L3

Symplectic field theory (SFT) can be viewed as TQFT approach to Gromov-Witten theory. As in Gromov-Witten theory, transversality for the Cauchy-Riemann operator is not satisfied in general, due to the presence of multiply-covered curves. When the underlying simple curve is sufficiently nice, I will outline that the transversality problem for their multiple covers can be elegantly solved using finite-dimensional obstruction bundles of constant rank. By fixing the underlying holomorphic curve, we furthermore define a local version of SFT by counting only multiple covers of this chosen curve. After introducing gravitational descendants, we use this new version of SFT to prove that a stable hypersurface intersecting an exceptional sphere (in a homologically nontrivial way) in a closed four-dimensional symplectic manifold must carry an elliptic orbit. Here we use that the local Gromov-Witten potential of the exceptional sphere factors through the local SFT invariants of the breaking orbits appearing after neck-stretching along the hypersurface.

Assuming the natural compactification X of a hypersurface in (C^*)^n is smooth, it can exhibit any Kodaira dimension depending on the size and shape of the Newton polyhedron of X. In a joint work with Mark Gross and Ludmil Katzarkov, we give a construction for the expected mirror symmetry partner of a complete intersection X in a toric variety which works for any Kodaira dimension of X. The mirror dual might be a reducible and is equipped with a sheaf of vanishing cycles. We give evidence for the duality by proving the symmetry of the Hodge numbers when X is a hypersurface. The leading example will be the mirror of a genus two curve. If time permits, I will explain relations to homological mirror symmetry and the Gross-Siebert construction.

The induced graph removal lemma states that for any fixed graph $H$ on $h$ vertices and any $e\textgreater 0$ there exists $d\textgreater0$ such that any graph $G$ with at most $d n^h$ induced copies of $H$ may be made $H$-free by adding or removing atmost $e n^2$ edges. This fact was originally proven by Alon, Fischer, Krivelevich and Szegedy. In this talk, we discuss a new proof and itsrelation to various regularity lemmas. This is joint work with Jacob Fox.

I will give an overview of some of the most interesting algebraic-lattice theoretical results on bilattices. I will focus in particular on the product construction that is used to represent a subclass of bilattices, the so-called 'interlaced bilattices', mentioning some alternative strategies to prove such a result. If time allows, I will discuss other algebras of logic related to bilattices (e.g., Nelson lattices) and their product representation.

I will discuss the dynamical emergence of IR conformal invariance describing the low energy excitations of near-extremal R-charged global AdS${}_5$ black holes. To keep some non-trivial dynamics in the sector of ${\cal N}=4$ SYM captured by the near horizon limits describing these IR physics, we are lead to study large N limits in the UV theory involving near vanishing horizon black holes. I will consider both near-BPS and non-BPS regimes, emphasising the differences in the local AdS${}_3$ throats emerging in both cases. I will compare these results with the predictions obtained by Kerr/CFT, obtaining a natural quantisation for the central charge of the near-BPS emergent IR CFT describing the open strings stretched between giant gravitons.

We compute three-point functions of single trace operators in planar N = 4 SYM. We consider the limit where one of the operators is much smaller than the other two. We find a precise match between weak and strong coupling in the Frolov-Tseytlin classical limit for a very general class of classical solutions. To achieve this match we clarify the issue of back-reaction and identify precisely which three-point functions are captured by a classical computation.

String compactifications incorporating non-vanishing H-flux have received increased attention over the past decade for their potential relevance to the moduli stabilization problem. Their internal spaces are in general not Kähler and, therefore, not Calabi-Yau. In the heterotic string an important technical problem is to construct gauge bundles on such spaces. I will present a method of how to explicitly construct gauge bundles over homogeneous nearly-Kähler manifolds of dimension six and discuss some of the arising implications for model building.

Four-dimensional (4d) supergravities with non-geometric terms in their potential are very promising models for phenomenology. Indeed, these terms, generated by so-called non-geometric fluxes, generically help to obtain de Sitter vacua, or to stabilise moduli. Unfortunately, deriving these theories from a compactified ten-dimensional (10d) supergravity has not been achieved so far. One reason is that non-geometric fluxes do not seem to match any 10d field, and another reason is the appearance of global issues in 10d non-geometric configurations.

After reviewing some background material, we present in this talk a solution to the two previous issues. Thanks to a field redefinition, we make the non-geometric Q-flux appear in a 10d action, which only differs from the NSNS action by a total derivative. In addition, this new action is globally well-defined, at least in some examples, and one can then perform the dimensional reduction to recover the 4d non-geometric potential. We also mention an application to the heterotic string.

Based on 1106.4015.