Past String Theory Seminar

Understanding the structure of hadrons in terms of their fundamental constituents requires an understanding of QCD at large distances, a vastly complex and unsolved dynamical problem. I will discuss in this talk a new approach to hadron structure based on superconformal quantum mechanics in the light-front and its holographic embedding in a higher dimensional gravity theory. This approach captures essential aspects of the confinement dynamics which are not apparent from the QCD Lagrangian, such as the emergence of a mass scale and confinement, the occurrence of a zero mode: the pion, universal Regge trajectories for mesons and baryons and precise connections between the light meson and nucleon spectra. This effective semiclassical approach to relativistic bound-state equations in QCD can be extended to heavy-light hadrons where heavy quark masses break the conformal invariance but the underlying dynamical supersymmetry holds.
 

A heterotic $G_2$ system is a quadruple $([Y,\varphi], [V, A], [TY,\theta], H)$ where $Y$ is a seven dimensional manifold with an integrable <br /> $G_2$ structure $\varphi$, $V$ is a bundle on $Y$ with an instanton connection $A$, $TY$ is the tangent bundle with an instanton connection $\theta$ and $H$ is a three form on $Y$ determined uniquely by the $G_2$ structure on $Y$. Further, H  is constrained so that it satisfies a condition that involves the Chern-Simons forms of $A$ and $\theta$, thus mixing the geometry of $Y$ with that of the bundles (this is the so called anomaly cancelation condition).  In this talk I will describe the tangent space of the moduli space of these systems. We first prove that a heterotic system is equivalent to an exterior covariant derivative $\cal D$ on the bundle ${\cal Q} = T^*Y\oplus {\rm End}(V)\oplus {\rm End}(TY)$ which satisfies $\check{\cal D}^2 = 0$ for some appropriately defined projection of the operator $\cal D$.  Remarkably, this equivalence implies the (Bianchi identity of) the anomaly cancelation condition. We show that the infinitesimal moduli space is given by the cohomology group $H^1_{\check{\cal D}}(Y, {\cal Q})$ and therefore it is finite dimensional.   Our analysis leads to results that are of relevance to all orders in $\alpha’$.  Time permitting, I will comment on work in progress about the finite deformations of heterotic $G_2$ systems and the relation to differential graded Lie algebras.

The gauged linear sigma model (GLSM) is a supersymmetric gauge theory in two dimensions which captures information about Calabi-Yaus and their moduli spaces. Recent result in supersymmetric localization provide new tools for computing quantum corrections in string compactifications. This talk will focus on the hemisphere partition function in the GLSM which computes the quantum corrected central charge of B-type D-branes. Several concrete examples of GLSMs and the application of the hemisphere partition function in the context of transporting D-branes in the Kahler moduli space will be given.

Calabi-Yau 3-folds play a large role in string theory.  Cohomology of sheaves on such varieties has many uses in string theory, including counting the number of particles or fields in a theory, as well as to help identify terms in the superpotential that determines the equations of motion of the corresponding string theory, and many other uses as well.  As a computational algebraic geometer, string theory provides a rich source of new computational problems to solve.

In this talk, we focus on the search for rigid divisors on these Calabi-Yau hypersurfaces of toric varieties.  We have had methods to compute sheaf cohomology on these varieties for many years now (Eisenbud-Mustata-Stillman, around 2000), but these methods fail for many of the examples of interest, in that they take a very long time, or the software (wisely) refuses to try!

We provide techniques and formulas for the sheaf cohomology of certain divisors of interest in string theory, that other current methods cannot handle.  Along the way, we describe a Macaulay2 package for computing with these objects, and show its use on examples.

This is joint work with Andreas Braun, Cody Long, Liam McAllister, and Benjamin Sung.

Localization and holography are powerful approaches to the computation of supersymmetric observables. The computations may, however, include divergences. Therefore, one needs renormalization schemes preserving supersymmetry. I will consider minimal gauged supergravity in five dimensions to demonstrate that the standard holographic renormalization scheme breaks supersymmetry, and propose a set of non-standard boundary counterterms that restore supersymmetry. I will then show that for a certain class of solutions the improved on-shell action correctly reproduces an intrinsic observable of four-dimensional SCFTs, the supersymmetric Casimir energy.

We begin by reviewing the “Gravity = Gauge x Gauge” paradigm that has emerged over the last decade. In particular, we will consider the origin of gravitational scattering amplitudes, symmetries and classical solutions in terms of the product of two Yang-Mills theories. Motivated by these developments we begin to address the classification of gravitational theories admitting a “factorisation” into a product of gauge theories. Progress in this direction leads us to twin supergravity theories - pair of supergravities with distinct supersymmetries, but identical bosonic sectors - from the perspective of Yang-Mills squared. 

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.

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.

A Fueter map between two hyperKaehler manifolds is a solution of a Cauchy-Riemann-type equation in the quaternionic context. In this talk I will describe relations between Fueter maps, generalized Seiberg-Witten equations, and Yang-Mills instantons on G2-manifolds (so called G2-instantons).

Calabi-Yau spaces provide well-understood examples of supersymmetric vacua in supergravity. The supersymmetry conditions on such spaces can be rephrased as the existence and integrability of a particular geometric structure. When fluxes are allowed, the conditions are more complicated and the analogue of the geometric structure is not well understood.
In this talk, I will review work that defines the analogue of Calabi-Yau geometry for generic D=4, N=2 supergravity backgrounds. The geometry is characterised by a pair of structures in generalised geometry, where supersymmetry is equivalent to integrability of the structures. I will also discuss the extension AdS backgrounds, where deformations of these geometric structures correspond to exactly marginal deformations of the dual field theories.

M5-branes on 4-manifolds M_4 realized as co-associatives in G_2 give rise to 2d (0,2) superconformal theories. In this talk I will propose a duality between these 2d (0,2) theories and 4d topological theories, which are sigma-models from M_4 into the Nahm moduli space. 

I will review the recently exposed connection between N=2 superconformal field theories in four dimensions and vertex operator algebras (VOAs). I will outline some general features of the VOAs that arise in this manner and describe the manner in which they reflect four-dimensional operations such as gauging and Higgsing. Time permitting, I will also touch on the modular properties of characters of these VOAs.

Pure spinor space, a cone over orthogonal Grassmannian OGr(5,10), is a central concept in the Berkovits formulation of string theory. The space of states of the beta-gamma system on pure spinors is tensor factor in the Hilbert space of string theory . This is why it would be nice to have a good definition of this space of states. This is not a straightforward task because of the conical singularity of the target. In the talk I will explain a strategy for attacking  conical targets. In the case of pure spinors the method gives a formula for partition function of pure spinors.

Generalised Geometry is a very powerful tool to study gravity duals of strongly coupled gauge theories. In this talk I will discuss how Exceptional Geometry can be used to study marginal deformations of N=1 SCFT's in 4 and 3 dimensions.

Heterotic vacua, defined with a holomorphic bundle and connection satisfying hermitian Yang-Mills, realise four-dimensional chiral gauge theories. We exploit the rich interplay between four-dimensional physics, supersymmetry and  geometry to construct a natural Kaehler metric for the moduli space, with a shockingly simple Kaehler potential. Along the way, we discover a natural geometric structure for the heterotic moduli.
 

Recently, there has been some progress in examining mirror symmetry beyond Calabi-Yau threefolds. I will discuss how this is related to flux vacua of type II supergravity on eight-dimensional manifolds equipped with SU(4)-structure. It will be shown that the natural framework to describe such vacua is generalized complex geometry. Two classes of type IIB solutions will be given, one of which is complex, the other symplectic, and I will describe in what sense these are mirror to one another.