Not a book, but an eloquent (and fun to read) summary has been given by Robbert Dijkgraaf:

The mathematics of string theory

String theory can be considered as a two-parameter deformation of classical geometry, where one parameter controls the generalization from points to loops, and the other parameter controls the quantization in terms of the sum over topologies of Riemann surfaces. The final formulation of non-perturbative string theory, which is not yet there, will have to bring together geometry, non-commutative algebra and loop spaces.

Not a book, but an informative and enjoyable summary has been given by Robbert Dijkgraaf:

The mathematics of string theory

String theory can be considered as a two-parameter deformation of classical geometry, where one parameter controls the generalization from points to loops, and the other parameter controls the quantization in terms of the sum over topologies of Riemann surfaces. The final formulation of non-perturbative string theory, which is not yet there, will have to bring together geometry, non-commutative algebra and loop spaces.