Index Theorems and Supersymmetry Uppsala University

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an fn (t); n=1 N The approximate paths coefficients {an }, thus the QN x (t) are functions of the Fourier (N ) measure is D a ∼ n=1 dan . i=1 Substituting the Chern character, the Todd class, and the Euler class into the index formula we arrive at the topological index (given by the integral in the far right hand side) Z index(d) = l(2l+1) (−1) l (−1) l Y i=1 M ! xi (T MC ) Z = e(T M ), M where in the first integral we used the following relation between the Euler class and the top Chern class cn (T MC ) = x1 x2 . 18) For d even, the space of Clifford forms can be decomposed into even and odd forms, C = C+ + C− , where C+ ≡ S ⊗ S+∗ = {ω ∈ S; ωγ5 = ω}, and C− ≡ S ⊗ S−∗ = {ω ∈ S; ωγ5 = −ω}. t0 t0 The measure is a product of Liouville measures; they are all classical quantities, N −1 dp00 Y dpi (t)dxi (t) := Dp(t)Dx(t). 4 The Supersymmetric Path Integral We derived one kind of path integral for bosons and another kind for fermions; except from commutativity and anti-commutativity of their variables, respectively, they differ by the boundary conditions imposed on their solutions. From a combinatorical point of view, there are 2n ways of swapping the indices, e. To be more explicit, there are also boundary terms that both depend on a total derivative in the integral, they do not contribute to the equations of motion and hence can be neglected. , · · · −→ Λp,q (M ) −→ Λp+1,q (M ) −→ The Dolbeault complex is obtained with p = 0: ∂¯ ∂¯ ∂¯ · · · −→ Λ0,q (M ) −→ Λ0,q+1 (M ) −→ Using similar arguments as in the de Rham case above, we have the characteristic classes cn/2 (T M ) = (−1)n/2 cn/2 (T M ) = (−1)n/2 e(T M ), td(T MC ) = td(T M ⊕ T M ) = td(T M )td(T M ), ch(σL ) = n/2 X ch(∧q T M ) = q=0 cn/2 (T M ) . Thus, the supersymmetric path integral is a product of three path integrals over the three given fields, respectively, that can be evaluated exactly by means of Gaussian integrals. Notice, however, that x̂(0) ∼ a+a† and p̂ ∼ −a+a† ; computing the expectation values hn|x̂(t)|ni and hn|p̂(t)|ni gives zero in both cases due to the orthogonality hn|n ± 1i = 0. 3 3 5 5 6 7 7 8 8 9 10 11 12 14 3 Path Integrals 3. The spin operator in the z-plane is equal to ~ [(|+ih+|) − (|−ih−|)] , 2 where the ket |+i (|−i) represents spin up (down). The Witten index determines whether it is not possible to spontaneously break the supersymmetry in a supersymmetric model. A broken supersymmetry implies that there is a mechanism that gives mass to supersymmetric particles (i. In chapter 5 below the same expressions of the index theorems presented here are derived using the supersymmetric path integral. Hence the non-Abelian field strength tensor is of the form Fµν = ∂µ Aν − ∂ν Aµ + [Aµ , Aν ]. 1) yields gµν (xα0 + εxα (t)) = gµν (xα0 ) + ∂λ gµν (xα0 )εxλ = gµν (xα0 ) + (Γκ λµ gκν + Γκ λν gκµ )εxλ ; Γµ %ν (xα0 + εxα (t)) = Γµ %ν (xα0 ) + ∂β Γµ %ν (xα0 )εxβ ; ẋ0% = εẋ% . fluctuations, can shift the potential in figure (5) in a negative direction, and thus restore supersymmetry. The infinitesimal translation from q to the upper-right corner r is equal to δxµ , hence the coordinate of r is xµ + εxµ + δxµ . Notice that the index is independent of the parameter β