Calendar

Kapustin and Witten have studied a one-parameter family of topological twists of 4d N=4 super Yang–Mills. They have shown that the categories of boundary conditions on a surface are exactly the categories participating in the geometric Langlands program of Beilinson and Drinfeld. Moreover, S-duality is manifested as a quantum geometric Langlands duality after the topological twist. In this talk I will describe some mathematical formalizations of Hilbert spaces of states on a 3-manifold. I will outline an equivalence between two such possible formalizations: complexified Floer homology of Abouzaid–Manolescu and skein modules. This is a report on work in progress joint with Sam Gunningham.

Zoom: https://harvard.zoom.us/j/94717938264

I will discuss the conditions under which Higgs and confining regimes in gauge theories with fundamental representation matter fields can be sharply distinguished. It is widely believed that these regimes are smoothly connected unless they are distinguished by the realization of global symmetries. However, I will show that when a U(1) global symmetry is spontaneously broken in both the confining and Higgs regimes, the two phases can be separated by a phase boundary. The phase transition between the two regimes may be detected by a novel topological vortex order parameter. I’ll illustrate these ideas by explicit calculations in gauge theories in three spacetime dimensions, and then explain the generalization to four dimensions. One important implication of our results is that nuclear matter and quark matter are sharply distinct phases of QCD with an approximate SU(3) flavor symmetry.

Zoom: https://harvard.zoom.us/j/977347126

The Eisenstein-Kronecker function is a useful object in number theory and physics. It is a tool for proving rationality for period integrals of cusp forms. It generates integration kernels of elliptic polylogarithms which express Feynman diagrams on the torus. It is the central object in the construction of elliptic R-matrices. We propose a new description in terms of iterated convolutions of the Eisenstein zeta function and discuss its features in the various settings.

Zoom: https://harvard.zoom.us/j/779283357

Kontsevich’s recursion, proved in the early 90s, is a recursion formula for the counts of rational holomorphic curves in complex manifolds. For complex fourfolds and sixfolds with a real structure (i.e. a conjugation), signed invariant counts of real rational holomorphic curves were defined by Welschinger in 2003. Solomon interpreted Welschinger’s invariants as holomorphic disk counts in 2006 and proposed Kontsevich-type recursions for them in 2007, along with an outline of a potential approach of proving them. For many symplectic fourfolds and sixfolds, these recursions determine all invariants from basic inputs. We establish Solomon’s recursions by re-interpreting his disk counts as degrees of relatively oriented pseudocycles from moduli spaces of stable real maps and lifting cobordisms from Deligne-Mumford moduli spaces of stable real curves (which is different from Solomon’s approach).

Zoom: https://harvard.zoom.us/j/94717938264

A given compact Lie group, G, admits many left-invariant Riemannian metrics. Typically, they form a finite dimension cone L(G). Up to a multiplicative constant, the Riemannian measure for such metrics is the Haar measure of the group. Because the group is compact, each metric g in L(G) has the property that there exists a constant C(G,g)—called the doubling constant—such that, for any radius r, the volume of the ball of radius 2r is at most C(G,g) times the volume of the ball of radius r. The title of this presentation asks the question: does there exist a constant C(G) such that, for all g in L(G), C(G,g) is bounded above by C(G). Is any compact Lie group uniformly doubling? We conjecture that this is the case. The only cases for which the conjecture is known are Riemannian tori and the group SU(2). The result for U(2) is work in progress. This reports on joint work with Maria Gordina (University of Connecticut) and Nathaniel Eldredge (University of Northern Colorado).

Zoom: https://harvard.zoom.us/j/779283357