03/17/2023: Enumerative invariants: what we did, and where to go next
01/10/2023: The structure of invariants counting coherent sheaves on complex surfaces
09/8/2022: Counting semistable coherent sheaves on surfaces
01/2021: Donaldson-Thomas theory of Calabi-Yau 3-folds
09/18/2020: Enumerative invariants in Algebraic Geometry and wall crossing formulae
06/01/2020: Universal structures in enumerative invariant theories
01/06/2020: Shifted symplectic Derived Algebraic Geometry for dummies
06/06/2019: Lagrangian MCF, Bridgeland stability and the Thomas-Yau conjecture
09/13/2018: A Ringel-Hall type construction of vertex algebras
06/05/2018: On Mirror Symmetry, Fukaya categories, and Bridgeland stability, with a view towards Lagrangian Mean Curvature Flow
01/09/2018: Lie brackets on the homology of moduli spaces, and wall-crossing formulae
09/14/2017: Conjectures on counting associative 3-folds in G₂ manifolds
06/06/2017: Constructing compact 8-manifolds with holonomy Spin(7)
06/05/2017: Constructing compact 7-manifolds with holonomy G2
01/11/2017: Counting problems for G2 manifolds
09/07/2016: Derived differential geometry and moduli spaces in differential geometry
March 17, 2023
TITLE: Enumerative invariants: what we did, and where to go next
I will very briefly review work done on enumerative invariants during the collaboration, including vertex algebra and Lie algebra structures on the homology of moduli spaces, enumerative invariants and wall-crossing formulae, invariants counting coherent sheaves on complex surfaces and Donaldson / Seiberg-Witten invariants, and counting coherent sheaves on curves, Fano 3-folds and Calabi-Yau 4-folds, orientability and canonical orientations on moduli spaces, and “orientation data” for Calabi-Yau 3-folds. Then I will propose some problems for future work.
January 10, 2023
TITLE: The structure of invariants counting coherent sheaves on complex surfaces
Let X be a complex projective surface with geometric genus pg > 0. We can form moduli spaces M(r,a,k)st ⊂ M(r,a,k)ss of Gieseker (semi)stable coherent sheaves on X with Chern character (r,a,k), where we take the rank r to be positive. In the case in which stable = semistable, there is a (reduced) perfect obstruction theory on M(r,a,k)ss, giving a virtual class [M(r,a,k)ss]virt in homology.
By integrating universal cohomology classes over this virtual class, one can define enumerative invariants counting semistable coherent sheaves on X. These have been studied by many authors, and include Donaldson invariants, K-theoretic Donaldson invariants, Segre and Verlinde invariants, part of Vafa-Witten invariants, and so on.
In my paper https://arxiv.org/abs/2111.04694, in a more general context, I extended the definition of the virtual class [M(r,a,k)ss]virt to allow strictly semistables, proved wall-crossing formulae for these classes and associated “pair invariants”, and gave an algorithm to compute the invariants [M(r,a,k)ss]virt by induction on the rank r, starting from data in rank 1, which is the Seiberg-Witten invariants of X and fundamental classes of Hilbert schemes of points on X. This is an algebro-geometric version of the construction of Donaldson invariants from Seiberg-Witten invariants; it builds on work of Mochizuki 2008.
This talk will report on a project to implement this algorithm, and actually compute the invariants [M(r,a,k)ss]virt for all ranks r > 0. I prove that the [M(r,a,k)ss]virt for fixed r and all a,k with a fixed mod r can be encoded in a generating function involving the Seiberg-Witten invariants and universal functions in infinitely many variables. I will spend most of the talk explaining the structure of this generating function, and what we can say about the universal functions, the Galois theory and algebraic numbers involved, and so on. This proves several conjectures in the literature by Lothar Göttsche, Martijn Kool, and others, and tells us, for example, the structure of U(r) and SU(r) Donaldson invariants of surfaces with b2+ > 1 for any rank r ≥ 2.
September 8, 2022
TITLE: Counting semistable coherent sheaves on surfaces
In arXiv:2111.04694 I set out a program which gives a common universal structure to many theories of enumerative invariants counting semistable objects in abelian or derived categories in Algebraic Geometry, for example, counting coherent sheaves on curves, surfaces, Fano 3-folds, Calabi-Yau 3- or 4-folds, or representations of quivers. I will outline the general programme briefly, and go into detail on my current project, which uses the results to compute invariants counting semistable vector bundles and coherent sheaves on complex projective surfaces.
Invariants of this kind have a long history, notably in Donaldson theory of 4-manifolds, Vafa-Witten invariants, and so on, and are the subject of many conjectures in mathematics and String Theory. It is common to combine the invariants in formal power series — “generating functions” — which may have interesting number-theoretic properties such as rationality or modularity. My results explain a lot about the general structure of such generating functions, resolving some conjectures in the literature, and I hope in future to make progress on deeper questions such as Vafa-Witten modularity as well.
Recorded lecture associated with the January 2021 meeting
TITLE: Donaldson-Thomas theory of Calabi-Yau 3-folds
Donaldson-Thomas (D-T) invariants are integers or rational numbers which count Gieseker (semi)stable coherent sheaves with fixed Chern character on a Calabi-Yau 3-fold X. They are unchanged under deformations of the complex structure of X, and change according to a known wall crossing formula under deformations of the Kahler class of X used to define the Gieseker stability condition. They were defined by Richard Thomas in 1998 for Chern characters in which stable=semistable, and by Joyce-Song in 2008 in the general case, using important results of Kai Behrend from 2005. Kontsevich-Soibelman 2008 also generalized D-T invariants to “motivic D-T invariants”, which contain more information than ordinary D-T invariants, and satisfy a known wall crossing formula under change of Kahler class, but are not unchanged under deformations of the complex structure of X. D-T invariants are important in String Theory, as “numbers of BPS states”, and are a major ingredient in Tom Bridgeland’s theory of “Joyce structures”, which is a theme of this conference.
This talk will summarize the theory of D-T invariants of Thomas and Joyce-Song, as background material for Tom Bridgeland’s talks. It is not new material, almost everything I say was known in 2008.
September 18, 2020
TITLE: Enumerative invariants in Algebraic Geometry and wall crossing formulae
ABSTRACT: At the June meeting I described a universal conjectural picture for enumerative invariants counting semistable objects in abelian categories/gauge theories, which claimed that under some assumptions:
(i) one can construct invariants, as virtual classes in the rational homology of the “projective linear” moduli stack, for all topological invariants (fixed Chern classes etc), including classes with strictly semistables;
(ii) these invariants satisfy a wall-crossing formula under change of stability condition, written in terms of a Lie bracket on the homology of the moduli stack, which came out of my vertex algebras project.
I also explained that we had proved the conjectures for quivers (joint with Gross-Tanaka).
In work in progress, I have now proved/am proving versions of the conjectures for a broad family of settings in Algebraic Geometry, in which invariants are formed using Behrend-Fantechi virtual classes (so not including the Calabi-Yau 4-fold case yet, but I hope to do this too if Oh-Thomas do the job properly). These include suitable quivers with relations, coherent sheaves on curves, surfaces and some 3-folds, and algebraic Seiberg-Witten invariants and Donaldson invariants of projective complex surfaces with b^2_+=1.
I have also understood how the picture should be modified to include algebraic Donaldson and Seiberg-Witten invariants of surfaces with b^2_+>1, and I have theorems on this. (Some of them even have proofs.)
The SW/Donaldson theory picture includes wall-crossing formulae which implicitly determine algebraic U(n) and SU(n) Donaldson invariants, of any rank, in terms of rank 1 Seiberg-Witten type invariants and invariants of Hilbert schemes of points, for any projective complex surface, without restriction on b^1, or b^2_+, or a simple type assumption.
June 1, 2020
TITLE: Universal structures in enumerative invariant theories
ABSTRACT: An enumerative invariant theory in Algebraic Geometry, Differential Geometry, or Representation Theory, is the study of invariants which ‘count’ -(semi)stable objects with fixed topological invariants in some geometric problem, by means of a virtual class of the moduli spaces of -(semi)stable objects in some homology theory. Examples include Mochizuki’s invariants counting coherent sheaves on surfaces, Donaldson-Thomas type invariants counting coherent sheaves on Calabi-Yau 3- and 4-folds and Fano 3-folds, and Donaldson invariants of 4-manifolds.
We make conjectures on new universal structures common to many enumerative invariant theories. Any such theory has two moduli spaces , where my big vertex algebras project http://people.maths.ox.ac.uk/~joyce/hall.pdf gives the structure of a graded vertex algebra, and a graded Lie algebra, closely related to . The virtual classes take values in . In most such theories, defining when (in gauge theory, when the moduli space contains reducibles) is a difficult problem. We conjecture that there is a natural way to define in homology over , and that the resulting classes satisfy a universal wall-crossing formula under change of stability condition , written using the Lie bracket on . We prove our conjectures for moduli spaces of representations of quivers without oriented cycles.
This is joint work with Jacob Gross and Yuuji Tanaka.
January 6, 2020
TITLE: Shifted symplectic Derived Algebraic Geometry for dummies
ABSTRACT: The Derived Algebraic Geometry (DAG) of Toën-Vezzosi and Lurie is the
study of derived schemes and derived stacks, enhancements of classical schemes
and stacks with a richer structure. For moduli spaces of coherent sheaves, etc., this
richer structure remembers (higher) obstructions to deformations. So derived
schemes include Behrend-Fantechi obstruction theories, and are good for virtual
cycles and enumerative invariants.
A secret of DAG is that one can treat (nice) derived stacks rather like smooth
manifolds, in a way not possible in classical algebraic geometry – despite the
complexity of the foundations, once the theory is set up, it is in some ways simpler
and better than the classical case.
In 2011, Pantev-Toën-Vaquié-Vezzosi introduced a version of symplectic
geometry in DAG, including k-shifted symplectic derived stacks for integer k, and
Lagrangians in them. They proved that if X is a Calabi-Yau m-fold, and M is the
derived moduli stack of coherent sheaves or complexes on X then M has a (2 – m)-
shifted symplectic structure. So, they found new geometric structures on Calabi-
Yau moduli spaces. These subsequently found applications in generalizations of
Donaldson-Thomas theory of Calabi-Yau 3-folds, and defining Donaldson-Thomas
type invariants counting (semi)stable coherent sheaves on Calabi-Yau 4-folds.
I think that String Theorists should care about the PTVV theory – it gives new
structures on moduli spaces of branes, and if it was translated into terms that
String Theorists understand, it might lead to new Physics. I will give an overview
of the subject, at a fairly basic level. Hopefully later talks will build on this.
Unfortunately, the first few minutes of the video are missing — it starts in the middle!
June 6, 2019
TITLE: Lagrangian MCF, Bridgeland stability and the Thomas-Yau conjecture
ABSTRACT: What Richard Thomas really meant to say when he made the Thomas-Yau Conjecture, was that if you start with a graded Lagrangian L in a Calabi-Yau manifold X, considered as an object in the derived Fukaya category DbF(X), and you follow Lagrangian MCF (L_t, t ∈ [0,∞)) with surgeries starting at L0 = L, then the limit L∞ determines the decomposition of L in a Bridgeland stability condition (Z,P) on DbF(X), whose semistable objects are special Lagrangians. I give an outline of this picture, and sketch a project to construct the Bridgeland stability condition (Z,P) by a different method when X is a Calabi-Yau 2-fold (work in progress, joint with Jason Lotay and Alexander Ritter).
September 13, 2018
TITLE: A Ringel-Hall type construction of vertex algebras
ABSTRACT: `Vertex algebras’ are complicated algebraic structures coming from physics, which arise in 2D conformal field theory and string theory, and also play an important role in mathematics, in areas such as monstrous moonshine and geometric Langlands. I will explain a new geometric construction of vertex algebras, which seems to be unknown. I discovered it by accident, while working on wall-crossing formulae for Donaldson-Thomas type invariants of Calabi-Yau 4-folds. The construction applies in many situations in algebraic geometry, differential geometry and representation theory, and produces vast numbers of new examples. It is also easy to generalize the construction in several ways to produce different types of vertex algebra, quantum vertex algebras and representations of vertex algebras. The construction seems to be closely related to, and is maybe the “correct” explanation for, a large body of work started by Grojnowski, Nakajima and others, which produces representations of interesting infinite-dimensional Lie algebras on the homology of moduli schemes, such as Hilbert schemes. Suppose A is a nice abelian category (such as coherent sheaves coh (X) on a smooth complex projective variety X, or representations mod-CQ of a quiver Q) or T is a nice triangulated category (such as Dbcoh (X) or Dbmod − CQ ) over C . Let M be the moduli stack of objects in A or T, as an Artin stack or higher stack. Consider the homology H*(M) over some ring R. Given a little extra data on M, for which there are natural choices in our examples, I will explain how to define the structure of a graded vertex algebra on H*(M). By a standard construction, one can then define a graded Lie algebra from the vertex algebra; roughly speaking, this is a Lie algebra structure on the homology H* (Mpl) of a projective linear version Mpl of the moduli stack M. For example, if we take T = Dbmod − CQ, the vertex algebra H*(M) is the lattice vertex algebra attached to the dimension vector lattice Z Q₀ of Q with the symmetrized intersection form. The degree zero part of the graded Lie algebra contains the associated Kac-Moody algebra. There is also a differential-geometric version: if X is a compact manifold equipped with an elliptic complex E (such as the de Rham complex or the Dirac operator), and M is the moduli stack (as a topological stack) of either all unitary connections on complex vector bundles on X , or all unitary connections on X satisfying a curvature condition depending on E (e.g., instantons on 4-manifolds, Hermitian-Einstein connections on Kahler manifolds, G₂-instantons or Spin(7)-instantons), then we can define a vertex algebra structure on H*(M). This should be part of the big picture into which other work in this collaboration on G₂-instantons and the Donaldson-Segal program fits. There must be a physical explanation for these vertex algebras, but so far, string theorists have not been able to give me one.
June 5, 2018
TITLE: On Mirror Symmetry, Fukaya categories, and Bridgeland stability, with a view towards Lagrangian Mean Curvature Flow
ABSTRACT: There will be no new research ideas in this talk, I will just cover well known background material. So experts (and indeed, the rest of the audience) are advised to go see the dinosaurs in the Natural History Museum instead.
My intended target audience is people who work on geometric flows, particularly (Lagrangian) Mean Curvature Flow, but may not be very familiar with areas of symplectic geometry around Homological Mirror Symmetry, Fukaya categories, etc.
I will briefly review the “big picture” of Homological Mirror Symmetry for Calabi-Yau manifolds M, M*, the derived Fukaya category DbF(M) of Lagrangians L in a Calabi-Yau manifold M, and how it is expected that there exists a Bridgeland stability condition on DbF(M) whose semistable objects are represented by special Lagrangian submanifolds L; also the role of obstructions to Lagrangian Floer cohomology, and how they determine whether a Lagrangian appears as an object in DbF(M).
There is a conjectural framework known as the “Thomas-Yau Conjecture” for how Lagrangian MCF fits into all this, quite similar to Ricci flow in the Poincare Conjecture and geometrization of 3-manifolds. The Thomas-Yau conjecture has nothing at all to do with Yau, and consists of what Richard Thomas really meant to say, although he actually said something different. I’ll describe a personal view of the Thomas-Yau Conjecture.
January 9, 2018
TITLE: Lie brackets on the homology of moduli spaces, and wall-crossing formulae
ABSTRACT: Let be a field, and be the “projective linear” moduli stack of objects in a suitable -linear abelian category (such as the coherent sheaves on a smooth projective ) or triangulated category (such as the derived category ). I will explain how to define a Lie bracket [ , ] on the homology (with a nonstandard grading), making into a graded Lie algebra. This is a new variation on the idea of Ringel-Hall algebra.
There is also a differential-geometric version of this: if is a compact manifold with a geometric structure giving instanton-type equations (e.g. oriented Riemannian 4-manifold, manifold, manifold) then we can define Lie brackets both on the homology of the moduli spaces of all or connections on for all , and on the homology of the moduli spaces of instanton or connections on for all .
All this is (at least conjecturally) related to enumerative invariants, virtual cycles, and wall-crossing formulae under change of stability condition.
Several important classes of invariants in algebraic and differential geometry — (higher rank) Donaldson invariants of 4-manifolds (in particular with ), Mochizuki invariants counting semistable coherent sheaves on surfaces, Donaldson-Thomas type invariants for CY 3-folds, Fano 3-folds, and CY 4-folds — are defined by forming virtual classes for moduli spaces of “semistable” objects, and integrating some cohomology classes over them. The virtual classes live in the homology of the “projective linear” moduli stack. Yuuji Tanaka and I are working on a way to define virtual classes counting strictly semistables, as well as just stables / stable pairs.
I conjecture that in all these theories, the virtual classes transform under change of stability condition by a universal wall-crossing formula (from my previous work on motivic invariants) in the Lie algebra (, [ , ]).
September 14, 2017
TITLE: Conjectures on counting associative 3-folds in G₂ manifolds
ABSTRACT: Riemannian 7-manifolds with holonomy G₂ are a special class of Ricci-flat Riemannian manifolds, which are of interest to physicists working in M-theory. Associative 3-folds are calibrated 3-submanifolds in 7-manifolds with holonomy G₂, so they are a special kind of minimal submanifold.
There is a well-known analogy between G₂ manifolds X in dimension 7 and Calabi-Yau 3-folds Y in dimension 6. Under this analogy one should compare associative 3-folds in X with J-holomorphic curves in Y. Much of symplectic geometry — Gromov-Witten theory, Lagrangian Floer theory, and so on — is concerned with “counting” J-holomorphic curves, to get an answer which is independent of the (almost) complex structure J up to deformation. So we can ask: might there be interesting geometry of G₂ manifolds concerned with “counting” associative 3-folds, which gives an answer unchanged under deformations of the G₂ structure?
This talk, based on arXiv:1610.09836, presents a conjectural answer to this question. It is connected to conjectures of Donaldson and Segal on defining invariants by “counting” G₂ instantons on X with “compensation terms” counting pairs of a G₂ instanton and an associative 3-fold on X. At the end we will briefly discuss a proposed modification to the Donaldson-Segal conjecture, to correct for wall-crossing behaviour of associative 3-folds we discover during our investigation.
June 6, 2017
TITLE: Constructing compact 8-manifolds with holonomy Spin(7)
June 5, 2017
TITLE: Constructing compact 7-manifolds with holonomy G2
January 11, 2017
TITLE: Counting problems for G2 manifolds
ABSTRACT: This talk(s) reviews my recent preprint arXiv:1610.0983.
Many important areas of geometry involve “counting” some kind of geometric object to define an “invariant” which is then shown to be unchanged under deformations of the base geometry. Examples include Donaldson invariants counting instantons on a compact oriented Riemannian 4-manifold (X,g) with b^2_+(X)>1 (which are unchanged under deformations of g), Gromov-Witten invariants of complex algebraic / symplectic manifolds (unchanged under deformations of the (almost) complex structure J), and so on.
Floer cohomology theories and Fukaya categories are similar: they involve “counting” objects such as J-holomorphic curves with boundary, and while the numbers are not deformation-invariant, one uses the numbers and some homological algebra to construct things which are deformation-invariant.
I will discuss whether there may be interesting “invariant” theories for compact -manifolds which are unchanged under deformations of the -structure (perhaps deformations preserving closure of 3- or 4-form), as considered by Donaldson and Segal in their 2009 paper “Gauge theory in higher dimensions II”.
The three obvious classes of objects in a -manifold one could try to “count” are associative 3-folds, coassociative 4-folds, and -instantons. I haven’t much to say about coassociatives.
For associative 3-folds, I argue that one cannot define deformation-invariant Gromov-Witten style invariants counting associatives, as there are singular behaviours of associatives which would change the numbers under deformation of the -structure. However, I suggest that there may still be interesting deformation-invariant information encoded in “numbers” of associatives, and I outline how one might define a supercommutative “ quantum cohomology algebra”, similar to quantum cohomology in symplectic geometry, but with features of a Floer theory.
On the way to this conclusion, I introduce some new ideas about associative 3-folds, including a way to orient moduli spaces of associatives, and make some conjectures on their singular behaviour.
For instantons, Donaldson and Segal proposed to define invariants, similar to Donaldson invariants of 4-manifolds, which “count” -instantons on the -manifold (X,). It is known that -instantons can “bubble” on an associative 3-fold under deformation of , and this would change the numbers of -instantons. So to make their invariants unchanged under deformation, they proposed to add “compensation terms” C(N,(P,A)) which count pairs of an associative N in X and a -instanton (P,A) on X with some weight, which would have the property that C(N,(P,A)) jumps by 1 when a -instanton bubbles on N leaving (P,A) after removal of singularities, so that the compensated sum remains unchanged. The precise definition of the weight C(N,(P,A)) has remained mysterious, though Walpuski and Haydys are working towards a definition.
I argue that for -instantons with group SU(2), it is not possible to define “compensation terms” C(N,(P,A)) with the required properties (for any conceivable definition, not just for current definitions), so that I do not believe that the Donaldson-Segal programme as currently formulated will succeed. The argument why not is actually a fairly simple thought-experiment using material from Donaldson and Segal’s paper, but it is based on the ideas about orienting associative moduli spaces mentioned earlier.
I won’t be able to say all this in an hour. So what I’ll probably actually do is give an 1-hour overview in three parts: 1. Introduction; 2. Counting associatives; 3. Counting -instantons; and then we can continue afterwards according to demand from those audience members who have not already run away screaming, if this is a nonempty set.
September 7, 2016
TITLE: Derived differential geometry and moduli spaces in differential geometry
ABSTRACT: Derived Differential Geometry (DDG) is the study of “derived manifolds” and “derived orbifolds”, where “derived” is in the sense of the Derived Algebraic Geometry of Jacob Lurie and Toen-Vezzosi. They include ordinary manifolds and orbifolds, but also many spaces which are singular at the classical level. There are several approaches to DDG, due to Spivak, Borisov-Noel and myself, all more-or-less equivalent. The “Kuranishi spaces” studied in symplectic geometry by Fukaya-Oh-Ohta-Ono are a prototype notion of derived orbifold.
I claim that moduli spaces M of solutions of a nonlinear elliptic p.d.e. on a compact manifold should naturally have the structure of a derived manifold (if solutions have no symmetries) or a derived orbifold (if solutions have finite symmetry groups, e.g. Deligne-Mumford stable J-holomorphic curves). This includes many very interesting problems — instantons on 4-manifolds and other gauge-theoretic moduli problems, J-holomorphic curves in symplectic geometry and so on. In particular, it includes moduli spaces used to define enumerative invariants (Donaldson invariants, Gromov-Witten invariants, etc.). This is because compact, oriented derived manifolds and derived orbifolds have virtual cycles in homology, and these virtual cycles may be used to define the invariants.
I also claim that many natural (partial) compactifications of such moduli spaces M, e.g. by including J-holomorphic curves with nodes, should naturally have the structure of derived manifolds or derived orbifolds with corners, where the boundary represents the extra singular solutions.
I will outline a method to prove the existence of natural derived manifold and derived orbifold structures on differential geometric moduli spaces by a method of “universal families”, based on Grothendieck’s representable functors in algebraic geometry. That is, given a moduli problem, we define a notion of family F of solutions over a base derived manifold or orbifold S. A “universal family” is a family with a universal property w.r.t. all other families. If a universal family exists (I claim it should, under reasonable conditions) it is unique up to equivalence, and the base M of the family is the moduli space, with a derived manifold/orbifold structure.