REPRESENTATION THEORY, REPRESENTATION VARIETIES AND APPLICATIONS.

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1 REPRESENTATION THEORY, REPRESENTATION VARIETIES AND APPLICATIONS Berkeley team. 1. Research Proposal (1) Tim Cramer Postdoc. I am interested in derived algebraic geometry and its application to representation theory and mathematical physics. Techniques from derived algebraic geometry have recently found applications in such areas as the representation theory of real and complex groups and the theory of D-modules. Another area where they have been applied, which I m particularly interested in, is Donaldson-Thomas theory and the theory of Hall algebras. Donaldson-Thomas invariants are defined using the virtual fundamental class and are best understood in the context of derived stacks. Recently, the notion of shifted symplectic structures has led to the categorification of Donaldson-Thomas invariants. In another direction, the Hall algebras of Joyce-Song and Kontsevich-Soibelman have been used to prove formal properties of Donaldson-Thomas invariants, such as their behavior with respect to wall-crossing. These properties generally reflect the behavior of the Hall algebra under derived equivalences. The construction of the Hall algebra of a derived category has been underway for many years with the most successful attempt being made by Toen through the use of dg categories. Like the understanding of the virtual fundamental class, this is a fundamental problem in derived algebraic geometry, and more generally, in the theory of higher categories. (2) Edward Frenkel One of the most exciting developments in mathematical physics in the last few years has been the work of Edward Witten and collaborators unifying the Langlands Program with dualities in Quantum Field Theory and String Theory. Physicists have the following elegant explanation of electromagnetic duality in terms of mysterious six-dimensional quantum field theories, known as (0,2) superconformal field theory (these are labeled by the ADE Dynkin diagrams): If this theory is compactified on a torus, we obtain a four-dimensional maximally supersymmetric gauge theory, and electromagnetic duality then corresponds to exchanging the radii of the basic circles on the torus. This raises the following question: What will happen if we replace the torus by an arbitrary Riemann surface X of genus g? This way, we obtain a myriad of dualities between various four-dimensional gauge theories, and it turns out that they are closely connected to confomal field theories on the Riemann surface X. Frenkel is studying these dualities and their generalizations obtained by introducing certain surface operators. (3) Michael Hutchings Professor. My research concerns invariants of contact manifolds and symplectic cobordisms between them, which are defined in terms of Reeb orbits (closed orbits of the Reeb vector field) on a contact manifold and holomorphic curves asymptotic to them. Different representations of the data of Reeb orbits and holomorphic curves give rise to different invariants, including linearized contact homology and rational symplectic field theory of contact manifolds of any dimension, and embedded contact homology of contact three-manifolds. I propose to develop such invariants and apply them to symplectic embedding problems and dynamical questions about Reeb vector fields. Specific projects: (1) Extend embedded contact homology to a field theory giving cobordism maps induced by four-dimensional strong symplectic cobordisms between contact three-manifolds. (2) Use these cobordism maps to define invariants of Lagrangian submanifolds of symplectic four-manifolds, and attempt to define related invariants of Legendrian knots in contact three-manifolds. (3) Calculate ECH capacities (certain numerical invariants of symplectic four-manifold with boundary defined using embedded contact homology that obstruct symplectic embeddings) for unit cotangent bundles of surfaces with Riemannian metrics with and without boundary and relate this to the dynamics of geodesics and billiards. (4) Explore new symplectic capacities defined using rational SFT and other representations of the data of Reeb orbits and holomorphic curves, and study the resulting symplectic embedding obstructions. (5) Extend the Weinstein conjecture for contact three-manifolds by improving the lower bound on the number of Reeb orbits or by establishing the existence of short Reeb orbits. (4) Martin Olsson. Professor. Olsson proposes to work on a variety of questions in algebraic and arithmetic geometry. More specifically on questions about moduli theory and log geometry, independence of l and other motivic aspects of varieties over finite fields, and algebraic stacks. The principal aim of the project on moduli theory and log geometry is to give modular interpretations of main components in moduli spaces. In past work Olsson and others have done this in a variety of cases. The main new case to be considered 1

2 2 REPRESENTATION THEORY, REPRESENTATION VARIETIES AND APPLICATIONS. in this project is that of spherical varieties. These are certain varieties with actions of reductive groups, and their moduli theory is in some ways similar to polarized toric varieties. The project on independence of l is a multifaceted project ultimately aimed at understanding the motivic aspects of sheaves on varieties. Currently Olsson has a large body of work in the subject which should be completed in the coming year. The main new areas of research here concern vanishing cycle functors and intersection cohomology, as well as p-adic cohomology theories. In the last project concerning stacks, Olsson is in particular interested in studying the notion of essential dimension. This notion is particularly interesting for classifying stacks of group where the notion can be phrased purely group theoretically, but studying it in the more general context of stacks enables one to use geometric techniques. The proposed work is well-suited for collaboration with members of the Tsinghua math community some of whom work in closely related areas. (5) Qi You Postdoc My interests in mathematics lie primarily in categorification, representation theory and applications to low dimensional topology. Categorification, initiated in the seminal paper by Crane and Frenkel, is a way of looking at some areas of mathematics from a higher point of view, with the hope that complicated phenomena come from shadows of simpler yet richer structures from above. Khovanov homology, which is a categorification of the Jones polynomial, is one of the first significant examples of this phenomenon. In general, constructing a categorification of some object needs a considerable amount of innovation, and geometric methods serve as an important guiding principle, with categories of sheaves leading to interesting categorifications. Currently, I am focusing on realizing the initial dream envisioned by Crane and Frenkel, on which we have made some initial progress. In particular, together with Khovanov and my other collaborators, we have provided a constructive of the small quantum group u q (sl 2 ) when q is a prime root of unity. I plan to further systematically lift this construction to a 4d-topological field theory, to investigate its relationship with Khovanov homology, as well as to for its algebro-geometric meanings. (6) Nicolai Reshetikhin. Professor. Currently he is working on mainly on two subjects. One of them is topological quantum field theory. In particular, he is interested questions related to quantum field theories with on space time with boundary and their semiclassical limit. Another direction is lies at the interface of solvable models in statistical mechanics, quantum integrable systems and representation theory of coideal subalgebras in quantum affine algebras. It is well known that the quantum Knizhnik-Zamilodchikov equation is closely related to the intertwines for quantum affine algebras (goes back to the work of I.Frenkel and N. Reshetikhin). In a similar way, intertwiners for co-ideal subalgebras of quantum affine algebras, related to reflecting boundary conditions for quantum integrable spin chains, satisfy the reflection qkz equation. N. Reshetikhin (with J. Stokman and B. Vlaar) obtained explicit formulae for solutions to these equations for quantum sl 2. They will be working on developing representation theoretical aspects of these results. This is closely related to vertex and q-vertex operators and the research at Tsinghua. (7) Vera Serganova. Professor. The methods of supersymmetry factor the questions interesting for physicists through the theory of representations of supergroups and superalgebras. There are still a lot of gaps in mathematical foundations in this subject: the methods familiar from representation theory of reductive groups get stuck early since the algebraic structure of representations is significantly more complicated. Below we list some projects in representation theory of superalgebras. 1. Categorification of Fock space via finite-dimensional representations of the orthosymplectc Lie supergroup. The main idea is to realize Fock space as the Grothendieck group of representations of osp(m, n) for all m and n and categorify the action of creation and annihilation operator by the natural functors of cohomological induction and restriction. 2. Lie supegroups and Deligne categories. In 2002 Deligne introduced the categories Rep GL(t) and Rep OSP (t). Those are pseudo-abelian tensor categories universal in the following sense: the categories of finite-dimensional representations of classical algebraic groups can be obtained from Deligne categories by specialization of the parameter t and taking a suitable quotient. Moreover, in the same way one can obtain categories of finite-dimensional representations of classical supergroups GL(m, n) and OSP (m, n). Deligne categories are semisimple for non-integral t, but when t is an integer it is an open problem to construct an abelian envelope for Rep GL(t) and Rep OSP (t). It seems possible to construct those abelian envelopes by taking a special version of inverse limit of the categories Rep GL(m, n) and Rep OSP (m, n), where t = m n. Another interesting problem is to develop a theory of Deligne categories for two other classical supergroups P (n) and Q(n). 3. Generalizing Borel Weil Bott theorem for supergroups. Let G be a classical supergroup. The theory of flag varieties for G was developed in 1980-s by Manin, Penkov, Skornyakov and Voronov. In

3 REPRESENTATION THEORY, REPRESENTATION VARIETIES AND APPLICATIONS. 3 particular, for a fixed G there are finitely many non-isomorphic flag supervarieties G/B (since not all Borel subgroups in G are conjugate). It was also known from this time that the naive analogue of the Borel Weil Bott theorem holds only for typical λ (thesis of I. Penkov). In general the cohomology H i (G/B, O(λ)) may not vanish for several i and may not be irreducible. Although partial results in this area were obtained almost 30 years ago, a complete answer is still unknown. We hope to solve this problem by categorifying odd reflections. (8) Constantin Teleman. Professor. Teleman is planning to investigate braided tensor deformations of the category of representations of a Lie algebra g. Long ago, Drinfeld constructed such formal deformations from invariant quadratic tensors in g. More interestingly, those deformations could be exponentiated, leading to the theory of quantum groups. At roots of unity, the representation categories have finite quotients which were used by Reshetikhin and Turaev to produce 3-manifold structures (quantum Chern-Simons theory). Now it is known (from homological mirror symmetry, for instance) that the wrong-degree part of the deformation complex of a category can be interpreted as deformation in the derived, Z/2 periodic world. It is therefore tempting to assert that all invariant symmetric tensors in g, of all degrees, give formal deformations for the braided tensor category of representations in this derived Z/2 periodic sense. Creatively interpreted, this is likely to be true. However, it is unknown whether these formal deformations extend to the quantum group. Specifically, one would like to specialize these derived deformations at roots of unity and use them to deform the 3-manifold structures. A hint for this is provided by the K-theoretic reduction of these structures (the K-theoretic Verlinde algebra) which was described by Freed, Hopkins and Teleman using twisted topological K-theory. It was noticed that more exotic higher twistings of K-theory, defined from representations of g, give rise to formal deformations of this K-theoretic Verlinde algebra. The project, therefore, is to investigate whether these K-theory deformations lift to braided tensor deformations of the category of representations. The information about bio-data of Berkeley participants can be found on the Departmental web site: Tsinghua team. Hall algebras of tame quivers and Fork space representations of quantum affine algebras (Bangming Deng, Jie Xiao): After the introduction of canonical basis, the study of Fock space representation has achieved an important progress. First, Leclerc and Thibon and some others studied the canonical basis of the Fock space. Second, Vasserot, Varagnolo and Schiffmann studied in detail the canonical basis of the Hall algebra of a cyclic quiver and related it to the canonical basis of the Fock space. By the work of Ringel and Lusztig, the Auslander-Reiten quiver of the nilpotent representations of a cyclic quiver is a tube, and its Hall algebra contains a proper subalgebra isomorphic to the quantum affine sl n. Moreover, the Hall algebra is a reductive extension of the quantum affine sl n. In this case, Hall algebra itself also admits the canonical basis. On the one hand, we want to extend the Fock space as a representation of the double Hall algebra of the cyclic quiver and to establish a relationship between Fock space representation with the basic representation. On the other hand, we want to generalize the above construction for the cyclic quiver case to a tame quiver of type Ã, D, or Ẽ. The difficulty is to find a good Hall algebra model in the general case. Hua and Xiao has proved that Hall algebras of tame quivers are reductive extensions of affine quantum groups. By a recent work of Lin, Xiao and Zhang, the Hall algebra generated by perprojective, preinjective modules and those from non-homogeneous tubes has the generic property. We wish to give a geometric construction of this algebra and provide an algebraic construction of its canonical basis. By a comparison of the global canonical basis with the canonical basis of affine quantum groups, we shall expect to obtain the canonical basis of Fock space representations in the general case. Calabi-Yau Spaces and Mirror Symmetry (Wenxuan Lu): Calabi-Yau spaces are important objects in geometry and mathematical physics. In differential geometry they are spaces with Kahler-Einstein metrics which are central objects of geometric analysis. In algebraic geometry Calabi-Yau varieties can be considered as generalizations of elliptic curves. In string theory Calabi-Yau spaces are important classes of spacetime backgrounds because of the special holonomies they have and the fact that they are solutions of the vacuum Einstein s equation.moreover Calabi-Yau spaces include Hyperkahler spaces. Some important moduli spaces in geometry such as Hitchin s moduli spaces are Hyperkahler. A guiding principle in the study of Calabi-Yau spaces is mirror symmetry. Mirror symmetry demands a duality of seemingly different pairs of families of Calabi-Yau spaces which exchanges geometric objects on them. These objects include bundles, sheaves and Lagrangian submanifolds. They have physical interpretations as topological D-branes in string theory. A deeper level study of mirror symmetry requires us to utilize the full Calabi-Yau structure and various conditions this structure can naturally pose on topological D-branes. This means we want to study bundles (or sheaves) endowed with some nice connections and Lagrangian submanifolds with some volume

4 4 REPRESENTATION THEORY, REPRESENTATION VARIETIES AND APPLICATIONS. minimizing properties (they are called special Lagrangians). They correspond to so-called BPS branes in string theory and are crucial in the study of realistic models, black holes and many other topics in string theory. My future research will be about the following two problems which are at the core of the study of Calabi-Yau geometry and mirror symmetry: 1 the problem of Calabi-Yau structures (in the framework of mirror symmetry); 2 the problem of stability conditions and related wall crossing phenomena. There are many ways to study Calabi-Yau structures (the Ricci flat metric). Besides Yau s continuity method establishing the existence one can use the following techniques: gluing constructions, (Ricci) flows, balanced metrics arising from projective embeddings (this has been used for numerical studies), some variational problems in the space of closed forms (Hitchin s functionals), etc. However none of the above listed approaches seems to be directly usable in the framework of mirror symmetry especially in SYZ s conjectural picture. I am trying to figure out bridges between the traditional ways of studying Calabi-Yau structure and the insights of mirror symmetry. The second problem arises from the fact that BPS conditions of branes are related to stability conditions. The prototype of the whole area is the equivalence between Hermitian Yang-Mills connections and slope stability conditions of holomorphic bundles established by Donaldson, Uhlenbeck and Yau. In general one expects that there is some categorical structure for branes and stability conditions should be formulated to satisfy the axioms of Douglas and Bridgeland. A stability condition depends on a charge (topological information) and some moduli (geometric information). As a result there are real codimensional one locus in the space of moduli (marginal stability walls) across which the set of stable objects (BPS branes) can change. This is known as wall crossing. This problem is closely related to the first problem because Calabi-Yau structures receive the so-called instanton corrections which also have wall crossings. Some of my previous works focused on this issue and revealed some nice structures and relations. I plan to investigate further implications of wall crossings on mirror symmetry and geometry of Calabi-Yau structures. Quantum Q-system and double affine Hecke algebra (Xiaoguang Ma): For any simple Lie algebra, and for many other Kac-Moody algebras, there exists an associated system of equations known as a Q-system. Solutions to the Q-system describe characters of special representations of the quantum affine algebra U q (ĝ) or of the Yangian Y (g). The quantum Q-system of type A was introduced in [FK1] and was generalised to simply-laced case in [FK2]. This q-deformation is related to the Feigin-Loktev fusion product. In [FK2], the authors define a generating function in the simply-laced case and use this generating function prove that the graded M-sum equals to the N-sum. One step in their proof is to calculate the constant term of the generating functions (with some modifications). The main object in this project is trying to understand the constant term of this generating function in a different way. We want to apply the techniques which are used in Cherednik s proof of Macdonald s constant term conjecture. We hope there are a family of functions so that this constant term defines an inner product among these functions. We could give an alternative proof of this constant term formula. Also some algebra may appear as the double affine Hecke algebra appearing in the proof of Macdonald s case. This algebra should be interesting. References [FK1] Philippe Di Francesco, Rinat Kedem, Non-commutative integrability, paths and quasideterminants. Adv. Math., 228:97-152, [FK2] Philippe Di Francesco, Rinat Kedem, Quantum cluster algebras and fusion products, IMRN Volume 2014, Issue 5. Derived Hall algebras and quantum cluster algebras (Jie Xiao, Fan Xu): We will focus on Hall algebras associated to triangulated categories. We study the interactive relations between Hall algebras and (quantum) cluster algebras. The key point is to define the proper map from derived Hall algebras to quantum cluster algebras, which should can be compared with the integration map from a derived Hall algebra to the corresponding quantum torus. Bridgeland recently defined an analogue of Ringel-Hall algebras and give the global realization of a quantum group. In order to define the map above, we try to construct (quantum) cluster algebras by an extensive application of Bridgelands approach to derived Hall algebra. The map is meaningful. It is an interface to relate canonical bases of Hall algebras to (cluster) algebras and vice versa. It also relates the stability condition of triangulated categories, derived Hall algebras and the Donaldson-Thomas invariant. Infinite dimensional Lie algebras and vertex algebras (Minxian Zhu): The Peter-Weyl decomposition allows one to identify the regular functions of a complex semisimple Lie group with the matrix coefficients of its irreducible representations. One can therefore define multiplication of regular functions in terms of the intertwining operators of its irreducible modules.

5 REPRESENTATION THEORY, REPRESENTATION VARIETIES AND APPLICATIONS. 5 In the infinite-dimensional setting, this process becomes more interesting. One can construct a vertex algebra arising from the (modified) regular representation of an infinite-dimensional Lie algebra using intertwining operators and the differential equations that govern the correlation functions of these intertwining operators. So far, the two main examples where this construction has been successfully carried out are for affine Lie algebras and the Virasoro algebra. In either case, we pair modules from two different categories of representations that have dual central charges. In the case of affine Lie algebras, the two central charges k and k add up to 2h where h is the dual Coxeter number of the simple Lie algebra, while in the Virasoro case, c and c add up to 26. The former class are also known as vertex algebras of differential operators on a complex semisimple Lie group. The representation structure of these algebras is very simple (a direct sum) when k is irrational. The rational case is more subtle, but can be understood by studying the regular representation of the corresponding quantum group at a root of unity. An ongoing project is to investigate further the structure of these families. One may ask if it is possible to extend the vertex algebra V c(κ), c(κ) to integral values of κ by analytic continuation where c(κ) = 13 6κ 6κ 1, c(κ) = κ + 6κ 1. One may try to construct a basis in V c, c such that the correlation functions of the vertex operators with respect to this basis have no poles at integral values of κ. This may also help answer the question whether there is a restricted version of V c, c when κ is an integer. Also, we can attempt to realize V 1,25 via a nice Fock space construction.