On string integrability. A journey through the two-dimensional hidden symmetries in the AdS/CFT dualities

One of the main topics in the modern String Theory are the AdS/CFT dualities. Proving such conjectures is extremely difficult since the gauge and string theory perturbative regimes do not overlap. In this perspective, the discovery of infinitely many conserved charges, i.e. the integrability, in the planar AdS/CFT has allowed us to reach immense progresses in understanding and confirming the duality. We review the fundamental concepts and properties of integrability in two-dimensional sigma-models and in the AdS/CFT context. The first part is focused on the AdS_5/CFT_4 duality, especially the classical and quantum integrability of the type IIB superstring on AdS_5 x S^5 are discussed in both pure spinor and Green-Schwarz formulations. The second part is dedicated to the AdS_4/CFT_3 duality with particular attention to the type IIA superstring on AdS_4 x CP^3 and its integrability. This review is based on a shortened and revised version of the author's PhD thesis, discussed at Uppsala University in September 2009.Comment: 116 pages, 11 figures, to be published in Advances in High Energy Physics, Special Issue on ''Gauge/String Duality'

Aspects of quantum integrability for pure spinor superstring in AdS(5)xS(5)

We consider the monodromy matrix for the pure spinor IIB superstring on $AdS_5\times S^5$ at leading order at strong coupling, in particular its variation under an infinitesimal and continuous deformation of the contour. Such variation is equivalent to the insertion of a local operator. Demanding the BRST-closure for such an operator rules out its existence, implying that the monodromy matrix remains contour-independent at the first order in perturbation theory. Furthermore we explicitly compute the field strength corresponding to the flat connections up to leading order and directly check that it is free from logarithmic divergences. The absence of anomaly in the coordinate transformation of the monodromy matrix and the UV-finiteness of the curvature tensor finally imply the integrability of the pure spinor superstring at the first order.Comment: 44 pages; 4 figure

Non-analyticity of holographic Rényi entropy in Lovelock gravity

We compute holographic Rényi entropies for spherical entangling surfaces on the boundary while considering third order Lovelock gravity with negative cosmological constant in the bulk. Our study shows that third order Lovelock black holes with hyperbolic event horizon are unstable, and at low temperatures those with smaller mass are favoured, giving rise to first order phase transitions in the bulk. We determine regions in the Lovelock parameter space in arbitrary dimensions, where bulk phase transitions happen and where boundary causality constraints are met. We show that each of these points corresponds to a dual boundary conformal field theory whose Rényi entropy exhibits a kink at a certain critical index n.This research was supported in part by the Icelandic Research Fund under contracts 163419-051 and 163422-051, and by grants from the University of Iceland Research Fund.Peer Reviewe

Quantized Strings and Instantons in Holography

We study worldsheet instantons in holographic type IIA backgrounds directly in string theory. The first background is a dimensional reduction of AdS$_7\times S^4$ and is dual to the maximally supersymmetric Yang-Mills theory on $S^5$. The second background is AdS$_4\times \mathbf{C}P^3$ dual to ABJM in the type IIA limit. We compute the one-loop partition function of the fundamental string in these backgrounds and show that the result is in exact agreement with field theory predictions. We argue that for higher rank instantons, the string partition function takes a product form of the single instanton partition function times the contribution of two orbifolds on the worldsheet. We determine the orbifold factor to be $n^{-3/2}$ where $n$ is the instanton rank. With this result, we reproduce the series of non-perturbative corrections in $\alpha'$ to the planar $S^5$ free energy. When studying the worldsheet instanton partition function on $\mathbf{C}P^3$, we encounter twelve fermionic and twelve bosonic zero modes. By deforming the ABJM theory, the zero-modes are lifted and consequently the tower of worldsheet instantons can be evaluated and matched to known results in the QFT. As a by-product, we determine a series of higher rank instanton corrections to the free energy of the mass-deformed and orbifolded ABJ(M) theory.Comment: 35 pages. v2: Minor correction

Finite-size corrections in the SU(2) x SU(2) sector of type IIA string theory on AdS_4 x CP^3

We consider finite-size corrections in the SU(2) x SU(2) sector of type IIA string theory on AdS_4 x CP^3, which is the string dual of the recently constructed N=6 superconformal Chern-Simons theory of Aharony, Bergman, Jafferis and Maldacena (ABJM theory). The string states we consider are in the R x S^2 x S^2 subspace of AdS_4 x CP^3 with an angular momentum J on CP^3 being large. We compute the finite-size corrections using two different methods, one is to consider curvature corrections to the Penrose limit giving an expansion in 1/J, the other by considering a low energy expansion in lambda'=lambda/J^2 of the string theory sigma-model, lambda being the 't Hooft coupling of the dual ABJM theory. For both methods there are interesting issues to deal with. In the near-pp-wave method there is a 1/\sqrt{J} interaction term for which we use zeta-function regularization in order to compute the 1/J correction to the energy. For the low energy sigma-model expansion we have to take into account a non-trivial coupling to a non-dynamical transverse direction. We find agreement between the two methods. At order lambda' and lambda'^2, for small lambda', our results are analogous to the ones for the SU(2) sector in type IIB string theory on AdS_5 x S^5. Instead at order lambda'^3 there are interactions between the two two-spheres. We compare our results with the recently proposed all-loop Bethe ansatz of Gromov and Vieira and find agreement.Comment: 21 pages. v2: typos fixed, refs. added. v3: misprints corrected, refs. adde

Entanglement entropy in generalised quantum Lifshitz models

Publisher's version (útgefin grein)We compute universal finite corrections to entanglement entropy for generalised quantum Lifshitz models in arbitrary odd spacetime dimensions. These are generalised free field theories with Lifshitz scaling symmetry, where the dynamical critical exponent z equals the number of spatial dimensions d, and which generalise the 2+1-dimensional quantum Lifshitz model to higher dimensions. We analyse two cases: one where the spatial manifold is a d-dimensional sphere and the entanglement entropy is evaluated for a hemisphere, and another where a d-dimensional flat torus is divided into two cylinders. In both examples the finite universal terms in the entanglement entropy are scale invariant and depend on the compactification radius of the scalar field.We acknowledge useful discussions with J. Bardarson, P. Di Vecchia, J. S. Dowker, D. Friedan, B. Gouteraux, K. Grosvenor, D. Medina-Rincon, R. Leigh, D. Seminara, W. Sybesma, S. Vandoren, and M. Zaletel. This research was supported in part by the Icelandic Research Fund under contracts 163419-053 and 163422-053, and by grants from the University of Iceland Research Fund.Peer Reviewe

Logarithmic negativity in quantum Lifshitz theories

Publisher's version (útgefin grein)We investigate quantum entanglement in a non-relativistic critical system by calculating the logarithmic negativity of a class of mixed states in the quantum Lifshitz model in one and two spatial dimensions. In 1+1 dimensions we employ a correlator approach to obtain analytic results for both open and periodic biharmonic chains. In 2+1 dimensions we use a replica method and consider spherical and toroidal spatial manifolds. In all cases, the universal finite part of the logarithmic negativity vanishes for mixed states defined on two disjoint components. For mixed states defined on adjacent components, we find a non-trivial logarithmic negativity reminiscent of two-dimensional conformal field theories. As a byproduct of our calculations, we obtain exact results for the odd entanglement entropy in 2+1 dimensions.Peer Reviewe

Operator Product Expansion for Pure Spinor Superstring on AdS(5)*S(5)

The tree-level operator product expansion coefficients of the matter currents are calculated in the pure spinor formalism for type IIB superstring in the AdS(5)*S(5) background.Comment: 18 pages, no figure, corrected typos and added acknowledgement

One-loop spectroscopy of semiclassically quantized strings: bosonic sector

We make a further step in the analytically exact quantization of spinning string states in semiclassical approximation, by evaluating the exact one-loop partition function for a class of two-spin string solutions for which quadratic fluctuations form a non-trivial system of coupled modes. This is the case of a folded string in the SU(2) sector, in the limit described by a quantum Landau–Lifshitz model. The same applies to the full bosonic sector of fluctuations over the folded spinning string in AdS5 with an angular momentum J in S5. Fluctuations are governed by a special class of fourth-order differential operators, with coefficients being meromorphic functions on the torus, which we are able to solve exactly

Finite-size corrections in the SU(2) x SU(2) sector of type IIA string theory on AdS_4 x CP^3

We consider finite-size corrections in the SU(2) x SU(2) sector of type IIA string theory on AdS_4 x CP^3, which is the string dual of the recently constructed N=6 superconformal Chern-Simons theory of Aharony, Bergman, Jafferis and Maldacena (ABJM theory). The string states we consider are in the R x S^2 x S^2 subspace of AdS_4 x CP^3 with an angular momentum J on CP^3 being large. We compute the finite-size corrections using two different methods, one is to consider curvature corrections to the Penrose limit giving an expansion in 1/J, the other by considering a low energy expansion in lambda'=lambda/J^2 of the string theory sigma-model, lambda being the 't Hooft coupling of the dual ABJM theory. For both methods there are interesting issues to deal with. In the near-pp-wave method there is a 1/\sqrt{J} interaction term for which we use zeta-function regularization in order to compute the 1/J correction to the energy. For the low energy sigma-model expansion we have to take into account a non-trivial coupling to a non-dynamical transverse direction. We find agreement between the two methods. At order lambda' and lambda'^2, for small lambda', our results are analogous to the ones for the SU(2) sector in type IIB string theory on AdS_5 x S^5. Instead at order lambda'^3 there are interactions between the two two-spheres. We compare our results with the recently proposed all-loop Bethe ansatz of Gromov and Vieira and find agreement.Comment: 21 pages. v2: typos fixed, refs. added. v3: misprints corrected, refs. adde