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Given a (2+1)D fermionic topological order and a symmetry fractionalization class for a global symmetry group G, we show how to construct a (3+1)D topologically invariant path integral for a fermionic G symmetry-protected topological state (G-FSPT) in terms of an exact combinatorial state sum. This provides a general way to compute anomalies in (2+1)D fermionic symmetry-enriched topological states of matter. Our construction uses the fermionic topological order (characterized by a super-modular tensor category) and symmetry fractionalization data to define a (3+1)D path integral for a bosonic theory that hosts a non-trivial emergent fermionic particle, and then condenses the fermion by summing over closed 3-form Z_2 background gauge fields. This procedure involves a number of non-trivial higher-form anomalies associated with Fermi statistics and fractional quantum numbers that need to be appropriately canceled off with a Grassmann integral that depends on a generalized spin structure. We show how our construction reproduces the Z_16 anomaly indicator for time-reversal symmetric topological superconductors with T^2=(−1)^F. Mathematically, with standard technical assumptions, this implies that our construction gives a combinatorial state sum on a triangulated 4-manifold that can distinguish all Z_16 Pin+ smooth bordism classes. As such, it contains the topological information encoded in the eta invariant of the pin+ Dirac operator, thus giving an example of a state sum TQFT that can distinguish exotic smooth structure.

Ref: arXiv:2104.14567

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

As a variant of Grothendieck’s Quot schemes, we introduce the moduli space of limit stable pairs. We show an example over a smooth projective algebraic surface where there is a virtual fundamental class. We are able to describe this class explicitly. We will also show an application towards moduli of sheaves.

Zoom: https://harvard.zoom.us/j/98248914765?pwd=Q01tRTVWTVBGT0lXek40VzdxdVVPQT09

(Password: 419419)

No definitive evidence of spacetime supersymmetry (SUSY) that transmutes fermions into bosons and vice versa has been revealed in nature so far. One may wonder whether SUSY can be realized in quantum materials. In this talk, I shall discuss how spacetime SUSY may emerge, in the sense of renormalization group flow, in the bulk of Weyl semimetals or at the boundary of topological insulators. Moreover, we have performed large-scale sign-problem-free quantum Monte Carlo simulations of various microscopic lattice models to numerically verify the emergence of spacetime SUSY at quantum critical points on the boundary of topological phases. I shall mention some experimental signatures such as optical conductivity which can be measured to test such emergent SUSY in candidate systems like the surface of 3D topological insulators.
References:
[1] Shao-Kai Jian, Yi-Fan Jiang, and Hong Yao, Phys. Rev. Lett. 114, 237001 (2015)
[2] Shao-Kai Jian, Chien-Hung Lin, Joseph Maciejko, and Hong Yao, Phys. Rev. Lett. 118, 166802 (2017)
[3] Zi-Xiang Li, Yi-Fan Jiang, and Hong Yao, Phys. Rev. Lett. 119, 107202 (2017)
[4] Zi-Xiang Li, Abolhassan Vaezi, Christian Mendl, and Hong Yao, Science Advances 4, eaau1463 (2018)

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