We discuss partition functions of 2d conformal field theories. Modular invariance is known to constrain the spectrum of such theories. We investigate these constraints systematically, using the linear functional method to put new improved upper bounds on the lowest gap in the spectrum. We also consider generalized partition functions of N = (2,2) superconformal theories and discuss the application of our results to Calabi-Yau compactifications. This talk is based on 1209.4649 with H.Ooguri and 1307.6562 with D.Friedan.

In this talk I will show that gauge-invariant derivatives of scalar fields grow asymptotically along the event horizon of extremal black holes. This growth originates from conservation laws on such horizons. The resulted conserved quantities can be seen as a natural generalization of the Newman-Penrose constants along null infinity. I will also present recent generalizations by Reall and Lucietti concerning electromagnetic and linearized gravitational perturbations of extremal black holes.

To the full order in fermions, we construct D = 10 type II supersymmetric double field theory. We spell the precise N = 2 supersymmetry transformation rules as for 32 supercharges. In terms of a stringy differential geometry beyond Riemann, the constructed action unifies type IIA and IIB supergravities in a manifestly covariant manner with respect to O(10, 10) T-duality and a ‘pair’ of local Lorentz groups, or Spin(1, 9) × Spin(9, 1), besides the usual general covariance of supergravities or the generalized diffeomorphism. The distinction of IIA and IIB may arise after a diagonal gauge fixing of the Lorentz groups. They are identified as two different types of ‘solutions’ rather than two different theories. References: arXiv:1210.5078 (N=2) arXiv:1206.3478 (bosonic N=2) arXiv:1112.0069 (N=1)

Computation of conformal dimensions in planar N=4 SYM using integrability techniques was a hot topic during the last decade, with more than thousand publications devoted to it. I will tell you about our new results in this domain: Instead of the Y-system used previously, we are now able to encode the conformal dimensions, at any value of the 't Hooft coupling, in much simpler way: through a Riemann-Hilbert problem. This appears to be not only a very beautiful mathematical setup, but also the most efficient approach to explicitly compute the dimensions. For instance, we've analytically computed the so called Konishi anomalous dimension up to 8 loops in perturbation theory.

In this talk we will describe the construction of F-Theory GUT models for elliptically fibred Calabi-Yau fourfolds admitting a non-trivial Mordell-Weil group. We work out the matter spectrum and Yukawa couplings, including singlets, for these geometries and present the fluxes corresponding to the U(1) symmetries.'

We give a division algebra R,C,H,O description of D = 3 Yang-Mills with N = 1,2,4,8 and hence, by tensoring left and right multiplets, a magic square RR, CR, CC, HR, HC, HH, OR, OC, OH, OO description of D = 3 supergravity with N = 2, 3, 4, 5, 6, 8, 9, 10, 12, 16.

This talk will have three parts 1. Past story - classical projective planes 2. Unfinished story - topology of the projective plane 3. Fairy story - physics and the projective plane

By generalizing the recent proof of the a-theorem, I derive constraints on the possible UV and IR asymptotics of 4D Lorentz invariant unitary quantum field theory. Within perturbation theory the only possible RG flow asymptotic is given by conformal field theory. I also give a non-perturbative argument that excludes theories with scale but not conformal invariance. This argument holds for theories in which the stress-energy tensor is sufficiently nontrivial in a definite technical sense.

I will describe how this number comes from quantum fluctuations during the earliest moments of the Big Bang to become the large scale structure of cosmology that we see today in the skies.

We construct six-dimensional (1,0) superconformal models with non-abelian gauge couplings for multiple tensor and hypermultiplets. The non-abelian gauge invariance for the two-form potentials is achieved by the introduction of a hierarchy of forms up to degree four, where the three and four forms are non-dynamical. This circumvents certain no-go theorems and turns out to be essential also for the formulation of supersymmetry. The inclusion of the hypermultiplets completes the field content to that of superconformal (2,0) theories of multiple M5 branes, though only (1,0) supersymmetry is manifestly realized. The interacting tensor and hypermultiplets form together with the other (non-dynamical) forms a new on-shell representation of the (1,0) supersymmetry algebra. In general these models provide only equations of motions. For a subclass also a Lagrangian formulation exists which captures certain aspects of the first order dynamics.

This is the first Polygon meeting, which is meant to be complementary to Triangles, with local speakers.

I will present a series of new results on classical and quantum properties of maximal supergravity in 4 dimensions, emphasizing the role of the recently discovered "new SO(8) theories". I will explain how this changes our understanding of some general properties of supergravity theories and comment on the impact of these results on the gauge/gravity correspondence.

We discuss various recent computations that come to bear on the conjectured holographic duality between Vasiliev's theory of 3d higher spin gravity and families of 2d CFTs with W symmetry, at both zero and finite temperature. These rely on novel, purely algebraic methods for computing bulk-boundary propagators for scalar fields in Vasiliev theory. Among other topics to be discussed, our results further clarify the physical content of the term "higher spin black hole" and the nature of the classical limit required for the duality.

Localization is a non-perturbative method to exactly compute path integrals in supersymmetric theories. I will describe how localization can be thus used to test holographic duality in a non-conformal setting, namely for N=2* theory, which is a massive deformation of N=4 super Yang-Mills, and for which the dual supergravity background is known explicitly.

We show that for every asymptotically AdS solution compactified on a torus there is a corresponding Ricci-flat solution obtained by replacing the torus by a sphere, performing a Weyl rescaling of the metric and appropriately analytically continuing the dimension of the torus/sphere (as in generalized dimensional reduction). This correspondence should allow us to develop a holographic dictionary for Ricci-flat spacetimes. In particular, it maps Minkowski spacetime to AdS on a torus, the holographic stress energy tensor of AdS to the stress energy tensor due to a brane localized in the interior of spacetime and AdS black branes to (asymptotically flat) Schwarzschild black branes. Applying it to the known solutions describing the hydrodynamic regime in AdS/CFT, we compute the dispersion relation of the Gregory-Laflamme unstable modes through cubic order in the wavenumber, finding remarkable agreement with numerical data. We further obtain the fluid dual to Rindler spacetime and show that its transport coefficients through second order follow from the AdS ones.