arXiv subjects

David M. Schmidtt

Publications and source records attributed to David M. Schmidtt.

16 recordsLinked to original sources

arXiv2025

Contact 4d Chern-Simons theory: Generalities

We refine and generalize the results of e-Print: 2307.10428 [hep-th], where evidence in favor of applying the non-Abelian localization method to handle the 4d Chern-Simons theory path integral formulation was presented. We show, via duality manipulations and invoking some symplectic geometry results, both inspired by the Beasley-Witten work e-Print: 0503126 [hep-th], that the path integral of a regularized version of the 4d Chern-Simons theory, formally takes the canonical symplectic form required by the method of non-Abelian localization. The new theory is defined on a deformed quotient space and interpolates between the conventional 3d Chern-Simons theory on a Seifert manifold M e-Print: 0503126 [hep-th], trivially embedded into $\mathbb{R}\times \text{M}$, and the Costello-Yamazaki e-Print: 1908.02289 [hep-th] 4d Chern-Simons theory defined on the same 4d manifold. It is also shown that the regularized theory is consistent, following an idea of Beasley e-Print: 0911.2687 [hep-th], with the insertion of coadjoint orbit defects of the 1d Chern-Simons theory type. This approach opens the possibility for using exact path integral methods to explore the quantum integrable structure of certain 2d integrable sigma models of the non-ultralocal type, which are widely known to be somehow immune to the use of more traditional quantization methods, like the algebraic Bethe ansatz.

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arXiv2023

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.

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arXiv2021

Symmetric space $λ$-model exchange algebra from 4d holomorphic Chern-Simons theory

We derive, within the Hamiltonian formalism, the classical exchange algebra of a lambda deformed string sigma model in a symmetric space directly from a 4d holomorphic Chern-Simons theory. The explicit forms of the extended Lax connection and R-matrix entering the Maillet bracket of the lambda model are explained from a symmetry principle. This approach, based on a gauge theory, may provide a mechanism for taming the non-ultralocality that afflicts most of the integrable string theories propagating in coset spaces.

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arXiv2020

Holomorphic Chern-Simons theory and lambda models: PCM case

In this note we consider the symplectic reduction of a four-dimensional holomorphic Chern-Simons theory recently introduced in arXiv:1908.02289 for describing integrable field theories. We work out explicitly the case of the lambda deformed Principal Chiral Model (PCM) and show that the symplectic reduction works as a localization mechanism. The reduced Chern-Simons theory restricts to the set of poles of the twist function underlying the theory, where the known classical integrability of the lambda deformed PCM can be reconstructed from the phase space data associated to this set of points in the spectral space.

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arXiv2019

$λ$-Deformation Of The $AdS_{5}\times S^{5}$ Pure Spinor Superstring

The lambda deformation of the pure spinor formalism of the superstring in the $AdS_{5}\times S^{5}$ background is introduced. It is shown that the deformation preserves the integrability as well as the one-loop conformal invariance of its parent theory. It is also shown that the effective action takes the standard form of the Berkovits-Howe action functional, allowing to calculate the deformed background supergeometry in a straightforward way. The background fields coincide with those of the lambda model of the Green-Schwarz formalism, hence satisfying the same set of supergravity equations of motion.

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arXiv2019

Lambda Models From Chern-Simons Theories

In this paper we refine and extend the results of arXiv:1701.04138, where a connection between the $AdS_{5}\times S^{5}$ superstring lambda model on $S^{1}=\partial D$ and a double Chern-Simons (CS) theory on $D$ based on the Lie superalgebra $\mathfrak{psu}(2,2|4)$ was suggested, after introduction of the spectral parameter $z$. 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 $z$-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 $z$-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 $λ\subset [0,1]$), showing how the lambda model classical integrable structure can be understood as a byproduct of the symplectic reduction process of the $z$-dependent CS theory.

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arXiv2017

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.

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arXiv2017

Integrable Lambda Models And Chern-Simons Theories

In this note we reveal a connection between the phase space of lambda models on $S^{1}\times \mathbb{R}$ and the phase space of double Chern-Simons theories on $D\times \mathbb{R}$ and explain in the process the origin of the non-ultralocality of the Maillet bracket, which emerges as a boundary algebra. In particular, this means that the (classical) $AdS_{5}\times S^{5}$ lambda model can be understood as a double Chern-Simons theory defined on the Lie superalgebra $\mathfrak{psu}(2,2|4)$ after a proper dependence of the spectral parameter is introduced. This offers a possibility for avoiding the use of the problematic non-ultralocal Poisson algebras that preclude the introduction of lattice regularizations and the application of the QISM to string sigma models. The utility of the equivalence at the quantum level is, however, still to be explored.

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arXiv2016

Exploring The Lambda Model Of The Hybrid Superstring

The purpose of this contribution is to initiate the study of integrable deformations for different superstring theory formalisms that manifest the property of (classical) integrability. In this paper we choose the hybrid formalism of the superstring in the background AdS_{2}xS^{2} and explore in detail the most immediate consequences of its lambda-deformation. The resulting action functional corresponds to the lambda-model of the matter part of the fairly more sophisticated pure spinor formalism, which is also known to be classical integrable. In particular, the deformation preserves the integrability and the one-loop conformal invariance of its parent theory, hence being a marginal deformation.

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arXiv2015

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 group symmetry.

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arXiv2014

An Integrable Deformation of the AdS5 x S5 Superstring

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.

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arXiv2014

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.

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arXiv2013

The Structure of Non-Abelian Kinks

We consider a class of integrable quantum field theories in 1+1 dimensions whose classical equations have kink solutions with internal collective coordinates that transform under a non-abelian symmetry group. These generalised sine-Gordon theories have been shown to be related to the world sheet theory of the string in the AdS/CFT correspondence. We provide a careful analysis of the boundary conditions at spatial infinity complicated by the fact that they are defined by actions with a WZ term. We go on to describe the local and non-local charges carried by the kinks and end by showing that their structure is perfectly consistent with the exact factorizable S-matrices that have been proposed to describe these theories.

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arXiv2011

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.

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arXiv2011

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.

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arXiv2010

Supersymmetry of Affine Toda Models as Fermionic Symmetry Flows of the Extended mKdV Hierarchy

We couple two copies of the supersymmetric mKdV hierarchy by means of the algebraic dressing technique. This allows to deduce the whole set of $(N,N)$ supersymmetry transformations of the relativistic sector of the extended mKdV hierarchy and to interpret them as fermionic symmetry flows. The construction is based on an extended Riemann-Hilbert problem for affine Kac-Moody superalgebras with a half-integer gradation. A generalized set of relativistic-like fermionic local current identities is introduced and it is shown that the simplest one, corresponding to the lowest isospectral times $t_{\pm 1}$ provides the supercharges generating rigid supersymmetry transformations in 2D superspace. The number of supercharges is equal to the dimension of the fermionic kernel of a given semisimple element $E \in \widehat{\mathfrak{g}}$ which defines both, the physical degrees of freedom and the symmetries of the model. The general construction is applied to the $N=(1,1)$ and $N=(2,2)$ sinh-Gordon models which are worked out in detail.

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