Aspects of Integrability in Gauge/String Correspondence

In this thesis, we present our work about some generalisations of ideas, techniques and physical interpretations typical for integrable models to one of the most outstanding advances in theoretical physics of nowadays: the AdS/CFT correspondences. We have undertaken the problem of testing this conjectured duality under various points of view, but with a clear starting point - the integrability - and with a clear ambitious task in mind: to study the finite-size effects in the energy spectrum of certain string solutions on a side and in the anomalous dimensions of the gauge theory on the other. Of course, the final desire woul be the exact comparison between these two faces of the gauge/string duality. In few words, the original part of this work consists in application of well known integrability technologies, in large parte borrowed by the study of relativistic (1+1)-dimensional integrable quantum field theories, to the highly non-relativisic and much complicated case of the thoeries involved in the recent conjectures of AdS5/CFT4 and AdS4/CFT3 corrspondences. In details, exploiting the spin chain nature of the dilatation operator of N = 4 Super-Yang-Mills theory, we concentrated our attention on one of the most important sector, namely the SL(2) sector - which is also very intersting for the QCD understanding - by formulating a new type of nonlinear integral equation (NLIE) based on a previously guessed asymptotic Bethe Ansatz. The solutions of this Bethe Ansatz are characterised by the length L of the correspondent spin chain and by the number s of its excitations. A NLIE allows one, at least in principle, to make analytical and numerical calculations for arbitrary values of these parameters. The results have been rather exciting. In the important regime of high Lorentz spin, the NLIE clarifies how it reduces to a linear integral equations which governs the subleading order in s, o(s0). This also holds in the regime with L ! 1, L/ ln s finite (long operators case). This region of parameters has been particularly investigated in literature especially because of an intriguing limit into the O(6) sigma model defined on the string side. One of the most powerful methods to keep under control the finite-size spectrum of an integrable relativistic theory is the so called thermodynamic Bethe Ansatz (TBA). We proposed a highly non-trivial generalisation of this technique to the non-relativistic case of AdS5/CFT4 and made the first steps in order to determine its full spectrum - of energies for the AdS side, of anomalous dimensions for the CFT one - at any values of the coupling constant and of the size. At the leading order in the size parameter, the calculation of the finite-size corrections is much simpler and does not necessitate the TBA. It consists in deriving for a nonrelativistc case a method, invented for the first time by L¨uscher to compute the finite-size effects on the mass spectrum of relativisic theories. So, we have formulated a new version of this approach to adapt it to the case of recently found classical string solutions on AdS4 × CP3, inside the new conjecture of an AdS4/CFT3 correspondence. Our results in part confirm the string and algebraic curve calculations, in part are completely new and then could be better understood by the rapidly evolving developments of this extremely exciting research field.

T-duality without isometry via extended gauge symmetries of 2D sigma models

Target space duality is one of the most profound properties of string theory. However it customarily requires that the background fields satisfy certain invariance conditions in order to perform it consistently; for instance the vector fields along the directions that T-duality is performed have to generate isometries. In the present paper we examine in detail the possibility to perform T-duality along non-isometric directions. In particular, based on a recent work of Kotov and Strobl, we study gauged 2D sigma models where gauge invariance for an extended set of gauge transformations imposes weaker constraints than in the standard case, notably the corresponding vector fields are not Killing. This formulation enables us to follow a procedure analogous to the derivation of the Buscher rules and obtain two dual models, by integrating out once the Lagrange multipliers and once the gauge fields. We show that this construction indeed works in non-trivial cases by examining an explicit class of examples based on step 2 nilmanifolds.

Dimensional reduction of the ABJM model

We dimensionally reduce the ABJM model, obtaining a two-dimensional theory that can be thought of as a 'master action'. This encodes information about both T- and S-duality, i.e. describes fundamental (F1) and D-strings (D1) in 9 and 10 dimensions. The Higgsed theory at large VEV, (v) over tilde, and large k yields D1-brane actions in 9d and 10d, depending on which auxiliary fields are integrated out. For N = 1 there is a map to a Green-Schwarz string wrapping a nontrivial circle in C(4)/Z(k).

Comments on the T-dual of the gravity dual of D5-branes on S-3

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Sigma models for genuinely non-geometric backgrounds

The existence of genuinely non-geometric backgrounds, i.e. ones without geometric dual, is an important question in string theory. In this paper we examine this question from a sigma model perspective. First we construct a particular class of Courant algebroids as protobialgebroids with all types of geometric and non-geometric fluxes. For such structures we apply the mathematical result that any Courant algebroid gives rise to a 3D topological sigma model of the AKSZ type and we discuss the corresponding 2D field theories. It is found that these models are always geometric, even when both 2-form and 2-vector fields are neither vanishing nor inverse of one another. Taking a further step, we suggest an extended class of 3D sigma models, whose world volume is embedded in phase space, which allow for genuinely non-geometric backgrounds. Adopting the doubled formalism such models can be related to double field theory, albeit from a world sheet perspective.

The closed-string 3-loop amplitude and S-duality

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)

String theory dualities and supergravity divergences

We demonstrate how duality invariance of the low energy expansion of the four-supergraviton amplitude in type II string theory determines the precise coefficients of multiloop logarithmic ultraviolet divergences of maximal supergravity in various dimensions. This is illustrated by the explicit moduli-dependence of terms of the form ¿2k R4, with k ¿ 3, in the effective action. Furthermore, we show that in the supergravity limit the perturbative contributions are swamped by an accumulation of non-perturbative effects of zero-action instantons.

Fermion perturbations in string theory black holes

In this paper we study fermion perturbations in four-dimensional black holes of string theory, obtained either from a non-extreme configuration of three intersecting five-branes with a boost along the common string or from a non-extreme intersecting system of two two-branes and two five-branes. The Dirac equation for the massless neutrino field, after conformal re-scaling of the metric, is written as a wave equation suitable to study the time evolution of the perturbation. We perform a numerical integration of the evolution equation, and with the aid of Prony fitting of the time-domain profile, we calculate the complex frequencies that dominate the quasinormal ringing stage, and also determine these quantities by the semi-analytical sixth-order WKB method. We also find numerically the decay factor of fermion fields at very late times, and show that the falloff is identical to those showing for massless fields in other four-dimensional black hole spacetimes.

T-duality of axial and vector dyonic integrable models

A general construction of affine nonabelian (NA)-Toda models in terms of the axial and vector gauged two loop WZNW model is discussed. They represent integrable perturbations of the conformal sigma -models (with tachyons included) describing (charged) black hole type string backgrounds. We study the off-critical T-duality between certain families of axial and vector type integrable models for the case of affine NA-Toda theories with one global U(1) symmetry. In particular we find the Lie algebraic condition defining a subclass of T-selfdual torsionless NA-Toda models and their zero curvature representation. (C) 2001 Academic Press.

Cubic twistorial string field theory

Witten has recently proposed a string theory in twistor space whose D-instanton contributions are conjectured to compute M = 4 super-Yang-Mills scattering amplitudes. An alternative string theory in twistor space was then proposed whose open string tree amplitudes reproduce the D-instanton computations of maximal degree in Witten's model. In this paper, a cubic open string field theory action is constructed for this alternative string in twistor space, and is shown to be invariant under parity transformations which exchange MHV and googly amplitudes. Since the string field theory action is gauge-invariant and reproduces the correct cubic super-Yang-Mills interactions, it provides strong support for the conjecture that the string theory correctly computes N-point super-Yang-Mills tree amplitudes. © SISSA/ISAS 2004.

Some aspects of abelian and nonabelian T-duality and the gauge/gravity correspondence

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Cosmic distance duality relation and the shape of galaxy clusters

Context. Observations in the cosmological domain are heavily dependent on the validity of the cosmic distance-duality (DD) relation, eta = D(L)(z)(1+ z)(2)/D(A)(z) = 1, an exact result required by the Etherington reciprocity theorem where D(L)(z) and D(A)(z) are, respectively, the luminosity and angular diameter distances. In the limit of very small redshifts D(A)(z) = D(L)(z) and this ratio is trivially satisfied. Measurements of Sunyaev-Zeldovich effect (SZE) and X-rays combined with the DD relation have been used to determine D(A)(z) from galaxy clusters. This combination offers the possibility of testing the validity of the DD relation, as well as determining which physical processes occur in galaxy clusters via their shapes. Aims. We use WMAP (7 years) results by fixing the conventional Lambda CDM model to verify the consistence between the validity of DD relation and different assumptions about galaxy cluster geometries usually adopted in the literature. Methods. We assume that. is a function of the redshift parametrized by two different relations: eta(z) = 1+eta(0)z, and eta(z) = 1+eta(0)z/(1+z), where eta(0) is a constant parameter quantifying the possible departure from the strict validity of the DD relation. In order to determine the probability density function (PDF) of eta(0), we consider the angular diameter distances from galaxy clusters recently studied by two different groups by assuming elliptical (isothermal) and spherical (non-isothermal) beta models. The strict validity of the DD relation will occur only if the maximum value of eta(0) PDF is centered on eta(0) = 0. Results. It was found that the elliptical beta model is in good agreement with the data, showing no violation of the DD relation (PDF peaked close to eta(0) = 0 at 1 sigma), while the spherical (non-isothermal) one is only marginally compatible at 3 sigma. Conclusions. The present results derived by combining the SZE and X-ray surface brightness data from galaxy clusters with the latest WMAP results (7-years) favors the elliptical geometry for galaxy clusters. It is remarkable that a local property like the geometry of galaxy clusters might be constrained by a global argument provided by the cosmic DD relation.

TESTING THE DISTANCE-DUALITY RELATION WITH GALAXY CLUSTERS AND TYPE Ia SUPERNOVAE

In this Letter, we propose a new and model-independent cosmological test for the distance-duality (DD) relation, eta = D(L)(z)(1 + z)(-2)/D(A)(z) = 1, where D(L) and D(A) are, respectively, the luminosity and angular diameter distances. For D(L) we consider two sub-samples of Type Ia supernovae (SNe Ia) taken from Constitution data whereas D(A) distances are provided by two samples of galaxy clusters compiled by De Filippis et al. and Bonamente et al. by combining Sunyaev-Zeldovich effect and X-ray surface brightness. The SNe Ia redshifts of each sub-sample were carefully chosen to coincide with the ones of the associated galaxy cluster sample (Delta z < 0.005), thereby allowing a direct test of the DD relation. Since for very low redshifts, D(A)(z) approximate to D(L)(z), we have tested the DD relation by assuming that. is a function of the redshift parameterized by two different expressions: eta(z) = 1 + eta(0)z and eta(z) = 1 +eta(0)z/(1 + z), where eta(0) is a constant parameter quantifying a possible departure from the strict validity of the reciprocity relation (eta(0) = 0). In the best scenario (linear parameterization), we obtain eta(0) = -0.28(-0.44)(+0.44) (2 sigma, statistical + systematic errors) for the De Filippis et al. sample (elliptical geometry), a result only marginally compatible with the DD relation. However, for the Bonamente et al. sample (spherical geometry) the constraint is eta(0) = -0.42(-0.34)(+0.34) (3 sigma, statistical + systematic errors), which is clearly incompatible with the duality-distance relation.

Aspects of a noncommutative scalar/tensor duality

We study the noncommutative massless Kalb-Ramond gauge field coupled to a dynamical U(1) gauge field in the adjoint representation together with a compensating vector field. We derive the Seiberg-Witten map and obtain the corresponding mapped action to first order in theta. The (emergent) gravity structure found in other situations is not present here. The off-shell dual scalar theory is derived and it does not coincide with the Seiberg-Witten mapped scalar theory. Dispersion relations are also discussed. The p-form generalization of the Seiberg-Witten map to order theta is also derived.

Duality linking standard and tachyon scalar field cosmologies

In this work we investigate the duality linking standard and tachyon scalar field homogeneous and isotropic cosmologies in N + 1 dimensions. We determine the transformation between standard and tachyon scalar fields and between their associated potentials, corresponding to the same background evolution. We show that, in general, the duality is broken at a perturbative level, when deviations from a homogeneous and isotropic background are taken into account. However, we find that for slow-rolling fields the duality is still preserved at a linear level. We illustrate our results with specific examples of cosmological relevance, where the correspondence between scalar and tachyon scalar field models can be calculated explicitly.