Forthcoming events in this series
12:45
Gauge Theory and Boundary Integrability
Abstract
Costello Yamazaki and Witten have proposed a new understanding of quantum integrable systems coming from a variant of Chern-Simons theory living on a product of two Riemann surfaces. I’ll review their work, and show how it can be extended to the case of integrable systems with boundary. The boundary Yang-Baxter Equations, twisted Yangians and Sklyanin determinants all have natural interpretations in terms of line operators in the theory.
12:45
The Laplacian flow in G_2 geometry
Abstract
Finding Riemannian metrics with holonomy G_2 is a challenging problem with links in mathematics to Einstein metrics and area-minimizing submanifolds, and to M-theory in theoretical physics. I will provide a brief survey on recent progress towards studying this problem using a geometric flow approach, including connections to calibrated fibrations.
12:45
String theory compactifications with sources
In recent years, more and more compactifications have emerged whose existence depends crucially on the presence of internal sources to the supergravity fields, such as D-branes and orientifold planes. I will review some solutions of this type in various dimensions, and illustrate their applications to holography and potentially to the problem of finding de Sitter solutions.
12:45
Large-N Non-Supersymmetric 6D CFTs: Hologram or Mirage?
Abstract
In this talk I will present a large class of non-supersymmetric AdS7 solutions of IIA supergravity, and their (in)stabilities. I will start by reviewing supersymmetric AdS7 solutions of 10D supergravity dual to 6D (1,0) SCFTs. I will then focus on their non-supersymmetric counterpart, discussing how they are related. The connection between supersymmetric and non-supersymmetric solutions leads to a hint for the SUSY breaking mechanism, which potentially allows to evade some of the assumptions of the Ooguri-Vafa Conjecture about the AdS landscape. A big subset of these solutions shows a curious pattern of perturbative instabilities whenever many open-string modes are considered. On the other hand an infinite class remains apparently stable.
12:45
Unveiling the mysteries of the E-string with Calabi-Yau geometry
Abstract
The E-string theory is usually considered as the simplest among 6D (1,0) superconformal field theories. Nonetheless, we still have little information about its spectrum of operators. In this talk, I'm going to describe our recent geometric approach using F-theory compactification on an elliptic Calabi-Yau threefold. The elliptic fibration is non-flat, which means that there are complex surface components in the fiber direction. From the geometry of non-flat fiber, we read out an infinite tower of particle states in the E-string theory. I will also discuss its relevance to 4D standard model building, which is a main motivation of this work.
12:45
SU(3) structures on Calabi-Yau manifolds
Abstract
In this talk, we show that a range of non-trivial SU(3) structures can be constructed on large classes of Calabi-Yau three-folds. Among the possible SU(3) structures we find Strominger-Hull systems, suitable for heterotic or type II string compactifications. These SU(3) structures of Strominger-Hull type have a non-vanishing and non-closed three-form flux which needs to be supported by source terms in the associated Bianchi identity. We discuss the possibility of finding such source terms and present first steps towards their explicit construction. Provided suitable sources exist, our methods lead to Calabi-Yau compactifications of string theory with a non Ricci-flat, physical metric which can be written down explicitly and in analytic form. The talk is based on the paper 1805.08499.
12:45
Periods, zeta-functions and attractor varieties
Abstract
The zeta-function of a manifold varies with the parameters and may be evaluated in terms of the periods. For a one parameter family of CY manifolds, the periods satisfy a single 4th order differential equation. Thus there is a straight and, it turns out, readily computable path that leads from a differential operator to a zeta-function. Especially interesting are the specialisations to singular manifolds, for which the zeta-function manifests modular behaviour. We are also able to find, from the zeta function, attractor points. These correspond to special values of the parameter for which there exists a 10D spacetime for which the 6D corresponds to a CY manifold and the 4D spacetime corresponds to an extremal supersymmetric black hole. These attractor CY manifolds are believed to have special number theoretic properties. This is joint work with Xenia de la Ossa, Mohamed Elmi and Duco van Straten.
16:00
General lessons on 4d SCFTs from Geometry
Abstract
The geometry of the moduli space of 4d N=2 moduli spaces, and in particular of their Coulomb branches (CBs), is very constrained. In this talk I will show that through its careful study, we can learn general and somewhat surprising lessons about the properties of N=2 super conformal field theories (SCFTs). Specifically I will show that we can prove that the scaling dimension of CB coordinates, and thus of the corresponding operator at the SCFT fixed point, has to be rational and it has a rank-dependent maximum value and that in general the moduli spaces of N=2 SCFTs can have metric singularities as well as complex structure singularities.
Finally I will outline how we can explicitly perform a classification of geometries of N>=3 SCFTs and carry out the program up to rank-2. The results are surprising and exciting in many ways.
12:45
Loop Amplitudes in the Scattering Equations Formalism
Abstract
I will describe recent progress in the study of scattering amplitudes in gauge theory and gravity at loop level, using the formalism of the scattering equations. The scattering equations relate the kinematics of the scattering of massless particles to the moduli space of the sphere. Underpinned by ambitwistor string theory, this formalism provides new insights into the relation between tree-level and loop-level contributions to scattering amplitudes. In this talk, I will describe results up to two loops on how loop integrands can be constructed as forward-limits of trees. One application is the loop-level understanding of the colour-kinematics duality, a symmetry of perturbative gauge theory which relates it to perturbative gravity.
12:45
Tinkertoys for E₈ (and related matters)
Abstract
I will review some recent progress on D=4, N=2 superconformal field theories in what has come to be known as "Class-S". This is a huge class of (mostly non-Lagrangian) SCFTs, whose properties are encoded in the data of a punctured Riemann surface and a collection (one per puncture) of nilpotent orbits in an ADE Lie algebra.
12:45
Tensionless Strings and Quantum Gravity Conjectures
Abstract
We test various conjectures on quantum gravity for general 6d string compactifications in the framework of F-theory. Starting with a gauge theory coupled to gravity, we first analyze the limit in Kähler moduli space where the gauge coupling tends to zero while gravity is kept dynamical. A key observation is made about the appearance of a tensionless string in such a limit. For a more quantitative analysis, we focus on a U(1) gauge symmetry and determine the elliptic genus of this string in terms of certain meromorphic weak Jacobi forms, of which modular properties allow us to determine the charge-to-mass ratios of certain string excitations. A tower of these asymptotically massless charged states are then confirmed to satisfy the (sub-)Lattice Weak Gravity Conjecture, the Completeness Conjecture, and the Swampland Distance Conjecture. If time permits, we interpret their charge-to-mass ratios in two a priori independent perspectives. All of this is then generalized to theories with multiple U(1)s.
12:45
Twisted BRST quantization and localization in supergravity
Abstract
Supersymmetric localization is a powerful technique to evaluate a class of functional integrals in supersymmetric field theories. It reduces the functional integral over field space to ordinary integrals over the space of solutions of the off-shell BPS equations. The application of this technique to supergravity suffers from some problems, both conceptual and practical. I will discuss one of the main conceptual problems, namely how to construct the fermionic symmetry with which to localize. I will show how a deformation of the BRST technique allows us to do this. As an application I will then sketch a computation of the one-loop determinant of the super-graviton that enters the localization formula for BPS black hole entropy.
12:45
Infrared enhancement of supersymmetry in four dimensions
Abstract
In this seminar I will discuss a recently-found class of RG flows in four dimensions exhibiting enhancement of supersymmetry in the infrared, which provides a lagrangian description of several strongly-coupled N=2 SCFTs. The procedure involves starting from a N=2 SCFT, coupling a chiral multiplet in the adjoint representation of the global symmetry to the moment map of the SCFT and turning on a nilpotent expectation value for this chiral. We show that, combining considerations based on 't Hooft anomaly matching and basic results about the N=2 superconformal algebra, it is possible to understand in detail the mechanism underlying this phenomenon and formulate a simple criterion for supersymmetry enhancement.
12:45
Higgs bundles, branes, and application
Abstract
Higgs bundles are pairs of holomorphic vector bundles and holomorphic 1-forms taking values in the endomorphisms of the bundle. Their moduli spaces carry a natural Hyperkahler structure, through which one can study Lagrangian subspaces (A-branes) or holomorphic subspaces (B-branes). Notably, these A and B-branes have gained significant attention in string theory. After introducing Higgs bundles and the associated Hitchin fibration, we shall look at natural constructions of families of different types of branes, and relate these spaces to the study of 3-manifolds, surface group representations and mirror symmetry.
12:45
Modular graph functions as iterated Eisenstein integrals
Abstract
Superstring scattering amplitudes in genus one have a low-energy expansion in terms of certain real analytic modular forms, called modular graph functions (D'Hoger, Green, Gürdogan and Vanhove). I will sketch the proof that these functions belong to a family of iterated integrals of modular forms (a generalization of Eichler integrals), recently introduced by Francis Brown, which explains many of their properties. The main tools are elliptic multiple polylogarithms (Brown and Levin), single-valued versions thereof, and elliptic multiple zeta values (Enriquez).
12:45
Twisted indices of 3d N=4 theories and moduli space of quasi-maps
Abstract
I will talk about the Witten index of supersymmetric quantum mechanics obtained from 3d gauge theories compacted on a Riemann surface. In particular, I will show that the twisted indices of 3d N=4 theories compute enumerative invariants of the moduli space, which can be identified as a space of quasi-maps to the Higgs branch. I will also discuss 3d mirror symmetry in this context which provides a non-trivial relation between a pair of generating functions of the invariants.
12:45
Supersymmetric partition functions on Seifert manifolds from line defects
Abstract
As we have learned over the last 10 years, many exact results for various observables in three-dimensional N=2 supersymmetric theories can be extracted from the computation of "supersymmetric partition functions" on curved three-manifold M_3, for instance on M_3= S^3 the three-sphere. Typically, such computations must be carried anew for each M_3 one might want to consider, and the technical difficulties mounts as the topology of M_3 gets more involved. In this talk, I will explain a different approach that allows us to compute the partition function on "almost" any half-BPS geometry. The basic idea is to relate different topologies by the insertion of certain half-BPS line defects, the "geometry-changing line operators." I will also explain how our formalism can be related to the Beem-Dimofte-Pasquetti holomorphic blocks. [Talk based on a paper to appear in a week, with Heeyeon Kim and Brian Willett.]
11:00
Goursat rank 4 rigid local systems
Abstract
In the late 1880's Goursat investigated what we now call rigid local systems, classically described as linear differential equations without accessory parameters. In this talk I will discuss some arithmetic and geometric aspects of certain particular cases of Goursat's in rank four. For example, I will discuss what are likely to be all cases where the monodromy group is finite. This is joint work with Danylo Radchenko.
12:45
(0,2) dualities and 4-simplices
Abstract
3d N=2 Chern-Simons-matter theories have a large variety of boundary conditions that preserve 2d N=(0,2) supersymmetry, and support chiral algebras. I'll discuss some examples of how the chiral algebras transform across dualities. I'll then explain how to construct duality interfaces in 3d N=2 theories, and relate dualities *of* duality interfaces to "Pachner moves" in triangulations of 4-manifolds. Based on recent and upcoming work with K. Costello, D. Gaiotto, and N. Paquette.
12:45
Modular properties of supersttring scattering amplitudes,
Abstract
The coefficients of the low energy expansion of closed string amplitudes transform as automorphic functions under En(Z) U-duality groups.
The seminar will give an overview of some features of the coefficients of low order terms in this expansion, which involve a fascinating interplay between multiple zeta values and certain elliptic and hyperelliptic generalisations, Langlands Eisenstein series for the En groups, and the ultraviolet behaviour of maximally supersymmetric supergravity.
12:45
Exotic Rational Conformal Field Theories and the Modular Bootstrap
Abstract
I will summarise old and recent developments on the classification and solution of Rational Conformal Field Theories in 2 dimensions using the method of Modular Differential Equations. Novel and exotic theories are found with small numbers of characters and simple fusion rules, one of these being the Baby Monster CFT. Correlation functions for many of these theories can be computed using crossing-symmetric differential equations.
12:45
Trace Anomalies and Boundary Conformal Field Theory
Abstract
The central charges “c” and “a” in two and four dimensional conformal field theories (CFTs) have a central organizing role in our understanding of quantum field theory (QFT) more generally. Appearing as coefficients of curvature invariants in the anomalous trace of the stress tensor, they constrain the possible relationships between QFTs under renormalization group flow. They provide important checks for dualities between different CFTs. They even have an important connection to a measure of quantum entanglement, the entanglement entropy. Less well known is that additional central charges appear when there is a boundary, four new coefficients in total in three and four dimensional boundary CFTs. While largely unstudied, these boundary charges hold out the tantalizing possibility of being as important in the classification of quantum field theory as the bulk central charges “a” and “c”. I will show how these charges can be computed from displacement operator correlation functions. I will also demonstrate a boundary conformal field theory in four dimensions with an exactly marginal coupling where these boundary charges depend on the marginal coupling. The talk is based on arXiv:1707.06224, arXiv:1709.07431, as well as work to appear shortly.
12:45
A Ringel-Hall type construction of vertex algebras
Abstract
Suppose A is a nice abelian category (such as coherent sheaves coh(X) on a smooth complex projective variety X, or representations mod-CQ of a quiver Q) or T is a nice triangulated category (such as D^bcoh(X) or D^bmod-CQ) over C. Let M be the moduli stack of objects in A or T. Consider the homology H_*(M) over some ring R.
Given a little extra data on M, for which there are natural choices in our examples, I will explain how to define the structure of a graded vertex algebra on H_*(M). By a standard construction, one can then define a graded Lie algebra from the vertex algebra; roughly speaking, this is a Lie algebra structure on the homology H_*(M^{pl}) of a "projective linear” version M^{pl} of the moduli stack M.
For example, if we take T = D^bmod-CQ, the vertex algebra H_*(M) is the lattice vertex algebra attached to the dimension vector lattice Z^{Q_0} of Q with the symmetrized intersection form. The degree zero part of the graded Lie algebra contains the associated Kac-Moody algebra.
The construction appears to be new, but is connected with a lot of work in Geometric Representation Theory, to do with Ringel-Hall-type algebras and their representations, such as the results of Grojnowski-Nakajima on Hilbert schemes. The vertex algebra construction is enormously general, and applies in huge classes of examples. There is a differential-geometric version too.
The question I am hoping someone in the audience will answer is this: what is the physical interpretation of these vertex algebras?
It is in some sense an "even Calabi-Yau” construction: when applied to coh(X) or D^bcoh(X), it is most natural for X a Calabi-Yau 2-fold or Calabi-Yau 4-fold, and is essentially trivial for X a Calabi-Yau 3-fold. I discovered it when I was investigating wall-crossing for Donaldson-Thomas type invariants for Calabi-Yau 4-folds. So perhaps one should look for an explanation in the physics of Calabi-Yau 2-folds or 4-folds, with M the moduli space of boundary conditions for the associated SCFT.
12:45
Algebraic systems biology: comparing models and data.
Abstract
I will overview my research for a general math audience.
First I will present the biological questions and motivate why systems biology needs computational algebraic biology and topological data analysis. Then I will present the mathematical methods I've developed to study these biological systems. Throughout I will provide examples.