By Verbaarschot J.

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Only the antisymmetrized part of the latter term contributes to the integral. Therefore, we have to antisymmetrize the exponential function before equating the coeﬃcients of the η(B). After multiplying out the squares in the exponent of both sides of the equation we obtain 1 dU e− 2t TrU XU −1 V Y V −1 = ct(n 2 −n)/2 1 det e− 2t xk yl , Δ(y)Δ(x) (16) which is the ﬁnal result for the Itzykson-Zuber integral. 2 Semiclassical calculation of the Itzykson-Zuber integral. In this section we calculate the Itzykson-Zuber integral by means of a saddle-point approximation.

Math. Phys. 19 (1978) 709. 4 8 Integration Theorems If an integrand is invariant under super-unitary transformations, there exist a number of powerful integration theorems. In character they are comparable to a complex contour integration. 1 The Parisi-Sourlas reduction Let us first consider the simplest case involving the graded vectors a θ p= , p† = (a∗ , θ ∗ ) . e. F (U p, U † P † ) = F (p, p† ), (2) for an arbitrary super-unitary transformation U . In this case the integration theorem is 1 π dada∗ dθdθ∗ F (p, p† ) = F (0, 0).

4 7 Grassmann integration In this lecture we introduce Grassmann variables and introduce integration over Grassmann variables. The basic objects that we will consider are vectors consisting out of m commuting components and n anti-commuting components. Such vectors will be called graded vectors or super-vectors, and the matrices acting on these vectors will be called (m|n) graded matrices or super-matrices. Our ﬁnal goal will be to establish an integration measure on manifolds based on this graded structure.