From world-sheet supersymmetry to super target spaces
Thomas Creutzig
2
Peter B. Rnne
0
1
0
DESY Theory Group, DESY Hamburg Notkestrasse 85, D-22607 Hamburg,
Germany
1
National Institute for Theoretical Physics and Centre for Theoretical Physics, University of the Witwatersrand
, Wits, 2050,
South Africa
2
Department of Physics and Astronomy, University of North Carolina
, Phillips Hall, CB 3255,
Chapel Hill, NC 27599-3255, U.S.A
We investigate the relation between N = (2, 2) super conformal Lie group WZNW models and Lie super group WZNW models. The B-twist of an exactly marginal perturbation of the world-sheet superconformal sigma model is the supergroup model. Moreover, the superconformal currents are expressed in terms of Lie superalgebra currents in the twisted theory. As applications, we find protected sectors and boundary actions in the supergroup sigma model. A special example is the relation between string theory on AdS3 S3 T4 in the RNS formalism and the U(1, 1|2) U(1|1) U(1|1) supergroup WZNW model.
1 Introduction 2 3 4
Superconformal and supergroup WZNW models
2.1 Supersymmetric WZNW models
2.2 A Sugawara-like construction of the superconformal algebra
2.3 Deformations
2.4 Topological conformal field theory
2.5 Supersymmetric deformations
2.6 Lie supergroup WZNW models
From world-sheet supersymmetry to supergroups
3.1 Some properties of the GL(N|N) WZNW model
3.2 The N = (2, 2) GL(N) GL(N) WZNW model
3.3 The boson-fermion interaction term
3.4 The principal chiral field as a D-term
Examples and applications
4.1 GL(2|2) and screening charges as chiral perturbations
4.2 Comparison to string theory on AdS3 S3 T4
4.3 Boundary actions and the Warner problem 5 Summary and outlook 1
Introduction
In this note we consider a relation between sigma models on bosonic groups with N = (2, 2)
world-sheet supersymmetry and models with supergroups as target spaces via topological
twisting. Our motivation to ask for such a relation and to understand it in detail comes
from two sides.
The first motivation to study the relation comes from boundary theories on
supergroups. For WZNW models on type I Lie supergroups there exists a nice prescription to
compute correlation functions in the bulk theory [14]. The WZNW model is equivalent
to a model consisting of the WZNW model of the bosonic subgroup, free fermions and an
interaction term that couples bosons and fermions. The first observation we make, is that
the action of the model without the interaction term resembles the topological twist of an
N = (2, 2) superconformal field theory.
Now, we would like to have a similar free fermion prescription for the boundary type I
supergroup WZNW model. So far only in the case of GL(1|1) this is known [5].1 There,
in addition to the bulk fermions, one had to introduce an additional fermionic boundary
degree of freedom. Moreover, the boundary screening charge looks like the square root of
the bulk interaction term. These two features are well-known in world-sheet
supersymmetric theories, i.e. in order to preserve N = 2 superconformal symmetry on the boundary
additional boundary fermions plus a factorization of the bulk superpotential into
boundary superpotentials is required [7]. We want to understand why we have such a similar
behaviour. Moreover, we would like to use techniques from world-sheet supersymmetry to
find boundary actions and hence a perturbative description involving free fermions to solve
boundary supergroup WZNW models.
The second hint of the relation came from non-trivial exact checks of the AdS3/CFT2
correspondence [8, 9]. Here correlation functions of chiral primary operators in the weak
coupling limit of string theory on AdS3S3 T4 were calculated, and precise agreement was
found with calculations done in the dual two-dimensional conformal field theory. Such an
agreement is at first sight surprising since the computations in the bulk and on the boundary
correspond to different points in moduli space, and some protection of the correlators must
be present. In [10] the explanation for the boundary side was given. The argument utilizes
that the dual conformal theory has a whole N = (4, 4) worth of supersymmetry. Using this
extended supersymmetry, the correlators, which correspond to an N = (2, 2) chiral ring,
can be shown to be covariantly constant over the total moduli space.
The question is whether we can now explain this from the string theory side, which only
has N = (2, 2) world-sheet supersymmetry, by finding some protected sectors. From [11]
we know that string theory on AdS3 S3 T4 in the hybrid formalism has a description
in terms of the PSU(1, 1|2) supergroup sigma model where RR-deformations correspond
to deformations away from the WZNW point. This lead us to the search for topological
sectors in PSU(1, 1|2). We, however, only found such sectors in U(1, 1|2), and in general
in GL(N |N ). Since a conformal topological sector correspond to the twist of a
worldsheet supersymmetric theory, this suggests a relation between the N = (...truncated)