Some mathematical aspects of quantum field theory

In recent years, physics especially Quantum Field Theory has had an enormous impact in mathematics. This thesis mainly contains two different parts of mathematical developments of problems inspired from physics.;Firstly, I study Topological Quantum Field Theory and its related topics.;Gromov-Witten theory of resolved conifold corresponds to the Chern-Simons theory of unknot. In a series of papers, Labastida, Marino, Ooguri, Vafa, proposed a conjectural description of Chern-Simons theory of special linear quantum group invariants of links. LMOV conjecture could be viewed as a counterpart of Gopakumar-Vafa conjecture. These are actually parts of the big picture, large N Chern-Simons/Topological string duality.;In the first chapter of this part, the orthogonal quantum group version of LMOV conjecture is rigorously formulated in mathematics by using the representation of Brauer centralizer algebra. We also obtain formulae of Lichorish-Millet type which could be viewed as the application in knot theory and topology. By using the cabling technique, we obtain a uniform formula of colored Kauffman polynomial for all torus links with all partitions. Combined these together, we are able to prove many interesting cases of this orthogonal LMOV conjecture. In particular we can apply this uniform formula to verify certain case of the conjecture at roots of unity. In fact, these integer coefficients appeared in the original (orthogonal) LMOV conjecture are called the BPS numbers in string theory.;In the second chapter of this part, graphic representations of the universal R-matrices has been used to discover the recursion formulae between various quantum group invariants. We study the recursion relations of R-matrices corresponding to the inclusions Uq( sln) ⊂ Uq(sl n+1), Uq(sl k) x Uq(sln--k ) ⊂ Uq(sln), Uq(sln) ⊂ U q(so2n), Uq(so2k) x Uq(sln--k) ⊂ Uq(so2n). As an application, we find the ODE recursion formulae for HOMFLY and Kauffman polynomials.;Secondly, I study the Modular Forms in Topology, Elliptic Genera, Loop Space and String Manifolds. In a series of papers, the following results are discovered.;By developing modular invariance on certain characteristic forms, several cancellation formulas emerge naturally as the generalization of the original gravity anomaly cancellation formulas obtained by L. Alvarez-Gaume and E. Witten in their celebrated paper [2] studying the string theory. These cancellation formulae directly imply the divisibility and the congruence phenomena of characteristic numbers by Atiyah-Singer Index theorem, which plays important roles both in topology and differential geometry. We recover the Hirzebruch divisibility of twist signature and obtain the twist higher Rokhlin congruence by applying the modular invariance properties on the elliptic forms and also prove that it is best possible by studying examples constructed from K3-surface and Bott manifold. We also obtain the divisibility results for the index of double twist signature operators.;By studying the "modular transgression" on elliptic forms, we obtain some modularly invariant secondary characteristic forms on odd dimensional manifolds. Also, by using this method, we heuristically calculate the Chern-Simons forms for flat bundles over free loop space. This direction is pioneered by Witten [94), who heuristically interpreted the Landweber-Stong elliptic genus as the index of the formal signature operator on free loop space as well as introduced the formal equivariant index of the Dirac operator on loop space, known as Witten operator.;We call a manifold to be string if its loop space is spin. It's known that a string manifold is a spin manifold with vanishing half first Pontryagin class. Using the arithmetic properties of Jacobi-Theta functions, we prove the vanishing of the Witten genus of certain nonsingular string complete intersections in products of complex projective spaces, which generalizes a known result of Landweber and Stong [54].