Cecotti, SergioFendley, PaulIntriligator, KenVafa, Cumrun2019-09-221992Cecotti, Sergio, Paul Fendley, Ken Intriligator, and Cumrun Vafa. 1992. “A New Supersymmetric Index.” Nuclear Physics B 386 (2): 405–52. https://doi.org/10.1016/0550-3213(92)90572-s.0169-68230550-32131873-1562http://nrs.harvard.edu/urn-3:HUL.InstRepos:41385051We show that Tr(-1)(F)F e(-betaH) is an index for N = 2 supersymmetric theories in two dimensions, in the sense that it is independent of almost all deformations of the theory. This index is related to the geometry of the vacua (Berry's curvature) and satisfies an exact differential equation as a function of beta. For integrable theories we can also compute the index thermodynamically, using the exact S-matrix. The equivalence of these two results implies a highly non-trivial equivalence of a set of coupled integral equations with these differential equations, among them Painleve III and the affine Toda equations.en-USJournal Article2019-09-2210.1016/0550-3213(92)90572-S