From Lattice Chern-Simons-Maxwell to Lattice Chern-Simons-Yang-Mills

by Jing-Yuan Chen (Tsinghua University)

Thursday 25 Sep 2025, 15:10 → 16:10 Asia/Shanghai

B105 (CHEP)

Chern-Simons theory has been very important in both the understanding of quantum Hall effect and the development of mathematical physics. Putting the simplest Chern-Simons theory, i.e. U(1) Chern-Simons, onto the lattice has been a problem of interest for some decades. Over time, there has been many scattered ideas, and we identify two as being crucial: 1) to correctly capture the topology of U(1), the U(1) lattice gauge field must be realized in the Villainized form, and 2) a highly non-trivial feature of the theory is that it is chiral, and a chiral theory cannot be fully topological, some non-topological terms such as the Maxwell term is needed. Putting these two ideas together, we defined and solved the U(1) Chern-Simons-Maxwell theory on the lattice, obtaining all the key features---including, most non-trivially, the chiral gravitational/framing anomaly. We consider this as completion of the problem of putting chiral U(1) Chern-Simons on the lattice.

One will then wonder how about non-abelian Chern-Simons. Again, a non-topological term, this time the Yang-Mills term, is needed for the chiral theory. On the other hand, the generalization of the Villaination of U(1) lattice gauge field to SU(N) is highly non-trivial---we must understand Villainization through the lens of higher category theory, so to generalize it into a multiplicative bundle gerbe realization of SU(N) lattice gauge field. We thus defined the lattice Chern-Simons-Yang-Mills theory (though unlike the U(1) case, this case is not analytically solvable). This also resolves the long standing problem in lattice QCD of defining instanton on lattice. More broadly, it leads to a more systematic picture of how continuum QFT and lattice QFT are related.

Bio:
Jing-Yuan Chen is a junior faculty at the Institute for Advanced Study at Tsinghua University. He received B.Sc. in physics and pure mathematics at the University of Michigan Ann Arbor in 2011, and Ph.D. in physics, supervised by Dam Thanh Son, at the University of Chicago in 2016. After that he worked as a Gordon and Betty Moore Postdoctoral Fellow at Stanford Institute for Theoretical Physics, until joining the current position in 2020.