We study D3 brane theories that are described as deformations of N=2 SCFTs. They arise at the self-intersection of a 7-brane in F-Theory. As we shall explain, the associated string junctions and their monodromies can be studied via torus knots or links. The monodromy reduces (potentially different) flavor algebras of dual deformations of N=2 theories and projects out charged states, leading to N=1 SCFTs. We propose an explanation for these effects in terms of an electron-monopole-dyon condensate.

# Past String Theory Seminar

Mirror symmetry predicts surprising geometric correspondences between distinct families of algebraic varieties. In some cases, these correspondences have arithmetic consequences. Among the arithmetic correspondences predicted by mirror symmetry are correspondences between point counts over finite fields, and more generally between factors of their Zeta functions. In particular, we will discuss our results on a common factor for Zeta functions alternate families of invertible polynomials. We will also explore closed formulas for the point counts for our alternate mirror families of K3 surfaces and their relation to their Picard–Fuchs equations. Finally, we will discuss how all of this relates to hypergeometric motives. This is joint work with: Charles Doran (University of Alberta, Canada), Tyler Kelly (University of Cambridge, UK), Steven Sperber (University of Minnesota, USA), John Voight (Dartmouth College, USA), and Ursula Whitcher (American Mathematical Society, USA).

Recently, millions of novel examples of compact G2 holonomy manifolds have been constructed as twisted connected sums of asymptotically cylindrical Calabi-Yau threefolds. In case these are K3 fibred, they can in turn be constructed from dual pairs of tops. This is analogous to Batyrev's construction of Calabi-Yau manifolds via reflexive polytopes. For compactifications of Type II superstrings on such G2 manifolds, we formulate a construction of the mirror.

Automorphic forms arise naturally when studying scattering amplitudes in toroidal compactifications of string theory. In this talk, I will summarize the conditions on four-graviton amplitudes from the literature required by U-duality, supersymmetry and string perturbation theory, which are satisfied by certain Eisenstein series on exceptional Lie groups. Physical information, such as instanton effects, are encoded in their Fourier coefficients on parabolic subgroups, which are, in general, difficult to compute. I will demonstrate a method for evaluating certain Fourier coefficients of interest in string theory. Based on arXiv:1511.04265, arXiv:1412.5625 and work in progress.

A conformal field theory is characterised by the CFT data, namely the spectrum of scaling dimensions and OPE coefficients. The idea of the conformal bootstrap is to use associativity of the operator algebra together with the symmetries of the theory to constraint the CFT data. For the sector of operators with large spin one can actually use these ideas to obtain analytical results. It was recently understood how to set up a systematic expansion around this sector, leading to analytic results to all orders in inverse powers of the spin. We will show how to use this large spin perturbation theory to obtain analytic results for vast families of CFTs. Some of the applications include vector models, weakly coupled gauge theories and the computation of loops for scalar theories in AdS.

The ambitwistor string of Mason and Skinner has been very successful in describing field theory amplitudes, at both loop and tree-level for a variety of theories. But the original action given by Mason and Skinner is already partially gauge-fixed, which obscures some issues related to modular invariance and the connection to conventional string theories. In this talk I will argue that the Null string is the ungauge-fixed version of the Ambitwistor string. This clarifies the geometry of the original Ambitwistor string and gives a road map to understanding modular invariance, and gives new formulas for loop amplitudes in which we expect that UV divergences will be easier to analyse.

In this talk I will argue that a systematic classification of 4d N=2 superconformal field theories is possible through a careful analysis of the geometry of their Coulomb branches. I will carefully describe this general framework and then carry out the classification explicitly in the rank-1, that is one complex dimensional Coulomb branch, case. We find that the landscape of rank-1 theories is still largely unexplored and make a strong case for the existence of many new rank-1 SCFTs, almost doubling the number of theories already known in the literature. The existence of 4 of them has been recently confirmed using alternative methods and others have an enlarged N=3, supersymmetry.

While our study focuses on Coulomb Branch geometries, we can extract much more information about these SCFTs. I will spend the last part of my talk outlining what else we can learn and the extent in which our study can be complementary to other method to study SCFTs (Conformal Bootstrap above all!).

In this talk I will introduce methods to use 2d gauge theories as a means to describe Calabi-Yau varieties and their moduli spaces. As I review, this description furnishes a natural framework to predict derived equivalences between pairs of (sometimes even non-birational) Calabi-Yau varieties. A prominent example of this kind is realized by the Rødland non-birational pair of Calabi-Yau threefolds.

Using the 2d gauge theory description, I will propose further examples of derived equivalences among non-birational Calabi-Yau varieties.

In 3d gauge theories, monopole operators create and destroy vortices. I will explore this idea in the context of 3d N = 4 supersymmetric gauge theories and explain how it leads to an exact calculation of quantum corrections to the Coulomb branch and a finite version of the AGT correspondence.

*In this talk I will discuss some features of interacting conformal higher spin theories in 2+1 dimensions. This is done in the context of Chern-Simons theory (giving e.g. the complete spin 2 covariant spin 3 sector) and a higher spin coupled unfolded equation for the scalar singleton. One motivation for studying these theories is that their non-linear properties are rather poorly understood contrary to the situation for the Vasiliev type theory in this dimension which is under much better control. Another reason for the interest in these theories comes from AdS4/CFT3 and the possibility that Neumann/mixed bc for bulk higher spin fields may lead to conformal higher spin fields governed by Chern-Simons terms on the boundary. These theories generalise the spin 2 gauged BLG-ABJ(M) theories found a few years ago to higher spins than 2.*