
Mikio Nakahara
Geometry, topology and physics
For those who have studied physics to undergraduate level. the abstract geometrical mathematics of research level theoretical physics might seem like a different language. Geometry, topology and physics by Mikio Nakahara helps to bridge that gap. The book starts with a couple of chapters going over undergraduate level physics and mathematics. This is followed by chapters on homology and homotopy groups. Much of the book looks at topics relating to differentiable manifolds, including Riemannian geometry, complex manifolds and fibre bundles, but with an emphasis on their use in quantum theory rather than general relativity.
The final two chapters look at anomalies in gauge field theories and at bosonic string theory.
The book has plenty of diagrams and examples of how the subject matter relates to physical systems. This book doesn't have too steep a learning curve  I felt that someone who had a sound understanding of undergraduate level physics would find it fairly straightforward to work through the first half of the book, but that after that it would become more challenging. Those looking for a gentler approach might prefer The geometry of physics by Theodore Frankel and move on to Nakahara's book when they want to get on to research level topics.