2. Lecture: Localization 2.1. Euler Class of Vector Bundle, Mathai-Quillen Form and Poincare-Hopf Theo- Rem

14 VASILY PESTUN

2. Lecture: Localization 2.1. Euler class of vector bundle, Mathai-Quillen form and Poincare-Hopf theorem. 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 ψµ = 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, ψµ) and (χi, hi) the definition of δ is δxµ = ψµ δχi = hi − Ai ψµχj jµ (2.1) δψµ = 0 i i µ j δh = δ(A jµψ 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 ∈ 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 ∈ 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 √ 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 ds| (2.12) (2π)n A (2π)dim X s x X ΠT (ΠTX) x∈s−1(0)⊂X

−1 where ds|x : Tx → Ex is the diﬀerential of the section s at a zero x ∈ s (0). The assumption that s is a generic section implies that det ds|x 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 ∈ 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 ∈ 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 → ∞ we ﬁnd

1 Z λ→∞ 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)] ∈ Hr(X) is Poincare dual to the homol- ogy class [F ] ∈ 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 ∈ 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 x∈XT

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) x∈XT

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 ∈ 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, ω) be a real symplectic manifold of dimR X = 2n with symplectic form ω 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 = −iaω (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 ωT ∈ Ω (X) ⊗ St defined by the equation a ωT = ω + µa (2.24) is a dT -closed equivariant differential form: a b a a dT ωT = (d + ia)(ω + µb) = dµa + iaω = 0 (2.25) This implies that the mixed-degree equivariant differential form α = eωT (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(ωT ) = ω 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 −. Suppose that T = SO(2) and that the set of fixed points XT is discrete. Then the Atiyah-Bott-Berline-Vergne localization formula (2.19) implies

Z a 1 n a X exp( µa) ω exp( µa) = (2.28) n! X eT (νx) x∈XT

T where νx is the normal bundle to a ﬁxed point x ∈ X in X and eT (νx) is the T -equivariant Euler class of the bundle νx. The rank of the normal bundle νx is 2n and the structure group is SO(2n). In notations of section 1.9 we evaluate the T -equivariant characteristic Euler class of the principal G- bundle for T = SO(2) and G = SO(2n) by equation (1.62) for the invariant polynomial on 1 g = so(2n) given by p = (2π)n Pf according to deﬁnition (1.81). 18 VASILY PESTUN

2.4. Gaussian integral example. To illustrate the localization formula (2.28) suppose that X = R2n with symplectic form n X i ω = dx ∧ dyi (2.29) i=1 and SO(2) action x cos w θ − sin w θ x i 7→ i i i (2.30) yi sin wiθ cos wiθ yi n where θ ∈ R/(2πZ) parametrizes SO(2) and (w1, . . . , wn) ∈ Z . T The point 0 ∈ X is the ﬁxed point so that X = {0}, and the normal bundle νx = T0X is an SO(2)-module of real dimension 2n and complex dimension n that splits into a direct sum of n irreducible SO(2) modules with weights (w1, . . . , wn). We identify Lie(SO(2)) with R with basis element {1} and coordinate function ∈ Lie(SO(2))∗. The SO(2) action (2.30) is Hamiltonian with respect to the moment map n 1 X µ = µ + w (x2 + y2) (2.31) 0 2 i i i i=1

Assuming that < 0 and all wi > 0 we ﬁnd by direct Gaussian integration 1 Z (2π)n ωn exp(µ) = exp(µ ) (2.32) n Qn 0 n! X (−) i=1 wi and the same result by the localization formula (2.28) because 1 e (ν ) = Pf(ρ(1)) (2.33) T x (2π)n according to the deﬁnition of the T -equivariant class (1.62) and the Euler characteristic class (1.81), and where ρ : Lie(SO(2)) → Lie(SO(2n)) is the homomorphism in (1.61) with

 0 −w1 ...... 0 0  w1 0 ...... 0 0    ......  ρ(1) =   (2.34) ......   0 0 ...... 0 −wn 0 0 ...... wn 0 according to (2.30). 2.5. Example of a two-sphere. Let (X, ω) be the two-sphere S2 with coordinates (θ, α) and symplectic structure ω = sin θdθ ∧ dα (2.35) Let the Hamiltonian function be H = − cos θ (2.36) so that ω = dH ∧ dα (2.37)

and the Hamiltonian vector ﬁeld be vH = ∂α. The diﬀerential form

ωT = ω + H = sin θdθ ∧ dα − cos θ EQUIVARIANT COHOMOLOGY AND LOCALIZATIONWINTER SCHOOL IN LES DIABLERETSJANUARY 201719

is dT -closed for dT = d + iα (2.38) Let α = etωT (2.39)

Locally there is a degree 1 form V such that ωT = dT V , for example V = −(cos θ)dα (2.40)

but globally V does not exist. The dT -cohomology class [α] of the form α is non-zero. The localization formula (2.27) gives Z 2π 2π exp(ωT ) = exp(−) + exp() (2.41) X − where the first term is the contribution of the T -fixed point θ = 0 and the second term is the contribution of the T -fixed point θ = π.