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The Moore-Tachikawa conjecture posits the existence of certain 2-dimensional topological quantum field theories (TQFTs) valued in a category of complex Hamiltonian varieties. Previous work by Ginzburg-Kazhdan and Braverman-Nakajima-Finkelberg has made significant progress toward proving this conjecture. In this talk, I will introduce a new approach to constructing these TQFTs using the framework of shifted symplectic geometry. This higher version of symplectic geometry, initially developed in derived algebraic geometry, also admits a concrete differential-geometric interpretation via Lie groupoids and differential forms, which plays a central role in our results. It provides an algebraic explanation for the existence of these TQFTs, showing that their structure comes naturally from three ingredients: Morita equivalence, as well as multiplication and identity bisections in abelian symplectic groupoids. It also allows us to generalize the Moore-Tachikawa TQFTs in various directions, raising interesting questions in Lie theory and Poisson geometry. This is joint work with Peter Crooks.

Condensation of topological defects is the foundation of the modern theory of bulk-boundary correspondence, also known as topological holography. In this talk, I discuss string condensation in 3+1D topological orders, which plays a role analogous to anyon condensation in 2+1D topological orders. I will demonstrate through examples how they correspond to 2+1D symmetry enrichd phases, including both gapped and gapless phases. Then I give a detailed analysis of string condensaiton in 3+1D discrete gauge theories. I compute the outcome of the condensation, namely the category of excitations surviving the condensation. The results suggest that a complete topological holography for 2+1D phases can only be established by taking into account all possible ways of condensing strings in the bulk 3+1D topological order.

The matter power spectrum on subgalactic scales is very weakly constrained so far. While inflation predicts a nearly scale-invariant primordial power spectrum down to very small scales, many new physics scenarios can lead to significantly different predictions, such as axion dark matter in the post-inflationary scenario, vector dark matter produced during inflation, early matter domination, kinetic misalignment axions, self-interacting dark matter, atomic dark matter, etc. Therefore, any successful measurement on the matter power spectrum tests inflation extensively and probes early universe dynamics and the nature of dark matter, making it a new frontier in cosmology and dark matter physics. We proposed observing fast radio bursts (FRB) with solar-system scale interferometry by sending radio telescopes to space, which allows us to greatly expand the sensitivity on the matter power spectrum from Mpc to AU scales. Two sightlines looking at the same FRB source can sample different regions of the Universe in the transverse direction and thus obtain an arrival time difference that depends on the matter power spectrum. Our calculations show that this setup will be sensitive to the scale-invariant power spectrum predicted by inflation on small scales and can also probe QCD axion miniclusters predicted in the post-inflationary scenario.

On April 4th, 2024, the Dark Energy Spectroscopic Instrument (DESI) released its first set of cosmological results based on measurements of the baryon acoustic oscillations (BAO) scale in the spatial distribution of galaxies and quasars, and in the Lyman-alpha forest. Those measurements constrain the expansion history of the Universe in the redshift range 0.1 < z < 4.16, with implications for studies of dark energy, neutrino cosmology and the Hubble constant. To make the most of the cosmological information content of the galaxy & quasar distributions, we now analyze the full shape of their power spectra, constraining the evolution of the large scale structure of the Universe over the range 0.1 < z < 2.1. In this talk, following a brief overview of the DESI instrument and observations, we will present the latest public results, discuss some of their main cosmological implications, highlight efforts towards the full shape results and expectations for the next data release.

It is commonly believed that logical states of quantum error-correcting codes have to be highly entangled such that codes capable of correcting more errors require more entanglement to encode a qubit. Here we show that this belief may or may not be true depending on a particular code. To this end, we characterize a tradeoff between the code distance d quantifying the number of correctable errors, and geometric entanglement of logical states quantifying their maximal overlap with product states or more general "topologically trivial" states. The maximum overlap is shown to be exponentially small in d for three families of codes: (1) low-density parity check (LDPC) codes with commuting check operators, (2) stabilizer codes, and (3) codes with a constant encoding rate. Equivalently, the geometric entanglement of any logical state of these codes grows at least linearly with d. On the opposite side, we also show that this distance-entanglement tradeoff does not hold in general. For any constant d and k (number of logical qubits), we show there exists a family of codes such that the geometric entanglement of some logical states approaches zero in the limit of large code length. This work was done by the collaboration with Sergey Bravyi, Zhi Li, and Beni Yoshida. (https://arxiv.org/abs/2405.01332)