Representation Theory, Calabi–Yau Manifolds, and Mirror Symmetry

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

Directions and Recommended Lodging

The conference was held in hybrid format, both in-person and online.

The workshop was partially supported by Simons and NSF Grant DMS-2227199.



  • 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)


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

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