# Research Programme

## Introduction

This year, the second Abel Prize has been awarded jointly to Atiyah and Singer "for their discovery and proof of the index theorem, bringing together topology, geometry and analysis, and their outstanding role in building new bridges between mathematics and theoretical physics."

This citation, concerning arguably the most prestigious prize in mathematics,
as well as the award of Fields Medals to e.g. Connes, Witten, and Kontsevich, confirms the remarkable fact that it is the frontiers of *pure* mathematics and *fundamental* physics that happen to be in close contact at the moment. This is, of course, not a new phenomenon, though it is significant that the three previous episodes where this happened marked some of the most significant revolutions in the history of science.
Indeed, modern mathematics and physics were born together in the 17th century through the work of Newton, who created both the calculus and classical mechanics in intimate relationship to each other. Subsequently, Einstein's general theory of relativity (which replaced Newton's concepts of space, time, and gravity) was formulated on the mathematical basis of Riemannian geometry, in turn inspiring Weyl and Cartan to reshape the latter from a local to a global theory; cf [23]. Third, through the work of Hilbert and von Neumann, quantum mechanics was an important source of the transition from classical analysis to its modern (abstract) form (see, e.g., [39]).

It appears, then, that we are currently witnessing another such episode, in which the autonomous development of geometry as a branch of pure mathematics (dating back at least to Euclid) is enriched by a remarkable flow of ideas from fundamental physics, notably quantum theory. It is primarily in this sense that we intend to realize the stated aim of NWO (the Dutch Research Council) and OOW (i.e., the combined Dutch inter-academic research schools in mathematics) to support research in the interface between mathematics and theoretical physics in The Netherlands. Furthermore, an important secondary effect of the cluster will undoubtedly be the enhancement of the opposite flow as well.
The present proposal focuses on the interplay between *geometry*, including algebraic, symplectic, and noncommutative geometry, and *quantum theory*, incorporating quantum field theory, string theory, and quantization. While at first sight geometry seems a vast and diverse field, even when restricted to the three areas mentioned, our cluster achieves its coherence largely from the recent insight that these areas are related in remarkable new ways, often initially suggested by quantum physics.

As will become clear throughout this proposal, at present the national situation seems quite favourable to participate in this development. We possess
considerable experience on all aspects of the present initiative, but
- and this is the *raison d'etre* for the present cluster - our joint expertise has on the whole been kept separate so far. Thus our basic goal is to join forces in order to create an infrastructure in which the
interplay between geometry and quantum theory can be exploited to maximal effect, primarily from a mathematical perspective.

## Proposed research

We now describe our concrete research plans. It should be taken into account that these plans incorporate ideas by 23 people, to be carried out by a sizeable additional group of PhD students and postdocs as well. Hence, rather than describing specific research problems in great detail, we have preferred to isolate a number of areas and emphasize their interrelations. For the sake of concreteness, however, certain topics have been marked in italics, for instance as the subject of PhD theses.

### Poisson geometry, quantization, and noncommutative geometry

As we have seen in the general overview, noncommutative geometry is closely related to Poisson geometry through the notion of quantization. One of our ambitions is to relate noncommutative geometry to algebraic geometry as well, in the following fashion. One of the original goals of noncommutative geometry was to provide new tools for the study of singular spaces, such as the K-theory and cyclic cohomology of an appropriate noncommutative algebra associated to a quotient space. Connes himself successfully applied his toolkit to foliated spaces, Penrose tilings, and certain other examples [6]. However, the application of noncommutative geometry to some other important classes of singular spaces, namely those that have traditionally been studied using algebraic geometry, is still in its infancy. Here we are thinking, for example, of orbifolds, certain types of moduli spaces, and
symplectic quotients (cf. [32]). Remarkably, it therefore seems that such spaces may alternatively be studied using either the tools of commutative algebra (in the setting of Grothendieck-style algebraic geometry), or of noncommutative algebra (in the context of noncommutative geometry). *The comparison of these methods (in the context of suitable examples like the ones listed) is bound to lead to new insights;* cf. [5]. This would combine the joint expertise of at least half of the cluster members.

Another pertinent interdisciplinary topic is the *functoriality of quantization*, in the sense recently proposed in [9]. The most immediate concrete consequence of this functoriality principle is an extension of the Guillemin--Sternberg conjecture in geometric quantization (which is a theorem now for compact Lie groups acting on compact symplectic manifolds, cf. [19]) to the noncompact case. Proving this, or else limiting the scope of the conjecture through the discovery of counterexamples, would combine the expertise of Duistermaat, Heckman, Landsman, Van den Ban, and others,
as it links symplectic and noncommutative geometry with index theory and representation theory. In the singular case, also stratification techniques from algebraic geometry will enter.
Furthermore, functoriality of quantization needs to be concretely developed through examples involving Lie groupoids and algebroids. This includes the establishment of a general index theorem for Lie groupoids, generalizing the ordinary and family index theorems of Atiyah and Singer, the index theorem for noncompact groups of Connes and Moscovici, as well as the index theorem for foliated spaces of Connes and Skandalis. Another necessary ingredient would be *the K-theory and representation theory of Lie groupoids*, which will be taken up from the perspective of a generalized orbit correspondence (identifying the coadjoint orbits in the dual of a Lie algebra with its
symplectic leaves, which notion immediately generalizes to Lie algebroids).

Parallel to this, we intend to study *deformations of Lie groupoids that are Hopf algebroids*, relating the subject to dynamical quantum groups and Yang-Baxter equations. In fact, the precise relationship between the quantum analogues of semisimple noncompact Lie groups and the concepts of noncommutative geometry remains to be clarified; here one might think of relating the Haar weight to the Dixmier trace, and the Duflo--Moore operators to the corresponding modular operators. In this effort, the combined expertise of Van den Ban, Crainic, Koelink, Opdam, Stokman, and Moerdijk will be relied upon.

Modern deformation theory heavily relies on the concept of an operad (originally invented in topology in the 1970s by Boardman--Vogt, May, Stasheff, and others), cf. [29]. In addition, operads relate to various other research topics in this cluster, notably to moduli spaces (cf. [18, 35]) and configuration spaces (in the sense of algebraic geometry). In the context of mirror symmetry for Calabi--Yau manifolds, the Fukaya category is a
so-called A_infty-category, which means that its composition structure is modeled on some operad and needs `higher compositions of morphisms' to compensate for a lack of straightforward associativity.
Operad structures also occur in various (topological or conformal) quantum field theories. Apart from developing their unifying role, we aim to address several important open problems, e.g., *the question to what extent the topological Boardman-Vogt resolution can be applied to non-topological operads*. The leading figures in this research will be Moerdijk.

Another notion that is central to the research topics mentioned so far is that of a gerbe. Gerbes were originally introduced in the 1960s by the Grothendieck school in algebraic geometry (in particular, by Giraud) in the context of non-abelian cohomology. In the 1990s, gerbes resurfaced in geometric quantization as well as in mirror symmetry, where they enter in the description of the mirror partner of a Calabi--Yau 3-fold in terms of the moduli space of Lagrangian submanifolds equipped with gerbes (cf.[24]). In addition, a gerbe over a manifold enables one to `twist' the K-theory of this manifold. First introduced in algebraic topology by Donovan and Karoubi in 1970 (and subsequently shown by Rosenberg to be a special case of C*-algebraic K-theory), twisted K-theory made a striking reappearance in 1998 in string theory [48]. Subsequently, Freed, Hopkins and Teleman observed that the Verlinde algebra of the Wess--Zumino--Witten model of conformal field theory (or, mathematically, the appropriate representation category of the underlying loop group LG) coincides with the suitably twisted K-theory of G.

Our plan to understand the representation theory and the K-theory of Lie groupoids by combining techniques from equivariant algebraic topology and from C*-algebra theory (cf. [44]), is partly motivated by examples coming from this development. Indeed, the gerbes occurring in this context can be described as extensions of Lie groupoids, and the K-theory of such a central extension is closely related to the twisted (by the gerbe) K-theory.

In a more categorical direction, we will attempt to relate the new approach to quantum probability and second quantization recently initiated by Guta and Maassen [20] to the general setting described in this section. Since their work is based on Joyal's combinatorial theory of so-called species of structure, which touches on a number of the themes discussed so far, this seems a realistic goal.

Interesting problems remain, of course, even at the purely classical level. We will focus on a generalization of the notion of a Poisson structure, called a Dirac structure (originating in Dirac's work on constrained systems). It was recently shown by Bursztyn and Crainic that Dirac structures are closely related to the group valued momentum maps of Alekseev et al, but in this relationship much remains to be understood (such as the precise relationship to Manin pairs and quasi-Poisson Lie groups). In addition, an exciting *link between this generalized Poisson geometry and mirror symmetry* was recently uncovered by Hitchin, who showed that complex versions of Dirac structures naturally appear in the theory of mirror symmetry and Calabi--Yau manifolds. This poses, of course, an attractive area of research in our cluster (Crainic, Dijkgraaf, Van der Geer, Looijenga, Stienstra).

### Integrable systems, Frobenius manifolds, and the geometric Langlands program

Integrable systems and representation theory (or Lie theory) are closely related to each other, as well as to algebraic geometry and quantization. Thus the area is ideally suited for the proposed cluster. The main researchers will be Cushman, Duistermaat, Heckman, Helminck, Koelink, Van de Leur, Opdam, and Stokman, relying on the knowledge of mirror symmetry, moduli spaces, conformal field theory, and quantization of practically all other cluster members.

As we have seen, some of the pertinent relationships are codified by the notion of a Frobenius manifold, others by the geometric Langlands program.
The starting point of the geometric Langlands correspondence is the
moduli space Bun(G) of principal G-bundles over a smooth projective curve C. In the context of the Langlands program, one associates a group
LG (the Langlands dual group)to a given complex semisimple algebraic group G.
We intend to study the recent conjecture of Hausel and Thaddeus [22] that the moduli spaces Bun(G) and Bun(LG) (with certain additional data) are in an appropriate sense relative
mirror partners (in the sense of Strominger--Yau--Zaslow, cf.[26]). This is
related to the conjectured existence of a general Fourier--Mukai
transform underlying the geometric Langlands duality (see
[16]).
We plan to *investigate whether there is a geometric Langlands correspondence for the moduli space
Bun(G,S) of G-bundles with parabolic structure at a finite
list S of marked points of C, and local systems with ramifications
at the elements of S* (this has been done in positive characteristic by Drinfeld for GL(2), and by Heinloth for GL(3)). This raises further questions about the ``categorification'' of the full Iwahori Hecke algebra, and is also related to the work of Varchenko and coworkers on the Bethe Ansatz, a subject well familiar to the researchers listed above.

The link between the geometric Langlands program and Hitchin's integrable systems (cf.[16,25]) beautifully fits in the cluster theme, and will be examined in detail. The point of departure is a remarkable result of Hitchin, which says that the symplectic space T^Bun(G) is a completely integrable system (assuming the curve C has genus g >=2; for g=2 and G=SL(2) the Hitchin system is related to the classical Neumann system, a relationship we plan to investigate for other low genus and small rank cases). In a monumental unpublished paper, Beilinson and Drinfeld [6] have recently proved a special case of the geometric Langlands correspondence through the quantization of the Hitchin system, involving infinite-dimensional Kac--Moody algebras, as well as the W-algebras first encountered in conformal field theory.

This breakthrough poses all sorts of questions, and suggests various generalizations. For example, *the relationship between
Beilinson and Drinfeld's notion of quantization and deformation or geometric quantization ought to be established*. As another example, deformation quantization suggests that one should be able to
*diagonalize the pertinent C*-algebra by means of a suitable spectral decomposition*. In an analytic setting, the semiclassical (WKB) approximation could be applied (here as well as in other integrable models). The geometric quantization of the Hitchin system, on the other hand, has at least two interesting aspects.
Firstly, the appropriate Guillemin--Sternberg conjecture should be proved; the classical reduction procedure leads to the well-known integrable systems named after Schlesinger. Secondly, the situation is analogous to the quantization of the Atiyah--Bott moduli space of flat connections over a curve [6], and leads to similar links with
conformal field theory (as established in detail by Laszlo).
A number of important open problems remain here, most notably *the unitarity of the representation of the mapping class group of C*, defined by either the geometric quantization procedure or the underlying conformal field theory. This is closely related to the construction of an appropriate braided tensor category describing, from the conformal field theory perspective, the charge sector structure of the model [6], or, from the loop group point of view, the pertinent representation category.

Quantization also provides a link with quantum (elliptic) Calogero--Moser--Sutherland
integrable models and the special functions related to these; in particular, the hypergeometric function for root systems belongs to this
family. There are many research issues related to
these integrable systems and their root system generalizations,
see for instance [40]. Similarly, cyclotomic Hecke algebras arise, with many open questions remaining; cf.[3].
The role of W-algebras in the construction of Beilinson and Drinfeld leads, via the classical Drinfeld-Sokolov Hamiltonian reduction procedure, to a direct link between Hitchin systems and integrable hierarchies of partial differential equations.
For example, the cases LG=SL_n give rise to the so-called
generalized KdV hierarchy. *The quantization procedure of Beilinson and Drinfeld then suggests a quantization of this hierarchy* (and its generalizations), which we plan to study in detail, again also in connection with the issue whether quantization commutes with reduction.

Integrable hierarchies will also be studied in connection with Frobenius manifolds, where we wish to relate four existing developments [36]: firstly, Barannikov's construction of Frobenius manifolds inspired by mirror symmetry, secondly, their construction from the KP hierarchy, thirdly, their origin in Saito's theory of isomonodromic transformations, and finally, the construction of `almost Frobenius manifolds' from generalized WDVV equations. The first three of these topics involve a geometric construction using admissible planes within an infinite-dimensional Grassmannian, and we propose to view these constructions on an equal footing. The third approach turns out to be closely related to the geometric Langlands program.

In this context, our main research questions are as follows.
Which of the various Frobenius manifolds constructed from integrable hierarchies
have a similar description like the ones of Barannikov, i.e., as a family of planes in the Grassmannian satisfying some additional restraints? To what extent can geometrical Darboux transformations be found that
relate Frobenius manifolds to each other? Another research issue involves *the
solutions of the generalized WDVV equations in the coordinate free setting
of the perturbative Seiberg-Witten prepotentials*. Finally, we intend to use deformations of connections in the construction of Frobenius manifolds.

To close this section, we announce a quite novel plan to relate the geometric Langlands program to noncommutative geometry. This will be done through the so-called the Baum--Connes conjecture (1982) in the latter field (cf.[6], Ch.II). This conjecture describes the K-theory of a (reduced) group C*-algebra K_0(C_r^*(G)) in terms of a 'topological' K-theory group K_top^0(G). (The underlying toolkit is heavily used in the study of the functoriality principle for quantization described in the preceding section, and, indeed, the conjecture itself may be formulated in terms of deformation quantization [6], [8].) The Baum--Connes conjecture was proved for a large class of groups in 1999 by V. Lafforgue. This class includes all reductive groups over a p-adic field, which implies that all discrete series representations of such groups can be realized as the index of an equivariant Fredholm operator defined on the Bruhat-Tits building of G.

Building on the expertise of Van den Ban, Heckman, Landsman, Opdam, and Stokman, our plan is *to examine the relation of K_top^0(G)*
to the structure of the Langlands dual group for G reductive and p-adic}.
A related problem is the study of so-called index functions, partly in connection with important open questions about the structure of the category of tempered representations. For example,
is it true that discrete series representations of G are
projective (and thus injective by duality) in the category of
tempered representations? Transposing these matters to the representation theory of the affine Hecke algebra [6] leads to interesting
formulas for index functions and to the following conjecture:
the K-theory of the reduced C*-algebra completion of an
affine Hecke algebra H is independent of the deformable parameters defining H.

### Moduli, mirrors, and topological strings

As mentioned in the general overview, algebraic geometry has greatly benefited from the input of physics, and our research themes reflect this. Our guiding idea is that the connections revealed by this input are merely the tip of an iceberg. Cluster members involved in what follows would be Cornelissen, Dijkgraaf, Van der Geer, Van der Kallen, Looijenga, Moonen, Steenbrink, and Stienstra, drawing on the expertise of other cluster members in relevant areas.

As a case in point, mirror symmetry will be an important
theme in our cluster. Although this subject initially dealt with complex manifolds, a number of
ingredients are well defined in a purely algebraic setting, like
counting of curves and variation of filtrations on the Rham
cohomology. *It is therefore tempting to ask to what extent the notion
of mirror symmetry is meaningful in a purely algebraic setting.*
A related question is, of course, what
consequences mirror symmetry might have in positive
characteristic. For example, is there such a notion for varieties
defined over finite fields and if so, what does it imply?
These questions are certainly difficult, but on the other hand, since
explicit computations are possible here they leave ample room for
experimentation. For example, moduli of Calabi-Yau varieties, both in characteristic zero and in positive characteristic, lend themselves for this purpose.
In particular, we would like to explore newly observed phenomena
in positive characteristic, like non-liftability.
For elliptic curves and K3-surfaces there is a beautiful theory of moduli in positive characteristic due to Serre, Tate and Dwork, which can easily be extended to Calabi--Yau 3-folds in positive characteristic. It
then shows remarkable analogies with what physicists have discovered about the space of complex moduli of Calabi-Yau 3-folds
near the large complex structure limit, such as p-adic integrality properties of the mirror map. In any case, any insight into these matters might contribute also to a better understanding of mirror
symmetry in characteristic zero, and might also have profound
applications to arithmetic geometry, for example for questions on
rational points on varieties defined over number fields.

As a second focus for study we propose the cohomology (and Chow rings)
of moduli spaces of stable maps. The cohomology of moduli spaces of abelian varieties, and possibly also those of curves, can be described in terms of automorphic forms.
For example, in recent work of Faber and Van der Geer moduli of
curves over finite fields were used to obtain information on
vector valued Siegel modular forms of genus 2. A geometric
study of the moduli spaces both in characteristic zero as well as in positive
characteristic could give concrete information on automorphic forms
in higher genus. *An interesting question is whether the cohomology of M_g
for g >= 4 can be described in terms of Siegel modular forms,
or whether other automorphic forms are needed.*
Concretely, we propose to work on the tautological rings of moduli
of stable maps; on stratifications on moduli spaces of stable
maps, both in characteristic zero and positive characteristic
and their implications for the cohomology (in positive characteristic
these stratifications are connected with subtle phenomena in the
de Rham cohomology, a largely unexplored territory).
Furthermore, we want to study the cohomology of local systems on these
moduli spaces and their relations with Siegel modular forms. For example,
*an enticing question is
what the zeta function of M_g over a finite field should be.*

Thirdly, a most interesting recent development has been the increased interest in non-archimedean aspects of algebraic/arithmetical geometry in connection with non-commutative geometry,
involving Connes, Manin, Marcolli, and others. This includes, for example, a reinterpretation of the correspondence between Mumford curves and the graph of their uniformizing group in the Bruhat-Tits tree as a holography correspondence in the sense of 't Hooft and S"{u}sskind, the association of spectral triples as defined in noncommutative geometry
to such Mumford curves, and the treatment of (enlarged) boundaries of classical modular curves as non-commutative space in the sense of Connes. Our cluster members seem well prepared to enter this game, as all expertise is at hand.
Concretely, we would like to *introduce and understand better orbifold versions of the
holography correspondence for Mumford `orbifold curves,' and explore their physical
meaning*. We wish to generalize holography and spectral aspects of the theory
to rigid analytic uniformization to higher dimensions, where the theory of
buildings will start to play an increasingly important role, and to the case
of positive characteristic. Building on work of Faber, Van der Geer, and Zagier, we are interested in *studying zeta
functions of curves over finite fields using modular forms*.

Finally, led by Dijkgraaf, we will study topological strings, of which
a comprehensive theory now seems close. For a large class of Calabi--Yau manifolds (basically including all toric cases), exact solutions of the B-model have been found in the form of matrix models. This gives a direct relation with integrable hierarchies such as the KP and Toda hierarchy. The corresponding A-models can be physically interpreted as quantum crystals, and mathematically there are promising relations with seven- and eight-dimensional manifolds with exceptional holonomy groups G_2 and Spin(7).
Particularly interesting is the *study of D-branes in the A-model and
B-model, leading to special Lagrangian and holomorphic calibrations
respectively*. Moreover, Kontsevich' derived category interpretation of mirror symmetry yields a powerful reformulation of
certain aspects of the geometric Langlands program in terms of quantum field theory, relating G-bundles and D-modules for the Langlands dual moduli space. This link is, of course, an ideal cluster theme.

The study of the mathematics of topological strings also has important implications for physics. It has been shown that these models compute the vacuum structure of various four-dimensional supersymmetric gauge theories. This gives a promising framework to settle longstanding open problems in the dynamics of gauge theories, perhaps even quark confinement. Recently, topological strings have been used to calculate the entropy of black holes in supergravity and string theory. This has profound implications for our understanding of quantum gravity.

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