as above, so below

I am a graduate student in the mathematics department studying mathematical physics. Currently I am trying to elucidate quantum mechanical aspects of symmetry.

sigma models and Berry phase

I am working on a project extending anomaly ideas we learned by studying crystals to studying anomalies on the boundary of topological sigma models. A paper should appear soon. Because any parameter may be promoted to a slowly varying field, the study of topological terms of sigma models is basically equivalent to the theory of Berry phase. I'm currently working out how this can be used to apply Berry phase ideas to higher dimensions.

surprising applications of anomalies

I had a couple recent papers giving some surprising applications to anomalies. The first, with Zohar Komargodski, Adar Sharon, and Xinan Zhou realized that higher symmetry anomalies can protect systems even at positive (may as well be infinite) temperature. This is a good way to prove that there is a confining scale in certain toy systems.

The second paper, with Max Metlitski concerned what happens when the true symmetry G acts on a low energy theory as a quotient group G/H such that G/H has an anomaly which is trivial when pulled back to G. The context and conclusions are very closely related to the previous paper. Basically there is a disordering perturbation but it needs to acheive a certain size. It needs to access degrees of freedom outside the low energy theory.


With Anton Kapustin we have recently finished a paper explaining how bosonization/fermionization work for gapped systems in 3+1D. We did this in order to provide a state sum formula for all spin cobordism invariants of BG and equivalently to give exactly solvable lattice models for all the fermionic SPT phases in 3+1D with unitary symmetry G not squaring to fermion parity or any other funny business. Along the way, we discovered a very interesting nonabelian topological field theory, which plays the role of the Ising category in fermionization in 3+1D. We called it the 2-Ising model. It is presented by a kind of 2-category state sum like Turaev-Viro or Levin-Wen. It is not a Crane-Yetter or Walker-Wang model. It contains nonabelian string operators. I am interested in whether this theory admits any interesting gapless boundary conditions.

spacetime symmetries and crystals

I recently finished a paper with Dominic Else where we learned how to gauge symmetries that act on spacetime, for the particular goal of studying how crystalline symmetries can protect short-range and long-range entanglement of quantum states. This culminated in a classification of so-called crystalline symmetry protected topological phases via equivariant cohomology (ordinary cohomology in the cases with no coupling to the geometry of the tangent bundle, cobordism cohomology in general). Here by gauging I mean coupling to background gauge field. We do not yet know how to make these things dynamical, but I am interested in studying this. Currently we are working on using our approach to study anomalies of discrete spacetime symmetries, and to figure out what happens on the boundary of these phases.

Here is an amusing toy for exploring the 17 wallpaper groups. Hold spacebar to draw, use c to toggle colors, and refresh to get a new wallpaper group. Try to guess which wallpaper group you're using! Open the dev console for the answer. (:

higher symmetries

I have done some work, primarily with Anton Kapustin but also with Curt von Keyserlingk on what Anton and I called higher symmetries in the original paper of this ``era" but what is often called generalized global symmetries (this is a very nice paper!). The term ``higher" is in reference to higher categories, as it was truly John Baez and collaborators who exposed the algebraic structure behind higher gauge theories. We simply ported their ideas into the study of global symmetries. I note that these higher symmetries exist in fluid systems, eg. the conservation of vorticity coincides with a 1-form symmetry, though I've never seen it in this language.

Here is a talk I gave at the Simons Center for Geometry and Physics about higher symmetries in the context of Abelian Chern-Simons theories, where I give a simple outline of the higher symmetry idea. The talk also contains an interpretation of certain fractional quantum Hall phases as spontaneous higher symmetry broken phases. In particular, the topological ground state degeneracy of U(1) level k is completely described by counting broken symmetry generators.

This paper with Curt von Keyserlingk explains how anomalies which are too severe to exist at the boundary of an SPT phase are always the 0-form part of a higher symmetry with non-trivial Postnikov class. When one appreciates this fact, one finds that the anomalies are always realizable on the boundary of the higher SPT phases described by Anton Kapustin and myself. The proof of these facts uses a pleasant physical interpretation of the Serre spectral sequence and includes an example of a discrete symmetry Noether's theorem.

Old stuff: In this paper, Anton Kapustin and I studied discrete topological terms, "theta angles" which may separate different confining phases of non-abelian gauge theory. It follows closely this paper of Sergei Gukov and Anton Kapustin. This one of mine and Anton's came soon after, where we used these ideas to propose higher symmetry protected topological phases and explained more about higher global symmetry.


Motivated by the connection between symmetry protected topological phases and group cohomology via Dijkgraaf-Witten topological gauge theory and the discovery of phases outside of this classification, I have studied Anton Kapustin's cobordism classification proposal in the case of fermionic systems with his student Alex Turzillo and Xie Chen's student Zitao Wang. The proposal is that the partition function of an invertible gapped phase, ie. one which a tensor-inverse, another phase with which one can couple to produce a trivial insulating ground state, is actually a cobordism invariant of spacetime. Further, the partition function, regarded as a homomorphism from the cobordism group of spacetime manifolds to the group U(1) of complex phases, determines the physical phase, and all homomorphisms are realized by physical phases.

Currently we are trying to prove these assertions. Dan Freed and Mike Hopkins have made a lot of progress in showing that unitary invertible TQFT amounts to the study of cobordism invariants. What is missing is a proof that from a lattice Hamiltonian one gets a TQFT (very non-trivial!) and that all cobordism invariants are realized by lattice Hamiltonians. The second has been satisfying worked out in spacetime dimensions up to three by Davide Gaiotto, his student Lakshya Bhardwaj, and Anton Kapustin. Anton and I have now worked out the four dimensional case for discrete symmetries not interacting with orientation or spin structure, as I discussed above. Surprisingly, higher symmetries show up in a crucial way! As for the first problem, I believe the solution lies in the work of Frank Verstraete's group via the theory of RG fixed point tensor networks and matrix product operators. This is only satisfying worked out in space dimension one.


More old stuff: Anomaly underlies basically all of my work. It is a very old story. The chiral anomaly allows the pion to decay into two photons, though this violates the symmetries of the Standard Model Lagrangian. It was long believed that chiral fermions, which only exist in even spacetime dimensions, were somehow the source of all internal symmetry anomalies. However, in our study of symmetric boundary conditions for SPT phases, Anton Kapustin and I found many such anomalies for discrete symmetries in odd dimensions. It's actually a nice story, where one can systematically from a Dijkgraaf-Witten anomaly polynomial produce a gapped theory with that anomaly. It remains an open question whether all cobordism anomalies are realized by gapped systems. I hope this question will be answered in the affirmative once more about the TQFT state sums is worked out (previous discussion).

more art

All of my coding art projects can be found on github. I owe much to my awesome collaborators there, especially data-doge/Eugene Lynch, who taught me javascript.

Here is a piece of a project with much more to come that was hacked together over several sleepless nights with Eugene Lynch in the wonderful atmosphere at the CCRMA in Stanford. To interact, draw by clicking and dragging in the sketchpad at the bottom left. This area represents the image of the base of the Hopf fibration. The 3d space you see is a stereographically projected 3-dimensional sphere which is the total space of the fibration. One navigates the 3d space using flight controls wasdqefr/arrows. I recommend closing the sketchpad before flying.

Here is one I also made with Eugene and which MUST BE USED IN HEADPHONES. Actually this art is rather dangerous and very much follows the point of view in Pascal Quignard's The Hatred of Music. In short, don't create feedback loops and you should be okay. Press spacebar to sever all connections, but remember that the effects have their own internal feedback loops and so will store large amounts of acoustical energy for long periods of time and which will release that acoustical energy when reconnected, blowing you headphones and possibly your membranes. Best to press spacebar and then refresh. You use flight controls, wasdqerf/arrows, to fly around. The cubes you see are audio effects. Flying up to a cube you should see at bottom left the parameters for that effect. Clicking the cube will cause it to spin. The effect is now primed. Clicking the same cube again it will stop spinning and no longer be primed. Clicking a different cube will cause an explosion of colored points from the first cube which will then seek the second cube. When the points reach it, an audio connection first->second has been acheived. The central geometry as a source represents your microphone and as a sink represents your headphones. DON'T CREATE FEEDBACK LOOPS!!

me with some self-organized criticality

I've been traveling a lot but you can find me about half the time in beautiful Berkeley, either in Lothlorien coop or in the hills around 37.906815/-122.23764.