**1. Introduction**

One of the things I find most fascinating is the interplay between mathematics and physics. One direction of this interplay is well-known i.e., how mathematics forms the basis of all physical concepts. In fact, this is now so ingrained in our minds that we naturally expect any new physical concept to have an appropriate mathematical description. Wigner calls this the “unreasonable effectiveness of mathematics” in his famous article. What is surprising, perhaps, is the other direction i.e., how physical intuition leads to new developments in mathematics. This is especially true with quantum physics and geometry.

Quantum field theories such as Yang-Mills and Chern-Simons theories provide a deep understanding of certain aspects of topology and geometry. Starting around in the 80s, these developments have led to the creation of the field of quantum topology and quantum algebra. Some of these results, which took mathematicians by surprise, were obtained from deep physical intuition. These developments are beautifully explained in a 2010 article by Atiyah, Dijkgraaf and Hitchin. For instance, understanding the space of the so called self-dual solutions of Yang-Mills theories leads to powerful invariants of four manifolds. Similarly, studying the holonomy of Wilson loop operators and their vacuum expectation values gives a new description of the Jones polynomial.

A basic and extremely important structure in mathematical physics and in the results mentioned above is that of a vector bundle. Several physical theories such as Yang-Mills theory and Chern-Simons theory can be cast in terms of vector bundles. Moreover, natural bundles on manifolds such as the tangent and cotangent bundles are useful in descriptions of general relativity and the phase space of classical mechanical systems (the latter, in fact, leads to the very useful concept of geometric quantization, which produces a quantum mechanical Hilbert space from the classical mechanical phase space). The concept of holonomy in a vector bundle, which measures the deviation produced in a standard vector when it is parallel transported around a loop, turns out to be the underlying mathematical description of the Aharonov-Bohm effect and the quantum field theoretic formulation of the Jones polynomial. These are just some examples of the use of vector bundles.

In this post, I want to discuss some aspects of vector bundles that are useful in the above applications. Aside from the definition and examples, I will talk about how finite dimensional vector bundles can be classified using homotopy. Continue reading