14 VASILY PESTUN 2. Lecture: Localization 2.1. Euler class of vector bundle, Mathai-Quillen form and Poincare-Hopf theo- rem. We will present the Euler class of a vector bundle can be presented in the form of an integral over fermionic variables. This presentation connects to Mathai-Quillen formalism for localization in topological or supersymmetric field theories. Let E be an oriented real vector bundle E of rank 2n over a manifold X. Let xµ be local coordinates on the base X, and let their differentials be denoted µ = dxµ. Let hi be local coordinates on the fibers of E. Let ΠE denote the superspace obtained from the total space of the bundle E by inverting the parity of the fibers, so that the coordinates in the fibers of i ΠE are odd variables χ . Let gij be the matrix of a Riemannian metric on the bundle E. i Let A µ be the matrix valued 1-form on X representing a connection on the bundle E. Using the connection A we can define an odd vector field δ on the superspace ΠT (ΠE), or, equivalently, a de Rham differential on the space of differential forms Ω•(ΠE). In local coordinates (xµ; µ) and (χi; hi) the definition of δ is δxµ = µ δχi = hi − Ai µχj jµ (2.1) δ µ = 0 i i µ j δh = δ(A jµ χ ) Here hi = Dχi is the covariant de Rham differential of χi, so that under the change of i i j i i i ~j framing on E given by χ = s jχ~ the h transforms in the same way, that is h = s jh . The odd vector field δ is nilpotent δ2 = 0 (2.2) and is called de Rham vector field on ΠT (ΠE). Consider an element Φ 2 Ω•(ΠE), i.e. Φ is a function on ΠT (ΠE), defined by the equation 1 Φ = exp(−tδV ) (2.3) (2π)2n where t 2 R>0 and 1 V = (g χihj) (2.4) 2 ij Notice that since hi has been defined as Dχi the definition (2.3) is coordinate independent. To expand the definition of Φ (2.3) we compute δ(χ, h) = (h − Aχ, h) − (χ, dAχ − A(h − Aχ)) = (h; h) − (χ, FAχ) (2.5) where we suppresed the indices i; j, the d denotes the de Rham differential on X and FA the curvature 2-form on the connection A FA = dA + A ^ A (2.6) The Gaussian integration of the form Φ along the vertical fibers of ΠE gives 1 Z 1 1 [dh][dχ] exp(− δ(χ, h)) = Pf(F ) (2.7) (2π)2n 2 (2π)n A which agrees with definition of the integer valued Euler class (1.81). The representation of the Euler class in the form (2.3) is called the Gaussian Mathai-Quillen representation of the Thom class. EQUIVARIANT COHOMOLOGY AND LOCALIZATIONWINTER SCHOOL IN LES DIABLERETSJANUARY 201715 The Euler class of the vector bundle E is an element of H2n(X; Z). If dim X = 2n, the number obtained after integration of the fundamental cycle on X Z e(E) = Φ (2.8) ΠT (ΠE) is an integer Euler characterstic of the vector bundle E. If E = TX the equation (2.8) provides the Euler characteristic of the manifold X in the form 1 Z t!0 1 Z e(X) = dim X exp(−tδV ) = dim X 1 (2.9) (2π) ΠT (ΠTX) (2π) ΠT (ΠTX) Given a section s of the vector bundle E, we can deform the form Φ in the same δ- cohomology class by taking 1 p V = (χ, h + −1s) (2.10) s 2 After integrating over (h; χ) the the resulting differential form on X has factor 1 exp(− s2) (2.11) 2t so it is concentraited in a neigborhood of the locus s−1(0) ⊂ X of zeroes of the section s. exercise: write down the computation more precisely In this way the Poincare-Hopf theorem is proven: given an oriented vector bundle E on an oriented manifold X, with rank E = dim X, the Euler characteristic of E is equal to the number of zeroes of a generic section s of E counted with orientation 1 Z 1 Z X e(E) = Pf(F ) = exp(−tδV ) = sign det dsj (2.12) (2π)n A (2π)dim X s x X ΠT (ΠTX) x2s−1(0)⊂X −1 where dsjx : Tx ! Ex is the differential of the section s at a zero x 2 s (0). The assumption that s is a generic section implies that det dsjx is non-zero. More generally, let r = rank E and d = dim X, with r ≤ d, take a generic section s of E ! X and consider its set of zeroes F ≡ s−1(0) ⊂ X. Then F is a subvariety of X of dimension d − r. Let α 2 Ωd−r(X) be a closed form on X, equivalently α is a function on ΠTX. Then the integral 1 Z hα; ΦE;si := r α exp(−tδVs) (2.13) (2π) ΠT (ΠE) does on deformations of section s to λs with a parameter λ 2 R. Then by scaling the section s to λs and sending λ ! 0 we find 1 Z λ!0 Z hα; ΦE;si = r α exp(−tδVλs) = α ^ e(E) (2.14) (2π) ΠT (ΠTX) X while sending λ ! 1 we find 1 Z λ!1 Z hα; ΦE;si = r α exp(−tδVλs) = α (2.15) (2π) ΠT (ΠTX) F 16 VASILY PESTUN The equality of two expressions for hα; ΦE;si can be interpreted as a localization formula Z Z α ^ e(E) = α (2.16) X F In this way we proved that cohomology class [e(E)] 2 Hr(X) is Poincare dual to the homol- ogy class [F ] 2 Hd−r(X) where F is the zero set of generic section of bundle E ! X. 2.2. Equivariant Atiyah-Bott-Berline-Vergne localization formula. Suppose that a compact abelian Lie group T acts equivariantly on the oriented vector bundle E ! X, and that α 2 ΩG(X) is a closed equivariant differential form on X in Cartan model, that is dT α = 0. Then equivariant version of (2.16) holds Z Z α ^ eT (E) = α (2.17) X F exercise: prove (2.17) in Cartan model for equivariant cohomology replacing Euler class by equivariant Euler class Now let X be an oriented real even-dimensional Riemannian manifold, E = TX ! X be the tangent bundle, and T be a compact group acting on X, and suppose that the set F = XT of T -fixed points has dimension 0, i.e. F is a union of discrete points. A section s of tangent bundle E is a vector field. Assume that there is a circle subgroup S ⊂ T that generates a vector field s on X whose set of zeroes coincide with XT , i.e. F = XS = XT . Let α be dT -closed T -equivariant differential form on X in Cartan model. Then equivariant Euler class localization formula (2.17) Z X α ^ eT (TX ) = αx (2.18) X x2XT Equivariant cohomologies HT (X) form a ring. Formally, we can consider the field of fractions of this ring, and multiply α on the left and right side of the above equality by a cohomology class which is inverse to eT (TX ), then we arrive to the equation Z X αx α = (2.19) X eT (Tx) x2XT where eT (Tx) := eT (TX )x is equivariant Euler class of the tangent bundle to X evaluated at the point x. Since x is a discrete fixed point of T -action on X, the fiber Tx of the tangent bundle at point x forms a T -module. Since T is compact real abelian Lie group, a real 1 T -module splits into a direct sum of 2 dimR Tx irreducible real two-dimensional modules 2 1 (Li ' R ' C )i=1:::n on which the weights of the T action are all non-zero. Then by (1.63), (1.81) and we find that the equivariant Euler class is 1 2 dim X − 1 dim X Y eT (Tx) = (2π) 2 wi (2.20) i=1 EQUIVARIANT COHOMOLOGY AND LOCALIZATIONWINTER SCHOOL IN LES DIABLERETSJANUARY 201717 _ α where wi 2 t are weights. In basis ( )α=1::: dim T of linear coordinate functions on t we can write explicitly 1 2 dim X − 1 dim X Y α eT (Tx) = (2π) 2 wiα (2.21) i=1 2.3. Duistermaat-Heckman localization. A particular example where the Atiyah-Bott- Berline-Vergne localization formula can be applied is a symplectic space on which a Lie group T acts in a Hamiltonian way. Namely, let (X; !) be a real symplectic manifold of dimR X = 2n with symplectic form ! and let compact connected Lie group T act on X in Hamiltonian way, which means that there exists a function, called moment map or Hamiltonian µ : X ! t_ (2.22) such that dµa = −ia! (2.23) in some basis (Ta) of t where ia is the contraction operation with the vector field generated by the Ta action on X. • ∗ The degree 2 element !T 2 Ω (X) ⊗ St defined by the equation a !T = ! + µa (2.24) is a dT -closed equivariant differential form: a b a a dT !T = (d + ia)(! + µb) = dµa + ia! = 0 (2.25) This implies that the mixed-degree equivariant differential form α = e!T (2.26) is also dT -closed, and we can apply the Atiyah-Bott-Berline-Vergne localization formula to the integral Z Z 1 n a exp(!T ) = ! exp( µa) (2.27) X n! X For T = SO(2) so that Lie(SO(2)) ' R the integral (2.27) is the typical partition function of a classical Hamiltonian mechanical system in statistical physics with Hamiltonian function µ : X ! R and inverse temperature parameter −.