Topological quantum field theories (TQFT) and conformal field theories (CFT) have found numerous  connections to mathematics. Physical analysis has led to advances in representation theory, knot theory, topology and many other mathematical fields; in turn, mathematical analysis has led 

to exact, high energy calculations in these quantum field theories. Holomorphic-topological (HT) field theory combines features of both TQFT and CFT, and supports new mathematical structures that generalize simultaneously those from both. Currently, there has been a lot of new developments, both  mathematically and physically, on the structure of HT theories, and especially their implications to representation theory.