4 VASILY PESTUN 2. Lecture: Localization 2.. 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. Let x µ be local coordinates on the base, and let their differentials be denoted ψ µ = dx µ. Let h i 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 ΠE are odd variables χ i. Let g ij be the matrix of a Riemannian metric on the bundle E. Let A i µ be the matrix valued -form on 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 (x µ, ψ µ ) and (χ i, h i ) the definition of δ is δx µ = ψ µ δψ µ = 0 δχ i = h i A i jµψ µ χ j δh i = δ(a i jµψ µ χ j ) (2.) Here h i = Dχ i is the covariant de Rham differential of χ i, so that under the change of framing on E given by χ i = s i j χ j the h i transforms in the same way, that is h i = s i j h j. The odd vector field δ is nilpotent δ 2 = 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 where t R >0 and Φ = exp( tδv ) (2.3) (2π) 2n V = 2 (g ijχ i h j ) (2.4) Notice that since h i has been defined as Dχ i 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, F A χ) (2.5) where we suppresed the indices i, j, the d denotes the de Rham differential on and F A the curvature 2-form on the connection A F A = da + A A (2.6) The Gaussian integration of the form Φ along the vertical fibers of ΠE gives [dh][dχ] exp( δ(χ, h)) = (2π) 2n 2 (2π) Pf(F A) (2.7) n which agrees with definition of the integer valued Euler class (.8). 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 207 5 The Euler class of the vector bundle E is an element of H 2n (, Z). If dim = 2n, the number obtained after integration of the fundamental cycle on e(e) = Φ (2.8) ΠT (ΠE) is an integer Euler characterstic of the vector bundle E. If E = T the equation (2.8) provides the Euler characteristic of the manifold in the form e() = exp( tδv ) t 0 = (2π) dim ΠT (ΠT ) (2.9) (2π) dim ΠT (ΠT ) Given a section s of the vector bundle E, we can deform the form Φ in the same δ- cohomology class by taking V s = 2 (χ, h + s) (2.0) After integrating over (h, χ) the the resulting differential form on has factor exp( 2t s2 ) (2.) so it is concentraited in a neigborhood of the locus s (0) 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, with rank E = dim, the Euler characteristic of E is equal to the number of zeroes of a generic section s of E counted with orientation e(e) = (2π) n Pf(F A ) = (2π) dim ΠT (ΠT ) exp( tδv s ) = x s (0) sign det ds x (2.2) where ds x : T x E x 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, with r d, take a generic section s of E and consider its set of zeroes F s (0). Then F is a subvariety of of dimension d r. Let α Ω d r () be a closed form on, equivalently α is a function on ΠT. Then the integral α, Φ E,s := α exp( tδv (2π) r s ) (2.3) 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 α, Φ E,s = α exp( tδv (2π) r λs ) λ 0 = α e(e) (2.4) while sending λ we find α, Φ E,s = (2π) r ΠT (ΠT ) ΠT (ΠT ) α exp( tδv λs ) λ = F α (2.5)
6 VASILY PESTUN The equality of two expressions for α, Φ E,s can be interpreted as a localization formula α e(e) = F α (2.6) In this way we proved that cohomology class [e(e)] H r () is Poincare dual to the homology class [F ] H d r () where F is the zero set of generic section of bundle E. 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, and that α Ω G () is a closed equivariant differential form on in Cartan model, that is d T α = 0. Then equivariant version of (2.6) holds α e T (E) = F α (2.7) exercise: prove (2.7) in Cartan model for equivariant cohomology replacing Euler class by equivariant Euler class Now let be an oriented real even-dimensional Riemannian manifold, E = T be the tangent bundle, and T be a compact group acting on, and suppose that the set F = T 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 whose set of zeroes coincide with T, i.e. F = S = T. Let α be d T -closed T -equivariant differential form on in Cartan model. Then equivariant Euler class localization formula (2.7) α e T (T ) = x T α x (2.8) Equivariant cohomologies H T () 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 e T (T ), then we arrive to the equation α = α x (2.9) e T (T x ) x T where e T (T x ) := e T (T ) x is equivariant Euler class of the tangent bundle to evaluated at the point x. Since x is a discrete fixed point of T -action on, the fiber T x of the tangent bundle at point x forms a T -module. Since T is compact real abelian Lie group, a real T -module splits into a direct sum of dim 2 R T x irreducible real two-dimensional modules (L i R 2 C ) i=...n on which the weights of the T action are all non-zero. Then by (.63), (.8) and we find that the equivariant Euler class is e T (T x ) = (2π) 2 dim 2 dim i= w i (2.20)
EQUIVARIANT COHOMOLOGY AND LOCALIZATIONWINTER SCHOOL IN LES DIABLERETSJANUARY 207 where w i t are weights. In basis (ɛ α ) α=... dim T of linear coordinate functions on t we can write explicitly e T (T x ) = (2π) 2 dim 2 dim i= w iα ɛ α (2.2) 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 (, ω) be a real symplectic manifold of dim R = 2n with symplectic form ω and let compact connected Lie group T act on in Hamiltonian way, which means that there exists a function, called moment map or Hamiltonian µ : t (2.22) such that dµ a = i a ω (2.23) in some basis (T a ) of t where i a is the contraction operation with the vector field generated by the T a action on. The degree 2 element ω T Ω () St defined by the equation is a d T -closed equivariant differential form: ω T = ω + ɛ a µ a (2.24) d T ω T = (d + ɛ a i a )(ω + ɛ b µ b ) = ɛ a dµ a + ɛ a i a ω = 0 (2.25) This implies that the mixed-degree equivariant differential form α = e ω T (2.26) is also d T -closed, and we can apply the Atiyah-Bott-Berline-Vergne localization formula to the integral exp(ω T ) = ω n exp(ɛ a µ a ) (2.27) n! 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 : R and inverse temperature parameter ɛ. Suppose that T = SO(2) and that the set of fixed points T is discrete. Then the Atiyah-Bott-Berline-Vergne localization formula (2.9) implies n! ω n exp(ɛ a µ a ) = exp(ɛa µ a ) e T (ν x ) x T (2.28) where ν x is the normal bundle to a fixed point x T in and e T (ν x ) is the T -equivariant Euler class of the bundle ν x. The rank of the normal bundle ν x is 2n and the structure group is SO(2n). In notations of section.9 we evaluate the T -equivariant characteristic Euler class of the principal G- bundle for T = SO(2) and G = SO(2n) by equation (.62) for the invariant polynomial on g = so(2n) given by p = (2π) n Pf according to definition (.8).
8 VASILY PESTUN 2.4. Gaussian integral example. To illustrate the localization formula (2.28) suppose that = R 2n with symplectic form n ω = dx i dy i (2.29) and SO(2) action ( xi i= ) ( ) ( ) cos wi θ sin w i θ xi y i sin w i θ cos w i θ y i (2.30) where θ R/(2πZ) parametrizes SO(2) and (w,..., w n ) Z n. The point 0 is the fixed point so that T = {0}, and the normal bundle ν x = T 0 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 (w,..., w n ). We identify Lie(SO(2)) with R with basis element {} and coordinate function ɛ Lie(SO(2)). The SO(2) action (2.30) is Hamiltonian with respect to the moment map µ = µ 0 + n w i (x 2 i + yi 2 ) (2.3) 2 i= Assuming that ɛ < 0 and all w i > 0 we find by direct Gaussian integration ω n (2π) n exp(ɛµ) = n! ( ɛ) n n i= w exp(ɛµ 0 ) (2.32) i and the same result by the localization formula (2.28) because e T (ν x ) = Pf(ɛρ()) (2.33) (2π) n according to the definition of the T -equivariant class (.62) and the Euler characteristic class (.8), and where ρ : Lie(SO(2)) Lie(SO(2n)) is the homomorphism in (.6) with 0 w...... 0 0 w 0...... 0 0.................. ρ() = (2.34).................. 0 0...... 0 w n 0 0...... w n 0 according to (2.30). 2.5. Example of a two-sphere. Let (, ω) be the two-sphere S 2 with coordinates (θ, α) and symplectic structure ω = sin θdθ dα (2.35) Let the Hamiltonian function be so that H = cos θ (2.36) ω = dh dα (2.37) and the Hamiltonian vector field be v H = α. The differential form ω T = ω + ɛh = sin θdθ dα ɛ cos θ
EQUIVARIANT COHOMOLOGY AND LOCALIZATIONWINTER SCHOOL IN LES DIABLERETSJANUARY 207 9 is d T -closed for d T = d + ɛi α (2.38) Let α = e tω T (2.39) Locally there is a degree form V such that ω T = d T V, for example V = (cos θ)dα (2.40) but globally V does not exist. The d T -cohomology class [α] of the form α is non-zero. The localization formula (2.27) gives exp(ω T ) = 2π ɛ exp( ɛ) + 2π exp(ɛ) (2.4) ɛ 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.