Past String Theory Seminar

I will present a new approach to study the RG flow in 3d N=4 gauge theories, based on an analysis of the Coulomb branch of vacua. The Coulomb branch is described as a complex algebraic variety and important information about the strongly coupled fixed points of the theory can be extracted from the study of its singularities. I will use this framework to study the fixed points of U(N) and Sp(N) gauge theories with fundamental matter, revealing some surprising scenarios at low amount of matter.

We consider the one parameter mirror families W of the Calabi-Yau 3-folds with Picard-Fuchs  equations of hypergeometric type. By mirror symmetry the  even D-brane masses of orginial Calabi-Yau manifolds M can be identified with four periods with respect to an integral symplectic basis of $H_3(W,\mathbb{Z})$ at the point of maximal unipotent monodromy. We establish that the masses of the D4 and D2 branes at the conifold are given by the two algebraically independent values of the L-function of the weight four holomorphic Hecke eigenform with eigenvalue one of $\Gamma_0(N)$. For the quintic in  $\mathbb{P}^4$ it this Hecke eigenform of $\Gamma_0(25)$ was as found by Chad Schoen.  It was discovered  by de la Ossa, Candelas and Villegas that  its  coefficients $a_p$ count the number of  solutions of  the mirror quinitic at the conifold over the finite number field $\mathbb{F}_p$ . Using the theory of periods and quasi-periods of $\Gamma_0(N)$ and the special geometry pairing on Calabi-Yau 3 folds we can fix further values in the connection matrix between the maximal unipotent monodromy point and the conifold point.  

Any four dimensional N=2 superconformal field theory (SCFT) contains a subsector of local operator which is isomorphic to a two dimensional chiral algebra.  If the 4d theory possesses N= 4 superconformal symmetry, the corresponding chiral algebra is an extension of the (small) N=4 super-Virasoro algebra.  In this talk I  will present some results on the classification of N=4 chiral algebras and discuss the conditions they should satisfy in order to correspond to a 4d theory. 
 

We discuss mathematical studies on the Vafa-Witten theory, one of topological twists of N=4 super Yang-Mills theory in four dimensions, from the viewpoints of both differential and algebraic geometry. After mentioning backgrounds and motivation, we describe some issues to construct mathematical theory of this Vafa-Witten one, and explain possible ways to sort them out by analytic and algebro-geometric methods, the latter is joint work with Richard Thomas.

I will talk about a generalisation of the Seiberg-Witten equations introduced by Taubes and Pidstrygach, in dimension 3 and 4 respectively, where the spinor representation is replaced by a hyperKahler manifold admitting certain symmetries. I will discuss the 4-dimensional equations and their relation with the almost-Kahler geometry of the underlying 4-manifold. In particular, I will show that the equations can be interpreted in terms of a PDE for an almost-complex structure on 4-manifold. This generalises a result of Donaldson. 

I will talk about three-dimensional N=2 supersymmetric gauge theories on a class of Seifert manifold. More precisely, I will compute the supersymmetric partition functions and correlation functions of BPS loop operators on M_{g,p}, which is defined by a circle bundle of degree p over a genus g Riemann surface. I will also talk about four-dimensional uplift of this construction, which computes the generalized index of N=1 gauge theories defined on elliptic fiberation over genus g Riemann surface. We will find that the partition function or the index can be written as a sum over "Bethe vacua” of two-dimensional A-twisted theory obtained by a circle compactification. With this framework, I will show how the partition functions on manifolds with different topologies are related to each other. We will also find that these observables are very useful to study the action of Seiberg-like dualities on co-dimension two BPS operators.

Compactifications of 6D Superconformal Field Theories (SCFTs) on four-manidolfds lead to novel interacting 2D SCFTs. I will describe the various Lagrangian and non-Lagrangian sectors of the resulting 2D theories, as well as their interactions. In general this construction can be embedded in compactifications of the physical superstring, providing a general template for realizing 2D conformal field theories coupled to worldsheet gravity, i.e. a UV completion for non-critical string theories.  

Nekrasov, Rosly and Shatashvili observed that the generating function of a certain space of SL(2) opers has a physical interpretation as the effective twisted superpotential for a four-dimensional N=2 quantum field theory. In this talk we describe the ingredients needed to generalise this observation to higher rank. Important ingredients are spectral networks generated by Strebel differentials and the abelianization method. As an example we find the twisted superpotential for the E6 Minahan-Nemeschansky theory. 
 

A K3 surface is called attractive if and only if its Picard number is 20: The maximal possible. Attractive K3 surfaces possess complex multiplication. This property endows attractive K3 surfaces with rich and well understood arithmetic. For example, the associated Galois representation turns out to be a product of well known two dimensional representations and the  Hasse-Weil L-function turns out to be a product of well known L-functions. On the other hand, attractive K3 surfaces show up as solutions of the attractor equations in type IIB string theory compactified on the product of a K3 surface with an elliptic curve. As such, these surfaces dictate the near horizon geometry of a charged black hole in this theory. We will try to see which arithmetic properties of the attractive K3 surfaces lend a stringy interpretation and use them to shed light on physical properties of the charged black hole. 
 

Transport properties of liquids and gases in the regime of weak coupling (or effective weak coupling) are determined by the solutions of relevant kinetic equations for particles or quasiparticles, with transport coefficients being proportional to the minimal eigenvalue of the linearized kinetic operator. At strong coupling, the same physical quantities can sometimes be determined from dual gravity, where quasinormal spectra enter as the eigenvalues of the linearized Einstein's equations. We discuss the problem of interpolating between the two regimes using results from higher derivative gravity.

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.  

The black hole information paradox comes about because of the classical no-hair theorems for black holes. I will discuss soft black hole hair in electrodynamics and in gravitation. Then some speculations on its relevance to the in formation paradox are presented.

We establish a correspondence between the ABC Conjecture and N=4 super-Yang-Mills theory. This is achieved by combining three ingredients:

(i) Elkies' method of mapping ABC-triples to elliptic curves in his demonstration that ABC implies Mordell/Faltings;

(ii) an explicit pair of elliptic curve and associated Belyi map given by Khadjavi-Scharaschkin; and

(iii) the fact that the bipartite brane-tiling/dimer model for a gauge theory with toric moduli space is a particular dessin d'enfant in the sense of Grothendieck. 
 

We explore this correspondence for the highest quality ABC-triples as well as large samples of random triples. The Conjecture itself is mapped to a statement about the fundamental domain of the toroidal compactification of the string realization of N=4 SYM.

I will present several new  3d N=2 dualities with super-potentials involving monopole operators. Some of the theories that I will discuss describe systems of D3 branes ending on pq-webs. In these cases  3d mirror symmetry is a consequence of S-duality.

 

It was discovered in the 1970s that black holes are thermodynamic objects carrying entropy, thus suggesting that they are really an ensemble of microscopic states. This idea has been realized in a remarkable manner in string theory, wherein one can describe these ensembles in a class of models. These ensembles are known, however, to contain configurations other than isolated black holes, and it remains an outstanding problem to precisely isolate a black hole in the microscopic ensemble. I will describe how this problem can be solved completely in N=4 string theory. The solution involves surprising relations to mock modular forms -- a class of functions first discovered by S. Ramanujan about 95 years ago. 

The possibility of a landscape of metastable vacua raises the question of what fraction of vacua are truly long lived. Naively any would-be vacuum state has many nearby decay paths, and all possible decays must be suppressed. An interesting model of this phenomena consists of N scalars with a random potential of fourth order. We show that the scaling of the typical minimal bounce action with N is readily understood. We discuss the extension to more realistic landscape models as well as the effects of gravity. 

A large number of examples of compact G2 manifolds, relevant to supersymmetric compactifications of M-Theory to four dimensions, can be constructed by forming a twisted connected sum of two appropriate building blocks times a circle. These building blocks, which are appropriate K3-fibred threefolds, are shown to have a natural and elegant construction in terms of tops, which parallels the construction of Calabi-Yau manifolds via reflexive polytopes.

I will consider higher derivative corrections to the graviton 3-point coupling within a weakly coupled theory of gravity. Lorentz invariance allows further structures beyond that of Einstein’s theory. I will argue that these structures are constrained by causality, and show that the problem cannot be fixed by adding conventional particles with spins J ≤ 2, but adding an infinite tower of massive particles with higher spins. Implications of this result in the context of AdS/CFT, quantum gravity in asymptotically flat space-times, and non-Gaussianity features of primordial gravitational waves are discussed.

It is unknown whether a bound on axion field ranges exists within quantum gravity. We study axion field ranges using extended supersymmetry, in particular allowing an analysis within strongly coupled regions of moduli space. We apply this strategy to Calabi-Yau compactifications with one and two Kähler moduli. We relate the maximally allowable decay constant to geometric properties of the underlying Calabi-Yau geometry. In all examples we find a maximal field range close to the reduced Planck mass (with the largest field range being 3.25 $M_P$). On this perspective, field ranges relate to the intersection and instanton numbers of the underlying Calabi-Yau geometry.

I will give an introductory account of the zeta-functions for one-parameter families of CY manifolds. The aim of the talk is to point out that the zeta-functions corresponding to singular manifolds of the family correspond to modular forms. In order to give this introductory account I will give a lightning review of finite fields and of the p-adic numbers.

It has been recently pointed out that maximal gauged supergravities in four dimensions often come in one-parameter families. The parameter measures the combination of electric and magnetic vectors that participate in the gauging. I will discuss the higher-dimensional origin of these dyonic gaugings, when the gauge group is chosen to be ISO(7). This gauged supergravity arises from consistent truncation of massive type IIA on the six-sphere, with its dyonically-gauging parameter identified with the Romans mass. The (AdS) vacua of the 4D supergravity give rise to new explicit AdS4 backgrounds of massive type IIA. I will also show that the 3D field theories dual to these AdS4 solutions are Chern-Simons-matter theories with a simple gauge group and level k also given by the Romans mass.

Two interesting properties of static curved space QFTs are Casimir Energies, and the Energy Gaps of fluctuations. We investigate what AdS/CFT has to say about these properties by examining holographic CFTs defined on curved but static spatially closed spacetimes. Being holographic, these CFTs have a dual gravitational description under Gauge/Gravity duality, and these properties of the CFT are reflected in the geometry of the dual bulk.  We can turn this on its head and ask, what does the existence of the gravitational bulk dual imply about these properties of the CFTs? In this talk we will consider holographic CFTs where the dual vacuum state is described by pure Einstein gravity with negative cosmological constant.  We will argue using the bulk geometry first, that if the CFT spacetime's spatial scalar curvature is positive there is a lower bound on the gap for scalar fluctuations, controlled by the minimum value of the boundary Ricci scalar. In fact, we will show that it is precisely the same bound as is satisfied by free scalar CFTs, suggesting that this bound might be something that applies more generally than just in a Holographic context. We will then show, in the case of 2+1 dimensional CFTs, that the Casimir energy is non-positive, and is in fact negative unless the CFT's scalar curvature is constant. In this case, there is no restriction on the boundary scalar curvature, and we can even allow singularities in the bulk, so long as they are 'good' singularities. If time permits, we will also describe some new results about the Hawking-Page transition in this context.