Jonathan Weitsman

Bohr-Sommerfeld orbits in the moduli space of flat connections and the Verlinde dimension formula

We show how the moduli space of flatSU(2) connections on a two-manifold can be quantized in the real polarization of [15], using the methods of [6]. The dimension of the quantization, given by the number of integral fibres of the polarization, matches the Verlinde formula, which is known to give the dimension of the quantization of this space in a Kähler polarization.

Index of a family of Dirac operators on loop space

We use methods of constructive field theory to generalize index theory to an infinite-dimensional setting. We study a family of Dirac operatorsQ on loop space. These operators arise in the context of supersymmetric nonlinear quantum field models with HamiltoniansH=Q2. In these modelsQ is self-adjoint and Fredholm. A natural grading operator Γ exists such that ΓQ+QΓ=0. We studyQ+=P−QP+, whereP±=1/2 (1±Γ) are the orthogonal projections onto the eigenspaces of Γ. We calculate the indexi(Q+) for Wess-Zumino models defined by a superpotentialV(ω). HereV is a polynomial of degreen≧2. We establish thati(Q+)=n−1=degδV. In particular, the field theory models have unbroken supersymmetry, and (forn≧3) they have degenerate vacua. We believe that this is the first index theorem for a Dirac operator that couples infinitely many degrees of freedom.

On semifree symplectic circle actions with isolated fixed points

Abstract Let M be a symplectic manifold, equipped with a semi-free symplectic circle action with a finite, non-empty fixed point set. We show that the circle action must be Hamiltonian, and M must have the equivariant cohomology and Chern classes of (P1)n.

Euler-Maclaurin with remainder for a simple integral polytope

We give an Euler Maclaurin formula with remainder for the sum of the values of a smooth function on the integral points in a simple integral polytope. This formula is proved by elementary methods.

The two-dimensional,

We construct a family of supersymmetric, two-dimensional quantum field models. We establish the existence of the HamiltonianH and the superchargeQ as self-adjoint operators. We establish the ultraviolet finiteness of the model, independent of perturbation theory. We develop functional integral representations of the heat kernel which are useful for proving estimates in these models. In a companion paper [1] we establish an index theorem forQ, an infinite dimensional Dirac operator on loop space. This paper and, another related one [2], provide the technical justification for our claim thatQ is Fredholm, and for our computation of its index by a homotopy onto quantum mechanics.

N=2

LET PpB denote the moduli space of flat connections on (the trivial SU(2) bundle on) a compact, oriented two-manifold P. The space s”, is a stratified space containing an open dense set Yg which is a symplectic manifold of (real) dimension 6g - 6. There exists a natural line bundle 9 -+ ,Fb, with hermitian metric ( , ), whose restriction to Y9 has a connection V with curvature given by the symplectic form o on Y9. If we equip the two-manifold Xg with a metric, the resulting Hodge star operator turns ,Yg into a Klhler manifold, with w as the Kahler form. Setting aside for the moment concern with the singularities of Pg,, we arrive at the usual setting for geometric quantization: we are given a symplectic manifold (qg, o), a line bundle 3’ + 9, with connection of curvature CO, and a polarization of the space of sections of 3. The quantization CYY~(C~) = H”(Pg, Yk) of this system has been of importance in attempts to construct a topological field theory. A topological field theory would assign to every two-manifold Cg the quantization XDk(Cg) of the system (Pg, CO, 9, V), which must be proved independent of the choice of Kahler structure on -Fg, coming from a choice of metric on X9. The space Xk(Xg) can be given the structure of a Hilbert space with the metric given by ((s1,s~)) = j(si>s~)~~~-~. A topological field theory would also assign an element sN of the quantization Xk(Xg) to every three-manifold N3 whose boundary is given by 8N3 = Cg. If this assignment satisfied the axioms of topological field theory, the space Zk(Eg) would be equipped with a representation of the group Diff(Xg) of diffeomorphisms of Eg. Furthermore, we would obtain invariants of closed three-manifolds as follows. Let N be a closed three-manifold, and let N = H be a

Wess-Zumino model on a cylinder

We present index theorey for Dirac operators Q on the loop space S1 → R. These Dirac operators are obtained from sypersymmetric quantum field models containing one real Bose field and one real (Majorana) Fermi field. The interactions of the model are described by a real polynomial V. We prove that Q is Fredholm and we compute its index, namely ± [(deg V + 1) mod 2].

Half density quantization of the moduli space of flat connections and witten's semiclassical manifold invariants☆

Abstract We develop geometric techniques to study the intersection ring of the moduli space g(t1, …, tn) of flat connections on a two-manifold Σg of genus g with n marked points p1, …, pn. We find explicit homology cycles dual to generators of this ring, which allow us to prove recursion relations in g and n for their intersection numbers. The recursion relations in the genus g are related to generalizations of the Newstead Conjecture and of some recursion relations due to Donaldson. The recursion relations in the number n of marked points yield analogs of the recursion relations appearing in the work of Witten and Kontsevich on moduli spaces of punctured curves.

The loop space S1 → R and supersymmetric quantum fields

We give a Euler–Maclaurin formula with remainder for the sum of a smooth function on the integral points in a simple integral lattice polytope. Our proof uses elementary methods.

Geometry of the intersection ring of the moduli space of flat connections and the conjectures of Newstead and Witten

On presente une theorie des pfaffiens relatifs sur un espace de Hilbert de dimension infinie