## Moduli spaces in mathematics and physics

EUROPEAN WOMEN IN MATHEMATICS

EWM Workshop on Moduli Spaces in Mathematics and Physics

Oxford, 2 and 3 July 1998

### Organising committee:

Frances Kirwan (Oxford), Sylvie Paycha (Clermont-Ferrand), Tsou Sheung Tsun (Oxford).

This interdisciplinary workshop was organized around 7 talks giving differenct points of view on the notion and use of moduli spaces. Various areas of mathematics and mathematical physics were represented: algebraic geometry, quantum field and gauge theory, and dynamical systems. The different perspectives presented here contributed to the richness of the meeting which was attended by about 20 mathematicians, including a good number of graduate students, from various countries in Europe and another 20 from Oxford and other universities in Britain. Special efforts were made by the speakers to present their topic in a form accessible to non-specialists.

Many participants expressed their wish for other such topical small scale meetings to take place in the future and some concrete proposals were made during the meeting.

The small scale of this meeting made possible many informal discussions among participants between the talks. At the end of the meeting, one of them nearly forgot her train, so engrossed was she with the discussions!

The contents of the talks will appear in proceedings which we hope will be readable by non-specialists who wish to have an idea what moduli spaces are.

### Abstracts of the talks

#### Frances Kirwan (Oxford): Introduction to Moduli Spaces

Classification problems in algebraic geometry (and other parts of geometry) often break down into two steps. The first step is to find as many discrete invariants as possible (for example, if we want to classify compact Riemann surfaces then the obvious discrete invariant is the genus). The second step is to fix values of the discrete invariants and to try to construct a moduli space; that is, an algebraic variety (or other appropriate space in other parts of geometry) whose points correspond to the equivalence classes of the objects to be classified, in some natural way. This talk will attempt to explain how this idea can be made more precise, and to describe some ways to construct moduli spaces.

#### Claire Voisin (Paris): Hodge theory and deformations of complex structure.

This talk will introduce to the theory of variations of Hodge structure, that is the way the Hodge decomposition of a projective or KÃ¤hler compact variety varies with the complex structure, and its applications: in one direction, the theory of periods helps understanding properties of the moduli space (Torelli type theorems, obstructions, curvature properties flatness…). In the other direction, deforming the variety allows to establish strong Hodge theoretic statements for the generic fiber (Noether-Lefschetz type theorems, (non)-triviality of the Abel-Jacobi map, Nori’s connectivity theorem).

#### Rosa-Maria Miro-Roig (Barcelona): Moduli Spaces of Vector Bundles on Algebraic Varieties.

Moduli spaces are one of the fundamental constructions of Algebraic Geometry and they arise in connection with classification problems. In my talk, I will restrict my attention to moduli spaces of stable vector bundles on smooth algebraic projective varieties. Roughly speaking a moduli space of stable vector bundles on an algebraic projective variety X is a scheme whose points are in “natural bijection” to isomorphic classes of stable vector bundles on X.

Once the existence of the moduli space is established, the question arises as what can be said about its local and global structure. More precisely, what does the moduli space look like, as an algebraic variety? Is it, for example, connected, irreducible, rational or smooth? What does it look as a topological space? What is its geometry? Until now, there is no a general answer to these questions.

The goal of my talk is to review some of the known results about moduli spaces of H-stable vector bundles on a smooth, irreducible, projective, algebraic variety (X,H). In particular, the properties which nicely reflect the general philosophy that moduli spaces inherit a lot of geometrical properties of the underlying variety.

#### Tsou Sheung Tsun (Oxford): Some Uses of Moduli Spaces in Particle and Field Theory.

In this talk I shall try to give an elementary introduction to certain areas of mathematical physics where the idea of moduli space is used to help solve problems or to further our understanding. In the wide area of gauge theory, I shall mention instantons, monopoles and duality. Then, under the general heading of string theory, I shall indicate briefly the use of moduli space in conformal field theory and M-theory.

#### Ragni Piene (Oslo): On the Use of Moduli Spaces in Curve Counting.

In enumerative algebraic geometry one works with various kinds of parameter spaces – Chow varieties, Hilbert schemes, moduli spaces of maps. We shall discuss these spaces and how they can be used to attack curve counting problems – in particular the problem of counting curves on a surface. This classical problem turns out to be of interest to theoretical physicists. Their interest has triggered quite a lot of work on the problem, in the context of both algebraic and symplectic geometry, but even more, their point of view has provided the mathematicians with new insight and new methods.

#### Mary Rees (Liverpool): TeichmÃ¼ller Distance and Meromorphic 1-forms.

This work arose out of a need to analyse a function of the form $d(x,\tau (x))$. (It uses some quite ancient theory, which was nontheless new to me.) Here, d is the TeichmÃ¼ller distance function on a TeichmÃ¼ller space ${\cal T}={\cal T}(S)$ of a surface S, and $\tau :{\calT}\rightarrow {\cal T}$ is a function. An example is given by $\tau(x)=x.g$, where g is an element of the modular group of S, which acts on ${\cal T}$. (I was actually motivated by a different, less classical example.) After introducing TeichmÃ¼ller space (mostly for marked spheres), and TeichmÃ¼ller distance, I shall show a connection with holomorphic and meromorphic 1-forms (on a different surface S’: a hyperelliptic curve in the case when S is a marked sphere). I shall look at bases of the first cohomology of a surface S’ in terms of homolomorphic and meromorphic 1-forms. I shall use this to show how to find the second derivative of the TeichmÃ¼ller distance function on ${\calT}(S)$.

#### Tatiana Ivanova (Dubna): Moduli Space of Self-Dual Gauge Fields, Holomorphic Bundles and Cohomology Sets.

The solution space of the self-dual Yang-Mills equations in Euclidean four-dimensional space R4 is considered. We discuss the Penrose-Ward correspondence between complex vector bundles over R4 with self-dual connections and holomorphic bundles over the twistor space of R4. The moduli space of self-dual Yang-Mills fields is described in terms of Cech and Dolbeault cohomology sets.

### Additional Info

- Place: Oxford University Oxford Great Britain
- Date or Period (start): Thursday, 02 July 1998
- Period (end): Friday, 03 July 1998