrability of the world sheet theory.
The world sheet theory in the Green-Schwarz formalism for strings on AdS5
written down by Metsaev and Tseytlin [3] is a generalized sigma model with a target that
is the semi-symmetric space [4] and can be described as the quotient of a Lie supergroup
by a bosonic subgroup:
scattering theory [6, 7]. The original motivation for considering this kind of deformation
was to understand whether the gauge xed world sheet theory could be re-formulated in a
manifestly relativistic way, as suggested by writing the equations of motion in relativistic
form using the Pohlmeyer reduction [8, 9]. The answer is that there is a relativistic world
sheet scattering theory, but this represents a deformation of the original string theory
rather than a re-formulation of it [6, 7].
There are two classes of these deformations depending on whether q is real, in which
case q = e
(at the classical level) known as the \
deformation", or q = ei =k for an
integer k, which we call the \k deformation".1 The two deformation are very di erent in
character. The former, investigated in [10{17] can be interpreted as a direct target space
deformation while the latter is more subtle being a discrete deformation. In the latter case,
the S-matrix is built from the R-matrix in the IRF/RSOS formulation. The k-deformation
has been investigated in [15, 16, 18{25].
The big question is whether the
or k deformations describe consistent string theories.
For the k deformation the evidence is as following:
1. As described above, the string world sheet theory has a consistent S-matrix [6, 7, 26]
that describes the scattering of a nite number of states which form the starting point
for the TBA [18, 19].
symmetries.
2. It is conjectured that the world sheet theory has a consistent Green Schwarz
Lagrangian formulation as a deformation of a gauged WZW theory associated to the
supergroup PSU(2; 2j4) [20]. This formulation has the requisite number of kappa
3. The target space geometry satis es the generalized Einstein equations when a suitable
ansatz is made for the RR
ux and dilaton based on the bosonic truncation of the
Green-Schwarz sigma model mentioned above [22].2
It is the purpose of the present work to add to this body of evidence by showing
that the continuous coupling of the world sheet theory is a marginal coupling in the one
loop approximation. This is a world sheet calculation that complements the target space
calculation listed above and provides additional evidence in favour of the Green-Schwarz
Lagrangian formulation.
This paper is organized as follows. In section 2, we lay the ground work for later
sections by calculating the beta functions of the integrable sigma models. We use the
1These latter deformations are also called \
deformations" where
is de ned in section 3. But in
the string theory context,
is not the deformation parameter that takes us away from the AdS5
2It is not entirely clear whether the solution actually corresponds to the world sheet action. This is
discussed [23].
{ 2 {
background eld method and choose a particularly simple kind of background eld. The
novelty of our approach is that we calculate the uctuations around the background eld
in terms of the currents of the theories rather than their fundamental eld directly. In
section 3 we turn to the k deformed versions of the integrable sigma models. After showing
how they have the same equations of (...truncated)