arXiv subjects

Stijn J. van Tongeren

Publications and source records attributed to Stijn J. van Tongeren.

34 records · Page 2Linked to original sources

arXiv2016

Introduction to the thermodynamic Bethe ansatz

We give a pedagogical introduction to the thermodynamic Bethe ansatz, a method that allows us to describe the thermodynamics of integrable models whose spectrum is found via the (asymptotic) Bethe ansatz. We set the stage by deriving the Fermi-Dirac distribution and associated free energy of free electrons, and then in a similar though technically more complicated fashion treat the thermodynamics of integrable models, focusing on the one dimensional Bose gas with delta function interaction as a clean pedagogical example, secondly the XXX spin chain as an elementary (lattice) model with prototypical complicating features in the form of bound states, and finally the SU(2) chiral Gross-Neveu model as a field theory example. Throughout this discussion we emphasize the central role of particle and hole densities, whose relations determine the model under consideration. We then discuss tricks that allow us to use the same methods to describe the exact spectra of integrable field theories on a circle, in particular the chiral Gross-Neveu model. We moreover discuss the simplification of TBA equations to Y systems, including the transition back to integral equations given sufficient analyticity data, in simple examples.

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arXiv2016

Yang-Baxter deformations, AdS/CFT, and twist-noncommutative gauge theory

We give an AdS/CFT interpretation to homogeneous Yang-Baxter deformations of the AdS_5 x S^5 superstring as noncommutative deformations of the dual gauge theory, going well beyond the canonical noncommutative case. These homogeneous Yang-Baxter deformations can be of so-called abelian or jordanian type. While abelian deformations have a clear interpretation in string theory and many already had well understood gauge theory duals, jordanian deformations appear novel on both counts. We discuss the symmetry structure of the deformed string from the uniformizing perspective of Drinfeld twists and indicate that this structure can be realized on the gauge theory side by considering theories on various noncommutative spaces. We then conjecture that these are the gauge theory duals of our strings, modulo subtleties involving singularities. We support this conjecture by a brane construction for two jordanian examples, corresponding to noncommutative spaces with [x^-,x^i] ~ x^i (i=1,2). We also discuss kappa-Minkowski type deformations of AdS_5 x S^5, one of which may be the gravity dual of gauge theory on spacelike kappa-Minkowski space.

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arXiv2016

Quantum deformations of the flat space superstring

We discuss a quantum deformation of the Green-Schwarz superstring on flat space, arising as a contraction limit of the corresponding deformation of AdS_5 x S^5. This contraction limit turns out to be equivalent to a previously studied limit that yields the so-called mirror model - the model obtained from the light cone gauge fixed AdS_5 x S^5 string by a double Wick rotation. Reversing this logic, the AdS_5 x S^5 superstring is the double Wick rotation of a quantum deformation of the flat space superstring. This quantum deformed flat space string realizes symmetries of timelike kappa-Poincare type, and is T dual to dS_5 x H^5, indicating interesting relations between symmetry algebras under T duality. Our results directly extend to AdS_2 x S^2 x T^6 and AdS_3 x S^3 x T^4, and beyond string theory to many (semi)symmetric space coset sigma models, such as for example a deformation of the four dimensional Minkowski sigma model with timelike kappa-Poincare symmetry. We also discuss possible null and spacelike deformations.

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arXiv2015

On classical Yang-Baxter based deformations of the AdS_5 x S^5 superstring

Interesting deformations of AdS_5 x S^5 such as the gravity dual of noncommutative SYM and Schödinger spacetimes have recently been shown to be integrable. We clarify questions regarding the reality and integrability properties of the associated construction based on R matrices that solve the classical Yang-Baxter equation, and present an overview of manifestly real R matrices associated to the various deformations. We also discuss when these R matrices should correspond to TsT transformations, which not all do, and briefly analyze the symmetries preserved by these deformations, for example finding Schrödinger superalgebras that were previously obtained as subalgebras of psu(2,2|4). Our results contain a (singular) generalization of an apparently non-TsT deformation of AdS_5 x S^5, whose status as a string background is an interesting open question.

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arXiv2015

Double Wick rotating Green-Schwarz strings

Via an appropriate field redefinition of the fermions, we find a set of conditions under which light cone gauge fixed world sheet theories of strings on two different backgrounds are related by a double Wick rotation. These conditions take the form of a set of transformation laws for the background fields, complementing a set of transformation laws for the metric and B field we found previously with a set for the dilaton and RR fields, and are compatible with the supergravity equations of motion. Our results prove that at least to second order in fermions, the AdS_5 x S^5 mirror model which plays an important role in the field of integrability in AdS/CFT, represents a string on `mirror AdS_5 x S^5', the background that follows from our transformations. We discuss analogous solutions for AdS_3 x S^3 x T^4 and AdS_2 x S^2 x T^6. The main ingredient in our derivation is the light cone gauge fixed action for a string on an (almost) completely generic background, which we explicitly derive to second order in fermions.

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arXiv2015

The $\mathrm{AdS}_5 \times \mathrm{S}^5$ mirror model as a string

Doing a double Wick rotation in the worldsheet theory of the light cone $\mathrm{AdS}_5 \times \mathrm{S}^5$ superstring results in an inequivalent, so-called mirror theory that plays a central role in the field of integrability in AdS/CFT. We show that this mirror theory can be interpreted as the light cone theory of a free string on a different background. This background is related to $\mathrm{dS}_5 \times \mathrm{H}^5$ by a double T duality, and has hidden supersymmetry. The geometry can also be extracted from an integrable deformation of the $\mathrm{AdS}_5 \times \mathrm{S}^5$ sigma model, and we prove the observed mirror duality of these deformed models at the bosonic level as a byproduct. While we focus on $\mathrm{AdS}_5 \times \mathrm{S}^5$, our results apply more generally.

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arXiv2014

Integrability of the AdS_5 x S^5 superstring and its deformations

This article reviews the application of integrability to the spectral problem of strings on AdS_5 x S^5 and its deformations. We begin with a pedagogical introduction to integrable field theories culminating in the description of their finite-volume spectra through the thermodynamic Bethe ansatz. Next, we apply these ideas to the AdS_5 x S^5 string and in later chapters discuss how to account for particular integrable deformations. Through the AdS/CFT correspondence this gives an exact description of anomalous scaling dimensions of single trace operators in planar N=4 supersymmetry Yang-Mills theory, its `orbifolds', and beta and gamma-deformed supersymmetric Yang-Mills theory. We also touch upon some subtleties arising in these deformed theories. Furthermore, we consider complex excited states (bound states) in the su(2) sector and give their thermodynamic Bethe ansatz description. Finally we discuss the thermodynamic Bethe ansatz for a quantum deformation of the AdS_5 x S^5 superstring S-matrix, with close relations to among others Pohlmeyer reduced string theory, and briefly indicate more recent developments in this area.

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arXiv2014

On the exact spectrum and mirror duality of the (AdS_5 x S^5)_eta superstring

We discuss the spectrum of a string propagating on eta-deformed AdS_5 x S^5 by treating its world-sheet theory as an integrable quantum field theory. The exact S-matrix of this field theory is given by a q-deformation of the AdS_5 x S^5 world-sheet S-matrix with real deformation parameter. By considering mirror (double Wick-rotated) versions of these world-sheet theories we give the Thermodynamic Bethe Ansatz description of their exact finite size spectra. Interestingly, this class of models maps onto itself under the mirror transformation. At the level of the string this appears to say that the light-cone worldsheet theories of strings on particular pairs of backgrounds are related by a double Wick rotation, a feature we call 'mirror duality'. We provide a partial check of these statements at the level of the sigma model by considering reduced actions and their corresponding (mirror) giant magnon solutions.

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arXiv2013

The Quantum Deformed Mirror TBA II

We discuss the description of generic excited states in the quantum deformed AdS_5 x S^5 mirror thermodynamic Bethe ansatz and derive the associated Y-system. This Y-system shows an interesting new feature; it depends explicitly on the excited state under consideration. Similarly, it also depends on twisted boundary conditions. We construct the asymptotic solution of these TBA and Y-system equations by deriving the twisted transfer matrix for the quantum deformed Hubbard model and finding the deformed mirror bound state dressing phase. This asymptotic construction is insensitive to the precise nature of the deformation, and thereby provides a nontrivial check of the interesting new features which arise precisely at roots of unity.

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arXiv2012

The Quantum Deformed Mirror TBA I

We derive the ground state thermodynamic Bethe ansatz equations for the quantum deformation of the AdS_5 x S^5 mirror model, taking the deformation parameter to be a root of unity. By virtue of the deformation, the resulting equations show an interesting structure between a finite number of Y-functions.

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arXiv2012

The spectral problem for strings on twisted AdS_5 x S^5

We discuss the spectral problem for integrable superstrings on generically twisted AdS_5 x S^5, meaning all its orbifolds and TsT transformed versions. We explicitly give the asymptotic description of these theories through a twisted transfer matrix, and carefully match the geometric deformations with twists allowed by integrability. We then discuss the mirror TBA equations that describe these theories at finite size. This unifies the treatment of various specific deformations previously considered in this setting, and extends it to completely general twists.

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arXiv2012

Lifting asymptotic degeneracies with the Mirror TBA

We describe a qualitative feature of the AdS_5 x S^5 string spectrum which is not captured by the asymptotic Bethe ansatz. This is reflected by an enhanced discrete symmetry in the asymptotic limit, whereby extra energy degeneracy enters the spectrum. We discuss how finite size corrections should lift this degeneracy, through both perturbative (Luscher) and non-perturbative approaches (the Mirror TBA), and illustrate this explicitly on two such asymptotically degenerate states.

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arXiv2011

Bound States in the Mirror TBA

The spectrum of the light-cone AdS_5 \times S^5 superstring contains states composed of particles with complex momenta including in particular those which turn into bound states in the decompactification limit. We propose the mirror TBA description for these states. We focus on a three-particle state which is a finite-size representative of a scattering state of a fundamental particle and a two-particle bound state and dual to an operator from the su(2) sector of N=4 SYM. We find that the analytic behavior of Y-functions differs drastically from the case of states with real momenta. Most importantly, Y_Q-functions exhibit poles in the analyticity strip which leads to the appearance of new terms in the formula for the energy of this state. In addition, the TBA equations are supplied by quantization conditions which involve Y_2. Considering yet another example of a three-particle state, we find that the corresponding quantization conditions do not even involve Y_1. Our treatment can be generalized to a wide class of states with complex momenta.

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arXiv2011

Orbifolded Konishi from the Mirror TBA

Starting with a discussion of the general applicability of the simplified mirror TBA equations to simple deformations of the AdS_5 x S^5 superstring, we proceed to study a specific type of orbifold to which the undeformed simplified TBA equations directly apply. We then use this set of equations, as well as Luscher's approach, to determine the NLO wrapping correction to the energy of what we call the orbifolded Konishi state, and show that they perfectly agree. In addition we discuss wrapping corrections to the ground state energy of the orbifolded model under consideration.

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arXiv2011

Twisting the Mirror TBA

We study finite-size corrections to the magnon dispersion relation in three models which differ from string theory on AdS5 x S5 in their boundary conditions. Asymptotically, this is accomplished by twisting the transfer matrix in a way which manifestly preserves integrability. In model I all world-sheet fields are periodic, whereas model II represents a particular orbifold of AdS5 x S5 and model III is a beta-deformed theory. For models I and II we construct the one-particle TBA equations and use them to determine the leading finite-size correction to the asymptotic Bethe equation. We also make some interesting observations concerning the quantization conditions for the momentum. For the same models we compute the leading and for model II the next-to-leading order finite-size corrections to the asymptotic magnon dispersion relation. Furthermore, we apply Lüscher's formulae to compute the leading finite-size corrections in beta-deformed theory. In addition to reproducing known results, we provide new predictions for two-particle states from the sl(2) sector, to be confronted with explicit field-theoretic calculations in the dual gauge theory. Finally, we prove that the leading finite-size correction to the energy of an sl(2) magnon in orbifold theory is the same as the one for an su(2) magnon in beta-deformed theory for special values of beta. We also speculate that for these values of beta our result for the next-to-leading order correction in the orbifold model might coincide with the corresponding correction to the energy of su(2) magnon in beta-deformed theory.

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arXiv2009

Fermionic reductions of the AdS4 x CP3 superstring

We discuss fermionic reductions of type IIA superstrings on AdS4 x CP3 in relation to the conjectured AdS4/CFT3 duality. The superstring theory is described by means of a coset model construction, which is classically integrable. We discuss the global light-cone symmetries of the action and related kappa-symmetry gauge choices, and also present the complete quartic action in covariant form with respect to these. Further, we study integrable (fermionic) reductions, in particular, a reduction yielding a quadratic action of two complex fermions on the string world-sheet. Interestingly, this model appears to be exactly the same as the corresponding integrable reduction found in the AdS5 x S5 case.

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