## John Voight : On Moduli of Nondegenerate Curves

- String Theory ( 244 Views )We study the conditions under which an algebraic curve can be modelled by a Laurent polynomial that is nondegenerate with respect to its Newton polytope. Such nondegenerate polynomials have become popular objects in explicit algebraic geometry, owing to their connection with toric geometry; however, despite their ubiquity, the intrinsic property of nondegeneracy has not seen much detailed study. We prove that every curve of genus $g \geq 4$ over an algebraically closed field is nondegenerate in the above sense. More generally, let $\mathcal{M}_g^{\textup{nd}}$ be the locus of nondegenerate curves inside the moduli space of curves of genus $g \geq 2$. Then we show that $\dim \mathcal{M}_g^{\textup{nd}} = \min(2g+1,3g-3)$, except for $g=7$ where $\dim \mathcal{M}_7^{\textup{nd}} = 16$; thus, a generic curve of genus $g$ is nondegenerate if and only if $g \geq 4$

## Brian Wecht : Towards a c-theorem in Four Dimensions

- String Theory ( 33 Views )Intuitively, renormalization group flow is irreversible, since one would expect the number of massless degrees of freedom of a theory to decrease as one goes to progressively larger distance scales. This has been proven rigorously in two dimensions (the "c-theorem"), but is otherwise an open problem. I will discuss recent efforts to prove an analogous theorem in four dimensions for supersymmetric theories, for which we can make exact statements not usually accessible in their non-supersymmetric counterparts.

## David Morrison : Introduction to F-Theory, V

- String Theory ( 32 Views )This is the fifth of a series of lectures on F-theory. The lectures will present the theory of elliptically fibered Calabi-Yau manifolds in considerable detail, explain how these manifolds are used to produce string vacua by means of the ``F-theory'' construction, and how various properties of these string vacua are determined by the corresponding Calabi-Yau manifolds.

## Ronen Plesser : `Toric duality is Seiberg duality

- String Theory ( 31 Views )We study four N=1 SU(N)^6 gauge theories, with bi-fundamental chiral matter and a superpotential. In the infrared, these gauge theories all realize the low-energy world-volume description of N coincident D3-branes transverse to the complex cone over a del Pezzo surface dP_3 which is the blowup of P^2 at three generic points. Therefore, the four gauge theories are expected to fall into the same universality class--an example of a phenomenon that has been termed "toric duality." However, little independent evidence has been given that such theories are infrared-equivalent. In fact, we show that the four gauge theories are related by the N=1 duality of Seiberg, vindicating this expectation. We also study holographic aspects of these gauge theories. In particular we relate the spectrum of chiral operators in the gauge theories to wrapped D3-brane states in the AdS dual description. We finally demonstrate that the other known examples of toric duality are related by N=1 duality, a fact which we conjecture holds generally.

## Andy Neitzke : Elements of Topological M-Theory

- String Theory ( 30 Views )I will discuss some action functionals constructed by Hitchin which lead to manifolds of special holonomy in six and seven dimensions, and explain how these actions could be related to a topological version of M-theory which deals with variations of G2 structures.

## Eric Sharpe : Kähler Cone Substructure

- String Theory ( 30 Views )To define a consistent perturbative geometric heterotic compactification one must specify not only a Calabi-Yau manifold M but also a bundle E on M. The bundle E is required to satisfy a subtle constraint known as ``stability,'' which depends upon the Kähler form. This dependence upon the Kähler form is highly nontrivial---the Kähler cone splits into subcones, with a distinct moduli space of bundles in each subcone---and has long been overlooked by physicists. In this talk we shall describe this behavior and its physical manifestation.

## Vijay Balasubramanian : Five Dimensions from Four: On Dimensional Transmogrification

- String Theory ( 26 Views )Recent developments have suggested that the classical physics of gravity in a given number of dimensions can be related to a non-gravitational theory in a lower number of dimensions. I will argue that the radial direction of five-dimensional AdS space can be generated by renormalization group (RG) flow from a dual four-dimensional theory. First, I will show that integrating out the physics outside finite balls in AdS spacetime generates a family of actions that satisfies an RG-like equation. Next, I will argue that the trace of the AdS spacetime stress tensor contains analogues of field theory beta functions and gives a gravitational analogue of an RG equation. Turning things around, I will show that the conformal anomaly computed from the four-dimensional field theory dual to AdS space completely encodes the gravitational physics of the skin of the five-dimensional spacetime. Finally, I will argue that recovering the deep interior of the five-dimensional spacetime from the four-dimensional field theory will require exotic new approaches to the renormalization group.

## Albion Lawrence : D-Branes on Calabi-Yau Threefolds

- String Theory ( 26 Views )D-branes on Calabi-Yau threefolds are interesting both as alternative probes of quantum geometry, and as nontrivial realizations of gauge theories with four supercharges. Of course, these issues are not independent. In this talk I will briefly discuss the former by describing and characterizing D-brane states at the Gepner point in the moduli space of the quintic. I will then use such D-branes to realize d=4 N=1 SUSY gauge theories, and discuss the computation of the superpotential of these theories, particularly for those realized by D6-branes wrapped around special Lagrangian submanifolds of the threefold. I will close with some speculations on the implications of mirror symmetry for these superpotentials.

## Rahul Pandharipande : Integers in the Gromov-Witten Theory of 3-folds

- String Theory ( 26 Views )Abstract: Gopakumar and Vafa have proposed a formula for the Gromov-Witten invariants of Calabi-Yau 3 folds in terms of numbers of BPS states. Part of this formula is explained by higher genus multiple covers and collapsed contributions in Gromov-Witten theory. The mathematical analysis leads to a direct generalization of Gopakumar and Vafa's formula to an integral structure for arbitrary 3 folds. I will discuss a mathematical approach to these questions via integrals over the moduli space of stable curves.