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.Comment: v5, published version up to formatting, 32 page

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.Comment: 24 pages, v3 as publishe

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.Comment: v2, updated discussion on target space interpretation, elaborated discussion on minor points, content matches published version, 28 pages, 3 figure

Quantum Spectral Curve for the eta-deformed AdS_5xS^5 superstring

The spectral problem for the ${\rm AdS}_5\times {\rm S}^5$ superstring and its dual planar maximally supersymmetric Yang-Mills theory can be efficiently solved through a set of functional equations known as the quantum spectral curve. We discuss how the same concepts apply to the $\eta$-deformed ${\rm AdS}_5\times {\rm S}^5$ superstring, an integrable deformation of the ${\rm AdS}_5\times {\rm S}^5$ superstring with quantum group symmetry. This model can be viewed as a trigonometric version of the ${\rm AdS}_5\times {\rm S}^5$ superstring, like the relation between the XXZ and XXX spin chains, or the sausage and the ${\rm S}^2$ sigma models for instance. We derive the quantum spectral curve for the $\eta$-deformed string by reformulating the corresponding ground-state thermodynamic Bethe ansatz equations as an analytic $Y$ system, and map this to an analytic $T$ system which upon suitable gauge fixing leads to a $\mathbf{P} \mu$ system -- the quantum spectral curve. We then discuss constraints on the asymptotics of this system to single out particular excited states. At the spectral level the $\eta$-deformed string and its quantum spectral curve interpolate between the ${\rm AdS}_5\times {\rm S}^5$ superstring and a superstring on "mirror" ${\rm AdS}_5\times {\rm S}^5$, reflecting a more general relationship between the spectral and thermodynamic data of the $\eta$-deformed string. In particular, the spectral problem of the mirror ${\rm AdS}_5\times {\rm S}^5$ string, and the thermodynamics of the undeformed ${\rm AdS}_5\times {\rm S}^5$ string, are described by a second rational limit of our trigonometric quantum spectral curve, distinct from the regular undeformed limit.Comment: 32+37 pages; 6 figures. v2: added reference

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.Comment: v3, published version, introduction slightly broadened, typos corrected, updates to outlook and references. Review based on author's PhD thesis, 214 pages, many figures. Partly based on arXiv:1009.4118, arXiv:1103.5853, arXiv:1111.0564, arXiv:1201.1451, arXiv:1208.3478, and arXiv:1210.818