While studying physics as a graduate student, I took a course at the University of Waterloo by Achim Kempf titled something like Advanced Mathematics for Quantum Physics. It was an extraordinary introduction to pure mathematics for physicists. For example, in that course we showed that by taking the Poisson bracket (used in Hamiltonian mechanics) and enforcing a specific type of non-commutativity on the elements, one will get Quantum Mechanics. This was Paul Dirac's discovery. After taking his course I left physics and went into graduate school in pure mathematics.
(I don't believe he published a book or lecture notes, unfortunately, though I just emailed him.)
In transitioning from physics to mathematics, I learned that the approach to mathematics is different in a pure setting than in a physics setting. Mathematicians define and prove everything. Nothing is left unsaid or stated. There is an incredible amount of clarity. Even in theoretical physics, I found there to be a lot of hand-waving and ill-defined statements and lack of rigor (which hilariously caused me a lot of anxiety). Overall, though, Mathematicians are focused on understanding and proving relationships between abstractions, whereas physicists are more interested in using these abstractions as tools. Therefore, the approach is very different: mathematicians don't care what the application is, they only want to understand the object under consideration.
Nevertheless, for a theoretical physicist looking to get a firm background in mathematics, you want to have the following core mathematical concepts, which will provide a foundation to explore any avenue:
- Linear Algebra
- Functional Analysis
- Topology
But the real list is something like:
- Set Theory
- Group and Ring Theory
- Linear Algebra
- Real Analysis
- Topology
- Functional Analysis
- Measure Theory
- Operator Algebra
Set, Group, and Ring theory are used extensively in physics, especially in Hamiltonian mechanics (see Poisson Bracket). Real Analysis and Linear Algebra are needed as a foundation for Functional Analysis. Functional Analysis could be described as an extension or marriage of Ring Theory, Group Theory, Linear Algebra, and Real Analysis. Therefore, many concepts in functional analysis are extended or used directly from Real Analysis and Linear Algebra. Measure Theory is important for the theory of integration, which is used extensively in applied physics and mathematics, probability theory (used in quantum mechanics), condensed matter physics, statistical physics, etc.
Topology and Operator Algebras are used extensively in advanced quantum mechanics and Relativity. Specifically, Algebraic Geometry is studied extensively in String Theory, whereas Topology is used extensively in General Relativity. Operator Algebras are an important area for understanding advanced Quantum Mechanics (ever heard someone talk about a Lie Group before?)
Some canonical text-books I would recommend:
- Linear Algebra: Advanced Linear Algebra by Steven Roman
- Real Analysis: Real Analysis by H. L. Royden
- Functional Analysis: A Course in Functional Analysis by John B. Conway
- Measure Theory: Measure Theory by Donald L. Cohn
Those are some decent text-books. I would say: give yourself two years to digest that material. Don't be hasty. Remember: mathematics is about definitions and proofs. Do not expect to see "applications" in any of those books. Just understand that the concepts are needed in advanced physics.
Unfortunately, though, I don't know of any text-book that forms a direct bridge between the two. If Achim Kempf had published his lecture notes, those may have worked, as essentially, he was doing just that.
Good luck!
First of all, Mirror Symmetry is huge. As you said, there are many fields involved. To know how much you need to know depends on where you're working. Roughly, one can divide the whole mathematical aspects of mirror symmetry into two categories. 1) Analytic and symplectic, (mainly (complex) differential geometry/symplectic geometry) 2) Algebraic (containing Algebraic geometry, homological algebra, etc.) I've been around with people who're doing Donaldson-Thomas theory (One Algebraic geometry side of Mirror symmetry) and personally willing to know more about homological mirror symmetry these days. Unfortunately, I don't know much about the analytic aspect which is related to Gromov-Witten theory.
The connections between these two categories are related to conjectures, one is called MNOP conjecture and the other interesting one is the homological mirror symmetry program.
As for the books and references, if you want to know just very little about what's going on, you may find Mirror symmetry written by leading mathematicians as well as mathematical physicist useful.However, Mirror symmetry and Algebraic geometry by Cox and Katz satisfies me more than the previous book (because obviously it's more mathematics.)
Best Answer
This got too long for a comment but is meant to be an extended one. I'm note quite the guy to say being mostly interested in it from a structural/mathematical perspective. Forgive me if I'm not telling you anything new.
You can definitely do TQFT within the confines of pure math. If what you want is the standard model you'll do well to understand your representation theory, as types of particles correspond to fundamental representations of Lie groups ($U(1)\times SU(2)\times SU(3)$ in the standard model, times the Poincaré group if you do the analysis.) From there a quantum field is a section of a vector bundle associated to the representation over space-time satisfying a variational principle (an extremal of an action) involving suitably equivariant connections (which are incidentally your bosons). Faria-Melo develops this and in fact exhibits the standard model in this framework.
They leave out a clear analysis of how representations tie in with types of particles, but this is done by Baez and Huerta in this text (http://math.ucr.edu/~huerta/guts/). Basically, elements in your fundamental representations are fermionic particle states, generators of the adjoint representation are bosons that act on your fermions in a way that can be represented by Feynmann diagrams.
Quantization is still fluffy to me, but it appears this is where quantum groups come in: You cannot deform a semi-simple Lie algebra and get a reasonable deformation of its representation theory (it's category of representations). You can however deform its universal enveloping algebra (which is a Hopf algebra, i.e., an object with a favourably interacting product and coproduct). There is a master class on this going on right now which talks about this for the purpose of studying 3-manifold invariants using 3-dimensional field theories. Notes about quantum groups may be found on its web page: http://www.math.ku.dk/english/research/conferences/2014/tqft/ They have incidentally a crash course on operator algebras as well, which is part of the theory that allows you to reasonably deal with infinite dimensional representations of the Poincaré group.
How the functor point of view on field theories relate to the "classical" one developed in Faria-Melo a bit fuzzy to me, but I suspect you may find some answers in Segal's article on conformal field theories (http://www.math.upenn.edu/~blockj/scfts/segal.pdf -- a pretty shitty scan but you'll find it in his 60th birth day thing).
Of course this leaves out nitty-gritty computational aspects of the kind a physicists would be able to tell you about, and I have never gotten close enough to what the physicists do to actually wanting to renormalize anything (something you apparently need to do because of self-interacting particles producing diverging integrals). This is definitely a pretty big part of QFT you'll be missing if you don't study the physicists approach as well.
It appears the big unifying idea in any case is that a physical system should be invariant under choice of presentation (gauge) up to a group or automorphisms (gauge transformation) and that this is true for classical systems (Lorentz or Poincaré invariance of space or space-time) as well as quantum systems (other Lie groups acting on a vector bundle of states) and that all of physics are more or less fall out as properties of stuff with the right symmetries. This appears to be what physicists and mathematicians agree on either way, so you can't go wrong studying representations.
Aside from Faria-Melo here are some notes I like to look at:
These notes are pretty explicit about the kind of mathematics they use math.lsa.umich.edu/~idolga/physicsbook.pdf
These notes on Lie groups and representation theory are very good. staff.science.uu.nl/~ban00101/lie2012/lie2010.pdf They come with video lectures. webmovies.science.uu.nl/WISM414