Review of localization in geometry

Review of localization in geometry: equivariant cohomology, characteristic classes, Atiyah-Bott formula, Atiyah-Singer equivariant index formula, Mathai-Quillen formalism

Foundations of equivariant de Rham theory have been laid in two papers by Henri Cartan [2] [3]. The book by Guillemin and Sternberg [4] covers Cartan's papers and treats equivariant de Rham theory from the perspective of supersymmetry. See also the book by Berline-Vergne [5], the lectures by Szabo [6] and by Cordes-Moore-Ramgoolam [7], and Vergne's review [8].

Equivariant cohomology
Let G be a compact connected Lie group. Let X be a G-manifold, which means that there is a defined action G × X → X of the group G on the manifold X.
If G acts freely on X (all stabilizers are trivial) then the space X/G is an ordinary manifold on which the usual cohomology theory H • (X/G) is defined. If the G action on X is free, the G-equivariant cohomology groups H • G (X) are defined to be the ordinary cohomology H • (X/G).
If the G action on X is not free, the naive definition of the equivariant cohomology H • G (X) fails because X/G is not an ordinary manifold. If non-trivial stabilizers exist, the corresponding points on X/G are not ordinary points but fractional or stacky points.
A proper topological definion of the G-equivariant cohomology H G (X) sets where the space EG, called universal bundle [9,10] is a topological space associated to G with the following properties (1) The space EG is contractible (2) The group G acts freely on EG Because of the property (1) the cohomology theory of X is isomorphic to the cohomology theory of X × EG, and because of the property (2) the group G acts freely on X × EG and hence the quotient space (X × G EG) has a well-defined ordinary cohomology theory.

Classifying space and characteristic classes
If X is a point pt, the ordinary cohomology theory H • (pt) is elementary where the quotient space BG = EG/G is called classifying space. The terminology universal bundle EG and classifying space BG comes from the fact that any smooth principal G-bundle on a manifold X can be induced by a pullback f * of the universal principal G-bundle EG → BG using a suitable smooth map f : X → BG.
The cohomology groups of BG are used to construct characteristic classes of principal G-bundles.
Let g = Lie(G) be the real Lie algebra of a compact connected Lie group G. Let R[g] be the space of real valued polynomial functions on g, and let R[g] G be the subspace of Ad G invariant polynomials on g.
For a principal G-bundle over a base manifold X the Chern-Weil morphism sends an adjoint invariant polynomial p on the Lie algebra g to a cohomology class [p(F A )] in H • (X) where F A = ∇ 2 A is the curvature 2-form of any connection ∇ A on the G-bundle. The cohomology class [p(F A )] does not depend on the choice of the connection A and is called the characteristic class of the G-bundle associated to the polynomial p ∈ R[g] G .
The main theorem of Chern-Weil theory is that the ring of characteristic classes R[g] G is isomorphic to the cohomology ring H • (BG) of the classifying space BG: the Chern-Weil morphism (2.3) is an isomorphism For the circle group G = S 1 ≃ U(1) the universal bundle ES 1 and classifying space BS 1 can be modelled as Then the Chern-Weil isomorphism is explicitly where ǫ ∈ g ∨ is a linear function on g = Lie(S 1 ) and C[ǫ] denotes the free polynomial ring on one generator ǫ. The ǫ ∈ H 2 (CP ∞ , C) is negative of the first Chern class c 1 of the universal bundle − c 1 (γ) = ǫ = 1 2π where tr 1 denotes trace of the curvature two-form F A = dA + A ∧ A in the fundamental complex 1-dimensional representation in which the Lie algebra of g = Lie(S 1 ) is represented by ıR. The cohomological degree of ǫ is deg ǫ = deg F A (γ) = 2 (2.8) Generally, for a compact connected Lie group G we reduce the Chern-Weil theory to the maximal torus T ⊂ G and identify where t is the Cartan Lie algebra t = Lie(T ) and W G is the Weyl group of G. For example, if G = U(n) the Weyl group W U (n) is the permutation group of n eigenvalues ǫ 1 , . . . ǫ n . Therefore The classifying space for G = U(n) is Gr n (C k+n ) (2.12) where Gr n (V ) denotes the space of n-planes in the vector space V . To summarize, if G is a connected compact Lie group with Lie algebra g = Lie(G), maximal torus T and its Lie algebra t = Lie(T ), and Weyl group W G , then it holds

Weil algebra
The cohomology H • (BG, R) of the classifying space BG can also be realized in the Weil algebra denotes shift of degree so that elements of g [1] are Grassmann. The space of polynomial functions R[g [1]] on g [1] is the anti-symmetric algebra Λg ∨ of g ∨ , and the space of polynomial functions R[g [2]] on g [2] is the symmetric algebra Sg ∨ of g ∨ .
The elements c ∈ g [1] have degree 1 and represent the connection 1-form on the universal bundle. The elements φ ∈ g [2] have degree 2 and represent the curvature 2-form on the universal bundle. An odd differential on functions on g [1] ⊕ g [2] can be described as an odd vector field δ such that δ 2 = 0. The odd vector field δ of degree 1 represents de Rham differential on the universal bundle which follows from the standard relations between the connection A and the curvature F A This definition implies δ 2 = 0. Indeed, Given a basis T α on the Lie algebra g with structure constants [T β , T γ ] = f α βγ T α the differential δ has the form The differential δ can be decomposed into the sum of two differentials The differential δ BRST is the BRST differential (Chevalley-Eilenberg differential for Lie algebra cohomology with coefficients in the Lie algebra module Sg ∨ ). The differential δ K is the Koszul differential (de Rham differential on Ω • (Πg)).
The field theory interpretation of the Weil algebra and the differential (3.6) was given in [11] and [12].
The Weil algebra W g = R[g [1] ⊕ g [2]] is an extension of the Chevalley-Eilenberg algebra CE g = R[g [1]] = Λg ∨ by the algebra R[g [2]] G = Sg ∨ of symmetric polynomials on g which is quasi-isomorphic to the algebra of differential forms on the universal bundle The duality between the Weil algebra W g and the de Rham algebra Ω • (EG) of differential forms on EG is provided by the Weil homomorphism after a choice of a connection 1-form A ∈ Ω 1 (EG) ⊗ g and its field strength F A ∈ Ω 2 (EG) ⊗ g on the universal bundle EG → BG. Indeed, the connection 1-form A ∈ Ω 1 (EG) ⊗ g and field strength F ∈ Ω 2 (EG) ⊗ g define maps g ∨ → Ω 1 (EG) and g ∨ → Ω 2 (EG) The cohomology of the Weil algebra is trivial corresponding to the trivial cohomology of Ω • (EG).
To define G-equivariant cohomology we need to consider G action on EG. To compute For any principal G-bundle π : P → P/G the differential forms on P in the image of the pullback π * of the space of differential forms on P/G are called basic Ω • (P ) basic = π * Ω • (P/G) (3.13) Let L α be the Lie derivative in the direction of a vector field α generated by a basis element T α ∈ g, and i α be the contraction with the vector field generated by T α .
An element ω ∈ Ω • (P ) basic can be characterized by two conditions (1) ω is invariant on P with respect to the G-action: L α ω = 0 (2) ω is horizontal on P with respect to the G-action: i α ω = 0 In the Weil model the contraction operation i α is realized as (3.14) and the Lie derivative L α is defined by the usual relation From the definition of Ω • (P ) basic for the case of P = EG we obtain

Weil model and Cartan model of equivariant cohomology
The isomorphism suggests to replace the topological model for G-equivariant cohomologies of real manifold X 3) or by the equivalent algebraic Weil model under the G-action induced from G-action on X and adjoint G-action on g.
It is convenient to think about (Ω • (X) ⊗ Sg ∨ ) as the space of smooth differential forms on X × g of degree 0 along g and polynomial along g.
In (T a ) basis on g, an element φ ∈ g is represented as φ = φ α T α . Then (φ α ) is the dual basis of g ∨ . Equivalently φ α is a linear coordinate on g.
The commutative ring R[g] of polynomial functions on the vector space underlying g is naturally represented in the coordinates as the ring of polynomials in generators {φ α } Hence, the space (4.5) can be equivalently presented as Given an action of the group G on any manifold M ρ g : m → g · m (4.8) the induced action on the space of differential forms Ω • (M) comes from the pullback by the map ρ g −1 ρ g : ω → ρ * g −1 ω, ω ∈ Ω • (M) (4.9) In particular, if M = g and ω ∈ g ∨ is a linear function on g, then (4.9) is the co-adjoint action on g ∨ .
The invariant subspace (Ω • (X) ⊗ R[g]) G forms a complex with respect to the Cartan differential is the de Rham differential, and i α : Ω • (X) → Ω •−1 (X) is the operation of contraction of the vector field on X generated by T α ∈ g with differential forms in Ω • (X).
The Cartan model of the G-equivariant cohomology H G (X) is To check that d 2 along vector field generated by T α . The infinitesimal action by a Lie algebra generator T a on an element ω ∈ Ω • (X) ⊗ R[g] is where L α ⊗ 1 is the geometrical Lie derivative by the vector field generated by T α on Ω • (X) and 1 ⊗ L a is the coadjoint action on R[g] by the antisymmetry of the structure constants f γ αβ = −f γ βα . Therefore d 2 The grading on Ω • (X) ⊗ R[g] is defined by the assignment Then In particular, if X = pt is a point then in agreement with (3.16). If x µ are coordinates on X, and ψ µ = dx µ are Grassman coordinates on the fibers of ΠT X, we can represent the Cartan differential (4.10) in the notations more common in quantum field theory traditions where v µ are components of the vector field on X generated by a basis element T α for the G-action on X. In quantum field theory, the coordinates x µ are typically coordinates on the infinite-dimensional space of bosonic fields, and ψ µ are typically coordinates on the infinite-dimensional space of fermionic fields.

4.2.
Weil model. The differential in Weil model can be presented in coordinate notations similar to (4.24) as follows In physical applications, typicallly c is the BRST ghost field for gauge symmetry, and Weil differential is the sum of a supersymmetry transformation and BRST transformation, for example see [13].

Equivariant characteristic classes in Cartan model
For a reference see [14] and [15]. Let G and T be compact connected Lie groups. We consider a T -equivariant G-principal bundle π : P → X. This means that an equivariant T -action is defined on P compatible with the G-bundle structure of π : P → X. One can take that G acts from the right and T acts from the left.
The compatibility means that T -action on the total space of P • commutes with the projection map π : P → X • commutes with the G action on the fibers of π : P → X Let D A = d + A be a T -invariant connection on a T -equivariant G-bundle P . Here the connection A is a g-valued 1-form on the total space of P (such a connection always exists by the averaging procedure for compact T ).
Then we define the T -equivariant connection and the T -equivariant curvature where ǫ a are coordinates on the Lie algebra t (like the coordinates φ a on the Lie algebra g in the previous section defining Cartan model of G-equivariant cohomology), which is in fact is an element of Ω 2 Let X T be the T -fixed point set in X. If the equivariant curvature F A,T is evaluated on X T , only the vertical component of i va contributes to the formula (5.3) and v a pairs with the vertical component of the connection A on the T -fiber of P given by g −1 dg. The T -action on G-fibers induces the homomorphism and let ρ(T a ) be the images of T a basis elements of t.

An ordinary characteristic class for a principal
Here F A is the curvature of any connection A on the G-bundle.
In the same way, a T -equivariant characteristic class for a principal G-bundle associated T (X). Here F A,T is the T -equivariant curvature of any T -equivariant connection A on the G-bundle.
Restricted to T -fixed points X T the T -equivariant characteristic class associated to poly- In particular, if V is a representation of G and p is the Chern character of the vector bundle V , then if X is a point, the equivariant Chern characters is an ordinary character of the space V as a G-module.

Standard characteristic classes
For a reference see the book by Bott and Tu [16].
6.1. Euler class. Let G = SO(2n) be the special orthogonal group which preserves a Rie- The Euler characterstic class is defined by the adjoint invariant polynomial of degree n on the Lie algebra so(2n) called Pfaffian and defined as follows. For an element For example, for the 2 × 2-blocks diagonal matrix For an antisymmetric (2n) × (2n) matrix x ′ , the definition implies that Pf(x) is a degree n polynomial of matrix elements of x which satisfies Let P be an SO(2n) principal bundle P → X.
In the standard normalization the Euler class e(P ) is defined in such a way that it takes values in H 2n (X, Z) and is given by For example, the Euler characteristic of an oriented real manifold X of real dimension 2n is an integer number given by where R denotes the curvature form of the tangent bundle T X .
In quantum field theories the definition (6.2) of the Pfaffian is usually realized in terms of a Gaussian integral over the Grassmann (anticommuting) variables θ which satisfy θ i θ j = −θ j θ i . The definition (6.2) is presented as By definition, the integral [dθ 2n . . . dθ 1 ] picks the coefficient of the monomial θ 1 . . . θ 2n of an element of the the Grassman algebra generated by θ.
6.2. Euler class of vector bundle and Mathai-Quillen form. See Mathai-Quillen [17] and Aityah-Jeffrey [18]. The Euler class of a vector bundle can be presented in a QFT formalism. Let E be an oriented real vector bundle E of rank 2n over a manifold X.
Let x µ be local coordinates on the base X, 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 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 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 jχ j the h i transforms in the same way, that is h i = s i jh j . The odd vector field δ is nilpotent δ 2 = 0 (6.9) and is called de Rham vector field on ΠT (ΠE).
Consider an element α of Ω • (ΠE) defined by the equation where t ∈ R >0 and Notice that since h i has been defined as Dχ i the definition (6.10) is coordinate independent.
To expand the definition of α (6.10) we compute where we suppresed the indices i, j, the d denotes the de Rham differential on X and F A the curvature 2-form on the connection A The Gaussian integration of the form α along the vertical fibers of ΠE gives which agrees with definition of the integer valued Euler class (6.5). The representation of the Euler class in the form (6.10) is called the Gaussian Mathai-Quillen representation of the Thom class. The Euler class of the vector bundle E is an element of H 2n (X, Z). If dim X = 2n, the number obtained after integration of the fundamental cycle on X is an integer Euler characterstic of the vector bundle E.
If E = T X the equation (6.15) provides the Euler characteristic of the manifold X in the form Given a section s of the vector bundle E, we can deform the form α in the same δcohomology class by taking After integrating over (h, χ) the the resulting differential form on X has factor so it is concentraited in a neigborhood of the locus s −1 (0) ⊂ X of zeroes of the section s.
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 where ds| x : T x → E x is the differential of the section s at a zero x ∈ s −1 (0). The assumption that s is a generic section implies that det ds| x is non-zero. For a short reference on the Mathai-Quillen formalism see [19].
6.3. Chern character. Let P be a principal GL(n, C) bundle over a manifold X. The Chern character is an adjoint invariant function ch : gl(n, C) → C (6.20) defined as the trace in the fundamental representation of the exponential map The exponential map is defined by formal series The eigenvalues of the gl(n, C) matrix x are called Chern roots. In terms of the Chern roots the Chern character is 6.4. Chern class. Let P be a principal GL(n, C) bundle over a manifold X. The Chern class c k for k ∈ Z >0 of x ∈ gl(n, C) is defined by expansion of the determinant t n c n (6.24) In particular c 1 (x) = tr x, c n (x) = det x (6.25) In terms of Chern roots c k is defined by elementary symmetric monomials Remark on integrality. Our conventions for characteristic classes of GL(n, C) bundles differ from the frequently used conventions in which Chern classes c k take value in H 2k (X, Z) by a factor of (−2π √ −1) k . In our conventions the characteristic class of degree 2k needs to be multiplied by where det is evaluated in the fundamental representation. The ratio evaluates to a series expansion involving Bernoulli numbers B k 6.6. TheÂ class. Let P be a principal GL(n, C) bundle over a manifold X. TheÂ class of x ∈ GL(n, C) is defined aŝ TheÂ class is related to the Todd class bŷ The localization theorem in K-theory gives the index formula of Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer relating the index to the Todd class Similarly, the index of Dirac operator / D : S + ⊗ E → S − ⊗ E from the positive chiral spinors S + to the negative chiral spinors S − , twisted by a vector bundle E, is defined as and is given by the Atiyah-Singer index formula Notice that on a Kahler manifold the Dirac complex consistently with the relation (6.30) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula Remark on 2π and √ −1 factors. The vector bundle E in the index formula (7.2) can be promoted to a complex We find where x i are Chern roots of the curvature of the n-dimensional complex bundle T 1,0 X . Hence, the Todd index formula (7.2) gives The above agrees with the Euler characteristic (6.6) provided it holds that det( √ −1x u(n) ) = Pf(x so(2n) ) (7.10) where x so(2n) represents the curvature of the 2n-dimensional real tangent bundle T X as 2n×2n antisymmetric matrices, and x u(n) represents the curvature of the complex holomorphic ndimensional tangent bundle T (1,0) X as n × n anti-hermitian matrices. That (7.10) holds is clear from the (2 × 2 representation of √ −1 √ −1 → 0 −1 1 0 (7.11)

Equivariant integration
See the paper by Atiyah and Bott [20].

Thom isomorphism and Atiyah-Bott localization. A map
f : F → X of manifolds induces a natural pushfoward map on the homology and pullback on the cohomology In the situation when there is Poincaré duality between homology and cohomology we can construct pushforward operation on the cohomology We can display the pullback and pushforward maps on the diagram For example, if F and X are compact manifolds and f : F ֒→ X is the inclusion, then for the pushforward map f * : where Φ F is the cohomology class in H • (X) which is Poincaré dual to the manifold F ⊂ X: for a form α on X we have If X is the total space of the orthogonal vector bundle π : X → F over the oriented manifold F then Φ F (X) is called the Thom class of the vector bundle X and f * : H • (F ) → H • (X) is the Thom isomorphism: to a form α on F we associate a form Φ ∧ π * α on X. The important property of the Thom class Φ F for a submanifold F ֒→ X is where e(ν F ) is the Euler class of the normal bundle to F in X. Combined with (8.3) the last equation gives More generally, if f : F ֒→ X is an inclusion of a manifold F into a manifold X the Poincaré dual class Φ F is isomorphic to the Thom class of the normal bundle of F in X.
Now we consider T -equivariant cohomologies for a compact abelian Lie group T acting on X. Let F = X T be the set of T fixed points in X. Then the equivariant Euler class e T (ν F ) is invertible, therefore the identity map on H • T (X) can be presented as Let π X : X → pt be the map from a manifold X to a point pt. The pushforward operator π X * : H • T (X) → H • T (pt) corresponds to the integration of the cohomology class over X. The pushforward is functorial. For maps F f → X π X → pt we have the composition π X * f * = π F * for F π F → pt. So we arrive to the Atiyah-Bott integration formula 8.2. Duistermaat-Heckman localization. A particular example where the Atiyah-Bott 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 dim R 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 such that dµ a = −i a ω (8.11) 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 X.
The degree 2 element ω T ∈ Ω • (X) ⊗ St * defined by the equation is a d T -closed equivariant differential form: This implies that the mixed-degree equivariant differential form α = e ω T (8.14) is also d T -closed, and we can apply the Atiyah-Bott localization formula to the integral X exp(ω T ) = 1 n! X ω n exp(ǫ a µ a ) (8.15) For T = SO(2) so that Lie(SO(2)) ≃ R the integral (8.15) is the typical partition function of a classical Hamiltonian mechanical system in statistical physics with Hamiltonian function µ : X → R and inverse temperature parameter −ǫ.
Suppose that T = SO (2) and that the set of fixed points X T is discrete. Then the Atiyah-Bott localization formula (8.9) implies where ν x is the normal bundle to a fixed point x ∈ X T in X 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 5 we evaluate the T -equivariant characteristic Euler class of the principal G-bundle for T = SO(2) and G = SO(2n) by equation (5.5) for the invariant polynomial on g = so(2n) given by p = 1 (2π) n Pf according to definition (6.5).

Gaussian integral example.
To illustrate the localization formula (8.16) suppose that X = R 2n with symplectic form and SO(2) action where θ ∈ R/(2πZ) parametrizes SO (2) and (w 1 , . . . , w n ) ∈ Z n . The point 0 ∈ X is the fixed point so that X T = {0}, and the normal bundle ν x = T 0 X 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 1 , . . . , w n ).
We identify Lie(SO (2)) with R with basis element {1} and coordinate function ǫ ∈ Lie(SO(2)) * . The SO(2) action (8.18) is Hamiltonian with respect to the moment map Assuming that ǫ < 0 and all w i > 0 we find by direct Gaussian integration

8.4.
Example of a two-sphere. Let (X, ω) be the two-sphere S 2 with coordinates (θ, α) and symplectic structure ω = sin θdθ ∧ dα (8.23) Let the Hamiltonian function be H = − cos θ (8.24) so that ω = dH ∧ dα (8.25) and the Hamiltonian vector field be v H = ∂ α . The differential form but globally V does not exist. The d T -cohomology class [α] of the form α is non-zero. The localization formula (8.15) gives 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.

Equivariant index formula (Dolbeault and Dirac)
Let G be a compact connected Lie group.
Suppose that X is a complex variety and E is a holomorphic G-equivariant vector bundle over X. Then the cohomology groups H • (X, E) form representation of G. In this case the index of E (7.1) can be refined to an equivariant index or character where ch G H i (X, E) is the character of a representation of G in the vector space H i (X, E). More concretely, the equivariant index can be thought of as a gadget that attaches to Gequivariant holomorphic bundle E a complex valued adjoint invariant function on the group G ind G (∂, E)(g) = n k=0 (−1) k tr H k (X,E) g (9.2) on elements g ∈ G. The sign alternating sum (9.2) is also known as the supertrace ind G (∂, E)(g) = str H • (X,E) g (9. 3) The index formula (7.2) is replaced by the equivariant index formula in which characteristic classes are promoted to G-equivariant characteristic classes in the Cartan model of G-equivariant cohomology with differential d G = d + φ a i a as in (4.10) Here φ a T a is an element of Lie algebra of G and e φ a Ta is an element of G, and T X denotes the holomorphic tangent bundle of the complex manifold X.
If the set X G of G-fixed points is discrete, then applying the localization formula (8.9) to the equivariant index (9.4) we find the equivariant Lefshetz formula The Euler character is cancelled against the numerator of the Todd character.
9.0.1. Example of CP 1 . Let X be CP 1 and let E = O(n) be a complex line bundle of degree n over CP 1 , and let G = U(1) equivariantly act on E as follows. Let z be a local coordinate on CP 1 , and let an element t ∈ U(1) ⊂ C × send the point with coordinate z to the point with coordinate tz so that where T 1,0 0 X denotes the fiber of the holomorphic tangent bundle at z = 0 and similarly T 1,0 ∞ X the fiber at z = ∞. Let the action of U(1) on the fiber of E at z = 0 be trivial. Then the action of U(1) on the fiber of E at z = ∞ is found from the gluing relation to be of weight −n, so that We can check against the direct computation. Assume n ≥ 0. The kernel of∂ is spanned by n + 1 holomorphic sections of O(n) of the form z k for k = 0, . . . , n, the cokernel is empty by Riemann-Roch. The section z k is acted upon by t ∈ T with weight t −k . Therefore ind T (∂, O(n), CP 1 ) = n k=0 t −k (9.10) Even more explicitly, for illustration, choose a connection 1-form A with constant curvature F A = − 1 2 inω, denoted in the patch around θ = 0 (or z = 0) by A (0) and in the patch around θ = π (or z = ∞) by A (π) The gauge transformation between the two patches is consistent with the defining E bundle transformation rule for the sections s (0) , s (π) in the patches around θ = 0 and θ = π The equivariant curvature F T of the connection A in the bundle E is given by as can be verified against the definition (5.3) F T = F + ǫi v A. Notice that to verify the expression for the equivariant curvature (9.14) in the patch near θ = π one needs to take into account contributions from the vertical component g −1 dg of the connection A on the total space of the principal U(1) bundle and from the T -action on the fiber at θ = π with weight −n.
If n ≥ −m we pick the contour of integration C to enclose all residues z = t j . The residue at z = 0 is zero and the sum of residues is (9.19). On the other hand, the same contour integral is evaluated by the residue at z = ∞ which is computed by expanding all fractions in inverse powers of z, and is given by the complete homogeneous polynomial in t i of degree n.
If n < −m we assume that the contour of integration is a small circle around the z = 0 and does not include any of the residues z = t j . Summing the residues outside of the contour, and taking that z = ∞ does not contribute, we get (9.19) with the (−) sign . The residue at z = 0 contributes by (9.20).
Also notice that the last line of (9.20) relates 1 to the first line by the reflection which is the consequence of the Serre duality on CP m .

Equivariant index and representation theory
The CP 1 in example (9.16) can be thought of as a flag manifold SU(2)/U(1), and (9.9) (9.16) as characters of SU(2)-modules. For index theory on general flag manifolds G C /B C , that is Borel-Weyl-Bott theorem 2 , the shift of the form (9.17) is a shift by the Weyl vector ρ = α>0 α where α are positive roots of g.
The index formula with localization to the fixed points on a flag manifold is equivalent to the Weyl character formula.
Let G be a compact simple Lie group. The Kirillov character formula equates the Tequivariant index of the Dirac operator ind T (D) on the G-coadjoint orbit of the element λ + ρ ∈ g * with the character χ λ of the G irreducible representation with highest weight λ.
The character χ λ is a function g → C determined by the representation of the Lie group G with highest weight λ as χ λ : X → tr λ e X , X ∈ g (10.1) Let X λ be an orbit of the co-adjoint action by G on g * . Such orbit is specified by an element λ ∈ t * /W where t is the Lie algebra of the maximal torus T ⊂ G and W is the Weyl group. The co-adjoint orbit X λ is a homogeneous symplectic G-manifold with the canonical symplectic structure ω defined at point x ∈ X ⊂ g * on tangent vectors in g by the formula The converse is also true: any homogeneous symplectic G-manifold is locally isomorphic to a coadjoint orbit of G or central extension of it. The minimal possible stabilizer of λ is the maximal abelian subgroup T ⊂ G, and the maximal co-adjoint orbit is G/T . Such orbit is called a full flag manifold. The real dimension of the full flag manifold is 2n = dim G − rk G, and is equal to the number of roots of g. If the stabilizer of λ is a larger group H, such that T ⊂ H ⊂ G, the orbit X λ is called a partial flag manifold G/H. A degenerate flag manifold is a projection from the full flag manifold with fibers isomorphic to H/T . Flag manifolds are equipped with natural complex and Kahler structure. There is an explicitly holomorphic realization of the flag manifolds as a complex quotient G C /P C where G C is the complexification of the compact group G and P C ⊂ G C is a parabolic subgroup. Let g = g − ⊕ h ⊕ g + be the standard decomposition of g into the Cartan h algebra and the upper triangular g + and lower triangular g − subspaces.
The minimal parabolic subgroup is known as Borel subgroup B C , its Lie algebra is conjugate to h ⊕ g + . The Lie algebra of generic parabolic subgroup P C ⊃ B C is conjugate to the direct sum of h ⊕ g + and a proper subspace of g − . Full flag manifolds with integral symplectic structure are in bijection with irreducible G-representations π λ of highest weight λ X λ+ρ ↔ π λ (10.3) This is known as the Kirillov correspondence in geometric representation theory. Namely, if λ ∈ g * is a weight, the symplectic structure ω is integral and there exists a line bundle L → X λ with a unitary connection of curvature ω. The line bundle L → X λ is acted upon by the maximal torus T ⊂ G and we can study the T -equivariant geometric objects. The Kirillov-Berline-Getzler-Vergne character formula equates the equivariant index of the Dirac operator / D twisted by the line bundle L → X λ+ρ on the co-adjoint orbit X λ+ρ with the character χ λ of the irreducible representation of G with highest weight λ This formula can be easily proven using the Atiyah-Singer equivariant index formula (10.5) and the Atiyah-Bott formula to localize the integral over X λ+ρ to the set of fixed points X T λ+ρ . The localization to X T λ+ρ yields the Weyl formula for the character. Indeed, the stabilizer of λ + ρ, where λ is a dominant weight, is the Cartan torus T ⊂ G. The co-adjoint orbit X λ+ρ is the full flag manifold. The T -fixed points are in the intersection X λ+ρ ∩ t, and hence, the set of the T -fixed points is the Weyl orbit of λ + ρ X T λ+ρ = Weyl(λ + ρ) (10.6) At each fixed point p ∈ X T λ+ρ the tangent space T X λ+ρ | p is generated by the root system of g. The tangent space is a complex T -module ⊕ α>0 C α with weights α given by the positive roots of g. Consequently, the denominator ofÂ T gives the Weyl denominator, the numerator ofÂ T cancels with the Euler class e T (T X ) in the localization formula, and the restriction of ch T (L) = e ω is e w(λ+ρ) We conclude that the localization of the equivariant index of the Dirac operator on X λ+ρ twisted by the line bundle L to the set of fixed points X T λ+ρ is precisely the Weyl formula for the character.
The Kirillov correspondence between the index of the Dirac operator of L → X λ+ρ and the character is closedly related to the Borel-Weyl-Bott theorem.
Let B C be a Borel subgroup of G C , T C be the maximal torus, λ an integral weight of T C . A weight λ defines a one-dimensional representation of B C by pulling back the representation on T C = B C /U C where U C is the unipotent radical of B C (the unipotent radical U C is generated by g + ). Let L λ → G C /B C be the associated line bundle, and O(L λ ) be the sheaf of regular local sections of L λ . For w ∈ Weyl G define the action of w on a weight λ by w * λ := w(λ + ρ) − ρ.
The Borel-Weyl-Bott theorem is that for any weight λ one has where R λ is the irreducible G-module with highest weight λ, the w is an element of Weyl group such that w * λ is dominant weight, and l(w) is the length of w. We remark that if there exists w ∈ Weyl G such that w * λ is dominant weight, then w is unique. There is no w ∈ Weyl G such that w * λ is dominant if in the basis of the fundamental weights Λ i some of the coordinates of λ + ρ vanish.
10.0.3. Example. For G = SU(2) the G C /B C = CP 1 , an integral weight of T C is an integer n ∈ Z, and the line bundle L n is the O(n) bundle over CP 1 . The Weyl weight is ρ = 1. The weight n ≥ 0 is dominant and the H 0 (CP 1 , O(n)) is the SL(2, C) module of highest weight n (in the basis of fundamental weights of SL (2)).
For weight n = −1 the H i (CP 1 , O(−1)) is empty for all i as there is no Weyl transformation w such that w * n is dominant (equivalently, because ρ + n = 0).
The relation between Borel-Weil-Bott theorem for G C /B C and the Dirac complex on G C /B C is that Dirac operator is precisely the Dolbeault operator shifted by the square root of the canonical bundle and consequently ind(X λ+ρ , / D ⊗ L λ+ρ ) = ind(G C /B C ,∂ ⊗ L λ ) (10.10) The Borel-Bott-Weyl theorem has a generalization for partial flag manifolds. Let P C be a parabolic subgroup of G C with B C ⊂ P C and let π : G C /B C → G C /P C denote the canonical projection. Let E → G C /P C be a vector bundle associated to an irreducible finite dimensional P C module, and let O(E) the the sheaf of local regular sections of E. Then O(E) is isomorphic to the direct image sheaf π * O(L) for a one-dimensional B C -module L and For application of Kirillov theory to Kac-Moody and Virasoro algebra see [24].

Equivariant index for differential operators
See the book by Atiyah [25]. Let E k be vector bundles over a manifold X. Let G be a compact Lie group acting on X and the bundles E k . The action of G on a bundle E induces canonically a linear action on the space of sections Γ(E). For g ∈ G and a section φ ∈ Γ(E) the action is (gφ)(x) = gφ(g −1 x), x ∈ X (11.1) Let D k be linear differential operators compatible with the G action, and let E be the complex (that is D k+1 • D k = 0) Since D k are G-equivariant operators, the G-action on Γ(E k ) induces the G-action on the cohomology H k (E). The equivariant index of the complex E is the virtual character 11.1. Atiyah-Singer equivariant index formula for elliptic complexes. If the set X G of G-fixed points is discrete, the Atiyah-Singer equivariant index formula is For the Dolbeault complex E k = Ω 0,k and D k =∂ : Ω 0,k → Ω 0,k+1 → Ω 0,•∂ → Ω 0,•+1 → (11.6) the index (11.5) agrees with (9.5) because the numerator in (11.5) decomposes as ch G E ch G Λ • T * 0,1 and the denominator as ch G Λ • T * 0,1 ch G Λ • T * 1,0 and the factor ch G Λ • T * 0,1 cancels out. For example, the equivariant index of∂ : where the denominator is the determinant of the operator 1 − t over the two-dimensional normal bundle to 0 ∈ C spanned by the vectors ∂ x and ∂x with eigenvalues t andt. In the numerator, 1 comes from the equivariant Chern character on the fiber of the trivial line bundle at x = 0 and −t comes from the equivariant Chern character on the fiber of the bundle of (0, 1) forms dx.
We can compare the expansion in power series in t k of the index with the direct computation. The terms t k for k ∈ Z ≥0 come from the local T -equivariant holomorphic functions x k which span the kernel of∂ on C x . The cokernel is empty by the Poincaré lemma. Compare with (9.10).
Similarly, for the∂ complex on C r we obtain where [] + means expansion in positive powers of t k .
For application to the localization computation on spheres of even dimension S 2r we can compute the index of a certain transversally elliptic operator D which naturally interpolates between the∂-complex in the neighborhood of one fixed point (north pole) of the r-torus T r action on S 2r and the∂-complex in the neighborhood of another fixed point (south pole). The index is a sum of two fixed point contributions where [] + and [] − denotes the expansions in positive and negative powers of t k .
11.2. Atiyah-Singer index formula for a free action G-manifold. Suppose that a compact Lie group G acts freely on a manifold X and let Y = X/G be the quotient, and let π : X → Y (11.10) be the associated G-principal bundle. Suppose that D is a G×T equivariant operator (differential) for a complex (E, D) of vector bundles E k over X as in (11.2). The G × T -equivariance means that the complex E and the operator D are pullbacks by π * of a T -equivariant complexẼ and operatorD on the base Y E = π * Ẽ , D = π * D (11.11) We want to compute the G × T -equivariant index ind G×T (D; X) for the complex (E, D) on the total space X for a G × T transversally elliptic operator D using T -equivariant index theory on the base Y . We can do that using Fourier theory on G (counting KK modes in G-fibers). Let R G be the set of all irreducible representations of G. For each irreducible representation α ∈ R G we denote by χ α the character of this representation, and by W α the vector bundle over Y associated to the principal G-bundle (11.10). Then, for each irrep α ∈ R G we consider a complexẼ ⊗ W α on Y obtained by tensoringẼ with the vector bundle W α over Y . The Atiyah-Singer formula is (11.12) 11.2.1. Example of S 2r−1 . We consider an example immediately relevant for localization on odd-dimensional spheres S 2r−1 which are subject to the equivariant action of the maximal torus T r of the isometry group SO(2r). The sphere π : S 2r−1 → CP r−1 is the total space of the S 1 Hopf fibration over the complex projective space CP r−1 . We will apply the equation (11.12) for a transversally elliptic operator D induced from the Dolbeault operatorD =∂ on CP r−1 by the pullback π * .
To compute the index of operator D = π * ∂ on π : S 2r−1 → CP r−1 we apply (11.12) and use (9.20) and obtain 11.3. General Atiyah-Singer index formula. The Atiyah-Singer index formula for the Dolbeault and Dirac complexes and the equivariant index formula (11.5) can be generalized to a generic situation of an equivariant index of transversally elliptic complex (11.2).
Let X be a real manifold. Let π : T * X → X be the cotangent bundle. Let {E • } be an indexed set of vector bundles on X and π * E • be the vector bundles over T * X defined by the pullback.
The symbol σ(D) of a differential operator D : Γ(E) → Γ(F ) (11.2) is a linear operator σ(D) : π * E → π * F which is defined by taking the highest degree part of the differential operator and replacing all derivatives ∂ ∂x µ by the conjugate coordinates p µ in the fibers of T * X.
For example, for the Laplacian ∆ : Ω 0 (X, R) → Ω 0 (X, R) with highest degree part in some coordinate system {x µ } given by ∆ = g µν ∂ µ ∂ ν where g µν is the inverse Riemannian metric, the symbol of ∆ is a Hom(R, R)-valued (i.e. number valued) function on T * X given by σ(∆) = g µν p µ p ν (11.14) where p µ are conjugate coordinates (momenta) on the fibers of T * X. A differential operator D : Γ(E) → Γ(F ) is elliptic if its symbol σ(D) : π * E → π * F is an isomorphism of vector bundles π * E and π * F on T * X outside of the zero section X ⊂ T * X.
The index of a differential operator D depends only on the topological class of its symbol in the topological K-theory of vector bundles on T * X. The Atiyah-Singer formula for the index of the complex (11.2) is Here T * X denotes the total space of the cotangent bundle of X with canonical orientation such that dx 1 ∧ dp 1 ∧ dx 2 ∧ dp 2 . . . is a positive element of Λ top (T * X).
Let n = dim R X. Let π * T X denote the vector bundle of dimension n over the total T * X obtained as pullback of T X → X to T * X. TheÂ G -character of π * T X iŝ A G (π * T X ) = det π * T X R G e R G /2 − e −R G /2 (11.16) where R G denotes the G-equivariant curvature of the bundle π * T X . Notice that the argument ofÂ is n × n matrix where n = dim R T X (real dimension of X) while if general index formula is specialized to Dirac operator on Kahler manifold X as in (7.4) the argument of theÂcharacter is an n × n matrix where n = dim C T 1,0 X (complex dimension of X). Even though the integration domain T * X is non-compact the integral (11.16) is welldefined because of the (G-transversal) ellipticity of the complex π * E.
For illustration take the complex to be E 0 D → E 1 . Since σ(D) : π * E 0 → π * E 1 is an isomorphism outside of the zero section we can pick a smooth connection on π * E 0 and π * E 1 such that its curvature on E 0 is equal to the curvature on E 1 away from a compact tubular neighborhood U ǫ X of X ⊂ T * X. Then ch G (π * E • ) is explicitly vanishing away from U ǫ X and the integration over T * X reduces to integration over the compact domain U ǫ X.
It is clear that under localization to the fixed points of the G-action on X the general formula (11.16) reduces to the fixed point formula (11.5). This is due to the fact that the numerator in theÂ-character det π * T X R G = Pf T T * X (R G ) is the Euler class of the tangent bundle T T * X to T * X which cancels with the denominator in (8.9), while the restriction of the denominator of (11.16) to fixed points is equal to (11.16) or (11.5), because det e R G = 1, since R G is a curvature of the tangent bundle T X with orthogonal structure group.

Equivariant cohomological field theories
Certain field theories first have been interpreted as cohomological and topological field theories by Witten, see [27], [28].
Often the path integral for supersymmetric field theories can be represented in the form where X is the superspace (usually of infinite dimension) of all fields of the theory. Moreover, the integrand measure α is closed with respect to an odd operator δ which is typically constructed as a sum of a supersymmetry algebra generator and a BRST charge The integrand is typically a product of an exponentiated action functional S, perhaps with insertion of a non-exponentiated observable O If X is a supermanifold, such as a total space ΠE of a vector bundle E (over a base Y ) with parity inversed fibers, the equivariant Euler characteristic class (Pfaffian) in the Atiyah-Bott formula (8.9) is replaced by the graded (super) version of the Pfaffian. The weights associated to fermionic components contribute inversely compared to the weights associated to bosonic components.
Typically, in quantum field theories the base Y of the bundle E → Y is the space of fields. Certain differential equations (like BPS equations) are represented by a section s : Y → E. The zero set of the section s −1 (0) ⊂ Y are the field configurations which solve the equations. For example, in topological self-dual Yang-Mills theory (Donaldson-Witten theory) the space Y is the infinite-dimensional affine space of all connections on a principal G-bundle on a smooth four-manifold M 4 . In a given framing, connections are represented by adjoint-valued 1-forms on M 4 , so Y ≃ Ω 1 (M 4 ) ⊗ ad g. A fiber of the vector bundle E at a given connection A on the G-bundle on M 4 is the space of adjoint-valued two-forms Ω 2+ (M 4 ) ⊗ ad g. The section s : Ω 1 (M 4 ) ⊗ ad g → Ω 2 is represented by the self-dual part of the curvature form The zeroes of the section s = 0 are connections A that are solutions of the equation F + A = 0. The integrand α is the Mathai-Quillen representative of the Thom class for the bundle E → Y like in (6.10) and (6.17). The integral over the space of all fields X = ΠE localizes to the integral over the zeroes s −1 (0) of the section , which in the Donaldson-Witten example is the moduli space of self-dual connections, called instanton moduli space.
The functional integral version of the localization formula of Atiyah-Bott has the same formal form X α = F f * α e(ν F ) (12.6) except that in the quantum field theory version the space X is an infinite-dimensional superspace of fields. The F denotes the localization locus in the space of fields. Let Φ F ⊂ H • (X) be the Poincaré dual class to F , or Thom class of the inclusion f : F ֒→ X which provides the isomorphism f * : H • (F ) → H • (X) (12.7) f * : 1 → Φ F (12.8) Let ν F be the normal bundle to F in X. In quantum feld theory language the space F is called the moduli space or localization locus, and ν F is the space of linearized fluctuations of fields transversal to the localization locus. The cohomology class of f * Φ F in H • (F ) is equal to the Euler class of the normal bundle ν F [f * Φ F ] = e(ν F ) (12.9) The localization (8.9) from X to F exists whenever the locus F is such that there exists an inverse to the Euler class e(ν F ) of its normal bundle in X. Two examples of such F have been considered above: (i) if X = ΠE is the total space of a vector bundle E → Y with parity inversed fibers, then F ⊂ Y ⊂ X can be taken to be the set of zeroes F = s − (0) of a generic section s : Y → E (ii) If X is a G-manifold for a compact group G, then F can be taken to be F = X G , the set of G-fixed points on X The formula (12.6) is more general than these examples. In practice, in quantum field theory problems, the localization locus F is found by deforming the form α to α t = α exp(−tδV ) (12.10) Here t ∈ R is a deformation parameter, and V is a fermionic functional on the space of fields, such that δV has a trivial cohomology class (the cohomology class δV is automatically trivial on effectively compact spaces, but on a non-compact space of fields, which usually appears in quantum field theory path integrals, one has to take extra care of the contributions from the boundary at infinity to ensure that δV has trivial cohomology class). If the even part of the functional δV is positive definite, then by sending the paratemeter t → ∞ we can see that the integral X α exp(−tδV ) (12.11) localizes to the locus F ⊂ X where δV vanishes. Such locus F has an invertible Euler class of its normal bundle in X and the localization formula (12.6) holds. In some quantum field theory problems, a compact Lie group G acts on X and δ is isomorphic to an equivariant de Rham differential in the Cartan model of G-equivariant cohomology of X, so that an element a of the Lie algeba of G appears as a parameter of the partition function Z.
Then the partition function Z(a) can be interpreted as an element of H • G (pt), and the Atiyah-Bott localization formula can be applied to compute Z(a).
There are are two types of equivariant partition functions.
In the partition functions of the first type Z(a), the variable a is a parameter of the quantum field theory such as a coupling constant, a background field, a choice of vacuum, an asymptotics of fields or a boundary condition. Such a partition function is typical for a quantum field theory on a non-compact space, such as the Nekrasov partition function of equivariant gauge theory on R 4 ǫ 1 ,ǫ 2 [29].
In the partition function of the second type, the variable a is actually a dynamical field of the quantum field theory, so that the complete partition function is defined by integration of the partial partition functionZ(a) ∈ H • G (pt) Z = a∈g µ(a)Z(a) (12.12) where µ(a) is a certain adjoint invariant volume form on the Lie algebra g. The partition function Z of second type is typical for quantum field theories on compact space-times reviewed in [1], such as the partition function of a supersymmetric gauge theory on S 4 [13] reviewed in Contribution [30], or on spheres of other dimensions, see summary of results in Contribution [26].