A rich source of applications can be found in string theory. Most straightforwardly, the partition function, or vacuum amplitude, of a conformal field theory on a Riemann surface are essentially given by theta functions (and their generalizations, e.g. Siegel modular forms). Various theta function identites have important physical consequences.
The simplest case to look at would be the torus, discussed at length for example in Polchinski's textbooks - the four spin structures on the torus lead to the four Jacobi theta functions, and the Green's function can also be expressed in terms of theta functions as a consequence. (By the way, bosonization also appears in this context as the worldsheet of a string is 1+1 dimensional (see e.g. Green-Schwarz-Witten's books)).
On the other hand, a very active recent topic would be the occurrence of mock modular forms and the "wall-crossing" phenomenon.
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