I'll attempt a different approach to your inquiry. I'll be referring to Peskin & Schroeder's QFT book, not because it is better than others, but because I'm familiar with P&S more than with others.
The authors, in their first chapter, mention this: "...we must first [...] make a serious detour into formalism. The three chapters [...] are almost completely formal..." Thus, I'll use those three chapters to outline the required math for reading on QFT.
I will also assume that there is already some solid knowledge of basic math stuff, like that gained in any undergrad curriculum; namely, calculus as well as vector calculus, linear algebra elements (matrix algebra, eigenvalues & eigenvectors, unitary matrices), and elementary complex functions; in general, what math one should expect from a graduating science student.
Here's a list of math topics that, according to P&S's book layout, are essential for reading QFT material. I'll give the list, roughly following their three chapters:
- Index notation for vectors and tensors.
- Dirac notation of state vectors.
- Calculus in the complex domain.
- Gaussian integrals.
- Special relativity mathematics, like 4-vector and tensor algebra.
- Asymptotic solutions of integrals, like the method of stationary phase.
- Action principle, Lagrange, and Hamilton approach, Euler-Lagrange equations, at the level of Goldstein 2nd edition's chapter 12. You cannot go a single step further in QFT without knowing that stuff.
- Vector and tensor transformations and invariances, particularly in the special relativity domain (Lorentz, etc.).
- Poisson brackets and Dirac's canonical commutator and anticommutator. It is understood that some basic familiarity with quantum mechanics topics is absolutely essential here.
- Fourier integrals, Fourier expansion, and the interplay between position and momentum spaces. I cannot overemphasize enough the importance of this topic.
- Commutator algebra; use of ladder operators in commutators (basically, a quantum mechanics topic).
- Unitary operators in connection with Lorentz transformations; operator algebra.
- Residue theorem and evaluation of contour integrals. Another topic of absolutely fundamental weight.
- Groups, generators, irreducibility rules, basic representation theory; emphasis on Lie groups, infinitesimal transformations, in connection with Lorentz groups and algebras. That's a huge topic, so I suggest some introductory text, like the relevant chapters from Zee's Gravity textbook.
- Perturbation theory, especially time-related; iterative integration, some familiarity with combinatorics.
- Understanding of the use of Kronecker and delta functions in the calculations of integrals and commutators.
- Multiple integrals and infinite-dimensional integrals is a must, of course.
- A lot of vital mathematics lies in physics texts. That is an old issue---should one read math first, or should one read physics and math methods concurrently? Most teachers follow the latter approach. So do P&S in their book; they teach more advanced topics like spinor algebra, gamma matrices, Wick's theorem, contractions, etc., along with physics topics.
- Finally, one last point, one that usually distinguishes physicists' viewpoint from mathematicians' viewpoint. Be ready to "evaluate" integrals, otherwise impossible to solve analytically, by applying purely physics concepts, like types of detectors used, rejection of particles' selected properties, and the like.
That's it, although the list is certainly far from complete. Anyway, I hope I have been of some help.
