On November 28 – Dec 1, 2022, the CMSA will host a Workshop on Representation Theory, Calabi-Yau Manifolds, and Mirror Symmetry.

For more information, please see: https://cmsa.fas.harvard.edu/event/representation-theory-calabi-yau-manifolds-and-mirror-symmetry/

The Motivated by the higher Chern bands of twisted graphene multilayers, we consider flat bands with

arbitrary Chern number C with ideal quantum geometry. While C>1 bands differ from Landau levels, we

show that these bands host exact fractional Chern insulator (FCI) ground states for short range

interactions. We show how to decompose ideal higher Chern bands into separate ideal bands with Chern

number 1 that are intertwined through translation and rotation symmetry. The decomposed bands admit

an SU(C) action that combines real space and momentum space translations. Remarkably, they also

allow for analytic construction of exact many-body ground states, such as generalized quantum Hall

ferromagnets and FCIs, including flavor-singlet Halperin states and Laughlin ferromagnets in the limit

of short-range interactions. In this limit, the SU(C) action is promoted to a symmetry on the ground state

subspace. While flavor singlet states are translation symmetric, the flavor ferromagnets correspond to

translation broken states and admit charged skyrmion excitations corresponding to a spatially varying

density wave pattern. We confirm our analytic predictions with numerical simulations of ideal bands of

twisted chiral multilayers of graphene, and discuss consequences for experimentally accessible systems

such as monolayer graphene twisted relative to a Bernal bilayer.

For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/quantum-matter-seminar/

 In this talk I will discuss the application of the Cobordism Conjecture to type IIB supergravity with non-trivial duality bundle. Calculating the relevant bordism groups we find that they are highly non-trivial and would predict the presence of various global symmetries in the underlying theory. Since quantum gravity theories do not allow for global symmetries, we discuss which defects need to be included to break them completely. Interestingly, we find many backgrounds that are well-known in the F-theory literature, such as [p,q]-7-branes, non-Higgsable clusters, as well as S-folds and their generalizations to higher codimensions. Further including worldsheet reflections, predicts the existence of a new non-supersymmetric 7-brane with interesting properties and applications, which I will discuss in more detail.

We explore generalized global symmetries in theories of physics beyond the Standard Model. Theories of Z′ bosons generically contain ‘non-invertible’ chiral symmetries, whose presence indicates a natural paradigm to break this symmetry by an exponentially small amount in an ultraviolet completion. For example, in models of gauged lepton family difference such as the phenomenologically well-motivated U(1)Lμ−Lτ, there is a non-invertible lepton number symmetry which protects neutrino masses. We embed these theories in gauged non-Abelian horizontal lepton symmetries, e.g. U(1)Lμ−Lτ⊂SU(3)H, where the generalized symmetries are broken nonperturbatively by the existence of lepton family magnetic monopoles. In such theories, either Majorana or Dirac neutrino masses may be generated through quantum gauge theory effects from the charged lepton Yukawas e.g. yν∼yτexp(−Sinst). These theories require no bevy of new fields nor ad hoc additional global symmetries, but are instead simple, natural, and predictive: the discovery of a lepton family Z′ at low energies will reveal the scale at which Lμ−Lτ emerges from a larger gauge symmetry.

For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/quantum-matter-seminar/

I will discuss various aspects of the geometry of Quot schemes of torsion quotients on a smooth projective curve. I will describe in particular results on the cohomology of tautological vector bundles over the Quot scheme, which mirror a parallel picture in the geometry of the Hilbert scheme of points over a surface. The subject has an interesting combinatorial and computational flavor. The talk is partly based on joint work with D. Oprea and S. Sam.

Cavity modification of material properties and phenomena is a novel research field motivated by the advances in strong light-matter interactions~[1]. For condensed matter systems it has been demonstrated experimentally that the transport properties of 2D materials can be modified via coupling to vacuum fields~[2,3]. While in polaritonic chemistry it has been shown that ground state chemical properties can be controlled with cavity fields~[4].  In the first part of my talk, I will present how the quantized cavity field can alter the conduction  properties of a condensed matter system by focusing on the paradigmatic Sommerfeld model of the free electron gas~[5]. The exact analytic solution of the Sommerfeld model in the cavity will be presented as well as its fundamental properties. Then, in the second part of the talk, I will focus on a many-particle system of cold ions in a harmonic trap coupled to the cavity field. I will show how this system couples collectively to the cavity and that hybrid states between light and matter, known as polaritons, emerge. The formation of polaritons leads to the modification of the properties of the cold ions and enhances the localization of the many-body wave function~[6]. Connections to experiments will be discussed as well.

[1] F. Garcia-Vidal, C. Ciuti, T. W. Ebbesen, Science, 373, 178 (2021)

[2] G. L. Paravicini-Bagliani et al., Nat. Phys. 15, 186-190 (2019)

[3] F. Appugliese et al., Science 375 (6584), 1030-1034 (2022)

[4] T. W. Ebbesen, Acc. Chem. Res. 49, 11, 2403–2412 (2016)

[5] V. Rokaj, M. Ruggenthaler, F. G. Eich, A. Rubio, Phys. Rev. Research 4, 013012 (2022)

[6] V. Rokaj, S.I. Mistakidis, H.R. Sadeghpour, arXiv:2207.03436 (2022)

For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/topological-quantum-matter-seminar/

The current “Transformer era” of deep learning is marked by the emergence of combinatorial and algorithmic reasoning capabilities in large sequence models, leading to dramatic advances in natural language understanding, program synthesis, and theorem proving. What is the nature of these models’ internal representations (i.e. how do they represent the states and computational steps of the algorithms they execute)? How can we understand and mitigate their weaknesses, given that they resist interpretation? In this work, we present some insights (and many further mysteries) through the lens of automata and their algebraic structure.

Specifically, we investigate the apparent mismatch between recurrent models of computation (automata & Turing machines) and Transformers (which are typically shallow and non-recurrent). Using tools from circuit complexity and semigroup theory, we characterize shortcut solutions, whereby a shallow Transformer with only o(T) layers can exactly replicate T computational steps of an automaton. We show that Transformers can efficiently represent these shortcuts in theory; furthermore, in synthetic experiments, standard training successfully finds these shortcuts. We demonstrate that shortcuts can lead to statistical brittleness, and discuss mitigations.

Joint work with Bingbin Liu, Jordan Ash, Surbhi Goel, and Akshay Krishnamurthy.

This seminar will be held in person and on Zoom. For more information on how to join, please see: https://live-hu-cmsa-222.pantheonsite.io/event_category/new-technologies-in-mathematics-seminar-series/

Based on joint work with Pravesh Kothari.

The nonlinear stability of the slowly-rotating Kerr-de Sitter family was first proven by Hintz and Vasy in 2016 using microlocal techniques. In my talk, I will present a novel proof of the nonlinear stability of slowly-rotating Kerr-de Sitter spacetimes that avoids frequency-space techniques outside of a neighborhood of the trapped set. The proof uses vector field techniques to uncover a spectral gap corresponding to exponential decay at the level of the linearized equation. The exponential decay of solutions to the linearized problem is then used in a bootstrap proof to conclude nonlinear stability.

For more information on how to join, please see: https://cmsa.fas.harvard.edu/event/general-relativity-2021-22/

Equilibrium statistical mechanics tells us how to control the self-assembly of passive materials by tuning the competition between energy and entropy to achieve desired states of organization. Out of equilibrium, no such principles apply and self organization principles are scarce. Active matter describes systems in which individual units dissipate energy to exert forces on their environment. Dissipation and injection of energy are then disconnected at the microscopic scale, hence driving the system strongly out of thermal equilibrium. This leads to a phenomenology markedly different from that of equilibrium systems, such as the emergence of dense phases in the absence of cohesive attractive forces, but it also leaves us without guiding principles to
understand the self-organization of active matter. In this talk I will review the progress which has been made over the past ten years to control the organization of self-propelled agents using motility control, either externally or through interactions. I will show that generic principles apply and illustrate the theoretical developments presented in the talk using recent experiments on the motility-induced self-organization of bacterial mixtures.

This seminar will be held in person and on Zoom. For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/active-matter-seminar/

The notion of global symmetry in quantum field theory (QFT) has witnessed dramatic generalizations in the past few years. One of the most exciting developments has been the identification of 4d QFTs possessing non-invertible symmetries, i.e. global symmetries whose generators exhibit fusion rules that are not group-like. In this talk, I will discuss realizations of non-invertible symmetries in string theory and holography. As a concrete case study, I will consider the Klebanov-Strassler setup for holographic confinement in Type IIB string theory. The global symmetries of the holographic 4d QFT (both invertible and non-invertible) can be accessed by studying the topological couplings of the low-energy effective action of the dual 5d supergravity theory. Moreover, non-invertible symmetry defects can be realized in terms of D-branes. The D-brane picture captures non-trivial aspects of the fusion of non-invertible symmetry defects, and of their action on extended operators of the 4d QFT.

For more information on how to join, please see: https://cmsa.fas.harvard.edu/event_category/quantum-matter-seminar/

The intersection theory of the moduli space of K3 surfaces polarized by a lattice is a subject of recent interest because of its deep connections with a wide variety of mathematics, including the intersection theory of moduli spaces of curves and the study of modular forms. Oprea and Pandharipande conjectured that the tautological rings of these moduli spaces of K3 surfaces are highly structured in a way that mirrors the picture for the moduli space of curves. I will discuss the proof of this conjecture in the case of K3 surfaces polarized by a degree 2 line bundle. This is joint work with Dragos Oprea and Rahul Pandharipande.

This seminar will be held on Monday, December 12th*

We study the phase fluctuations in the normal state of a general two-dimensional (2d) superconducting system with s-wave pairing. The effect of phase fluctuations of the pairing fields can be dealt with perturbatively using disorder averaging, after we treat the local superconducting order parameter as a static disordered background. It is then confirmed that the phase fluctuations above the 2d Berenzinskii-Kosterlitz-Thouless (BKT) transition give birth to the pseudogap phenomenon, leading to a significant broadening of the single-particle spectral functions. Quantitatively, the broadening of the spectral weights at the BCS gap is characterized by the ratio of the superconducting coherence length and the spatial correlation length of the superconducting pairing order parameter. Our results are tested on the attractive-U fermion Hubbard model on the square lattice, using unbiased determinant quantum Monte Carlo method and stochastic analytic continuation. We also apply our method to 2d superconductors with d-wave pairing and observe that the phase fluctuations may lead to Fermi-arc phenomenon above the BKT transition.

For more information on how to join, please see: CMSA quantum matter in mathematics and physics