Which are the good introductory books on modern mathematical physics? Which are the good advanced books?
I read Whittaker's Analytical Dynamics, and I am reading Arnold's Mathematical Methods of Classical Mechanics. However, I am not very interested in books on classical mechanics, nor books for engineers. The related questions offer mostly books on classical mechanics, or books for engineers or the public. I am aware of Spivak's and of Landau’s books. I appreciate rigor and I am not interested in popular books. For instance, A Road to Reality is not appropriate but Lectures on Quantum Mechanics for Mathematics Students is.
I am asking in particular about books on quantum theories, gravity and on cosmology. I am also asking about unfalsifiable theories. I mean anything from string theory thorough conformal cyclic cosmology to loop quantum gravity.
To state my background I am a master student of mathematics. I took courses in classical mechanics, continuum mechanics, mathematical models of physics, quantum mechanics, field theory, etc. I audited courses on calculating conformal Feynman amplitudes in $\phi^4$ and in string theory, both of which assumed knowledge in conformal field theories, that I lack.
I have taken or partially audited diverse courses in mathematics. I will make these explicit if need be. I have some knowledge of group theory, representation theory, Lie groups, operator algebras, symplectic geometry, analysis on manifolds, complex manifolds, differential topology, etc.
So, what are good books for a young mathematician who wants to dabble in physics?