Academic Year 2012/2013

In the present talk I will describe a new approach to the study of U(r)-gauge theories on ALE spaces of type A_n. After a brief introduction about instantons and gauge theories on an ALE space X of type A_n, I will describe a conjectural relation between instantons on X and framed sheaves on a "stacky compactification" of X. In the second part of the talk, I will describe how this new approach can give new hints to solve the Alday-Gaiotto-Tachikawa conjecture for (pure) U(1)-gauge theories.

We discuss the exact computation of the partition function of a D1-D5 brane system on a resolved A_1 singularity via supersymmetric localisation. We show how the D1-brane perspective can be used to describe the equivariant Gromov-Witten invariants of the ADHM moduli space in terms of Givental's formalism. We finally discuss the D5 brane viewpoint and its relation with higher rank Donaldson-Thomas theory.

We construct a supersymmetic Yang Mills Higgs theory on H^3, and we show that half of the supersymmetry is preserved by the supersymmetric hyperbolic monopole configurations. Then, through studying the low energy dynamics of these solutions we obtain a multiplet of zero modes that will yield the defining equations of the geometry of the hyperbolic monopoles via the on shell closure of its algebra. Finally, knowing the geometry of the moduli space will allow us to tell something about the dynamics of the hyperbolic monopoles.

We discuss symmetry protected topological phases in one-dimensional quantum spin chains. Focusing on on-site symmetries only, the topological classes are shown to be in one-to-one correspondence with projective representations of the symmetry group. After having introduced the general setup, we focus on the implications of spin chains with PSU(N) symmetry. These are shown to admit N different topological phases. Each of these may be characterized unambiguously using a non-local string order parameter. Indeed, analytical and numerical results confirm that this order parameter may be used to extract a quantized topological invariant. If time permits we also comment on the possible realization of non-trivial topological phases in cold atom systems.

It was recently proposed a mechanism by which standard General Relativity in five-dimensional spacetime may indeed emerge at a special critical point of a particular Chern-Simons (CS) action. To achieve this result, both the Lie algebra and the symmetric invariant tensor that define the CS Lagrangian are constructed by means of the Lie algebra S-expansion method with a suitable finite abelian semigroup. An interesting question that emerge is the following one: are there other Lie algebras that, constructed also by means of the S-expansion method, permits to construct a CS lagrangian that in a certain critical point also lead to standard General Relativity in d=5? To answer this question the general properties of the expansion method are studied. This permits to perform the S-expansion at the level of the CS Lagrangian (for the AdS algebra) with an arbitrary-unknown semigroup. Then we establish the conditions that the semigroup must satisfy in order to lead the desired connection to standard General Relativity. Finally we implement those conditions by means of computer programs which give us the solution. We propose this procedure as a useful tool in problems where it is important to find a physical theory or a family of them that, on a certain limit, converge to some particular theory. For example, to study if a similar relation can be generalized to the supersymmetric case.

Entanglement entropies of spatial regions are useful quantities for characterizing the ground states of quantum field theories. However, their usefulness depends partly on the assumption that they are universal and duality-invariant for a given theory, and that they distinguish between different theories. We will explore whether these two assumptions are correct, focusing on two very simple free two-dimensional CFTs, the Dirac fermion and the compact boson. Explicit calculations of entanglement entropies in the literature suggest that these quantities may fail to agree when calculated in theories that are known to be dual, and may also fail to distinguish between theories that are known not to be dual. While getting at the root of these puzzling discrepancies, we will discover a new kind of duality between these two well-studied theories.

TBA

TBA

The Brauer loop model is a statistical lattice model with each face decorated by loops that are allowed to cross. In 2005 de Gier and Nienhuis noticed a connection between the ground state of the periodic Brauer loop model and the degrees of some algebraic varieties as calculated by Knutson in 2003. This connection was explored further by Di Francesco and Zinn-Justin in 2006, and proved shortly after by Knutson and Zinn-Justin. In these works a special role was played by the ground state components in the `permutation sector', that is, the components that can be viewed as permutations between points 1,..,n and points n+1,...,2n. Around the same time Di Francesco calculated the ground state of the Brauer model with reflecting boundaries, including closed-form expressions for the sum of all ground state components and the sum of all components in the permutation sector. In this talk we present a work-in-progress in collaboration with Paul Zinn-Justin on similar calculations for the model with non-reflecting boundaries. As in the earlier works, we use the transfer matrix approach to build a possible solution, which we then attempt to prove using recursions.

Charged black holes in planar AdS can be unstable to the formation of charged scalar hair. Through holography, one hopes to use this fact to uncover general principles governing unconventional superconductors. It has been shown however that the behaviour of `holographic superconductors' depends strongly on their embedding into string/M-theory. In this talk I will discuss two aspects of this issue in which lessons can be learned. First I will demonstrate how the ground states of these `top-down' systems may be found by blowing up stars in global AdS, then I will hunt for the instability of highest temperature in a wider consistent truncation of supergravity. Both aspects raise concerns that a new approach to the problem is needed.

Preseminar for Students/Postdocs

Quivers are directed graphs which encode information about the gauge groups and matter content of a large class of gauge theories, many of which have AdS/CFT duals. The counting of local gauge invariant operators and the computation of their correlators (in the free field limit) can be done by simple diagrammatic manipulations of the quiver, with the help of permutation group theory data. This data includes Young diagrams, Littlewood-Richardson numbers and branching coefficients of permutation groups. Riemann surfaces obtained by thickening the quivers are intimately related to these computations.

In this talk we will discuss how giant gravitons and their open string interactions emerge from super Yang-Mills Theory. This is accomplished by diagonalizing the one loop dilatation operator on a class of operators with bare dimension of order N. From the result of this diagonalization, the Gauss Law governing the allowed open string excitations of giant gravitons is clearly visible. In addition, we show that this sector of the theory is integrable.

Preseminar for Students/Postdocs

We will present a proof of the following result: any background of ten- or eleven-dimensional supergravity which preserves more than half of the supersymmetry is locally homogeneous.

In this talk we consider families of charged rotating asymptotically AdS5 Extremal black holes with Vanishing Horizon (EVH black holes) whose near horizon geometries develop locally AdS3 throats. Using the AdS3 /CFT2 duality, we propose an EVH/CFT2 correspondence to describe the near-horizon low energy IR dynamics of near-EVH black holes involving a speci?c large N limit of the 4d N = 4 SYM. We give a map between the UV and IR near-EVH excitations, showing that the ?UV ?rst law? of thermodynamics reduces to the ?IR ?rst law? satis?ed by the near horizon BTZ black holes in this near-EVH limit. We also discuss the connection between our EVH/CFT proposal and the Kerr/CFT correspondence in the cases where the two overlap.

I will review my recent work on integrability of M-brane configurations and the description of M-brane models in higher gauge theory. In particular, I will discuss categorified analogues of instantons and present superconformal equations of motion for the non-abelian tensor multiplet in six dimensions.

Even a cursory inspection of the Hodge plot associated with Calabi-Yau threefolds that are hypersurfaces in toric varieties reveals striking structures. These patterns correspond to webs of elliptic-K3 fibrations whose mirror images are also elliptic-K3 fibrations. Such manifolds arise from reflexive polytopes that can be cut into two parts along slices corresponding to the K3 fibers. Any two half-polytopes over a given slice can be combined into a reflexive polytope. This fact, together with a remarkable relation on the additivity of Hodge numbers, explains much of the structure of the observed patterns.

A lower bound is derived for the boundary entropy s = ln g of a 1+1d quantum critical system with boundary, under the conditions that the bulk conformal central charge c is >=1 and the most relevant bulk scaling dimension is >(c-1)/12. This is the first general restriction on the possible values of g for bulk critical systems with c >= 1.

The construction of rigid minimal supersymmetry multiplets in curved spacetime will be reviewed. For conformal supermultiplets, we will show how correction terms in curved spacetime are fixed by the compatibility of conformal and spin structure, leading to rigid supermultiplets on lorentzian manifolds admitting twistor spinors. For a particular class of twistor spinors, we will also show how to incorporate supersymmetric gauge couplings in dimensions not equal to four and describe rigid supersymmetric gauge theories on bosonic supergravity vacua in dimensions six and ten.

In this talk we will attempt to give an description of the nonassociative geometry probed by closed strings in flat non-geometric R-flux backgrounds. Starting from a suitable Courant sigma-model on an open membrane we will derive a twisted Poisson sigma-model on the boundary of the membrane. The corresponding boundary correlation functions reproduce Kontsevich's deformation quantization formula. For constant R-flux, we will show how to derive closed formulas for the corresponding nonassociative star product and its associator. We will then demonstrate how our approach leads to a consistent quantization of Nambu-Poisson 3-brackets and develop various versions of the Seiberg-Witten map which relate our nonassociative star products to associative ones. Finally, we will show that the Kontsevich formula coincides with the star product obtained by quantizing the dual of a Lie 2-algebra via convolution in an integrating Lie 2-group.

We discuss the description of flux compactifications in the context of matrix models for superstrings. These include conventional compactifications with geometric fluxes, as well as unconventional non-geometric string compactifications. The former are related to the study of nilmanifolds, whose geometry dictates the conditions which describe a matrix model compactification on them. Furthermore, we formulate the conditions which describe compactifications with non-geometric fluxes. It is argued that the geometric and non-geometric fluxes exchange their properties when going from position space to momentum space thus providing a duality among the two. Moreover, the operations which connect solutions with different fluxes are described and their relation to T-duality is discussed. Finally, we discuss flux quantization in this framework as well as the possibility to describe cases with coexistent fluxes.

Recently it has been shown that various extreme black holes are unstable under linear perturbations at the horizon. I will begin by describing a result due to Aretakis, which shows that a massless scalar field at the horizon of an extreme Kerr or Reissner-Nordstrom black hole is generically unstable. I will then present recent work which extends this in two ways. Firstly, such a scalar field instability exists for all known extreme black holes in any dimension. Secondly, a similar instability occurs for linearized gravitational, and electromagnetic, perturbations of an extreme Kerr black hole.

The Jones polynomial invariant in knot theory and the Potts model in statistical mechanics are closely related through the bracket state sum model - a partition function defined on knot diagrams that specializes to the Jones polynomial and can, by different specialization, represent the dichromatic and Tutte polynomials for plane graphs. Via this connection, one can use knot and link diagrams to represent the partition function for the Potts model. The loops in the bracket expansion then correspond to boundaries of regions of constant spin in the Potts model. These states (loop collections) in the bracket model are elevated to a category whose homology is Khovanov homology, an invariant more powerful than the Jones polynomial. From the point of view of the physics of the Potts model it is natural to ask for a physical interpretation of this homology theory based on states delineating regions of constant spin. We will raise these questions and discuss how Khovanov homology and its graded Euler characteristic look from the point of view of the Potts model. We will also point out how this way of thinking leads to a quantum-information theoretic reformulation of Khovanov homology and the Jones polynomial.

Preseminar for Students and Postdocs

The quark-antiquark potential in nonabelian gauge theories is captured by an infinite rectangular Wilson loop. In this talk I will concentrate on N=4 supersymmetric YM theory and explain some recent progress in calculating this potential exactly. To achieve this one first generalizes the problem to Wilson loops with angles and to Wilson loop with local operator insertions. It is then possible to reformulate the question in terms of an open spin chain: The operator insertions are the spins and the angle is related to the boundary condition on the spin-chain. It is widely believed that spin chain models arising from N=4 SYM are integrable and hence can give exact results for all values of the coupling. I will show how this formalism allows to reproduce the weak coupling expansion and also mention all-loop results derived from it.

The BLG and ABJM theories are conjectured to describe two and arbitrarily many M2-branes, respectively, the downside to ABJM being a lack of manifest maximal SUSY. Both can be formulated in terms of 3-algebras. Furthermore the 3-algebra of functions on a 3-sphere (with Nambu-Poisson bracket) is thought to describe an infinite number of M2-branes. When these are stretched between two M5-branes the configuration is similar to a 'magnetic bag', which has an infinite number of D1-branes stretched between two D3-branes. All these 3-algebras sit in a wider class of structures known as differential crossed modules. This is also the language used for non-abelian gerbes, with differential crossed modules taking the role of Lie algebras for principal bundles.

Our understanding of the Holographic Principle is mainly restricted to Anti-De Sitter spacetimes and is based essentially on the AdS/CFT correspondence. In this talk, I review recent progress in understanding a dual of flat-spacetimes. My discussion would remain confined to the 3d bulk and the road to the elusive flat holography would be constructed driven by the simple observation that flat-spacetimes can be obtained by a large radius limit of AdS. Symmetry structures would dictate our formalism.

TBA

Preseminar for Students/Postdocs

Lovelock theory is the natural extension of general relativity to higher dimensions. It can be also thought of as a toy model for ghost-free higher curvature corrections in gravitational theories. It generically admits a family of AdS vacua, most (but not all) of them supporting black hole solutions that display interesting features. This provides an appealing arena to explore different holographic aspects in the context of the AdS/CFT and the fluid/gravity correspondences. I will elaborate on many of these features and will discuss a new type of phase transition possibly arising in the realm of quantum corrected gravitational theories.

We consider Euclidean functional integrals involving actions which are not exclusively real. Writing the action in terms of only real fields (which is always possible), such terms appear as explicitly imaginary terms in the Euclidean action. The usual quanization procedure which involves finding the critical points of the action and then quantizing the spectrum of fluctuations about these critical points fails. In the case of complex actions, there do not exist, in general, any critical points of the action on the space of real fields, the critical points are in general complex. The proper definition of the function integral then requires the analytic continuation of the functional integration into the space of complex fields so as to pass through the complex critical points according to the method of steepest descent. We show a simple example where this procedure can be carried out explicitly. The procedure of finding the critical points of the real part of the action and quantizing the corresponding fluctuations, treating the (exponential of the) complex part of the action as a bounded integrable function is shown to fail in our explicit example, at least perturbatively. Application to tunnelling in spin systems is also studied.