BIMSA-Tsinghua Quantum Symmetry Seminar

This talk will explain factorization homology, which is intended to abstract and organize the observables of a TQFT. Factorization homology is a construction that associates a chain complex to a (framed) n-manifold M and a (rigid) n-category C. One can rightfully think of C as the domain of a topological QFT, and C as an organization of point/line/surface/… observables of the QFT as they interact with one another.

I will explain several pleasant features of factorization homology, and outline how these features alone can be used to work with factorization homology. I will identify a few values of factorization homology, which recover some familiar invariants of quantum topology (ie, the Jones polynomial and Skein modules). Much of this theory has yet to be fully developed. I will be clear about which aspects can be found in literature and which are more speculative.

All of this work is joint with John Francis.

For a long time, the Alexander polynomial was the only easily computable link invariant to be known. But in 1984, Jones discovered his well known polynomial link invariant, and that gave birth to the vast theory of quantum link invariants. However, unlike for the Alexander invariant, it is in general hard to deduce precise topological properties on a knot or link from the value quantum invariants take on that link. For instance, no genus bound is known for the Jones polynomial.

The Links-Gould invariants of oriented links \(LG^{m,n}(L,t\_{0},t\_{1})\) are two variable quantum invariants obtained by the Reshetikhin-Turaev construction applied to Hopf superalgebras \(U_{q}\mathfrak{gl}(m \vert n)\). These invariants are known to be generalizations of the Alexander invariant.

Using representation theory of \(U_{q}\mathfrak{gl}(2 \vert 1)\), we proved in recent work with Guillaume Tahar that the degree of the Links-Gould polynomial \(LG^{2,1}\) provides a lower bound on the Seifert genus of any knot, therefore improving the bound known as the Seifert inequality in the case of the Alexander invariant.

Reference: arXiv:2310.15617

Speaker Intro:

I was born in France and grew up between France and the United States. I studied math at ENS de Cachan (Paris), Université Paris 7 and Université de Bourgogne, where I obtained my PhD (directed by P. Schauenburg and E. Wagner). My mathematical interests are related to low dimensional topology. I have been studying knot and link theory, and more precisely connections that exist between classical and quantum link invariants. On a more personal level, I enjoy spending time with my three children Côme, Aliocha and Madeleine.

Classification of SPT/SET orders: boundary-bulk relation and higher categories

Abstract: It is known that 2d (spatial dimension) symmetry protected topological (SPT) orders and symmetry enriched topological (SET) orders with finite onsite symmetries can be characterized by using the idea of gauging the symmetry and minimal modular extensions. In this talk, I will introduce another characterization of SPT/SET orders in all dimensions based on the boundary-bulk relation. In 1d, this result recovers the well-known classification of 1d SPT/SET orders. For 2d SPT/SET orders, this result gives a 2-categorical point of view of minimal modular extensions. In higher dimensions, we also obtain many precise mathematical conjectures on higher categories. If time permits, I will briefly introduce a new characterization of SPT/SET orders in the language of enriched categories.

Modular categories are important algebraic structures that appear in a diverse quantity of applications, including topological quantum field theory, representation theory of braid quantum groups, von Neumann algebras, conformal field theory, and vertex operator algebras. They also appear in the study of topological phases of matter and topological quantum computation. An important class of modular categories are those of odd dimension. In this talk, we will present a classification of low-rank odd-dimensional modular categories and the methods we used to do so, including a computer algorithm and other properties of odd-dimensional modular categories.

Reference: arXiv:2305.14542 (with Agustina Czenky and Julia Plavnik)

Speaker Intro:

I am a rising high school senior at Bellevue High School in Bellevue, Washington and a participant in the MIT PRIMES-USA program, where I conducted this research. I am also interested in computer science, both algorithmically and through web development.

We study the interplay of classical and quantum topology, in two-dimensional topological orders which spontaneously break continuous symmetries. In particular, we present a framework that addresses how the topologically ordered quantum ground states affect the universal properties of topological point defects and textures in the classical long range order. Mathematically, based on the long exact sequence of homotopy groups, we show how to classify universal properties of point defects and textures using the inflation-restriction exact sequence. We illustrate the framework using many examples, where point defects can permute anyons in the topological orders and/or exhibit a ‘’defect fractionalization’’ phenomenon.

Reference: https://arxiv.org/abs/2211.13207

Understanding the symmetries of algebraic objects can help decrease their complexity by several fold, and this reduction in complexity is intensified when there are fewer and fewer fixed-points.  Typically one requires the unit or neutral element to be fixed by any symmetry, so a symmetry for which this is the only fixed-point has been called “fixed-point-free”.  These are the most active of symmetries and have strong implications for the objects they act upon.  For example, a finite group with a fixed-point-free automorphism must be solvable; if the automorphism is order 2 then the group is abelian and the symmetry is inversion.   In this talk, we will attempt to generalize classical results on fixed-point-free automorphisms of finite groups to fusion rings and categories.  Of particular interest is understanding when/if combinatorial symmetries of fusion rings can be lifted to categorical symmetries of fusion categories.  This talk will be self-contained, aimed toward a general audience, and will include copious amounts of examples. 

Reference: arXiv:2306.01666

Speaker Intro:

I grew up in a rural area in northern Michigan and did not consider mathematics as a career until I was in my 20’s; I cooked professionally for about 10 years.  I’ve since earned degrees in the states of Michigan, Oregon, and Washington, and done postdoctoral research in Australia, Canada, and the United States.  My research will probably stay related to tensor categories and their many applications, but I will always think about whatever problems seem interesting to me in the moment.  I am a year-round alpine climber, and an avid wildlife photographer.

Consider the exterior algebra of the tensor product of two complex vector spaces of dimension n and k. This space could be regarded as a bimodule for the action of dual pairs of Lie groups. For example, for GL(n) x GL(k) - case this exterior algebra decomposes into direct sum of bimodules parametrised by conjugate partitions inside the n x k rectangle. This is the skew Howe duality. On the level of characters the skew Howe duality yields the dual Cauchy identity for the Schur functions.

We interpret the skew Howe duality as a natural consequence of lattice paths on lozenge tilings of certain partial hexagonal domains. This combinatorial approach also allows to obtain product formulas for the q-deformations of multiplicities or different dual pairs of Lie groups . We consider the corresponding probability measures on Young diagrams and prove the uniform convergence to the limit shape of Young diagrams in the limit when n and k go to infinity. (Joint work with A.Nazarov and T.Schrimshaw.)

Vaughan Jones initiated the theory of subfactors which deals with the relative position of a subfactor inside an ambient factor. It is a very natural and fundamental question to consider the relative positions of multiple subfactors. In general, the theory becomes quite complicated. In the same vein, Jones also initiated a systematic study of a pair of subfactors. In this talk, I shall attempt to explain some of our contributions to this ongoing project.

Speaker Intro:

I am an Assistant Professor in the Department of Mathematics and Statistics at Indian Institute of Technology, Kanpur (https://www.iitk.ac.in/math/). I have received my PhD from the Institute of Mathematical Sciences (HBNI) under the supervision of Prof. V. S. Sunder. I work with topics in C*-algebras and von Neumann algebras (more precisely, Jones’ subfactor theory).

Let \(G\) be a group and \(k\) an algebraically closed field of positive characteristic \(p\). Let \(V\) be a finite dimensional representation of \(G\) over \(k\). Then by the classical Krull-Schmidt theorem, the tensor power \(V^{\otimes n}\) can be uniquely decomposed into a direct sum of indecomposable representations. But we know very little about this decomposition, even for very small groups, such as \(G = (Z/2)^3\) for \(p = 2\) or \(G = (Z/3)^2\) for \(p = 3\). For example, what can we say about the number \(d\_n(V)\) of such summands of dimension coprime to \(p\)? It is easy to show that there exists a finite limit \(d(V ) := \lim\_{n \to \infty} d\_n(V )^{1/n}\), but what kind of number is it? For example, is it algebraic or transcendental? Until recently, there was no techniques to solve such questions (and in particular the same question about the sum of dimensions of these summands is still wide open). Remarkably, a new subject which may be called “Lie theory in tensor categories” gives methods to show that \(d(V)\) is indeed an algebraic number, which moreover has the form \[d(V)= \sum\_{1 \leq j \leq p/2} n_j(V) \[j]_q\] where \(n_j(V) \in \mathbb{N}\), \(q := \exp(\pi i/p)\), and \(\[j]_q := (q^j-q^{-j})/(q-q^{-1})\). Moreover, \(d\) is a character of the Green ring of \(G\) over \(k\). Furthermore, \[d_n(V) \leq C_V d(V)^n\] for some positive \(C_V \leq 1\) gre and we can give lower bounds for \(C_V\). In the talk I will explain what Lie theory in tensor categories is and how it can be applied to such problems.

This is joint work with K. Coulembier and V. Ostrik.

References:

https://arxiv.org/abs/2107.02372 (to appear in Annals of Math)

https://arxiv.org/abs/2301.09804

Speaker Intro:

Professor Pavel Etingof is a professor of mathematics at MIT. He works on many aspects of representation theory and mathematical physics, and has published many influential papers in top journals. In 1994, he received his PhD in mathematics at Yale University and became Benjamin Peirce Assistant Professor at Harvard University after that. In 1998, he was an assistant professor at MIT, and became a professor there since 2005. In 1999, he was a Fellow of the Clay Mathematics Institute. In 2002, he was an invited speaker at the International Congress of Mathematicians in Beijing. He is a Fellow of the American Mathematical Society, and in 2016, he became a fellow of the American Academy of Arts and Sciences. His editorial work includes Editor-in-Chief of Selecta Mathematica and Managing Editor of Journal of the AMS.