## Final Report Summary - STACKSCATS (Stacks and Categorification)

This project was a huge success. The project outcomes were slightly different than the original objectives but still agreed with the spirit of the objectives. The outcomes of the project fall into two overall categories. The first category was shifted symplectic geometry and application to the categorification of Donaldson--Thomas invariants. This resulted in three articles, two of which have already been accepted for publication. The authors examined the geometry of Lagrangians and a linearization of Lagrangian intersection using perverse sheaves. The second category is the development of a new approach to analytic geometry. There are four articles completed or being completed on this approach.

In more detail, the original proposal involved two separate collaborations:

(a) between the fellow Dr Oren Ben-Bassat and Professor Dominic Joyce at Oxford, working on Objective 1, which concerned the study of moduli spaces M of coherent sheaves on a Calabi-Yau 3-fold X; and

(b) between the fellow Dr Oren Ben-Bassat and Professor Kobi Kremnizer at Oxford, working on Objective 2, which concerned the categorification of the Verlinde algebra.

Significant progress was made on both Objectives during the project.

Starting with (a), a little background: for Objective 1, "Calabi-Yau 3-folds" X are geometric spaces of 6 real dimensions which have a rich geometric structure and are important in Differential and Algebraic Geometry in Mathematics, and in String Theory in Theoretical Physics. "Coherent sheaves" E on X are natural objects much studied in Algebraic Geometry, roughly speaking they are representations of the (sheaf of) holomorphic functions on X. Holomorphic vector bundles on X such as the tangent bundle TX are examples of coherent sheaves, and general coherent sheaves E may be thought of as (singular) vector bundles on a submanifold of X.

Given a class of mathematical objects, such as the family of all coherent sheaves E on a fixed Calabi-Yau 3-fold X, geometers often seek to construct a "moduli space", a geometric space whose points parametrize (isomorphism classes of) the objects in question, and whose geometric structure captures properties of families of such objects. For any smooth projective scheme (such as a Calabi-Yau 3-fold), it is known that the moduli space of coherent sheaves is an "Artin stack". We can roughly speaking think of an Artin stack M as (locally) being a quotient M=[Y/G], where Y is a generally singular "scheme", and G is an algebraic group (such as a group of matrices). The scheme Y is called the "atlas".

Objective 1 concerns the special properties of moduli stacks M of coherent sheaves on a Calabi-Yau 3-fold X. The idea is that if X is a Calabi-Yau 3-fold (as opposed to some more general smooth projective scheme), then M should have some additional special geometric structure / geometric properties, in addition to just being an Artin stack. Quite a lot about this was already known: "Donaldson-Thomas theory" uses special properties of Calabi-Yau 3-fold moduli stacks M to define numbers called "Donaldson-Thomas invariants", which "count" the points in M. These have beautiful mathematical properties, and are important in String Theory as "numbers of BPS states". The "counting" has to be done in a subtle way (since the actual number is almost always infinite). There is much current interest in extending Donaldson-Thomas theory in several directions -- for instance, to realize the Donaldson-Thomas invariants (numbers) as the dimension of natural vector spaces (thought of as a kind of "cohomology" of the moduli space M), or to generalize them from ordinary numbers to taking values in a larger ring of "motives", so that the generalized invariants would remember more information.

The goal of Objective 1(a) was to show that if M is a Calabi-Yau 3-fold moduli stack, locally modelled on [Y/G] for Y an atlas, then the atlas Y could be written as the "critical locus" of a holomorphic function f : U --> C, where U is a complex manifold. This was known near a single point of M by old work of Professor Joyce, and the proposal envisaged extending the results to larger subsets of M.

This objective was achieved very successfully, in fact we proved a theorem rather stronger than we believed would be possible at the outset, in a paper now published in the journal Geometry and Topology. We showed that rather than working in the complex category, we could model the atlas Y as an algebraic critical locus, that is, we can take U and f to be algebraic objects, rather than complex manifolds and holomorphic functions. This means the result works over other fields. Also, since algebraic open sets are generally very large (usually they are "dense", that is, almost the whole of the space M), we achieved our objective of extending Joyce's original pointwise result to larger regions of M.

This basic result on the local structure of Calabi-Yau 3-fold moduli stacks should have many future applications. During the project, the fellow Br Ben-Bassat worked on two such follow-on projects, one available at arXiv:1309.0596 and published in Contemporary Mathematics, and one joint with Dr Lino Amorim, available at arXiv:1601.01536.

Next we discuss (b) and Objective 2. "Classical Representation Theory" seeks to understand the representations of groups on vector spaces. "Geometric Representation Theory" is a recent, exciting area which generalizes classical representation theory in several ways. One is that one seeks to produce representations of (generally infinite and geometric) groups by geometric constructions -- often through the study of moduli spaces of coherent sheaves, as in Objective 1 -- and another is through the use of methods from category theory, and "categorification", a complicated idea, but broadly to do with lifting some interesting classical groups and representations (a group acting on a vector space) to a new mathematical level (e.g. a monoidal category acting on an abelian category).

Objective 2 was to categorify the "Verlinde algebra", an important topic in classical representation theory, using Geometric Representation Theory, and (twisted) equivariant coherent sheaves on algebraic groups, and in particular complex analytic rather than algebraic methods.

Once Dr Ben-Bassat and Professor Kremnizer started to work seriously on Objective 2, they realized that to make progress they would require a theory of "derived complex analytic geometry". However, very little work had been done on this, as most of "derived geometry" is done in a strictly algebraic context. They decided to redirect their efforts on Objective 2 to the development of a theory of derived complex algebraic geometry, which should hopefully be used to solve Objective 2 in some years time, after the end of the fellowship.

This redirected project has been very successful, resulting in three foundational papers (i) arXiv:1312.0338 by Dr Ben-Bassat and Dr Kremnizer, accepted for publication in the Annales de la Faculté des Sciences de Toulouse, (ii) arXiv:1502.01401 by F. Bambozzi and Dr Ben-Bassat, published in the Journal of Number Theory, and (iii) arXiv:1511.09045 by Bambozzi, Ben-Bassat and Kremnizer, submitted to a journal. A fourth article is in progress.

The broad approach is to show that one can do "derived complex analytic geometry" within the Derived Algebraic Geometry theory of Toen and Vezzosi. To do this one has to produce as input a suitable category of complex analytic objects, with a "topology" (in a Grothendieck sense) satisfying a long list of necessary properties required by Toen-Vezzosi's theory; they do this by working in certain categories of Banach spaces. They characterize topologies used in four main settings: affinoid, dagger affinoid, Stein, and the derived analytic geometry setting which encompasses all of the others.

In more detail, the original proposal involved two separate collaborations:

(a) between the fellow Dr Oren Ben-Bassat and Professor Dominic Joyce at Oxford, working on Objective 1, which concerned the study of moduli spaces M of coherent sheaves on a Calabi-Yau 3-fold X; and

(b) between the fellow Dr Oren Ben-Bassat and Professor Kobi Kremnizer at Oxford, working on Objective 2, which concerned the categorification of the Verlinde algebra.

Significant progress was made on both Objectives during the project.

Starting with (a), a little background: for Objective 1, "Calabi-Yau 3-folds" X are geometric spaces of 6 real dimensions which have a rich geometric structure and are important in Differential and Algebraic Geometry in Mathematics, and in String Theory in Theoretical Physics. "Coherent sheaves" E on X are natural objects much studied in Algebraic Geometry, roughly speaking they are representations of the (sheaf of) holomorphic functions on X. Holomorphic vector bundles on X such as the tangent bundle TX are examples of coherent sheaves, and general coherent sheaves E may be thought of as (singular) vector bundles on a submanifold of X.

Given a class of mathematical objects, such as the family of all coherent sheaves E on a fixed Calabi-Yau 3-fold X, geometers often seek to construct a "moduli space", a geometric space whose points parametrize (isomorphism classes of) the objects in question, and whose geometric structure captures properties of families of such objects. For any smooth projective scheme (such as a Calabi-Yau 3-fold), it is known that the moduli space of coherent sheaves is an "Artin stack". We can roughly speaking think of an Artin stack M as (locally) being a quotient M=[Y/G], where Y is a generally singular "scheme", and G is an algebraic group (such as a group of matrices). The scheme Y is called the "atlas".

Objective 1 concerns the special properties of moduli stacks M of coherent sheaves on a Calabi-Yau 3-fold X. The idea is that if X is a Calabi-Yau 3-fold (as opposed to some more general smooth projective scheme), then M should have some additional special geometric structure / geometric properties, in addition to just being an Artin stack. Quite a lot about this was already known: "Donaldson-Thomas theory" uses special properties of Calabi-Yau 3-fold moduli stacks M to define numbers called "Donaldson-Thomas invariants", which "count" the points in M. These have beautiful mathematical properties, and are important in String Theory as "numbers of BPS states". The "counting" has to be done in a subtle way (since the actual number is almost always infinite). There is much current interest in extending Donaldson-Thomas theory in several directions -- for instance, to realize the Donaldson-Thomas invariants (numbers) as the dimension of natural vector spaces (thought of as a kind of "cohomology" of the moduli space M), or to generalize them from ordinary numbers to taking values in a larger ring of "motives", so that the generalized invariants would remember more information.

The goal of Objective 1(a) was to show that if M is a Calabi-Yau 3-fold moduli stack, locally modelled on [Y/G] for Y an atlas, then the atlas Y could be written as the "critical locus" of a holomorphic function f : U --> C, where U is a complex manifold. This was known near a single point of M by old work of Professor Joyce, and the proposal envisaged extending the results to larger subsets of M.

This objective was achieved very successfully, in fact we proved a theorem rather stronger than we believed would be possible at the outset, in a paper now published in the journal Geometry and Topology. We showed that rather than working in the complex category, we could model the atlas Y as an algebraic critical locus, that is, we can take U and f to be algebraic objects, rather than complex manifolds and holomorphic functions. This means the result works over other fields. Also, since algebraic open sets are generally very large (usually they are "dense", that is, almost the whole of the space M), we achieved our objective of extending Joyce's original pointwise result to larger regions of M.

This basic result on the local structure of Calabi-Yau 3-fold moduli stacks should have many future applications. During the project, the fellow Br Ben-Bassat worked on two such follow-on projects, one available at arXiv:1309.0596 and published in Contemporary Mathematics, and one joint with Dr Lino Amorim, available at arXiv:1601.01536.

Next we discuss (b) and Objective 2. "Classical Representation Theory" seeks to understand the representations of groups on vector spaces. "Geometric Representation Theory" is a recent, exciting area which generalizes classical representation theory in several ways. One is that one seeks to produce representations of (generally infinite and geometric) groups by geometric constructions -- often through the study of moduli spaces of coherent sheaves, as in Objective 1 -- and another is through the use of methods from category theory, and "categorification", a complicated idea, but broadly to do with lifting some interesting classical groups and representations (a group acting on a vector space) to a new mathematical level (e.g. a monoidal category acting on an abelian category).

Objective 2 was to categorify the "Verlinde algebra", an important topic in classical representation theory, using Geometric Representation Theory, and (twisted) equivariant coherent sheaves on algebraic groups, and in particular complex analytic rather than algebraic methods.

Once Dr Ben-Bassat and Professor Kremnizer started to work seriously on Objective 2, they realized that to make progress they would require a theory of "derived complex analytic geometry". However, very little work had been done on this, as most of "derived geometry" is done in a strictly algebraic context. They decided to redirect their efforts on Objective 2 to the development of a theory of derived complex algebraic geometry, which should hopefully be used to solve Objective 2 in some years time, after the end of the fellowship.

This redirected project has been very successful, resulting in three foundational papers (i) arXiv:1312.0338 by Dr Ben-Bassat and Dr Kremnizer, accepted for publication in the Annales de la Faculté des Sciences de Toulouse, (ii) arXiv:1502.01401 by F. Bambozzi and Dr Ben-Bassat, published in the Journal of Number Theory, and (iii) arXiv:1511.09045 by Bambozzi, Ben-Bassat and Kremnizer, submitted to a journal. A fourth article is in progress.

The broad approach is to show that one can do "derived complex analytic geometry" within the Derived Algebraic Geometry theory of Toen and Vezzosi. To do this one has to produce as input a suitable category of complex analytic objects, with a "topology" (in a Grothendieck sense) satisfying a long list of necessary properties required by Toen-Vezzosi's theory; they do this by working in certain categories of Banach spaces. They characterize topologies used in four main settings: affinoid, dagger affinoid, Stein, and the derived analytic geometry setting which encompasses all of the others.