A summary of the research programme for our EPSRC Programme Grant on 'Motivic invariants and Categorification'
What 'motivic invariants' and 'categorification' mean
The proposal aims to generalize ‘classical’ theories from a broad class of problems in Geometry or Algebra in two directions: we can make them motivic, or we can categorify them. Both generalizations should yield a richer theory than the classical situation, with more structure and more information, and may be used to prove new facts about the classical case. Examples of suitable classical theories are enumerative invariants in Geometry, and representations of some algebra in Representation Theory.
In Algebraic Geometry, an invariant of varieties Φ is motivic if it is additive over cutting into pieces. The most basic is the Euler characteristic χ, but there are many others. The aim of ‘motivic’ generalization is to replace Euler characteristics by more general motivic invariants Φ. This gives a new, richer theory, and it can also make new things possible. For example, for a quotient space (stack) X/G one expects Φ(X/G) = Φ(X)/Φ(G). But when Φ=χ this fails, as χ(G) = 0 for most groups G, and you cannot divide by 0. So for stacks, motivic techniques often work better than Euler characteristic ones.
There is a natural sequence of classes of mathematical objects, of increasing complexity: integers, vector spaces, categories, 2-categories, . . . . Moving leftwards in this sequence is easy, and loses information: to a vector space we associate its dimension, and so on. Categorification roughly means moving rightwards in this sequence in some problem, adding information, turning integer invariants into vector spaces, etc.
Both motivic and categoric generalizations have strong connections with Physics. The ‘Quantum’ in subjects like Quantum Groups can often be interpreted (for instance, in the construction of Quantum Groups as Ringel–Hall algebras) as a motivic generalization of a classical situation, in which Euler characteristics are replaced by Poincaré polynomials. The word ‘categorification’ was coined in Physics, and is connected with passing from a Field Theory in n dimensions to one in n+1 dimensions.
Higher representation theory
Classical representation theory studies symmetries of sets with extra structures, such as vector spaces, whereas higher representation theory studies symmetries of (higher) categories with extra structures. Given a Kac-Moody algebra g, Rouquier constructed an associated 2-category A(g). Its integrable actions on additive categories can be studied in a way similar to the integrable representations of g: there are Jordan-Hölder series and a counterpart of simple representations, which admit a construction similar to that of simple representations as quotients of Verma modules. There is also a description of simple representations as categories of sheaves over a quantized version of Nakajima quiver varieties.
Various categories of algebraic and geometric origin are known to admit extra symmetries that correspond to an action of a 2-category A(g). This gives important information on the category and has been used to construct equivalences of derived categories, for example to settle conjectures of Broué on representations of symmetric groups, and Rickard on rational representations of general linear groups.
The 2-category of 2-representations can be endowed with a tensor structure that should lead to a braided 2-category, and give rise to a 4-dimensional topological quantum eld theory that can be computed algebraically. A shadow of this is the connection with the Khovanov-Rozansky homology of knots.
Much current work in geometric representation theory deals with situations which are at heart 1- or 2-dimensional. For example, categories of quiver representations mod-KQ behave very much like categories coh(C) of coherent sheaves on a (1-dimensional) curve C, and Grojnowski and Nakajima study Hilbert schemes Hilbn(S) of a (2-dimensional) surface S. One question we want to ask is whether there is interesting representation theory coming from 3-dimensional situations. We believe that Donaldson-Thomas theory of Calabi-Yau 3-folds is a good place to start to answer this.
Donaldson-Thomas invariants of Calabi-Yau 3-folds
A Calabi-Yau 3-fold is a smooth projective 3-fold X over a field K, usually C, with trivial canonical bundle KX, usually with H1(OX) = 0. They are interesting from many points of view in mathematics, and are the key to building realistic models of the universe in String Theory. Coherent sheaves on X are generalizations of holomorphic vector bundles on X, and form an abelian category coh(X). Given a stability condition τ on coh(X), such as Gieseker stability, one can form coarse moduli schemes Mstα(τ), Mssα(τ) of τ-(semi)stable coherent sheaves with Chern character α in Heven(X;Q). Donaldson-Thomas invariants DTα(τ) are numbers which 'count' τ-(semi)stable coherent sheaves on X in class α. They are unchanged under deformations of X. The first definition by Richard Thomas worked only when Mstα(τ) = Mssα(τ). Recently there has been much activity in the field:
- Maulik, Nekrasov, Okounkov and Pandharipande showed how to use Donaldson-Thomas invariants to count curves on X, and stated the MNOP Conjecture relating Donaldson-Thomas and Gromov-Witten invariants.
- Behrend showed that DTα(τ) can be written as a weighted Euler characteristic DTα(τ) = χ(Mstα(τ),ν), where ν is the Behrend function, a Z-valued constructible function on Mstα(τ). This is important to us, as it means that D-T invariants are in a sense motivic.
- Joyce and Song generalized the definition of DTα(τ) to all classes α, and proved wall-crossing formulae for DTα(τ) under change of stability condition τ.
- Kontsevich and Soibelman discuss 'motivic' Donaldson-Thomas invariants. Their 'cohomological Hall algebras' categorify Donaldson-Thomas type invariants for quivers with relations. Both of these rely on unproved conjectures.
- Donaldson and Segal sketched how to categorify Donaldson-Thomas invariants using G2-manifolds.
Field theories in terms of higher category theory
Two dimensional (2d) conformal field theories (CFTs) are basic building blocks of String Theory. Mirror symmetry conjectures can be phrased in terms of 2d CFTs and related 2d topological field theories (TFTs). Mathematical work done on mirror symmetry is inspired by predictions made by physicists. Many of these predictions are made by the use of tools from CFT. A natural question is whether these tools can be made mathematically rigorous.
The mathematical understanding of 2d CFT has advanced much in the past 20 years. There are now several mathematical points of view on this subject. One is the notion of a modular functor defined by Segal. In this approach a CFT is a symmetric monoidal functor from a bordism category, whose objects are circles and morphisms are Riemann surfaces, to the category of topological vector spaces.The modular functor definition can be also extended to supersymmetric conformal field theories (SCFTs) using super Riemann surfaces. Similarly one can dene topological conformal field theories (TCFTs), again as a symmetric monoidal functor, but now the source category has objects circles, and morphisms the homologies of spaces of Riemann surfaces.
Another approach is by using vertex algebras. This is an algebraic notion that formalizes the operator product expansion of CFTs (the pair of pants operation in the modular functor approach). Supersymmetry can be introduced for vertex algebras as well. Recently Beilinson and Drinfeld have developed chiral algebras and factorization algebras which are a geometric approach to vertex algebras.
The third approach is modular tensor categories, which are braided monoidal categories with finiteness conditions. The three approaches are related, and equivalent under some assumptions, e.g. the representations of a vertex algebra satisfying finiteness conditions form a modular tensor category.
Homological mirror symmetry deals with extended (open/closed) N = 2 SCFTs and their topological twists. Different approaches have been developed to this. On the topological side, Costello has shown that the notion of an extended TCFT is equivalent to that of an A∞ Calabi-Yau category.
Using all these developments it is possible to give a mathematically precise formulation of mirror symmetry conjectures in terms of CFTs. From the point of view of geometric representation theory, this can be seen as a vast generalization of the geometric Langlands programme, in which affine Lie algebras are replaced by other vertex algebras. Bridgeland stability can also be included in this picture, in terms of the moduli space of N = 2 SCFTs, and its projection to the moduli of TCFTs.
Our research programme
Our programme is divided into the following four linked Strands:
Strand 1. Extensions of Donaldson-Thomas theory.
Motivic generalizations of Donaldson-Thomas invariants, following Kontsevich and Soibelman (2008, 2010). Prove integrality conjecture for Donaldson-Thomas invariants due to Joyce-Song (2008). Categorication of Donaldson-Thomas theory, e.g. using perverse sheaves, or Saito's mixed Hodge modules. Detailed study of examples. Extensions to dimensions m > 3: for X a Calabi-Yau m-fold, study ways to 'count' Hilbert schemes Hilbn(X), and moduli schemes of coherent sheaves on X, which preserve good properties of the Donaldson-Thomas case.
Strand 2. Donaldson-Thomas theory and representations of algebras.
Realize categorification in Strand 1 as representation of some interesting algebra, e.g. a W1+∞-algebra. Predict new structure for Donaldson-Thomas generating functions using representation theory, e.g. modularity.
Strand 3. Higher representation theory.
Extend definition of 2-Kac-Moody algebras to a wider class, e.g. Heisenberg algebras, gl∞, W1+∞. Provide actions on categories of sheaves over Hilbert schemes of points on C3, and more generally over moduli spaces of representations of 3-Calabi-Yau algebras. Define and provide an action of 2-cluster algebras, and relate these constructions to Donaldson-Thomas invariants.
Strand 4. Representation theory and superconformal field theories.
Use mathematically rigorous definition of SCFTs to make physical ideas into precise mathematics, in particular topological twists and BPS states, and relate to Bridgeland stability. Predict structure for categorification of Donaldson-Thomas theory. Use representation theory to explicitly compute the open/closed SCFT for an abelian variety X. Verify that the A and B topological twists of this SCFT are equivalent to DbFuk(X) and Dbcoh(X), and deduce homological mirror symmetry in this example. Extend to other classes of Calabi-Yau m-folds.