CMSA

Lipschitz properties of transport maps under a log-Lipschitz condition
2022-11-30 15:00 - 16:00 (America/New_York) | Harvard Science Center, 1 Oxford Street, Cambridge, MA, 02138
Probability Seminar

Location: Room 109, Harvard Science Center, 1 Oxford Street, Cambridge MA 02138
Speaker: Dan Mikulincer (MIT)

Title: Lipschitz properties of transport maps under a log-Lipschitz condition
Abstract: Consider the problem of realizing a target probability measure as a push forward, by a transport map, of a given source measure. Typically one thinks about the target measure as being ‘complicated’ while the source is simpler and often more structured. In such a setting, for applications, it is desirable to find Lipschitz transport maps which afford the transfer of analytic properties from the source to the target. The talk will focus on Lipschitz regularity when the target measure satisfies a log-Lipschitz condition.
I will present a construction of a transport map, constructed infinitesimally along the Langevin flow, and explain how to analyze its Lipschitz constant. The analysis of this map leads to several new results which apply both to Euclidean spaces and manifolds, and which, at the moment, seem to be out of reach of the classically studied optimal transport theory.
Joint work with Max Fathi and Yair Shenfeld.

Representation Theory, Calabi–Yau Manifolds, and Mirror Symmetry
2022-11-28 09:00 - 2022-12-01 15:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Videos are available on the CMSA Youtube Playlist.
On November 28 – Dec 1, 2022, the CMSA hosted a Workshop on Representation Theory, Calabi-Yau Manifolds, and Mirror Symmetry.
Organizers: An Huang (Brandeis University) | Siu-Cheong Lau (Boston University) | Tsung-Ju Lee (CMSA, Harvard) | Andrew Linshaw (University of Denver)
Scientific Advisor: Shing-Tung Yau (Harvard, Tsinghua)
Location: Room G10, CMSA, 20 Garden Street, Cambridge MA 02138

The conference was held in hybrid format, both in-person and online.
The workshop was partially supported by Simons and NSF Grant DMS-2227199.

Speakers:

Tomoyuki Arakawa (Kyoto)
Thomas Creutzig (Edmonton)
Jonathan Mboyo Esole (Northeastern)
Fei Han (National University of Singapore)
Shinobu Hosono (Gakushuin University)
Flor Orosz Hunziker (Colorado)
Cuipo Jiang (Shanghai)
Shashank Kanade (Denver)
Matt Kerr (Washington University in St. Louis)
Carl Lian (Humboldt-Universität zu Berlin)
Nai-Chung Conan Leung (CUHK)
Ivan Loseu (Yale)
Robert McRae (Tsinghua University)
Anne Moreau (Université Paris-Saclay, Orsay)
Tony Pantev (University of Pennsylvania)
Mauricio Romo (Tsinghua University)
Bailin Song (USTC)
Cumrun Vafa (Harvard University)
Chin-Lung Wang (National Taiwan University)
Weiqiang Wang (Virginia)
Yaping Yang (University of Melbourne)
Shing-Tung Yau (Tsinghua University)
Chenglong Yu (Tsinghua University)
Gufang Zhao (University of Melbourne)

Schedule (Eastern Time)
Schedule (pdf)
11/28 (Monday)

08:30am – 08:55am
Refreshments

08:55am – 09:00am
Opening remarks by Horng-Tzer Yau

09:00am – 09:45am
Shing-Tung Yau*
Title: The Hull-Strominger system through conifold transitions
Abstract: In this talk I discuss the geometry of C-Y manifolds outside of the Kähler regime and especially describe the Hull-Strominger system through the conifold transitions.

10:00am – 10:45am
Chenglong Yu*
Title: Commensurabilities among Lattices in PU(1,n)
Abstract: In joint work with Zhiwei Zheng, we study commensurabilities among certain subgroups in PU(1,n). Those groups arise from the monodromy of hypergeometric functions. Their discreteness and arithmeticity are classified by Deligne and Mostow. Thurston also obtained similar results via flat conic metrics. However, the classification of the lattices among them up to conjugation and finite index (commensurability) is not completed. When n=1, it is the commensurabilities of hyperbolic triangles. The cases of n=2 are almost resolved by Deligne-Mostow and Sauter’s commensurability pairs, and commensurability invariants by Kappes-Möller and McMullen. Our approach relies on the study of some higher dimensional Calabi-Yau type varieties instead of complex reflection groups. We obtain some relations and commensurability indices for higher n and also give new proofs for existing pairs in n=2.

11:00am – 11:45am
Thomas Creutzig*
Title: Shifted equivariant W-algebras
Abstract: The CDO of a compact Lie group is a family of VOAs whose top level is the space of functions on the Lie group. Similar structures appear at the intersections of boundary conditions in 4-dimensional gauge theories, I will call these new families of VOAs shifted equivariant W-algebras. I will introduce these algebras, construct them and explain how they can be used to quickly prove the GKO-coset realization of principal W-algebras.

11:45am – 1:30 pm
Lunch

01:30pm – 02:15pm
Cumrun Vafa
Title: Reflections on Mirror Symmetry
Abstract: In this talk I review some of the motivations leading to the search and discovery of mirror symmetry as well as some of the applications it has had.

02:30pm – 03:15pm
Jonathan Mboyo Esole
Title: Algebraic topology and matter representations in F-theory
Abstract: Recently, it was observed that representations appearing in geometric engineering in F-theory all satisfy a unique property: they correspond to characteristic representations of embedding of Dynkin index one between Lie algebras. However, the reason why that is the case is still being understood. In this talk, I will present new insights, giving a geometric explanation for this fact using K-theory and the topology of Lie groups and their classifying spaces. In physics, this will be interpreted as conditions on the charge of instantons and the classifications of Wess-Zumino-Witten terms.

03:15pm – 03:45 pm
Break

03:45pm – 04:30pm
Weiqiang Wang
Title: A Drinfeld presentation of affine i-quantum groups
Abstract: A quantum symmetric pair of affine type (U, U^i) consists of a Drinfeld-Jimbo affine quantum group (a quantum deformation of a loop algebra) U and its coideal subalgebra U^i (called i-quantum group). A loop presentation for U was formulated by Drinfeld and proved by Beck. In this talk, we explain how i-quantum groups can be viewed as a generalization of quantum groups, and then we give a Drinfeld type presentation for the affine quasi-split i-quantum group U^i. This is based on joint work with Ming Lu (Sichuan) and Weinan Zhang (Virginia).

04:45pm – 05:30pm
Tony Pantev
Title: Decomposition, anomalies, and quantum symmetries
Abstract: Decomposition is a phenomenon in quantum physics which converts quantum field theories with non-effectively acting gauge symmetries into equivalent more tractable theories in which the fields live on a disconnected space. I will explain the mathematical content of decomposition which turns out to be a higher categorical version of Pontryagin duality. I will examine how this duality interacts with quantum anomalies and secondary quantum symmetries and will show how the anomalies can be canceled by homotopy coherent actions of diagrams of groups. I will discuss in detail the case of 2-groupoids which plays a central role in anomaly cancellation, and will describe a new duality operation that yields decomposition in the presence of anomalies. The talk is based on joint works with Robbins, Sharpe, and Vandermeulen.

11/29 (Tuesday)

Refreshments

09:00am – 09:45am
Robert MacRae*
Title: Rationality for a large class of affine W-algebras
Abstract: One of the most important results in vertex operator algebras is Huang’s theorem that the representation category of a “strongly rational” vertex operator algebra is a semisimple modular tensor category. Conversely, it has been conjectured that every (unitary) modular tensor category is the representation category of a strongly rational (unitary) vertex operator algebra. In this talk, I will describe my results on strong rationality for a large class of affine W-algebras at admissible levels. This yields a large family of modular tensor categories which generalize those associated to affine Lie algebras at positive integer levels, as well as those associated to the Virasoro algebra.

10:00am – 10:45am
Bailin Song*
Title: The global sections of chiral de Rham complexes on compact Calabi-Yau manifolds
Abstract: Chiral de Rham complex is a sheaf of vertex algebras on a complex manifold. We will describe the space of global sections of the chiral de Rham complexes on compact Calabi-Yau manifolds.

11:00am – 11:45am
Carl Lian*
Title: Curve-counting with fixed domain
Abstract: The fixed-domain curve-counting problem asks for the number of pointed curves of fixed (general) complex structure in a target variety X subject to incidence conditions at the marked points. The question comes in two flavors: one can ask for a virtual count coming from Gromov-Witten theory, in which case the answer can be computed (in principle) from the quantum cohomology of X, or one can ask for the “honest” geometric count, which tends to be more subtle. The answers are conjectured to agree in the presence of sufficient positivity, but do not always. I will give an overview of some recent results and open directions. Some of this work is joint with Alessio Cela, Gavril Farkas, and Rahul Pandharipande.

11:45am – 01:30pm
Lunch

01:30pm – 02:15pm
Chin-Lung Wang
Title: A blowup formula in quantum cohomology
Abstract: We study analytic continuations of quantum cohomology $QH(Y)$ under a blowup $phi: Y to X$ of complex projective manifolds along the extremal ray variable $q^{ell}$. Under $H(Y) = phi^* H(X) oplus K$ where $K = ker phi_*$, we show that (i) the restriction of quantum product along the $phi^*H(X)$ direction, denoted by $QH(Y)_X$, is meromorphic in $x := 1/q^ell$, (ii) $K$ deforms uniquely to a quantum ideal $widetilde K$ in $QH(Y)_X$, (iii) the quotient ring $QH(Y)_X/widetilde K$ is regular over $x$, and its restriction to $x = 0$ is isomorphic to $QH(X)$. This is a joint work (in progress) with Y.-P. Lee and H.-W. Lin.

02:30pm – 03:15pm
Ivan Loseu
Title: Quantizations of nilpotent orbits and their Lagrangian subvarieties
Abstract: I’ll report on some recent progress on classifying quantizations of the algebras of regular functions of nilpotent orbits (and their covers) in semisimple Lie algebras, as well as the classification of quantizations of certain Lagrangian subvarieties. An ultimate goal here is to understand the classification of unitary representations of real semisimple Lie groups.

03:15pm – 03:45pm
Break

03:45pm – 04:30pm
Matt Kerr*
Title: $K_2$ and quantum curves
Abstract: The basic objects for this talk are motives consisting of a curve together with a $K_2$ class, and their mixed Hodge-theoretic invariants.
My main objective will be to explain a connection (recently proved in joint work with C. Doran and S. Sinha Babu) between (i) Hodge-theoretically distinguished points in the moduli of such motives and (ii) eigenvalues of operators on L^2(R) obtained by quantizing the equations of the curves.
By local mirror symmetry, this gives evidence for a conjecture in topological string theory (due to M. Marino, A. Grassi, and others) relating enumerative invariants of toric CY 3-folds to spectra of quantum curves.

04:45pm – 05:30pm
Flor Orosz Hunziker
Title: Tensor structures associated to the N=1 super Virasoro algebra
Abstract: We have recently shown that there is a natural category of representations associated to the N=1 super Virasoro vertex operator algebras that have braided tensor structure. We will describe this category and discuss the problem of establishing its rigidity at particular central charges. This talk is based on joint work in progress with Thomas Creutzig, Robert McRae and Jinwei Yang.

11/30 (Wednesday)

08:30am – 09:00am
Refreshments

09:00am – 09:45am
Tomoyuki Arakawa
Title: 4D/2D duality and representation theory
Abstract: This talk is about the 4D/2D duality discovered by Beem et al. rather recently in physics. It associates a vertex operator algebra (VOA) to any 4-dimensional superconformal field theory, which is expected to be a complete invariant of thl theory. The VOAs appearing in this manner may be regarded as chiralization of various symplectic singularities and their representations are expected to be closely related with the Coulomb branch of the 4D theory. I will talk about this remarkable 4D/2D duality from a representation theoretic perspective.

10:00am – 10:45am
Shashank Kanade
Title: Combinatorics of principal W-algebras of type A
Abstract: The combinatorics of principal W_r(p,p’) algebras of type A is controlled by cylindric partitions. However, very little seems to be known in general about fermionic expressions for the corresponding characters. Welsh’s work explains the case of Virasoro minimal models W_2(p,p’). Andrews, Schilling and Warnaar invented and used an A_2 version of the usual (A_1) Bailey machinery to give fermionic characters (up to a factor of (q)_infty) of some, but not all, W_3(3,p’) modules. In a recent joint work with Russell, we have given a complete set of conjectures encompassing all of the remaining modules for W_3(3,p’), and proved our conjectures for small values of p’. In another direction, characters of W_r(p,p’) algebras also arise as appropriate limits of certain sl_r coloured Jones invariants of torus knots T(p,p’), and we expect this to provide further insights on the underlying combinatorics.

11:00am – 11:45am
Gufang Zhao
Title: Quasimaps to quivers with potentials
Abstract: This talk concerns non-compact GIT quotient of a vector space, in the presence of an abelian group action and an equivariant regular function (potential) on the quotient. We define virtual counts of quasimaps from prestable curves to the critical locus of the potential. The construction borrows ideas from the theory of gauged linear sigma models as well as recent development in shifted symplectic geometry and Donaldson-Thomas theory of Calabi-Yau 4-folds. Examples of virtual counts arising from quivers with potentials are discussed. This is based on work in progress, in collaboration with Yalong Cao.

11:45am – 01:30pm
Group Photo, Lunch

01:30pm – 02:15pm
Yaping Yang
Title: Cohomological Hall algebras and perverse coherent sheaves on toric Calabi-Yau 3-folds
Abstract: Let X be a smooth local toric Calabi-Yau 3-fold. On the cohomology of the moduli spaces of certain sheaves on X, there is an action of the cohomological Hall algebra (COHA) of Kontsevich and Soibelman via “raising operators”. I will discuss the “double” of the COHA that acts on the cohomology of the moduli space by adding the “lowering operators”. We associate a root system to X. The double COHA is expected to be the shifted Yangian of this root system. We also give a prediction for the shift in terms of an intersection pairing. We provide evidence of the aforementioned expectation in various examples. This is based on my joint work with M. Rapcak, Y. Soibelman, and G. Zhao

02:30pm – 03:15pm
Fei Han
Title: Graded T-duality with H-flux for 2d sigma models
Abstract: T-duality in string theory can be realised as a transformation acting on the worldsheet fields in the two-dimensional nonlinear sigma model. Bouwknegt-Evslin-Mathai established the T-duality in a background flux for the first time upon compactifying spacetime in one direction to a principal circle by constructing the T-dual maps transforming the twisted cohomology of the dual spacetimes. In this talk, we will describe our recent work on how to promote the T-duality maps of Bouwknegt-Evslin-Mathai in two aspects. More precisely, we will introduce (1) graded T-duality, concerning the graded T-duality maps of all levels of twistings; (2) the 2-dimensional sigma model picture, concerning the double loop space of spacetimes. This represents our joint work with Mathai.

03:15pm – 3:45pm
Break

03:45pm – 04:30pm
Mauricio Romo
Title: Networks and BPS Counting: A-branes view point
Abstract: I will review the countings of BPS invariants via exponential/spectral networks and present an interpretation of this counting as a count of certain points in the moduli space of A-branes corresponding to degenerate Lagrangians.

04:45pm – 05:30pm
Shinobu Hosono
Title: Mirror symmetry of abelian fibered Calabi-Yau manifolds with ρ = 2
Abstract: I will describe mirror symmetry of Calabi-Yau manifolds fibered by (1,8)-polarized abelian surfaces, which have Picard number two. Finding a mirror family over a toric variety explicitly, I observe that mirror symmetry of all related Calabi-Yau manifods arises from the corresponding boundary points, which are not necessarily toric boundary points. Calculating Gromov-Witten invariants up to genus 2, I find that the generating functions are expressed elliptic (quasi-)modular forms, which reminds us the modular anomaly equation found for elliptic surfaces. This talk is based on a published work with Hiromichi Takaki (arXiv:2103.08150).

06:00pm
Banquet @ Royal East Restaurant, 782 Main St, Cambridge, MA 02139

12/1 (Thursday)

08:30am – 09:00am
Refreshments

09:00am – 09:45am
Conan Nai Chung Leung*
Title: Quantization of Kahler manifolds
Abstract: I will explain my recent work on relationships among geometric quantization, deformation quantization, Berezin-Toeplitz quantization and brane quantization.

10:00am – 10:45am
Cuipo Jiang*
Title: Cohomological varieties associated to vertex operator algebras
Abstract: We define and examine the cohomological variety of a vertex algebra, a notion cohomologically dual to that of the associated variety, which measures the smoothness of the associated scheme at the vertex point. We study its basic properties. As examples, we construct a closed subvariety of the cohomological variety for rational affine vertex operator algebras constructed from finite dimensional simple Lie algebras. We also determine the cohomological varieties of the simple Virasoro vertex operator algebras. These examples indicate that, although the associated variety for a rational $C_2$-cofinite vertex operator algebra is always a simple point, the cohomological variety can have as large a dimension as possible. This talk is based on joint work with Antoine Caradot and Zongzhu Lin.

11:00am – 11:45am
Anne Moreau*
Title: Action of the automorphism group on the Jacobian of Klein’s quartic curve
Abstract: In a joint work with Dimitri Markouchevitch, we prove that the quotient variety of the 3-dimensional Jacobian of the plane Klein quartic curve by its full automorphism group of order 336 is isomorphic to the 3-dimensional weighted projective space with weights 1,2,4,7.
The latter isomorphism is a particular case of the general conjecture of Bernstein and Schwarzman suggesting that a quotient of the n-dimensional complex space by the action of an irreducible complex crystallographic group generated by reflections is a weighted projective space.
In this talk, I will explain this conjecture and the proof of our result. An important ingredient is the computation of the Hilbert function of the algebra of invariant theta-functions on the Jacobian.

11:45am – 11:50am
Closing remarks

11:50am
Free discussions and departure

* = Online speaker
CMSA COVID-19 Policies

Friday after Thanksgiving
2022-11-25 | MA
Holiday: Friday after Thanksgiving
The CMSA will be closed on Friday, November 25, 2022.

Continuum field theory of graphene bilayer system
2022-11-23 09:00 - 10:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar
Speaker: Jian Kang, School of Physical Science and Technology, ShanghaiTech University, Shanghai, China
Title: Continuum field theory of graphene bilayer system
Abstract: The Bistritzer-MacDonald (BM) model predicted the existence of the narrow bands in the magic-angle twisted bilayer graphene (MATBG), and nowadays is a starting point for most theoretical works. In this talk, I will briefly review the BM model and then present a continuum field theory [1] for graphene bilayer system allowing any smooth lattice deformation including the small twist angle. With the gradient expansion to the second order, the continuum theory for MATBG [2] produces the spectrum that almost perfectly matches the spectrum of the microscopic model, suggesting the validity of this theory. In the presence of the lattice deformation, the inclusion of the pseudo-vector potential does not destroy but shift the flat band chiral limit to a smaller twist angle. Furthermore, the continuum theory contains another important interlayer tunneling term that was overlooked in all previous works. This term non-negligibly breaks the particle-hole symmetry of the narrow bands and may be related with the experimentally observed particle-hole asymmetry.
1. O. Vafek and JK, arXiv: 2208.05933.
2. JK and O. Vafek, arXiv: 2208.05953.

3D gravity and gravitational entanglement entropy
2022-11-22 09:30 - 11:00 | Virtual
Quantum Matter Seminar
Speaker: Gabriel Wong (Harvard CMSA)
Title: 3D gravity and gravitational entanglement entropy
Abstract: Recent progress in AdS/CFT has provided a good understanding of how the bulk spacetime is encoded in the entanglement structure of the boundary CFT. However, little is known about how spacetime emerges directly from the bulk quantum theory. We address this question in an effective 3d quantum theory of pure gravity, which describes the high temperature regime of a holographic CFT. This theory can be viewed as a $q$-deformation and dimensional uplift of JT gravity. Using this model, we show that the Bekenstein-Hawking entropy of a two-sided black hole equals the bulk entanglement entropy of gravitational edge modes. These edge modes transform under a quantum group, which defines the data associated to an extended topological quantum field theory. Our calculation suggests an effective description of bulk microstates in terms of collective, anyonic degrees of freedom whose entanglement leads to the emergence of the bulk spacetime. Finally, we give a proposal for obtaining the Ryu Takayanagi formula using the same quantum group edge modes.

Light states in the interior of CY moduli spaces
2022-11-18 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Damian van de Heisteeg
Title: Light states in the interior of CY moduli spaces
Abstract: In string theory one finds that states become massless as one approaches boundaries in Calabi-Yau moduli spaces. In this talk we look in the opposite direction, that is, we search for points where the mass gap for these light states is maximized — the so-called desert. In explicit examples we identify these desert points, and find that they correspond to special points in the moduli space of the CY, such as orbifold points and rank two attractors.

Dynamic and multicolor electron microscopy
2022-11-17 13:00 - 14:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Active Matter Seminar

Speaker: Max Prigozhin (Harvard)
Title: Dynamic and multicolor electron microscopy
Abstract: My lab is developing biophysical methods to achieve multicolor and dynamic biological imaging at the molecular scale. Our approach to capturing the dynamics of cellular processes involves cryo-vitrifying samples after known time delays following stimulation using custom cryo- plunging and high-pressure freezing instruments. To achieve multicolor electron imaging, we are exploring the property of cathodoluminescence—optical emission induced by the electron beam. We are developing nanoprobes (“cathodophores”) that will be used as luminescent protein tags in electron microscopy. We are applying these new methods to study G-protein- coupled receptor signaling and to visualize the formation of biomolecular condensates.

Ringdown and geometry of trapping for black holes
2022-11-17 09:30 - 10:30 | Hybrid
General Relativity Seminar

Speaker: Semyon Dyatlov (MIT)
Title: Ringdown and geometry of trapping for black holes
Abstract: Quasi-normal modes are complex exponential frequencies appearing in long time expansions of solutions to linear wave equations on black hole backgrounds. They appear in particular during the ringdown phase of a black hole merger when the dynamics is expected to be driven by linear effects. In this talk I give an overview of various results in pure mathematics which relate asymptotic behavior of quasi-normal modes at high frequency to the geometry of the set of trapped null geodesics, such as the photon sphere in Schwarzschild (-de Sitter). These trapped geodesics have two kinds of behavior: the geodesic flow is hyperbolic in directions normal to the trapped set (a feature stable under perturbations) and it is completely integrable on the trapped set. It turns out that normal hyperbolicity gives information about the rate of decay of quasi-normal modes, while complete integrability gives rise to a quantization condition.

Outlier-Robust Algorithms for Clustering Non-Spherical Mixtures
2022-11-16 15:30 - 16:30 | MA
Probability Seminar

Speaker: Ainesh Bakshi (MIT)
Title: Outlier-Robust Algorithms for Clustering Non-Spherical Mixtures
Abstract: In this talk, we describe the first polynomial time algorithm for robustly clustering a mixture of statistically-separated, high-dimensional Gaussians. Prior to our work this question was open even in the special case of 2 components in the mixture. Our main conceptual contribution is distilling analytic properties of distributions, namely hyper-contractivity of degree-two polynomials and anti-concentration of linear projections, which are necessary and sufficient for clustering.

Noether’s Learning Dynamics: Role of Symmetry Breaking in Neural Networks
2022-11-16 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Colloquium
Speaker: Hidenori Tanaka (NTT Research at Harvard)
Title: Noether’s Learning Dynamics: Role of Symmetry Breaking in Neural Networks
Abstract: In nature, symmetry governs regularities, while symmetry breaking brings texture. In artificial neural networks, symmetry has been a central design principle, but the role of symmetry breaking is not well understood. Here, we develop a Lagrangian formulation to study the geometry of learning dynamics in neural networks and reveal a key mechanism of explicit symmetry breaking behind the efficiency and stability of modern neural networks. Then, we generalize Noether’s theorem known in physics to describe a unique symmetry breaking mechanism in learning and derive the resulting motion of the Noether charge: Noether’s Learning Dynamics (NLD). Finally, we apply NLD to neural networks with normalization layers and discuss practical insights. Overall, through the lens of Lagrangian mechanics, we have established a theoretical foundation to discover geometric design principles for the learning dynamics of neural networks.

Vacuum fluctuations in cavities: breakdown of the topological protection in the integer Quantum Hall effect
2022-11-16 10:00 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar
Speaker: Jérôme Faist (ETH Zurich)
Title: Vacuum fluctuations in cavities: breakdown of the topological protection in the integer Quantum Hall effect
Abstract: When a collection of electronic excitations are strongly coupled to a single mode cavity, mixed light-matter excitations called polaritons are created. The situation is especially interesting when the strength of the light-matter coupling ΩR is such that the coupling energy becomes close to the one of the bare matter resonance ω0. For this value of parameters, the system enters the so-called ultra-strong coupling regime, in which a number of very interesting physical effects were predicted caused by the counter-rotating and diamagnetic terms of the Hamiltonian.
In a microcavity, the strength of the electric field caused by the vacuum fluctuations, to which the strength of the light-matter coupling ΩR is proportional, scales inversely with the cavity volume. One very interesting feature of the circuit-based metamaterials is the fact that this volume can be scaled down to deep subwavelength values in all three dimension of space.1 Using metamaterial coupled to two-dimensional electron gases under a strong applied magnetic field, we have now explored to which extend this volume can be scaled down and reached a regime where the stability of the polariton is limited by diffraction into a continuum of plasmon modes2.
We have also used transport to probe the ultra-strong light-matter coupling3, and show now that the latter can induce a breakdown of the integer quantum Hall effect4. The phenomenon is explained in terms of cavity-assisted hopping, an anti-resonant process where an electron can scatter from one edge of the sample to the other by “borrowing” a photon from the cavity5. We are also evaluating a proposal suggesting that the value of the quantization voltage can be renormalized by the cavity6.

Scalari, G. et al. Ultrastrong Coupling of the Cyclotron Transition of a 2D Electron Gas to a THz Metamaterial. Science 335, 1323–1326 (2012).
Rajabali, S. et al. Polaritonic Nonlocality in Light Matter Interaction. Nat Photon 15, 690–695 (2021).
Paravicini-Bagliani, G. L. et al. Magneto-Transport Controlled by Landau Polariton States. Nat. Phys. 15, 186–190 (2019).
Appugliese, F. et al. Breakdown of topological protection by cavity vacuum fields in the integer quantum Hall effect. Science 375, 1030–1034 (2022).
Ciuti, C. Cavity-mediated electron hopping in disordered quantum Hall systems. Phys. Rev. B 104, 155307 (2021).
Rokaj, V., Penz, M., Sentef, M. A., Ruggenthaler, M. & Rubio, A. Polaritonic Hofstadter butterfly and cavity control of the quantized Hall conductance. Phys. Rev. B 105, 205424 (2022).

The Emergence Proposal in Quantum Gravity and the Species Scale
2022-11-15 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Swampland Seminar
Speaker: Alvaro Herraez (Saclay)
Title: The Emergence Proposal in Quantum Gravity and the Species Scale
Abstract: The Emergence Proposal claims that in Quantum Gravity the kinetic terms of the fields in the IR emerge from integrating out (infinite) towers of particles up to the QG cutoff. After introducing this proposal in the context of the Swampland Program, I will explain why it is natural to identify this QG cutoff with the Species Scale, motivating it by direct computation in the presence of the relevant towers. Then, I will present evidence for this proposal by directly studying how it is realized in different string theory setups, where the kinetic terms of scalars, p-forms and even scalar potentials can be shown to emerge after integrating out such towers.

Topology of the Fermi sea: Ordinary metals as topological materials
2022-11-15 09:30 - 11:00 | MA
Quantum Matter Seminar
Speaker: Pok Man Tam (University of Pennsylvania)
Title: Topology of the Fermi sea: Ordinary metals as topological materials
Abstract: It has long been known that the quantum ground state of a metal is characterized by an abstract manifold in momentum space called the Fermi sea. Fermi sea can be distinguished topologically in much the same way that a ball can be distinguished from a donut by counting the number of holes. The associated topological invariant, i.e. the Euler characteristic (χ_F), serves to classify metals. Here I will survey two recent proposals relating χ_F to experimental observables, namely: (i) equal-time density/number correlations [1], and (ii) Andreev state transport along a planar Josephson junction [2]. Moreover, from the perspective of quantum information, I will explain how multipartite entanglement in real space probes the Fermi sea topology in momentum space [1]. Our works not only provide a new connection between topology and entanglement in gapless quantum matters, but also suggest accessible experimental platforms to extract the topology in metals.
[1] P. M. Tam, M. Claassen, C. L. Kane, Phys. Rev. X 12, 031022 (2022)
[2] P. M. Tam and C. L. Kane, arXiv:2210.08048

Quantum trace and length conjecture for hyperbolic knot
2022-11-11 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Mauricio Romo
Title: Quantum trace and length conjecture for hyperbolic knot
Abstract: I will define the quantum trace map for an ideally triangulated hyperbolic knot complement on S^3. This map assigns an operator to each element L of the Kauffman Skein module of knot complement. Motivated by an interpretation of this operator in the context of SL(2,C) Chern-Simons theory, one can formulate a ‘length conjecture’ for the hyperbolic length of L.

Schwarzschild-like Topological Solitons in Gravity
2022-11-10 09:30 - 10:30 | Virtual
General Relativity Seminar

Speaker: Pierre Heidmann (Johns Hopkins)
Title: Schwarzschild-like Topological Solitons in Gravity
Abstract: We present large classes of non-extremal solitons in gravity that are asymptotic to four-dimensional Minkowski spacetime plus extra compact dimensions. They correspond to smooth horizonless geometries induced by topology in spacetime and supported by electromagnetic flux, which characterize coherent states of quantum gravity. We discuss a new approach to deal with Einstein-Maxwell equations in more than four dimensions, such that they decompose into a set of Ernst equations. We generate the solitons by applying different techniques associated with the Ernst formalism. We focus on solitons with zero net charge yet supported by flux, and compare them to Schwarzschild black holes. These are also ultra-compact geometries with very high redshift but differ in many aspects. At the end of the talk, we discuss the stability properties of the solitons and their gravitational signatures.

Liouville quantum gravity from random matrix dynamics
2022-11-09 15:30 - 16:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Probability Seminar
Speaker: Hugo Falconet (Courant Institute, NYU)
Title: Liouville quantum gravity from random matrix dynamics
Abstract: The Liouville quantum gravity measure is a properly renormalized exponential of the 2d GFF. In this talk, I will explain how it appears as a limit of natural random matrix dynamics: if (U_t) is a Brownian motion on the unitary group at equilibrium, then the measures $|det(U_t – e^{i theta}|^gamma dt dtheta$ converge to the 2d LQG measure with parameter $gamma$, in the limit of large dimension. This extends results from Webb, Nikula and Saksman for fixed time. The proof relies on a new method for Fisher-Hartwig asymptotics of Toeplitz determinants with real symbols, which extends to multi-time settings. I will explain this method and how to obtain multi-time loop equations by stochastic analysis on Lie groups.
Based on a joint work with Paul Bourgade.

CMSA/Tsinghua Math-Science Literature Lecture: Large cardinals and small sets: The AD+ Duality Program
2022-11-09 09:30 - 11:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
CMSA/Tsinghua Math-Science Literature Lecture

Prof. Hugh Woodin will present a lecture in the CMSA/Tsinghua Math-Science Literature Lecture Series.
Date: Wednesday, November 9, 2022
Time: 9:30 – 11:00 am ET
Location: Via Zoom Webinar and Room G10, CMSA, 20 Garden Street, Cambridge MA 02138

Title: Large cardinals and small sets: The AD+ Duality Program
Abstract: The determinacy axiom, AD, was introduced by Mycielski and Steinhaus over 60 years ago as an alternative to the Axiom of Choice for the study of arbitrary sets of real numbers. The modern view is that determinacy axioms concern generalizations of the borel sets, and deep connections with large cardinal axioms have emerged.
The study of determinacy axioms has led to a specific technical refinement of AD, this is the axiom AD+. The further connections with large axioms have in turn implicitly led to a duality program, this is the AD+ Duality Program.
The main open problems here are intertwined with those of the Inner Model Program, which is the central program in the study of large cardinal axioms.
This has now all been distilled into a series of specific conjectures.

Talk chair: Horng-Tzer Yau (Harvard Mathematics & CMSA)
Moderator: Alejandro Poveda Ruzafa (Harvard CMSA)

Beginning in Spring 2020, the CMSA began hosting a lecture series on literature in the mathematical sciences, with a focus on significant developments in mathematics that have influenced the discipline, and the lifetime accomplishments of significant scholars.

CMSA COVID-19 Policies

Topological symmetry in field theory
2022-11-08 11:30 - 13:00 | Virtual
Quantum Matter Seminar
Speaker: Daniel S. Freed (U Texas)
Title: Topological symmetry in field theory
Abstract: Recently there has been lots of activity surrounding generalized notions of symmetry in quantum field theory, including “categorical symmetries,” “higher symmetries,” “noninvertible symmetries,” etc. Inspired by definitions of abstract (finite) groups and algebras and their linear actions, we introduce a framework for these symmetries in field theory and a calculus of topological defects based on techniques in topological field theory. This is joint work with Constantin Teleman and Greg Moore.

EFT strings and emergence
2022-11-07 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Swampland Seminar
Speaker: Fernando Marchesano (IFT Madrid)
Title: EFT strings and emergence
Abstract: We revisit the Emergence Proposal in 4d N=2 vector multiplet sectors that arise from type II string Calabi-Yau compactifications, with emphasis on the role of axionic fundamental strings, or EFT strings. We focus on large-volume type IIA compactifications, where EFT strings arise from NS5-branes wrapping internal four-cycles, and consider a set of infinite-distance moduli-space limits that can be classified in terms of a scaling weight w=1,2,3. It has been shown before how one-loop threshold effects of an infinite tower of BPS particles made up of D2/D0-branes generate the asymptotic behaviour of the gauge kinetic functions along limits with $w=3$. We extend this result to w=2 limits, by taking into account D2-brane multi-wrapping numbers. In w=1 limits the leading tower involves EFT string oscillations, and one can reproduce the behaviour of both weakly and strongly-coupled U(1)’s independently on whether the EFT string is critical or not, by assuming that charged modes dominate the light spectrum.

Force transmission informs the collective behavior of active cell layers
2022-11-03 13:00 - 14:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Active Matter Seminar
Speaker: Siavash Monfared, Niels Bohr Institute, Copenhagen
Title: Force transmission informs the collective behavior of active cell layers
Abstract: Collective cell migration drives numerous physiological processes such as tissue morphogenesis, wound healing, tumor progression and cancer invasion. However, how the interplay of mechanical interactions and the modes of collective self-organization among cells informs such processes is yet to be established. In this talk, I will focus on the role of three-dimensional force transmission, from a theoretical and computational perspective, on two phenomena: (1) cell extrusion from a cellular monolayer and (2) density-independent solid-like to fluid-like transition of active cell layers. For the first topic, I will focus on how increasing cell-cell adhesion relative to cell-substrate adhesion enables cells to collectively exploit distinct mechanical pathways – leveraging defects in nematic and hexatic phases associated with cellular arrangement – to eliminate an unwanted cell. For the second topic, I will show how solid-like to fluid-like transition in active cell layers is linked to the percolation of isotropic stresses. This is achieved via two distinct and independent paths to model this transition by increasing (a) cell-cell adhesion and (b) active traction forces. Additionally, using finite-size scaling analyses, the phase transition associated with each path is mapped onto the 2D site percolation universality class. Our results highlight the importance of force transmission in informing the collective behavior of living cells and opens the door to new sets of questions for those interested in connecting the physics of cellular self-organization to the dynamics of biological systems.

Asymptotic geometry of null hypersurface in Schwarzschild spacetime and null Penrose inequality
2022-11-03 10:30 - 11:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar

Speaker: Pengyu Le (BIMSA)
Title: Asymptotic geometry of null hypersurface in Schwarzschild spacetime and null Penrose inequality
Abstract: Null Penrose inequality is an important case of the well-known Penrose inequality on a null hypersurface. It conjectures the relation between the area of the outmost marginally trapped surface and the Bondi mass at null infinity. Following the proposal of Christodoulou and Sauter, we employ the perturbation method to study the asymptotic geometry of null hypersurfaces at null infinity in a perturbed vacuum Schwarzshild spacetime. We explain how to apply this perturbation theory to prove null Penrose inequality on a nearly spherically symmetric null hypersurface in a perturbed vacuum Schwarzschild spacetime.

Doping and inverting Mott insulators on semiconductor moire superlattices
2022-11-02 12:45 - 13:45 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Liang Fu (MIT)

Title: Doping and inverting Mott insulators on semiconductor moire superlattices
Abstract: Semiconductor bilayer heterostructures provide a remarkable platform for simulating Hubbard models on an emergent lattice defined by moire potential minima. As a hallmark of Hubbard model physics, the Mott insulator state with local magnetic moments has been observed at half filling of moire band. In this talk, I will describe new phases of matter that grow out of the canonical 120-degree antiferromagnetic Mott insulator on the triangular lattice. First, in an intermediate range of magnetic fields, doping this Mott insulator gives rise to a dilute gas of spin polarons, which form a pseudogap metal. Second, the application of an electric field between the two layers can invert the many-body gap of a charge-transfer Mott insulator, resulting in a continuous phase transition to a quantum anomalous Hall insulator with a chiral spin structure. Experimental results will be discussed and compared with theoretical predictions.

Optical axion electrodynamics
2022-11-02 09:00 - 10:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar
Speaker: Junyeong Ahn (Harvard)
Title: Optical axion electrodynamics
Abstract: Electromagnetic fields in a magneto-electric medium behave in close analogy to photons coupled to the hypothetical elementary particle, the axion. This emergent axion electrodynamics is expected to provide novel ways to detect and control material properties with electromagnetic fields. Despite having been studied intensively for over a decade, its theoretical understanding remains mostly confined to the static limit. Formulating axion electrodynamics at general optical frequencies requires resolving the difficulty of calculating optical magneto-electric coupling in periodic systems and demands a proper generalization of the axion field. In this talk, I will introduce a theory of optical axion electrodynamics that allows for a simple quantitative analysis. Then, I will move on to discuss the issue of the Kerr effect in axion antiferromagnets, refuting the conventional wisdom that the Kerr effect is a measure of the net magnetic moment. Finally, I will apply our theory to a topological antiferromagnet MnBi2Te4.
References:
[1] Theory of Optical Axion Electrodynamics, J. Ahn, S.Y. Xu, A.Vishwanath, arXiv:2205.06843

Kardar-Parisi-Zhang dynamics in integrable quantum magnets
2022-11-01 09:00 - 10:30 | Virtual
Quantum Matter Seminar
Speaker: Francisco Machado (Berkeley/Harvard)
Title: Kardar-Parisi-Zhang dynamics in integrable quantum magnets
Abstract: Although the equations of motion that govern quantum mechanics are well-known, understanding the emergent macroscopic behavior that arises from a particular set of microscopic interactions remains remarkably challenging. One particularly important behavior is that of hydrodynamical transport; when a quantum system has a conserved quantity (i.e. total spin), the late-time, coarse-grained dynamics of the conserved charge is expected to follow a simple, classical hydrodynamical description. However the nature and properties of this hydrodynamical description can depend on many details of the underlying interactions. For example, the presence of additional dynamical constraints can fundamentally alter the propagation of the conserved quantity and induce slower-than-diffusion propagation. At the same time, the presence of an extensive number of conserved quantities in the form of integrability, can imbue the system with stable quasi-particles that propagate ballistically through the system.
In this talk, I will discuss another possibility that arises from the interplay of integrability and symmetry; in integrable one dimensional quantum magnets with complex symmetries, spin transport is neither ballistic nor diffusive, but rather superdiffusive. Using a novel method for the simulation of quantum dynamics (termed Density Matrix Truncation), I will present a detailed analysis of spin transport in a variety of integrable quantum magnets with various symmetries. Crucially, our analysis is not restricted to capturing the dynamical exponent of the transport dynamics and enables us to fully characterize its universality class: for all superdiffusive models, we find that transport falls under the celebrated Kardar-Parisi-Zhang (KPZ) universality class.
Finally, I will discuss how modern atomic, molecular and optical platforms provide an important bridge to connect the microscopic interactions to the resulting hydrodynamical transport dynamics. To this end, I will present recent experimental results, where this KPZ universal behavior was observed using atoms confined to an optical lattice.
[1] Universal Kardar-Parisi-Zhang dynamics in integrable quantum systems
B Ye†, FM*, J Kemp*, RB Hutson, NY Yao
(PRL in press) – arXiv:2205.02853
[2] Quantum gas microscopy of Kardar-Parisi-Zhang superdiffusion
D Wei, A Rubio-Abadal, B Ye, FM, J Kemp, K Srakaew, S Hollerith, J Rui, S Gopalakrishnan, NY Yao, I Bloch, J Zeiher
Science (2022) — arXiv:2107.00038

Some non-concave dynamic optimization problems in finance
2022-10-28 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Member Seminar
Speaker: Shuaijie Qian (Harvard CMSA)
Title: Some non-concave dynamic optimization problems in finance
Abstract: Non-concave dynamic optimization problems appear in many areas of finance and economics. Most of existing literature solves these problems using the concavification principle, and derives equivalent, concave optimization problems whose value functions are still concave. In this talk, I will present our recent work on some non-concave dynamic optimization problems, where the concavification principle may not hold and the resulting value function is indeed non-concave.
The first work is about the portfolio selection model with capital gains tax and a bitcoin mining model with exit options and technology innovation, where the average tax basis and the average mining cost serves as an approximation, respectively. This approximation results in a non-concave value function, and the associated HJB equation problem turns out to admit infinitely many solutions. We show that the value function is the minimal (viscosity) solution of the HJB equation problem. Moreover, the penalty method still works and converges to the value function.
The second work is about a non-concave utility maximization problem with portfolio constraints. We find that adding bounded portfolio constraints, which makes the concavification principle invalid, can significantly affect economic insights in the existing literature. As the resulting value function is likely discontinuous, we introduce a new definition of viscosity solution, prove the corresponding comparison principle, and show that a monotone, stable, and consistent finite difference scheme converges to the solution of the utility maximization problem.

2-Categories and the Massive 3d A-Model
2022-10-28 09:30 - 10:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Algebraic Geometry in String Theory Seminar
Speaker: Ahsan Khan, IAS
Title: 2-Categories and the Massive 3d A-Model
Abstract: I will outline the construction of a 2-category associated to a hyperKahler moment map. The construction is based on partial differential equations in one, two, and three dimensions combined with a three-dimensional version of the Gaiotto-Moore-Witten web formalism.

Gravitational Wave, Angular Momentum, and Supertranslation Ambiguity
2022-10-27 10:30 - 11:30 | Virtual
General Relativity Seminar

Speaker: Naqing Xie (Fudan University)
Title: Gravitational Wave, Angular Momentum, and Supertranslation Ambiguity

Abstract: The supertranslation ambiguity of angular momentum is a long-standing and conceptually important issue in general relativity. Recently, there appeared the first definition of angular momentum at null infinity that is supertranslation invariant. However, in the compact binary coalescence community, supertranslation ambiguity is often ignored. We have shown that, in the linearised theory of gravitational wave, the new angular momentum coincides with the classical definition at the quadrupole level. This talk is based on a recent joint work with Xiaokai He and Xiaoning Wu.

From Engine to Auto
2022-10-26 14:00 - 15:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
New Technologies in Mathematics Seminar
Speakers: João Araújo, Mathematics Department, Universidade Nova de Lisboa and Michael Kinyon, Department of Mathematics, University of Denver

Title: From Engine to Auto

Abstract: Bill McCune produced the program EQP that deals with first order logic formulas and in 1996 managed to solve Robbins’ Conjecture. This very powerful tool reduces to triviality any result that can be obtained by encoding the assumptions and the goals. The next step was to turn the program into a genuine assistant for the working mathematician: find ways to help the prover with proofs; reduce the lengths of the automatic proofs to better crack them; solve problems in higher order logic; devise tools that autonomously prove results of a given type, etc.

In this talk we are going to show some of the tools and strategies we have been producing. There will be real illustrations of theorems obtained for groups, loops, semigroups, logic algebras, lattices and generalizations, quandles, and many more.

Clique listing algorithms
2022-10-26 12:30 - 13:30 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Virginia Vassilevska Williams (MIT)
Title: Clique listing algorithms
Abstract: A k-clique in a graph G is a subgraph of G on k vertices in which every pair of vertices is linked by an edge. Cliques are a natural notion of social network cohesiveness with a long history.
A fundamental question, with many applications, is “How fast can one list all k-cliques in a given graph?”.
Even just detecting whether an n-vertex graph contains a k-Clique has long been known to be NP-complete when k can depend on n (and hence no efficient algorithm is likely to exist for it). If k is a small constant, such as 3 or 4 (independent of n), even the brute-force algorithm runs in polynomial time, O(n^k), and can list all k-cliques in the graph; though O(n^k) time is far from practical. As the number of k-cliques in an n-vertex graph can be Omega(n^k), the brute-force algorithm is in some sense optimal, but only if there are Omega(n^k) k-cliques. In this talk we will show how to list k-cliques faster when the input graph has few k-cliques, with running times depending on the number of vertices n, the number of edges m, the number of k-cliques T and more. We will focus on the case when k=3, but we will note some extensions.
(Based on joint work with Andreas Bjorklund, Rasmus Pagh, Uri Zwick, Mina Dalirrooyfard, Surya Mathialagan and Yinzhan Xu)

Kähler bands—Chern insulators, holomorphicity and induced quantum geometry
2022-10-26 09:00 - 10:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Topological Quantum Matter Seminar

Speaker: Bruno Mera, Tohoku University

Title: Kähler bands—Chern insulators, holomorphicity and induced quantum geometry

Abstract: The notion of topological phases has dramatically changed our understanding of insulators. There is much to learn about a band insulator beyond the assertion that it has a gap separating the valence bands from the conduction bands. In the particular case of two dimensions, the occupied bands may have a nontrivial topological twist determining what is called a Chern insulator. This topological twist is not just a mathematical observation, it has observable consequences—the transverse Hall conductivity is quantized and proportional to the 1st Chern number of the vector bundle of occupied states over the Brillouin zone. Finer properties of band insulators refer not just to the topology, but also to their geometry. Of particular interest is the momentum-space quantum metric and the Berry curvature. The latter is the curvature of a connection on the vector bundle of occupied states. The study of the geometry of band insulators can also be used to probe whether the material may host stable fractional topological phases. In particular, for a Chern band to have an algebra of projected density operators which is isomorphic to the W∞ algebra found by Girvin, MacDonald and Platzman—the GMP algebra—in the context of the fractional quantum Hall effect, certain geometric constraints, associated with the holomorphic character of the Bloch wave functions, are naturally found and they enforce a compatibility relation between the quantum metric and the Berry curvature of the band. The Brillouin zone is then endowed with a Kähler structure which, in this case, is also translation-invariant (flat). Motivated by the above, we will provide an overview of the geometry of Chern insulators from the perspective of Kähler geometry, introducing the notion of a Kähler band which shares properties with the well-known ideal case of the lowest Landau level. Furthermore, we will provide a prescription, borrowing ideas from geometric quantization, to generate a flat Kähler band in some appropriate asymptotic limit. Such flat Kähler bands are potential candidates to host and realize fractional Chern insulating phases. Using geometric quantization arguments, we then provide a natural generalization of the theory to all even dimensions.

References:
[1] Tomoki Ozawa and Bruno Mera. Relations between topology and the quantum metric for Chern insulators. Phys. Rev. B, 104:045103, Jul 2021.
[2] Bruno Mera and Tomoki Ozawa. Kähler geometry and Chern insulators: Relations between topology and the quantum metric. Phys. Rev. B, 104:045104, Jul 2021.
[3] Bruno Mera and Tomoki Ozawa. Engineering geometrically flat Chern bands with Fubini-Study Kähler structure. Phys. Rev. B, 104:115160, Sep 2021.
