Scientific Series

Idempotent (aka Karoubi) completion is used throughout mathematics: for instance, it is a common step when building a Fukaya category. I will explain the n-category generalization of idempotent completion. We call it "condensation completion" because it answers the question of classifying the gapped phases of matter that can be reached from a given one by condensing some of the chemicals in the matter system. From the TFT side, condensation preserves full dualizability. In fact, if one starts with the n-category consisting purely of ℂ in degree n, its condensation completion is equivalent both to the n-category of n-dualizable ℂ-linear (n-1)-categories and to an n-category of lattice condensed matter systems with commuting projector Hamiltonians. This establishes an equivalence between large families of TFTs and of gapped topological phases. Based on joint work with D. Gaiotto.

In order to solve the problem of time in quantum gravity, various approaches to a relational quantum dynamics have been proposed. In this talk, I will exploit quantum reduction maps to illustrate a previously unknown equivalence between three of the well-known ones: (1) relational observables in the clock-neutral picture of Dirac quantization, (2) Page and Wootters’ (PW) Schrödinger picture formalism, and (3) the relational Heisenberg picture obtained via symmetry reduction. Constituting three faces of the same dynamics, we call this equivalence the trinity. To establish the equivalence, a quantization procedure for relational Dirac observables is developed using covariant POVMs which encompass non-ideal clocks. The quantum reduction maps reveal this procedure as the quantum analog of the classical method of gauge-invariantly extending gauge-fixed quantities. The quantum reduction maps also allow one to extend a recent ‘clock-neutral’ approach to changing temporal reference frames, transforming relational observables and states between different clock choices, and demonstrate a clock dependent temporal nonlocality effect. Using the trinity, I will discuss how Kuchar's three fundamental criticisms against the PW formalism, namely that its conditional probabilities would (i) yield the wrong localization probabilities for relativistic particles, (ii) violate the constraints, and (iii) produce incorrect transition probabilities, can be resolved. Given the trinity, these resolutions also apply to approaches (1) and (3) and corroborate the PW formalism, if done correctly, as a viable approach to the problem of time. Time permitting, I will explain, however, why the slogan `time from entanglement' in the PW formalism is misleading.

It is expected that the Betti version of the geometric Langlands program should ultimately be about the equivalence of two 4-dimensional topological field theories. In this talk I will give an overview of ongoing work in categorified sheaf theory and explain how one can use it to describe the categories of boundary conditions arising on the spectral side.

I'll explain the TFT perspective on holomorphic-topological twists of 3d N=4 and 4d N=2 theories, and outline some connections between the topics discussed in Justin and Davide's previous lectures, and various ongoing work of Justin, Philsang, Kevin, Davide, Tudor, myself, etc.

The area law for entanglement provides one of the most important connections between information theory and quantum many-body physics. It is not only related to the universality of quantum phases, but also to efficient numerical simulations in the ground state (i.e., the lowest energy state).  Various numerical observations have led to a strong belief that the area law is true for every non-critical phase in short-range interacting systems [1]. The so-called area-law conjecture states that the entanglement entropy is proportional to the surface region of subsystem if the ground state is non-critical (or gapped). 

However, the area law for long-range interacting systems is still elusive as the long-range interaction results in correlation patterns similar to the ones in critical phases. Here, we show that for generic non-critical one-dimensional ground states, the area law robustly holds without any corrections even under long-range interactions [2]. Our result guarantees an efficient description of ground states by the matrix-product state in experimentally relevant long-range systems, which justifies the density-matrix renormalization algorithm. In the present talk, I will give an overview of the results, and show ideas of the proof if the time allows. 

[1] J. Eisert, M. Cramer, and M. B. Plenio, ``Colloquium: Area laws for the entanglement entropy,'' Rev. Mod. Phys. 82, 277–306 (2010).

[2] T. Kuwahara and K. Saito, ``Area law of non-critical ground states in 1d long-range interacting systems,'' arXiv preprint arXiv:1908.11547 (2019),

As our understanding of the universe and its fundamental building blocks extends to shorter and shorter distances, experiments capable of probing these scales are becoming increasingly difficult to construct. Fundamental particle physics faces a potential crisis: an absence of data at the shortest possible scales. Yet remarkably, even in the absence of experimental data, the requirement of theoretical consistency puts stringent constraints on viable models of fundamental particles and their interactions. In this talk I’ll discuss a variety of criteria that constrain theories of particles in flat spacetime and de Sitter. Such criteria have the possibility to address questions such as: What low energy theories admit consistent UV completions? Which massive particles are allowed in an interacting theory? Is string theory the unique weakly coupled UV completion of General Relativity?

I will explain that a geometric theory built upon the theory of complex surfaces can be used to understand wide variety of phenomena in five-dimensional supersymmetric theories, which includes the following:

1. Classification of 5d superconformal field theories (SCFTs).
2. Enhanced flavor symmetries of 5d SCFTs.
3. 5d N=1 gauge theory descriptions of 5d and 6d SCFTs.
4. Dualities between 5d N=1 gauge theories.
5. T-dualities between 6d N=(1,0) little string theories.

This relationship between geometry and 5d theories is based on M-theory and F-theory compactifications.

In our four-dimensional world supersymmetry is the only extension of the classical Poincaré invariance which laid the foundation of modern physics in the beginning of the 20th century. Supersymmetry, a new geometric symmetry extending Poincaré, was discovered in 1970 –– it was overlooked for decades because of its quantum nature. In the next 10 years or so supersymmetry 

assumed the role of a universal framework in which new models for natural phenomena and regularities (e.g. the concept of naturalness) have been developed. It gave rise to a powerful stream of theoretical phenomenology.

The fact that LHC at CERN produced no evidence for low-energy supersymmetry (and naturalness as well) was a powerful blow. However, despite its absence in experiments the less known second mission of supersymmetry is highly successful, with remarkable advances occurring on a regular basis. Supersymmetry proved its power and uniqueness for those who address hard questions in strongly coupled field theories, including Yang-Mills. Some supersymmetry-based exact results obtained in four dimensions are the main topics of my talk. In the past one could hardly dream that such results are possible.

Theories beyond the Standard Model of particle physics often predict new, light, feebly interacting particles whose discovery requires novel search strategies. A light particle, the QCD axion, elegantly solves the outstanding strong-CP problem of the Standard Model; cousins of the QCD axion can also appear, and are natural dark matter candidates.  First, I will discuss my experimental proposal based on thin films, in which dark matter can efficiently convert to detectable single photons. A prototype experiment is underway, and current techniques promise to reach significant new dark matter parameter space.

Second, I will show how rotating black holes turn into axionic and gravitational wave beacons, creating nature's laboratories for ultralight bosons. When an axion's Compton wavelength is comparable to a black hole size, energy and angular momentum from the black hole source exponentially-growing bound states of particles, forming `gravitational atoms'.  These `gravitational atoms' emit monochromatic gravitational wave signals, enabling current searches at gravitational wave observatories to discover ultralight axions. If the axions interact with one another, instead of gravitational waves, black holes populate the universe with axion waves.

Stacking two graphene layers twisted by the ‘magic angle’ 1.1º generates flat energy bands, which in turn catalyzes various strongly correlated phenomena depending on filling and sample details. While this system is most famous for the superconducting and insulating states observed at fractional fillings, I argue that charge neutrality presents an interesting interplay of disorder and interactions.

In scanning tunnelling microscopy (STM), the most striking signature of interactions occurs close to charge neutrality, where the splitting between the flat bands increases dramatically. In analogy with quantum Hall ferromagnetism, I show that this effect may be qualitatively understood as the result of an exchange energy gain. A low-energy manifold of gapped, symmetry-breaking states is identified, one of which possesses quantum valley Hall order. Transport measurements yield ostensibly conflicting information at charge neutrality: while some samples reveal semimetallicity (as expected when correlations are weak), yet others exhibit robust insulation. I reconcile these observations and those of STM by arguing that strong interactions supplemented by weak, smooth disorder stabilize a network of locally gapped quantum valley Hall domains with spatially varying Chern numbers determined by the disorder landscape—--even when an entirely different order is favored in the clean limit. I conclude with a discussion of experimental tests of this proposal via local probes and transport.

In this talk, we present a new outlook on canonical quantum gravity and its coupling to matter.

We will show how this fresh perspective combines the critical elements of holography, loop quantum gravity, and relative locality. I will first focus on the consequences of cutting open a portion of space and show that new symmetry charges and new degrees of freedom reveal themselves. 
I will explain the nature of this boundary symmetry algebra in metric gravity and then first-order gravity. We will see that a rich structure appears that explains from the continuum perspective the non-commutativity of geometric flux, the quantization of the area spectra, the nature of the simplicity constraints but also reveals the dual momentum observables and finally allow to reconcile the elements of canonical gravity with Lorentz invariance.
I will discuss the issue of quantization as a challenge of finding a representation of the boundary algebra and will give clues about where we are in this process.

Let G be a complex reductive group, and X be any smooth projective G-variety. In this talk, we will construct an algebra homomorphism from the G-equivariant homology of the affine Grassmannian Gr_G to the G-equivariant quantum cohomology of X. The construction uses shift operators in quantum cohomologies. We will also discuss the possible extension to the loop rotation equivariant setting and the relation with the Peterson isomorphism when X is the flag variety associated with G. This is based on joint work with Alexander Braverman.