Geometry and Physics


In recent years we have witnessed a remarkable renaissance in the interaction between geometry and physics. A turning point was in 1982 when Donaldson used the moduli space of instanton solutions of the Yang-Mills equations to prove that smooth four-manifolds with definite intersection form must be standard. The moduli spaces were later used to define new invariants for smooth four-manifolds---detecting distinct differentiable structures on, for example, complex algebraic surfaces.

Investigation of moduli spaces of instantons, magnetic monopoles or solutions of Nahm's equations made contact with the twistor theory of Penrose and various aspects of differential geometry such as Einstein manifolds. The solutions of the field equations can often be described in terms of holomorphic geometry and the moduli spaces have special geometrical structures such as hyperKähler metrics.

Employing the moduli spaces of flat connections, Casson obtained an invariant for homology three-spheres and showed a connection with the theory of knots and links---a theory for which a remarkable new invariant was discovered only a year before by Jones. Then Floer used moduli spaces of instantons over a four-dimensional tube, asymptotic to flat connections at the ends, to define the instanton homology groups for homology three-spheres and Taubes proved that the Casson invariant equals one half of the Euler characteristic of these groups.

The new feature of the present interaction with physics is that quantum theory is involved and has significant links with topology. Witten provided a (3 + 1)-dimensional topological quantum field theoretic interpretation of the Donaldson-Floer invariants. He also used the Chern-Simons action to give a path integral definition of new invariants of three-dimensional manifolds as well as an intrinsic three-dimensional description of the Jones polynomial of links.

This deep relationship between geometry and physics is demonstrated by the work of Yau, going back to his use of minimal surfaces in the proof of positive total energy and the positivity of the Bondi mass. Yau's work also makes contact with three-dimensional topology via minimal surfaces and themes such as Kähler-Einstein geometry and mirror symmetry are of increasing importance.

In 1995, we organised a six month Special Session on Geometry and Physics based on the above topics at the University of Aarhus, Denmark. The main activities were centred around four workshops and a ten-day conference. In addition we, together with Hans Jørgen Munkholm, organised a summer school at Odense University shortly before the main conference for Ph.D. students. All told there were over 350 participants.

This volume contains invited contributions from the participants of the above activities. This includes both lectures notes from the summer school and research papers from the workshops and conference. All the articles have been refereed.

The opening talk at the conference was given by Sir Michael Atiyah, who presented a survey of the interactions between geometry and physics. Whilst we were planning the special session, one major development that occurred was the introduction of the Seiberg-Witten invariants, which enabled the solution of various open problems arising in Donaldson's theory. Atiyah's talk, reproduced in the first article of this volume, puts the new developments in a historical perspective and gives a more balanced account of the interaction between geometry and physics than we were able to give in the short space above. Various speakers during the special session described many applications of these new invariants (see, for example, the articles by LeBrun, Ohta & Ono, Bradlow & García-Prada in this volume). In particular, Taubes gave two talks describing his fundamental results relating the Seiberg-Witten invariants and Gromov-Witten invariants of four-manifolds. This discussion is presented in the penultimate article of these proceedings. Also, at the summer school, Zoltán Szabó gave an introduction to this new theory.

The other lectures at the summer school were given by Kenji Fukaya, Lisa Jeffrey, Michael Thaddeus, Dror Bar-Natan and Gang Tian. Their notes all appear in this volume, with the exception of Tian's lectures on quantum cohomology which are to appear in the proceedings on ``Current Developments in Mathematics'', Cambridge, 1995. For the non-specialist, the lecture notes from the summer school should form a good basis for the understanding of the subsequent papers in this volume.

The first workshop was on Moduli Spaces of Semi-Stable Bundles over Riemann Surfaces held from the 16th to the 29th May. These moduli spaces may be used to give a geometrical construction of Witten's (2+1)-dimensional quantum field theory. The workshop itself is represented in this volume by the articles of Kumar and Ueno. Conformal field theory provides the abstract existence of a tensor product on the category of integrable representations of affine Kac-Moody algebras of a fixed level. Kumar proposes a concrete geometrical way of constructing this product. At this workshop Ueno gave a series of talks on his influential joint work with Tsuchiya and Yamada on conformal field theory. The resulting set of notes, which give new ideas correcting the earlier work and provide a detailed discussion at a level accessible to graduate students, close this volume. The summer school lecture notes of Jeffrey and Thaddeus in this volume give an introduction to the moduli spaces of semi-stable bundles from two different and complementary viewpoints.

The workshop in June (19th to 30th) was organised in collaboration with Uffe Haagerup (Odense). The theme was Operator Algebras and Topology and the participants included Vaughan Jones, Ocneanu and Voiculescu. This workshop is represented here by the paper of Dietmar Bisch & Vaughan Jones computing dimensions of irreducible representations of semi-simple Fuss-Catalan algebras and fusion rules for the associated subfactors. This paper clearly illustrates the planar topological aspects of subfactors, which Jones strongly emphasised in his lectures.

The ten-day conference was entitled ``Quantum Invariants and Low-Dimensional Topology including Special Holonomy and Twistor Theory''. This was the main event of the special session with over 170 participants and took place from the 18th to the 27th July.

Taking the second topic first, the themes of symplectic geometry and Einstein metrics were to the fore. Tian presented results that showed that the Calabi conjecture for the existence of Kähler-Einstein metrics on manifolds with c_1>0 is false even when there are no holomorphic symmetries. As mentioned above, applications of the Seiberg-Witten invariants were given by various speakers, but the S-duality underlying these invariants was also discussed by Segal in relation to the topology of the space of magnetic monopoles. This duality relates to mirror symmetry, which appears to be present in the examples Joyce gave of the first compact manifolds with holonomy G_2 or Spin(7). A deformation theory/twistor theory approach to holonomy is explained by Merkulov. (During the conference Merkulov & Schwachhöfer showed that Berger's affine holonomy classification is incomplete; this result will appear elsewhere.) Freed's article discusses new ways of computing curvature and holonomy for the determinant line bundle.

Fukaya's conference talk carried on the Floer homology theme of his summer school course using the ideas to show how S-cobordism might be applied to symplectic geometry. Yau & Zaslow describe how new symplectic invariants may be created using string theory. Moments maps and symplectic geometry are also discussed by Karshon and Alekseev. Other articles on special geometries include those by Biquard & Gauduchon, Poon, Tod, Dancer & Swann. Many of these structures are Einstein and the general equations for cohomogeneity-one Einstein metrics are discussed by Eschenburg & Wang. Fino & Salamon study the derived Euler characteristic in relation to these special geometries and in particular symmetric spaces.

The theme of Quantum Invariants was carried through to the workshop from the 1st to the 22nd of August, which concentrated on the relatively new topic of finite type invariants of three-manifolds and relations with non-perturbative quantum invariants. The workshop brought together many of the active researchers in this area for the first time. Garoufalidis & Ohtsuki compare Ohtsuki's definition of finite type invariant with others on rational homology three-spheres. Ohtsuki shows the existence of a rational series whose r-adic reduction gives the Reshetikhin-Turaev invariant at an r'th root of unity, explains how the Casson invariant is essentially the second coefficient of his series and describes the series for a lens space as a rational function. Lawrence gives a holomorphic interpretation of the Witten-Reshetikhin-Turaev invariants for a certain family of three-manifolds, thereby producing an expression for Ohtsuki's series. In this case the series determines all of the invariant and she asks whether this is true in general. Another construction of a three-manifold invariant, which when appropriately evaluated gives the Reshetikhin-Turaev invariant, is given in Lê's article.

Various contributors discuss invariants for three-manifolds with extra data. Using techniques from the Feynman expansion of path integrals, Axelrod gives a rigorous definition of perturbative invariants for three-manifolds with a flat connection. By using the universal Vassiliev-Kontsevich invariant, J. Murakami describes a generalised Casson invariant for a three-manifold containing a knot. Goryunov shows that finite type invariants for knots in the solid torus are equivalent to finite type invariants in Arnold's J^+-theory of plane curves. An introduction to some of the many different approaches for finite type invariants of knots is given by Bar-Natan's summer school notes.

On the subject of quantum invariants, H. Murakami & Ohtsuki show how Kuperberg's graphical approach to quantum Sp(2)-invariants of links can be used to define quantum Sp(2)-invariants for three-manifolds at certain levels. Kauffman discusses the non-triviality of Hennings' invariants and how they differ from Reshetikhin and Turaev's invariant. Spin TQFT's and the corresponding spin mapping class group representations are described by Masbaum. On a more abstract level of TQFT's, Yetter shows how Hopf algebras are implicitly involved and Kerler discusses the relationship between some different invariants coming from quantum groups by using category theory. Deguchi & Tsurusaki describe an intriguing practical application of quantum invariants to random knotting in certain polymers.

To end the special session, in September a workshop on Quantum Cohomology and Mirror Symmetry was held. This workshop is represented by Bott's article where he describes recent work with Taubes on a purely topological approach to some of the above-mentioned physics-inspired invariants.

The special session was supported financially by several organisations. Most of the initial funding came from The Danish Research Council, both directly and via various grants aimed at the three areas of Operator Algebras, Algebra and Topology. In this connection we are particularly grateful for the work done by Ib Madsen in securing this funding, and to Uffe Haagerup, Henning Haahr Andersen and Ib Madsen for providing support via the above mentioned special grants. In addition, we thank Vagn Lundsgaard Hansen who represents Mathematics on the Research Council.

We also received financial assistance from the Universities of Aarhus and Odense. We thank the chairmen, Hans Anton Salomonsen and Niels Jørgen Nielsen, of the respective mathematics departments for their financial and practical support making facilities available.

The summer school in Odense was organised with the help of Hans Jørgen Munkholm (Odense) and principally financed by the The Danish Research Academy, whom we also thank.

Financial support was also forthcoming from the European grants for ``Global Analysis and Differential Geometry'' GADGET and ``K-theorie et ses applications en géométrie et artihmétique''.

We had a lot of practical and secretarial help, and we particularly wish to thank Karen Damgaard, Oddbjørg Wethelund, Søren Kold and Flemming Lindblad Johansen in Aarhus and Lisbeth Larsen, Margit Christiansen and Jeppe Buk in Odense, as well as all the Ph.D. students who helped us out.

Finally, we thank the advisory committee (S. K. Donaldson, N. J. Hitchin, V. G. Turaev and S.-T. Yau) for their support and advice, the referees for their speedy work, Arthur Greenspoon of the American Mathematical Society for help with proof reading and Marcel Dekker for agreeing to publish this volume.

Jørgen Ellegaard Andersen, Johan Dupont,
Henrik Pedersen and Andrew Swann
April, 1996

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