These lecture notes are devoted to the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories. Introduction. Lecture 1. WDVV equations and Frobenius manifolds. {Appendix A.} Polynomial solutions of WDVV. {Appendix B.} Symmetriies of WDVV. Twisted Frobenius manifolds. {Appendix C.} WDVV and Chazy equation. Affine connections on curves with projective structure. Lecture 2. Topological conformal field theories and their moduli. Lecture 3. Spaces of isomonodromy deformations as Frobenius manifolds. {Appendix D.} Geometry of flat pencils of metrics. {Appendix E.} WDVV and Painlev\'e-VI. {Appendix F.} Branching of solutions of the equations of isomonodromic deformations and braid group. {Appendix G.} Monodromy group of a Frobenius manifold. {Appendix H.} Generalized hypergeometric equation associated to a Frobenius manifold and its monodromy. {Appendix I.} Determination of a superpotential of a Frobenius manifold. Lecture 4. Frobenius structure on the space of orbits of a Coxeter group. {Appendix J.} Extended complex crystallographic groups and twisted Frobenius manifolds. Lecture 5. Differential geometry of Hurwitz spaces. Lecture 6. Frobenius manifolds and integrable hierarchies. Coupling to topological gravity.

http://cds.cern.ch/record/265648/files/9407018.pdf

Geometry of 2D topological field theories

To Professor Shoshichi Kobayashi on his 60th birthday 1. Introduction. In this paper we shall introduce a new family of varieties, which we call quiver varieties, and study their geometric structures. They have close relation to the singularity theory and the representation theory of the Kac-Moody algebras. Our original motivation was to study solutions of the anti-self-dual Yang-Mills equations on a particular class of 4-dimensional noncompact complete manifolds, the so-called ALE spaces (or the ALE gravitational instantons), which were constructed by Kronheimer [Krl]. In [KN] we gave a description of the framed moduli space of all solutions in terms of solutions of a system of quadratic equations (called the ADHM equations) for representations of a quiver on an affine, simply laced Dynkin graph. It is an analogue of the description, given by Atiyah, Drinfeld, Hitchin, and Manin [ADHM], of the moduli space for IR 4 (or S4) in terms of solutions of a quadratic equation for certain finite-dimensional matrices. Once we set aside their gauge-theoretic origin, there is no longer reason to restrict ourselves to affine Dynkin graphs. Definitions can be generalized to arbitrary finite graphs. We get what we call quiver varieties. We study geometric structures of quiver varieties in this paper. In [Nal] it was noticed that the moduli space of anti-self-dual connections on ALE spaces has a hyper-K/ihler structure, namely a Riemannian metric equipped with three endo-morphisms I, J, K of the tangent bundle which satisfy the relations of quaternion algebra and are covariant constant with respect to the Levi-Civita connection: The same holds for general quiver varieties. In particular, quiver varieties have holomorphic symplectic forms. We study further properties of the quiver variety, such as a natural *-action, symplectic geometry, topology, and so on. As ALE spaces closely related to simple singularities, quiver varieties have very special kinds of singularities that enjoy very nice properties. Surprisingly, the ADHM equation appears in a very different context. In [L3] Lusztig used it to construct \"canonical bases\" of the part U-of the quantized enveloping algebra U associated by Drinfeld and Jimbo to the graph. Motivated by his results, we give a geometric construction of irreducible highest-weight integrable representations of the Kac-Moody algebra associated to the graph (Theorem 10.14). The weight space of the representation space will be given as a vector space consisting of constructible functions on a Lagrangian subvariety of a quiver variety. The action of the Kac-Moody …

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Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras

Contents: Matrix Eigenvalue Methods.- Double Bracket Isospectral Flows.- Singular Value Decomposition.- Linear Programming.- Approximation and Control.- Balanced Matrix Factorizations.- Invariant Theory and System Balancing.- Balancing via Gradient Flows.- Sensitivity Optimization.- Linear Algebra.- Dynamical Systems.- Global Analysis.

Optimization and Dynamical Systems

On decrit la construction d'une famille particuliere de 4-varietes hyper-Kahler: les espaces asymptotiquement localement euclidiens ALE On decrit une 4-variete de Riemann avec juste une extremite qui a l'infini ressemble au quotient R 4 /Γ de l'espace euclidien R 4 par un groupe fini Γ d'identifications

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The construction of ALE spaces as hyper-Kähler quotients

On its original publication, this book provided the first elementary treatment of representation theory of finite groups of Lie type in book form. This second edition features new material to reflect the continuous evolution of the subject, including entirely new chapters on Hecke algebras, Green functions and Lusztig families. The authors cover the basic theory of representations of finite groups of Lie type, such as linear, unitary, orthogonal and symplectic groups. They emphasise the Curtis–Alvis duality map and Mackey's theorem and the results that can be deduced from it, before moving on to a discussion of Deligne–Lusztig induction and Lusztig's Jordan decomposition theorem for characters. The book contains the background information needed to make it a useful resource for beginning graduate students in algebra as well as seasoned researchers. It includes exercises and explicit examples.

Representations of finite groups of Lie type

Regular group actions.- Deformation theory.- The quotient of the adjoint action.- The resolution of the adjoint quotient.- Subregular singularities.- Simple singularities.- Nilpotent elements in simple lie algebras.- Deformations of simple singularities.

Simple singularities and simple algebraic groups

Four lectures on simple groups and singularities

We study the normalized elliptic genera Φ(X)=ϕ(X)/e
k/2 for 4k-dimensional homogeneous spin manifolds X and show that they are constant as modular functions. The basic tool is a reduction formula relating Φ(X) to that of the self-intersection of the fixed point set of an involution γ on X. When Φ(X) is a constant it equals the signature of X. We derive a general formula for sign(G/H), G⊃H compact Lie groups, and determine its value in some cases by making use of the theory of involutions in compact Lie groups.

Peter Slodowy

Papers

Simple singularities and simple algebraic groups

Four lectures on simple groups and singularities

Elliptic genera, involutions, and homogeneous spin manifolds

Vorlesungen über das Ikosaeder

Platonic solids, Kleinian singularities, and Lie groups