19 research outputs found
Integrability vs Supersymmetry: Poisson Structures of The Pohlmeyer Reduction
We construct recursively an infinite number of Poisson structures for the
supersymmetric integrable hierarchy governing the Pohlmeyer reduction of
superstring sigma models on the target spaces AdS_{n}\times S^n, n=2,3,5. These
Poisson structures are all non-local and not relativistic except one, which is
the canonical Poisson structure of the semi-symmetric space sine-Gordon model
(SSSSG). We verify that the superposition of the first three Poisson structures
corresponds to the canonical Poisson structure of the reduced sigma model.
Using the recursion relations we construct commuting charges on the reduced
sigma model out of those of the SSSSG model and in the process we explain the
integrable origin of the Zukhovsky map and the twisted inner product used in
the sigma model side. Then, we compute the complete Poisson superalgebra for
the conserved Drinfeld-Sokolov supercharges associated to an exotic kind of
extended non-local rigid 2d supersymmetry recently introduced in the SSSSG
context. The superalgebra has a kink central charge which turns out to be a
generalization to the SSSSG models of the well-known central extensions of the
N=1 sine-Gordon and N=2 complex sine-Gordon model Poisson superalgebras
computed from 2d superspace. The computation is done in two different ways
concluding the proof of the existence of 2d supersymmetry in the reduced sigma
model phase space under the boost invariant SSSSG Poisson structure.Comment: 33 pages, Published versio
Lambda Models From Chern-Simons Theories
In this paper we refine and extend the results of arXiv:1701.04138, where a
connection between the superstring lambda model on
and a double Chern-Simons (CS) theory on based on the
Lie superalgebra was suggested, after introduction of
the spectral parameter . The relation between both theories mimics the
well-known CS/WZW symplectic reduction equivalence but is non-chiral in nature.
All the statements are now valid in the strong sense, i.e. valid on the whole
phase space, making the connection between both theories precise. By
constructing a -dependent gauge field in the 2+1 Hamiltonian CS theory it is
shown that: i) by performing a symplectic reduction of the CS theory the
Maillet algebra satisfied by the extended Lax connection of the lambda model
emerges as a boundary current algebra and ii) the Poisson algebra of the
supertraces of -dependent Wilson loops in the CS theory obey some sort of
spectral parameter generalization of the Goldman bracket. The latter algebra is
interpreted as the precursor of the (ambiguous) lambda model monodromy matrix
Poisson algebra prior to the symplectic reduction. As a consequence, the
problematic non-ultralocality of lambda models is avoided (for any value of the
deformation parameter ), showing how the lambda model
classical integrable structure can be understood as a byproduct of the
symplectic reduction process of the -dependent CS theory.Comment: Published version+Erratum (of typos), 57 page
A generalized 4d Chern-Simons theory
A generalization of the 4d Chern-Simons theory action introduced by Costello
and Yamazaki is presented. We apply general arguments from symplectic geometry
concerning the Hamiltonian action of a symmetry group on the space of gauge
connections defined on a 4d manifold and construct an action functional that is
quadratic in the moment map associated to the group action. The generalization
relies on the use of contact 1-forms defined on non-trivial circle bundles over
Riemann surfaces and mimics closely the approach used by Beasley and Witten to
reformulate conventional 3d Chern-Simons theories on Seifert manifolds. We also
show that the path integral of the generalized theory associated to integrable
field theories of the PCM type, takes the canonical form of a symplectic
integral over a subspace of the space of gauge connections, turning it a
potential candidate for using the method of non-Abelian localization.
Alternatively, this new quadratic completion of the 4d Chern-Simons theory can
also be deduced in an intuitive way from manipulations similar to those used in
T-duality. Further details on how to recover the original 4d Chern-Simons
theory data, from the point of view of the Hamiltonian formalism applied to the
generalized theory, are included as well.Comment: 54 pages. Comments are welcom
Integrable deformations of strings on symmetric spaces
A general class of deformations of integrable sigma-models with symmetric
space F/G target-spaces are found. These deformations involve defining the
non-abelian T dual of the sigma-model and then replacing the coupling of the
Lagrange multiplier imposing flatness with a gauged F/F WZW model. The original
sigma-model is obtained in the limit of large level. The resulting deformed
theories are shown to preserve both integrability and the equations-of-motion,
but involve a deformation of the symplectic structure. It is shown that this
deformed symplectic structure involves a linear combination of the original
Poisson bracket and a generalization of the Faddeev-Reshetikhin Poisson bracket
which we show can be re-expressed as two decoupled F current algebras. It is
then shown that the deformation can be incorporated into the classical model of
strings on R x F/G via a generalization of the Pohlmeyer reduction. In this
case, in the limit of large sigma-model coupling it is shown that the theory
becomes the relativistic symmetric space sine-Gordon theory. These results
point to the existence of a deformation of this kind for the full Green-Schwarz
superstring on AdS5 x S5.Comment: 41 pages, typos corrected, references adde
Supersymmetry Flows, Semi-Symmetric Space Sine-Gordon Models And The Pohlmeyer Reduction
We study the extended supersymmetric integrable hierarchy underlying the
Pohlmeyer reduction of superstring sigma models on semi-symmetric superspaces
F/G. This integrable hierarchy is constructed by coupling two copies of the
homogeneous integrable hierarchy associated to the loop Lie superalgebra
extension f of the Lie superalgebra f of F and this is done by means of the
algebraic dressing technique and a Riemann-Hilbert factorization problem. By
using the Drinfeld-Sokolov procedure we construct explicitly, a set of 2D spin
\pm1/2 conserved supercharges generating supersymmetry flows in the phase space
of the reduced model. We introduce the bi-Hamiltonian structure of the extended
homogeneous hierarchy and show that the two brackets are of the
Kostant-Kirillov type on the co-adjoint orbits defined by the light-cone Lax
operators L_\pm. By using the second symplectic structure, we show that these
supersymmetries are Hamiltonian flows, we compute part of the supercharge
algebra and find the supersymmetric field variations they induce. We also show
that this second Poisson structure coincides with the canonical
Lorentz-Invariant symplectic structure of the WZNW model involved in the
Lagrangian formulation of the extended integrable hierarchy, namely, the
semi-symmetric space sine-Gordon model (SSSSG), which is the Pohlmeyer reduced
action functional for the transverse degrees of freedom of superstring sigma
models on the cosets F/G. We work out in some detail the Pohlmeyer reduction of
the AdS_2xS^2 and the AdS_3xS^3 superstrings and show that the new conserved
supercharges can be related to the supercharges extracted from 2D superspace.
In particular, for the AdS_2xS^2 example, they are formally the same.Comment: V2: Two references added, V3: Modifications in section 2.6, V4:
Published versio
S-matrices and quantum group symmetry of k-deformed sigma models
Recently, several kinds of integrable deformations of the string world sheet theory in the gauge/gravity correspondence have been constructed. One class of these, the k deformations associated to the more general q deformations but with q=exp(i pi/k) a root of unity, has been shown to be related to a particular discrete deformation of the principal chiral models and (semi-)symmetric space sigma models involving a gauged WZW model. We conjecture a form for the exact S-matrices of the bosonic integrable field theories of this type. The S-matrices imply that the theories have a hidden infinite dimensional affine quantum group symmetry. We provide some evidence, via quantum inverse scattering techniques, that the theories do indeed possess the finite-dimensional part of this quantum grou
An integrable deformation of the AdS5×S5superstring
The S-matrix on the world-sheet theory of the string in AdS5 x S5 has
previously been shown to admit a deformation where the symmetry algebra is
replaced by the associated quantum group. The case where q is real has been
identified as a particular deformation of the Green-Schwarz sigma model. An
interpretation of the case with q a root of unity has, until now, been lacking.
We show that the Green-Schwarz sigma model admits a discrete deformation which
can be viewed as a rather simple deformation of the F/F_V gauged WZW model,
where F=PSU(2,2|4). The deformation parameter q is then a k-th root of unity
where k is the level. The deformed theory has the same equations-of-motion as
the Green-Schwarz sigma model but has a different symplectic structure. We show
that the resulting theory is integrable and has just the right amount of
kappa-symmetries that appear as a remnant of the fermionic part of the original
gauge symmetry. This points to the existence of a fully consistent deformed
string background.Comment: 23 pages, improved and expanded discussion of metric and B fiel
Giant magnons of string theory in the lambda background
The analogues of giant magnon configurations are studied on the string world
sheet in the lambda background. This is a discrete deformation of the
AdS(5)xS(5) background that preserves the integrability of the world sheet
theory. Giant magnon solutions are generated using the dressing method and
their dispersion relation is found. This reduces to the usual dyonic giant
magnon dispersion relation in the appropriate limit and becomes relativistic in
another limit where the lambda model becomes the generalized sine-Gordon theory
of the Pohlmeyer reduction. The scattering of giant magnons is then shown in
the semi-classical limit to be described by the quantum S-matrix that is a
quantum group deformation of the conventional giant magnon S-matrix. It is
further shown that in the small g limit, a sector of the S-matrix is related to
the XXZ spin chain whose spectrum matches the spectrum of magnon bound states.Comment: 53 pages, 6 figures, final version to appear in JHE