## Western Hemisphere Colloquium on Geometry and Physics (WHCGP)

This biweekly online colloquium features geometers and physicists presenting current research on a wide range of topics in the interface of the two fields. The talks are aimed at a broad audience. They will take place via Zoom on alternate Mondays at 3pm Eastern, noon Pacific, 4pm BRT. Each session features a 60 minute talk, followed by 15 minutes for questions and discussion. You may join the meeting 15 minutes in advance. Questions and comments may be submitted to the moderator via the chat interface during the talk, or presented in person during the Q&A session. These colloquia will be recorded and will be available (linked from this page) asap after the event.

As an alternative to Zoom, you may watch a live stream of the lecture at our: YouTube streaming site.

Organizing committee: Tudor Dimofte, Ron Donagi, Dan Freed, Sheldon Katz, Dave Morrison, Andy Neitzke.

(Indexed at researchseminars.org.)

Go to Past Talks.

November 2, 2020

### Christopher Beem (Oxford)

November 16, 2020

### Sergei Gukov (Cal Tech)

November 30, 2020

### Sourav Chatterjee (Stanford)

December 14, 2020

April 13, 2020

April 27, 2020

### Topological strings, twistors, and Skyrmions

Abstract: It has long been known that holomorphic field theories on twistor space lead to "physical" field theories on Minkowski space. In this talk I will discuss a type I (unoriented) version of the topological B model on twistor space. The corresponding theory on Minkowski space is a sigma-model with target the group SO(8). This is a variant of the Skyrme model that appears as the low-energy effective theory of mesons in QCD. (The group SO(8) appears because of the Green-Schwarz mechanism in the topological string). The origin of this model in the topological string implies many remarkable properties. For one thing, the model is, in a certain sense, integrable. Further, although the Lagrangian is power-counting non-renormalizable, counter-terms at all loops can be uniquely fixed.

May 11, 2020

### Intrinsic Mirror Symmetry

Abstract: I will talk about joint work with Bernd Siebert, proposing a general mirror construction for log Calabi-Yau pairs, i.e., a pair (X,D) with D a "maximally degenerate" boundary divisor and K_X+D=0, and for maximally unipotent degenerations of Calabi-Yau manifolds. We accomplish this by constructing the coordinate ring or homogeneous coordinate ring respectively in the two cases, using certain kinds of Gromov-Witten invariants we call "punctured invariants", developed jointly with Abramovich and Chen.

May 18, 2020

### Quantum Modularity from 3-Manifolds

Abstract: Quantum modular forms are functions on rational numbers that have rather mysterious weak modular properties. Mock modular forms and false theta functions are examples of holomorphic functions on the upper-half plane which lead to quantum modular forms. Inspired by the 3d-3d correspondence in string theory, a new topological invariants named homological blocks for (in particular plumbed) three-manifolds have been proposed a few years ago. My talk aims to explain the recent observations on the quantum modular properties of the homological blocks, as well as the relation to logarithmic vertex algebras. The talk will be based on a series of work in collaboration with Sungbong Chun, Boris Feigin, Francesca Ferrari, Sergei Gukov, Sarah Harrison, and Gabriele Sgroi.

June 1, 2020

### Integrable Kondo problems and affine Geometric Langlands

Abstract: I will present some work on integrable line defects in WZW models and their relation to 4d CS theory, the IM/ODe correspondence and affine generalizations of Geometric Langlands constructions.

June 15, 2020

### Space-time analyticity in QFT

Abstact: I will talk on a joint work with Graeme Segal. We propose a new axiomatics for unitary quantum field theory which includes both Lorentzian and Euclidean signatures for curved space-time manifolds. The key to the definition is certain open domain in the space of complex-valued symmetric bilinear forms on a real vector space. The justification comes from holomorphic convexity (lower bound) and from higher gauge theories (upper bound).

June 22, 2020

### From gapped phases of matter to Topological Quantum Field Theory and back again

Abstract: I will review the connection between gapped phases of matter and Topological Quantum Field Theory (TQFT). Conjecturally, this connection becomes 1-1 correspondence if one restricts to a special class of phases and TQFTs (namely, invertible ones). A related conjecture is that the space of all lattice Hamiltonians describing Short-Range Entangled phases of matter is an infinite loop space. These conjectures predict that the space of lattice Hamiltonians has non-trivial cohomology in particular dimensions. We test this by constructing closed differential forms on the space of gapped lattice Hamiltonians following a suggestion by Kitaev. These differential forms can be regarded as a higher-categorical generalization of the curvature of the Berry connection and correspond to Wess-Zumino-Witten forms in field theory.

June 29, 2020

July 6, 2020

### Spacetime, Quantum Mechanics and Clusterhedra at Infinity

Abstract: Elementary particle scattering is perhaps the most basic physical process in Nature. The data specifying the scattering process defines a "kinematic space", associated with the on-shell propagation of particles out to infinity. By contrast the usual approach to computing scattering amplitudes, involving path integrals and Feynman diagrams, invokes auxilliary structures beyond this kinematic space--local interactions in the interior of spacetime, and unitary evolution in Hilbert space. This description makes space-time locality and quantum-mechanical unitarity manifest, but hides the extraordinary simplicity and infinite hidden symmetries of the amplitude that have been uncovered over the past thirty years. The past decade has seen the emergence of a new picture, where scattering amplitudes are seen as the answer to an entirely different sort of mathematical question involving "positive geometries" directly in the kinematic space, making surprising connections to total positivity, combinatorics and geometry of the grassmannian, and cluster algebras. The hidden symmetries of amplitudes are made manifest in this way, while locality and unitarity are seen as derivative notions, arising from the "factorizing" boundary structure of the positive geometries. This was first see in the story of "amplituhedra" and scattering amplitudes in planar N=4 SYM theory. In the past few years, a similar structure has been seen for non-superysmmetric "bi-adjoint" scalar theories with cubic interactions, in any number of dimensions. The positive geometries through to one-loop order are given by "cluster polytopes"--generalized associahedra for finite-type cluster algebras--with a simple description involving "dynamical evolution" in the kinematic space. Extending these ideas involves understanding cluster algebras associated with triangulations of general Riemann surfaces. These cluster algebras are infinite, reflecting the infinite action of mapping class group. One of the manifestations of this infinity is that the "g-vector fan" of the cluster algebra is not space-filling, making it impossible to define cluster polytopes, and obstructing the connection with positive geometries and scattering amplitudes. Remarkably, incorporating non-cluster variables, associated with closed loops in the Riemann surfaces, suggests a natural way of modding out by the mapping class group, canonically compactifying the cluster complex, and associating it with "clusterhedron" polytopes. Clusterhedra are conjectured to exist for all surfaces, providing the positive geometry in kinematic space for scattering amplitudes in the bi-adjoint scalar theory to all loop orders and all orders in the 1/N expansion. In this talk I will give a simple, self-contained overview of this set of ideas, assuming no prior knowledge of scattering amplitudes or cluster algebras.

July 13, 2020

### Knot categorification from mirror symmetry, via string theory

Abstract: I will describe two approaches to categorifying quantum link invariants which work uniformly for all simple Lie algebras, and originate from geometry and string theory. A key aspect of both approaches is that it is manifest that decategorification gives the quantum link invariants one set out to categorify. Many ingredients that go into the story have been found by mathematicians earlier, but string theory spells out how they should be put together for a uniform framework for knot categorification. The first approach is based on derived categories of coherent sheaves on resolutions of slices in affine Grassmannians. Some elements of it have been discovered by mathematicians earlier and others are new. The second approach is perhaps more surprising. It uses symplectic geometry and is related to the first by two dimensional (equivariant) mirror symmetry. Unlike previous symplectic geometry based approaches, it produces a bi-graded homology theory. In both cases, mirror symmetry, and techniques developed to understand it play a crucial role. I will explain the string theory origin of the two approaches, and the relation to another string theory based approach, due to Witten.

July 20, 2020

### Breaking News About, Topologically Twisted Rank One N=2* Supersymmetric Yang-Mills Theory On Four-Manifolds, Without Spin

Abstract: I will report on work in progress with Jan Manschot. We generalize previous results concerning a topological theory in four dimension that generalizes both the Donaldson invariants and the Vafa-Witten invariants. In contrast with previous studies we include an arbitrary background spin-c structure with connection. The Coulomb branch measure involves non-holomorphic topological couplings to the background spin-c connection. (This violates some folklore). Using some novel identities for the $N=2*$ prepotential, the Coulomb branch integral can be evaluated explicitly using the theory of mock modular and Jacobi forms. For $b_2^+>1$ the path integral can be written explicitly in terms of Seiberg-Witten invariants and modular functions of the ultraviolet coupling. We discuss the orbit of partition functions of the three rank one $N=2*$ theories under the action of S-duality.

July 27, 2020

August 3, 2020

### 5d SCFTs: Symmetries and Moduli Spaces

Abstract: I will report on recent developments in 5d SCFTs, studying their global symmetries, 0- and higher-form, M-theory on a canonical singularity. We provide a geometric characterization of the Coulomb and Higgs branch moduli spaces and connect this to recent work on magnetic quivers in 3d.

August 10, 2020

### 3d mirror symmetry and its discontents

Abstract: One of the central topics of the interaction between QFT and math is mirror symmetry for 2d theories. This theory has a more mysterious and exotic friend one dimension higher, sometimes called 3d mirror symmetry, which relates two 3-dimensional theories with N=4 supersymmetry. For roughly a decade, I struggled to understand this phenomenon without understanding what most of the words in the previous sentence meant. Eventually, I wised up and based on work of Braverman, Finkelberg, Nakajima, Dimofte, Gaiotto, Hilburn and others, I actually did learn a little bit, and will now try to explain to you what I learned. This knowledge has some interesting payoffs in the mathematics related to 3d theories, such as an understanding of Bezrukavnikov and Kaledin's noncommutative resolutions of the Coulomb branch, and explaining a lot of interesting Koszul dualities between category O's.

August 17, 2020

### Branes in symplectic groupoids

Abstract: After reviewing coisotropic A-branes in symplectic manifolds and their role in mirror symmetry and geometric quantization, I will explain how the problem of holomorphic quantization of Poisson brackets may be recast, and in some cases solved, as a problem of computing morphisms between coisotropic branes in symplectic groupoids. This is joint work with Francis Bischoff and Joshua Lackman.

August 24, 2020

### The asymptotic geometry of the Hitchin moduli space

Abstract: Hitchin's equations are a system of gauge theoretic equations on a Riemann surface that are of interest in many areas including representation theory, Teichmuller theory, and the geometric Langlands correspondence. The Hitchin moduli space carries a natural hyperkahler metric. An intricate conjectural description of its asymptotic structure appears in the work of physicists Gaiotto-Moore-Neitzke and there has been a lot of progress on this recently. I will discuss some recent results using tools coming out of geometric analysis which are well-suited for verifying these extremely delicate conjectures. This strategy often stretches the limits of what can currently be done via geometric analysis, and simultaneously leads to new insights into these conjectures.

September 14, 2020

### Quantum geometric Langlands as a fully extended TFT

I will survey several recent works realizing Betti geometric Langlands and its quantization as fully extended TFT's. In physics terms this is most closely related to the Kapustin-Witten twist of N=4 d=4 SYM at generic values of \Psi.

I will outline numerous applications to quantum topology, most notably to a conjecture of Witten on finite-dimensionality of skein modules, a conjecture of Bonahon-Wong concerning skein modules at root-of-unity parameters, a proposal of Ben-Zvi concerning cluster varieties and factorization homology, and an appearance of the type-A spherical DAHA from a once-marked torus.

Finally, I will present a novel conjectural appearance of S-duality in the study of skein modules of 3-manifolds.

September 21, 2020

### How is the hypersimplex related to the amplituhedron?

Abstract: In 1987, Gelfand-Goresky-MacPherson-Serganova made a beautiful connection between the geometry of the Grassmannian and convex polytopes, via the moment map; the moment map image of the Grassmannian Gr(k,n) is a polytope known as the hypersimplex Delta(k,n). In 2013, motivated by the desire to give a geometric basis for the computation of scattering amplitudes in N=4 SYM, Arkani-Hamed and Trnka introduced the amplituhedron A(n,k,m) as the image of the positive Grassmannian Gr+(k,n) under a linear map Z from R^n to R^{k+m} which is totally positive. While the case m=4 is most relevant to physics, the amplituhedron makes sense for any m. In my talk I will explain some strange parallels between the positroidal subdivisions of the hypersimplex Delta(k+1,n) and the m=2 amplituhedron A(n,k,2). One link is provided by the positive tropical Grassmannian. Attributions: based on joint works with Tomek Lukowski, Matteo Parisi, and David Speyer.

Disclaimer: I'm neither a geometer nor a physicist.

October 5, 2020

### Borcherds-Kac-Moody algebras, 2d strings, & other curiosities

Abstract: In this talk we will extol the virtues of compactifying critical string theory down to few noncompact spacetime dimensions (particularly two). These string vacua possess rich groups of dualities. BPS-saturated quantities, which mathematically are described by automorphic forms, are invariant under such duality transformations. Further, such BPS states can furnish representations of interesting algebras, such as infinite-dimensional Lie algebras. In this talk, we explore some particularly nice, concrete examples, which employ holomorphic super vertex operator algebras in our `compactification' theories. The BPS states in these models organize into representations of algebras, which we prove are (new) examples of Borcherds-Kac-Moody superalgebras.

October 19, 2020