Academic Year 2009/2010

Solvmanifolds, in particular nilmanifolds, commonly known as twisted tori, provide several examples of internal manifolds in flux compactifications towards de Sitter, Minkowski or Anti de Sitter. The properties of these manifolds, their relation to the six-dimensional torus, and the string vacua obtained on them are the main interests of this talk. We will first present some properties of these manifolds. We will give a generic construction of their Maurer-Cartan forms out of the six-dimensional torus, via a transformation called the twist. This transformation actually encodes most the properties of these manifolds, in particular their compactness. We will then describe several Minkowski flux backgrounds of type II supergravity obtained on these manifolds. Thanks to the generalized complex geometry approach, we will show that one can obtain those solutions from solutions on the torus, via the twist transformation. The latter then acts as a solution generating technique, being able to relate backgrounds which are not T-duals. Finally, we will apply this twist transformation technique to relate Kahler/non-Kahler solutions of the heterotic string.

After briefly reviewing the progress that has been made in using the methods of integrable systems to compute the spectrum of planar N=4 gauge theory, I will discuss various ways in which integrable models with boundaries enter the picture. In particular, I will discuss boundary conditions arising from the addition of fundamental matter to the gauge theory. Finally, I will argue that the open-boundaries setup provides a useful laboratory for testing ideas about finite-size effects, which have been the subject of much recent interest.

In this talk we will explore the properties of a flux tube formed in the confined phase of the Yang-Mills vacuum. In addition we will look into the question of how the properties of the flux tube compares with the properties of a string. The main tool for looking at these properties is numerical simulations in lattice gauge theories.

I will discuss classes of supersymmetric solutions of Type IIB supergravity, characterised by "interpolating" parameters. I will then focus on a particular solution corresponding to fivebranes wrapped on the S^2 of the resolved conifold. By changing a parameter the solution interpolates between the deformed conifold with flux and the resolved conifold with branes. Therefore, it displays a geometric transition, purely in the supergravity context. The solution is a simple example of torsional geometry and may be thought of as a non-Kahler analog of the conifold. I will explain that applying a solution generating method valid for general SU(3) structure solutions, D3 branes can be added and the solution of Butti et al may be recovered in this way. This describes the baryonic branch of the Klebanov-Strassler theory. The presence of a B-field makes the S^2 non-commutative and this can be seen in the weakly coupled field theory as a fuzzy two-sphere far along the baryonic branch of the Klebanov-Strassler theory. Finally, I will discuss how this story is modified by the addition of ``flavour'' D5 branes to the set up.

I will discuss a gradient formula for beta functions of two-dimensional quantum field theories. The gradient formula has the form \partial_{i}c = - (g_{ij}+\Delta g_{ij} +b_{ij})\beta^{j} where \beta^{j} are the beta functions, c and g_{ij} are the Zamolodchikov c-function and metric, b_{ij} is an antisymmetric tensor introduced by H. Osborn and \Delta g_{ij} is a certain metric correction. The formula is derived under the assumption of stress-energy conservation and certain conditions on the infrared behaviour the most significant of which is the condition that the large distance limit of the field theory does not exhibit spontaneously broken global conformal symmetry. Being specialized to non-linear sigma models this formula implies a one-to-one correspondence between renormalization group fixed points and critical points of c. The talk is based on a joint work with Dan Friedan.

AdS_2/CFT_1 correspondence leads to a prescription for computing the degeneracy of black hole states in terms of path integral over string fields living on the near horizon geometry of the black hole. In this talk, I will discuss about how to make use of the enhanced supersymmetries of the near horizon geometry and localization techniques to argue that the path integral receives contribution only from a special class of string field configurations which are invariant under a subgroup of the supersymmetry transformations. I will identify saddle points which are invariant under this subgroup. I will also use this analysis to show that the integration over infinite number of zero modes generated by the asymptotic symmetries of AdS_2 generate a finite contribution to the path integral.

Monopoles, or topological solutions of the Yang-Mills equations in three spacial dimensions, have been an interesting subject of study for many years; despite this, very few exact solutions are in fact known. In this seminar we use methods from integrable systems which allow to describe monopoles in algebraic-geometric terms via their spectral curve, an algebraic curve satisfying certain constraints. We focus on one class of symmetric monopoles, namely charge 3 with cyclic symmetry, and we show how to use this symmetry to simplify and solve the constraints: we are hence able to prove the existence of a one parameter family of such monopoles.

In three spacetime dimensions, one can describe the effects of both the cosmological constant and of quantum gravity in terms of deformations of spacetime symmetries. The cosmological constant deforms the commutation relations (e.g. from the Poincare Lie algebra to the de Sitter or anti-de Sitter Lie algebras) while the Planck length deforms the co-commutators. I will explain these ideas in a Euclidean setting and discuss, in some detail, an interesting relation between the cosmological deformation of the Euclidean group to SO(4) and its quantum-gravitational deformation to the quantum double of SU(2). The talk is based on (parts of) the paper `q-deformation and semi-dualisation in 3d quantum gravity' with Shahn Majid J. Phys. A 42 (2009) 425402, arXiv:0806.2587

One-dimensional defect lines have become an important tool in studying two-dimensional conformal field theories and their perturbations. I shall review the notion of topological defect lines and explain how they can be used to obtain information on renormalisation group flows. As an application, I shall discuss recent results on coupled bulk and boundary flows in minimal models.

I will discuss some aspects of multiple M2-branes ending on an M5-brane. One motivation is to study multiple self-dual strings within the M5-brane worldvolume, which arise as the boundary of the M2-branes. This can be studied using the ABJM model for multiple M2-branes. This is a Chern-Simons matter theory. As such it needs to be modified in the presence of a boundary to preserve gauge invariance. I will first show how a pure Chern-Simons theory can be suitably modified by adding boundary degrees of freedom, resulting in a WZW model. This result is equivalent to the standard procedure of imposing boundary conditions on the Chern-Simons gauge potential, but is much more convenient when generalising to the ABJM model. Time permitting, I will briefly describe how the BLG model for multiple M2-branes can be used to derive a generalisation of non-commutative geometry on the M5-brane worldvolume theory in the presence of a constant background 3-form potential.

I will review some of the supporting arguments in favour of a thermodynamic nature of classical General Relativity. In the case of black holes, I will review some clues pointing to recent fuzzball ideas, outline some of the puzzles appearing there and finally, comment on extremal black holes/CFT correspondence. I will end (if time permits) with some remarks on holography in Killing horizons and emergent IR CFTs.

Classical integrable equations are known to exist in quantum integrable problems as exact relations even for non-zero Planck's constant. One face of this general phenomenon is appearance of discrete integrable classical dynamics governed by the Hirota equation in the space of commuting transfer matrices (integrals of motion) for integrable quantum spin chains. The spectrum of the quantum spin model can be found by solving this discrete dynamical system via a chain of Backlund transformations, which appears to be equivalent to the nested Bethe ansatz procedure. Bethe equations themselves acquire then the meaning of dynamical equations for Ruijsenaars-type many-body problems in discrete time. This nontrivial interplay between quantum and classical integrability becomes even more meaningful and natural for supersymmetric quantum spin chains. In the talk, we outline the construction of quantum transfer matrices for GL(K|M)-invariant spin chains and Baxter's Q-operators and show how their eigenvaluescan be found in terms of the discrete Hirota dynamics.

Conformally invariant sigma-models on superspaces are two-dimensional supersymmetric quantum field theories which play a prominent role in a number of recent developments in mathematical physics. Apart from their applications in string theory and condensed matter physics (especially disordered systems) they also provide a geometric road towards logarithmic conformal field theories. Last but not least, they arise as critical continuum limits of certain super spin chains. In my talk I will review recent progress on this subject, with special emphasis on supergroups and supercosets as superspaces. It will first be sketched how superspace sigma models arise in string theory in the context of the AdS/CFT correspondence and the quantization of strings on flux backgrounds. Employing the examples of superspheres and projective superspaces (the supertwistor space employed by Witten being a special case) it will then be indicated how exact spectra of anomalous critical dimensions can be calculated as a function of some geometric modulus. These results are then used to argue for new and highly non-trivial dualities between geometric and non-geometric supersymmetric conformal field theories such as supersphere sigma-models and OSP Gross-Neveu models.

Many approaches to quantum gravity introduce a discretization of the underlying manifold as an uv cutoff in order to obtain well defined models. Typically such a discretization leads however to a breaking of diffeomorphism symmetry. This represents a severe problem for lattice approaches to quantum gravity as diffeomorphism symmetry is deeply intertwined with the dynamics of general relativity. Perfect actions, which can be constructed by a renormalization block transformation from the continuum, might remedy the situation. These actions, although defined on the lattice, capture the continuum physics of the model, and hence should capture also the symmetries of the continuum. In this talk I will give an overview of the situation and describe recent progress.

We extend Alday and Maldacena's work on N=4 scattering amplitudes using AdS/CFT correspondence to more general kinematical configurations, including wordsheets which can be embed in AdS4 but not AdS3. We find the Lax pair for the associated Pohlmeyer reduced system of equations. We notice that there is again a natural hyperKahler structure, which can be used to compute the area of the minimal surface bounded by the null polygons. We also make some attempts to directly construct solutions to the reduced system of equations we obtain, which are integrable.

I will discuss finding signs of the minimal models in a certain family of differential equations.

Every minimal model in conformal field theory can be viewed as the scaling limit of a restricted-solid-on-solid (RSOS) model at criticality. States in irreducible modules of the minimal model $M(p,p')$ can be described combinatorially by paths that represent configurations in the corresponding RSOS model. We present an elementary introduction to this path-state correspondence and show how these paths can be dressed with a natural particle interpretation. These particles, in turn, can be interpreted in terms of the kinks and breathers of the restricted sine-Gordon model. This is a key step toward the reformulation of a conformal field theory in a (quasi)-particle formalism, suitable for the analysis of its integrable (off-critical) deformation.

I describe four dimensional N=2 supersymmetric gauge theory in the Omega-background with the two dimensional N=2 super-Poincare invariance. I explain how this gauge theory provides the quantization of the classical integrable system underlying the moduli space of vacua of the ordinary four dimensional N=2 theory. This four dimensional gauge theory in its low energy description has two dimensional twisted superpotential which becomes the Yang-Yang function of the integrable system. I present the thermodynamic-Bethe-ansatz like formulae for this Yang-Yang function and for the spectra of commuting Hamiltonians following the direct computation in gauge theory. Particular examples of the many-body systems include the periodic Toda chain, the elliptic Calogero-Moser system, and their relativistic versions. Gauge theory gives a complete characterization of the L^2-spectrum for these integrable systems.

In this talk, I will discuss work in progress with Bertrand Eynard, in which we derive the BKMP "remodelling the B-model" conjecture, in the large radius limit. This is the claim that Gromov-Witten invariants of any toric Calabi-Yau 3-fold coincide with the spectral invariants of the mirror curve. Our method consists in explicitly constructing a matrix model which reproduces the topological string partition function obtained via the vertex formalism, and then demonstrating that the spectral curve of this matrix model coincides with the mirror geometry.

I will discuss a number of inter-connected issues pertaining to (i) quaternionic-Kahler structures, (ii) D-instanton quantum corrections to hypermultiplet moduli spaces in type II string compactifications on Calabi-Yau 3-folds as well as (iii) the wall-crossing phenomena arising in relation to the BPS spectrum of N=2 supersymmetric theories and the Donaldson-Thomas type invariants for Calabi-Yau categories endowed with a stability structure recently proposed by Kontsevich and Soibelman, within the framework of a twistor-theoretic approach centered on holomorphic (pre)potentials.

Solvmanifolds, in particular nilmanifolds, commonly known as twisted tori, provide several examples of internal manifolds in flux compactifications towards de Sitter, Minkowski or Anti de Sitter. The properties of these manifolds, their relation to the six-dimensional torus, and the string vacua obtained on them are the main interests of this talk. We will first present some properties of these manifolds. We will give a generic construction of their Maurer-Cartan forms out of the six-dimensional torus, via a transformation called the twist. This transformation actually encodes most the properties of these manifolds, in particular their compactness. We will then describe several Minkowski flux backgrounds of type II supergravity obtained on these manifolds. Thanks to the generalized complex geometry approach, we will show that one can obtain those solutions from solutions on the torus, via the twist transformation. The latter then acts as a solution generating technique, being able to relate backgrounds which are not T-duals. Finally, we will apply this twist transformation technique to relate Kahler/non-Kahler solutions of the heterotic string.

After briefly reviewing the progress that has been made in using the methods of integrable systems to compute the spectrum of planar N=4 gauge theory, I will discuss various ways in which integrable models with boundaries enter the picture. In particular, I will discuss boundary conditions arising from the addition of fundamental matter to the gauge theory. Finally, I will argue that the open-boundaries setup provides a useful laboratory for testing ideas about finite-size effects, which have been the subject of much recent interest.