By transforming to momentum space, the integrals used to compute the Feynman weight of a graph can be simplified. After carrying out this process, I will compute the partition function of two simple systems, quantum mechanics on a circle and circle-valued quantum mechanics. Finally I will discuss how these methods easily generalize to quantum field theory in the case of a free scalar bosonic field.

Let $(W,\Pi)$ be a Riemann domain over a complex manifold $M$ and $w_0$ be a point in $W$. Let $\mathbb D$ be the unit disk in $\mathbb C$ and $\mathbb T=\partial\mathbb D$. Consider the space ${\mathcal S}_{1,w_0}({\overline {\mathbb D}},W,M)$ of continuous mappings $f$ of $\mathbb T$ into $W$ such that $f(1)=w_0$ and $\Pi\circ f$ extends to a holomorphic on $\mathbb D$ mapping $\hat f$. Mappings $f_0,f_1\in{\mathcal S}_{1,w_0}({\overline{\mathbb D}},W,M)$ are called {\it holomorphically homotopic or $h$-homotopic} if there is a continuous mapping $f_t$ of $[0,1]$ into ${\mathcal S}_{1,w_0}({\overline{\mathbb D}},W,M)$. Clearly, the $h$-homotopy is an equivalence relation and the equivalence class of $f\in{\mathcal S}_{1,w_0}({\overline{\mathbb D}},W,M)$ will be denoted by $[f]$ and the set of all equivalence classes by $\eta_1(W,M,w_0)$. \par There is a natural mapping $\iota_1:\,\eta_1(W,M,w_0)\to\pi_1(W,w_0)$ generated by assigning to $f\in{\mathcal S}_{1,w_0}({\overline{\mathbb D}},W,M)$ its restriction to $\mathbb T$. We introduce on $\eta_1(W,M,w_0)$ a binary operation $\star$ which induces on $\eta_1(W,M,w_0)$ a structure of a semigroup with unity and show that $\eta_1(W,M,w_0)$ is an algebraic biholomorphic invariant of Riemann domains. Moreover, $\iota_1([f_1]\star[f_2])=\iota_1([f_1])\cdot\iota_1([f_2])$, where $\cdot$ is the standard operation on $\pi_1(W,w_0)$. Then we establish standard properties of $\eta_1(W,M,w_0)$ and provide some examples. When $W$ is a finitely connected domain in $M=\mathbb C$ and $\Pi$ is the identity, we show that $\iota_1$ is an isomorphism of $\eta_1(W,M,w_0)$ onto the minimal subsemigroup of $\pi_1(W,w_0)$ containing holomorphic generators and invariant with respect to the inner automorphisms. In particular, we show for a general domain $W\subset\mathbb C$ that $[f_1]=[f_2]$ if and only if $\iota_1([f_1])=\iota_1([f_2])$. This is a joint work with Evgeny Poletsky.

The path integral formula for the index of the Dirac operator can be interpreted as a localization formula for U(1)-equivariant cohomology of the free loop space of the manifold. In this lecture I shall first recall the Cartan model of equivariant differential forms of a finite dimensional manifold and the localization formula of Berline-Vergne. We shall then see that the loop space analogue of this result will give the A hat genus. This can be regarded as the bosonic component of the index formula. The corresponding localization formula in the supersymmetric case gives the full index formula.

Following the previous talks on the Atiyah-Singer index theorem by Masoud, we will prove another important special case, namely the Hirzebruch-Riemann-Roch theorem. This theorem gives the holomorphic Euler characteristic of a holomorphic vector bundle over a compact Kähler manifold in terms of the Todd class of the manifold and the Chern character of the vector bundle. It will be shown how in the case of a holomorphic line bundle over a Riemann surface this reduces to the classical Riemann-Roch theorem.