LECTURES ON EQUIVARIANT LOCALIZATION VASILY PESTUN Abstract. These are informal notes of the lectures on equivariant localization given at the program Geometry of Strings and Fields at The Galileo Galilei Institute in September, 203. All remarks and corrections are welcomed from the participants. Contents 0.. References. Cartan model of equivariant cohomology 2.. Cartan model 2.2. Equivariant characteristic classes in Cartan model 4.3. Frequently used characteristic classes 5.4. Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer index formula 6 2. Equivariant integration 7 2.. Thom isomorphism, the Euler class and the Atiyah-Bott- Berline-Vergne integration formula 7 3. Equivariant index theory 3.. Kirillov character formula 3.2. Index of a complex 4 3.3. Atiyah-Singer index formula 4 4. Equivariant cohomological field theories 7 4.. Four-dimensional gauge theories 7 4.2. The Ω-background 8 0.. References. The are multiple sources for these educational lectures. Some of them are to the original papers, some are pedagogical exposition, or a combination of both. We assemble a list of references here and excuse us for omitting references in the course Berline, Getzler, Vergne Heat kernels and Dirac operators (Springer, 99) Szabo Equivariant Localization of Path Integrals hep-th/9608068
2 VASILY PESTUN Cordes, Moore, Ramgoolam Lectures on 2D Yang-Mills Theory, Equivariant Cohomology and Topological Field Theories hep-th/9420 Atiyah Elliptic Operators and Compact Groups (Lecture Notes in Math, 974) Bott, Tu Equivariant characteristic classes in the Cartan model, math/00200 Vergne Applications of equivariant cohomology math/0607389 Atiyah-Bott The moment map and equivariant cohqmology, 982 Berline-Vergne The equivariant Chern character and index of G-invariant operators. (Lectures at CIME, Venise 992) Witten Introduction to cohomological field theories Int. Jour. Mod. Phys. 99 Witten Topological Quantum Field Theory, 988 Kirillov, Lectures on the theory of group representations, 97. Konstant, Quantization and Unitary representations, 970. Kirillov, Merits and Demerits of the orbit method, 997 Mathai, Quillen Superconnections, Thom classes and equivariant differential forms 986 Atiyah, Jeffrey Topological Lagrangians and cohomology, 990 Nekrasov, Okounkov Seiberg-Witten Theory and Random Partitions hep-th/0306238 Pestun Localization of gauge theory on a four-sphere and supersymmetric Wilson loops 072.2824 Kamnitzer Lectures on geometric constructions of the irreducible representations of GL n 092.0569. Cartan model of equivariant cohomology Let be a manifold and T be a Lie group acting on. Often we are interested in the geometry of the quotient space /T. The equivariant cohomology H T () is a proper definition of H (/T ) when the quotient /T is not smooth... Cartan model. The Cartan model of the equivariant cohomology H T () uses the algebra of functions on t valued in differential forms on, that is Ω () C(t). The action of group element t T on a
LECTURES ON EQUIVARIANT LOCALIZATION 3 form α Ω () C(t) is defined by (t α)() = t (α(t t)) t (.2) Let T a be the basis in t and ɛ a be respective coordinate functions on g, so that an element of t be written as ɛ a T a, and fab c be the structure constants [T a, T b ] = fab c T c. Infinitesimal action by a Lie algebra generator T a on an element α Ω () C(g) is then given by T a α = L a L a (.3) where L a is the geometrical Lie derivative by the vector field associated to the generator T a on Ω () and L a is the coadjoint action on C(g): for α C(g) we have L a α = f c abɛ b α ɛ c (.4) The T -invariant subspace in Ω () C(g) is called the algebra of equivariant differential forms which means that Ω T () = (Ω () T (t )) T (.5) α Ω T () (L a + L a )α = 0 (.6) {eq:lie-geom-alg} Define the equivariant Cartan differential d T by the formula d T = d ɛ a i va (.7) where i va is the contraction with the vector field v a associated to the action by the T a generator on. The square of d T is the geometric Lie derivative d 2 T = ɛ a L va (.8) On equivariant forms α Ω T () the action by geometric Lie derivative L a is equivalent to the coadjoint action by L a by (.6). Therefore, on α Ω T () the d 2 T acts as d 2 T α = ɛ a ( L a )α = ɛ a fabɛ c b α ɛ = 0 (.9) c by the antisymmetry of the structure constants fab c = f ba c. We conclude that d 2 T = 0 when acted on equivariant differential form Ω T (). We can consistently define grading on Ω T () by assignment deg d = deg i va = deg ɛ a = 2 (.0) {eq:grading} And we remind that the T -action on differential forms α Ω (M) is defined by the pullback t α = ρ t α (.) where ρ t : M M is the map defining the T -action on M
4 VASILY PESTUN and in then deg d T = so that d T : Ω p T () Ωp+ T () where Ω p () defines the subspace of the degree p in Ω T () according to the grading (.0). Since d 2 T = 0 on graded space Ω T (), and d T increases the grading by, we have the differential complex dt Ω p T () d T Ω p+ T () d T... and we defind the cohomology of the (Ω T (), d T ) complex in the standard fashion H T () Ker d T /Im d T (.) If = pt is a point then d T = 0, therefore the T -cohomology of a point H T (pt) is the algebra of T -invariant functions on t H T (pt) = (T (t )) T (.2).2. Equivariant characteristic classes in Cartan model. Let E be a T -equivariant G-principal bundle and let D A = d + A be the T -invariant connection and A is the connection g-valued -form on the total space of E (such connection always exists by the averaging procedure for compact T ). Then we define the T -equivariant connection DA T = D A ɛ a i va (.3) and the T -equivariant curvature which is in fact is an element of Ω 2 T () F T A = (D t A) 2 + ɛ a L va (.4) F T A = F A + ɛ a L va [ɛ a i va, D A ] = F A ɛ a i va A (.5) {eq:equivariant-curva The connection A is the g-valued -form on the total space of the principal bundle E. Let T be the T -fixed point set in. If the equivariant curvature F T is evaluated on T, only vertical component of i va contributes to the formula (.5) and v a pairs with the vertical component of the connection A on the T -fiber of E given by g dg. The T -action on G-fiber induces the homomorphism ρ : t g (.6) and let ρ(t a ) be the images of T a basis elements. An differential form representing an ordinary characteristic class for a G-bundle with a connection A can be taken to be P (F A ) where P is an Ad-invariant polynomial on g and F A is the curvature. In the same fashion a T -equivariant differential form representing T -equivariant characteristic class is an element in Ω 2 T () defined as P (FA T ) where F A T is the equivariant curvature.
LECTURES ON EQUIVARIANT LOCALIZATION 5 Above the T -fixed points with the T -action on G-fibers specified by ρ : t g the characteristic class is.3. Frequently used characteristic classes. P (F A ɛ a ρ(t a )) (.7).3.. Chern character. For U(n) bundle with curvature two-form F valued in the Lie algebra u(n) define the Chern character by trace in the fundamental representation n ch(f ) = tr e F = e x i (.8) where x i are eigenvalues of F. Remark. Our conventions for characteristic classes differs from the i frequently used conventions where x i are eigenvalues of F by factor 2π of 2πi. In our conventions the characteristic class of degree 2n needs to be multiplied by /( 2πi) n to be integral..3.2. Chern classes. The Chern classes c n are generated by the formula n det( + tf ) = t n c n (.9).3.3. Euler class. For principal SO(2n) bundle with curvature F the Euler class is e = Pf( if ) (.20) so that the integral Euler characteristic χ() is χ() = e(t ( 2πi) n ) (.2) Example. Let = R 2 C z and T = U() acting by z tz where t is the fundamental character of T. The T -equivariant U() curvature is F T (T ) = iɛ (.22) and the T -equivariant Euler class is n=0 i= e T (T ) = iɛ (.23) Example 2. For a two-sphere = S 2 the curvature of the tangent SO(2) bundle in the local orthonormal frame (e, e 2 ) is R 22 = and Pf( ir) = i ω S 2 where ω S 2 is the standard volume form χ(s 2 ) = ( i)ω S 2 = 2 (.24) 2πi
6 VASILY PESTUN Remark 2. If U(n) curvature is presented as SO(2n) curvature by embedding ( ) 0 i (.25) 0 then e(f so(n) ) = Pf ( ) n if so(2n) = det Fu(n) = c n = x i (.26).3.4. The Todd class and A-roof class. The Todd class is defined as n x i td = (.27) e x i and  class is defined as i=  = e 2 xi td = n i= Notice that on Kahler manifold the Dirac complex D : S + S is isomorphic to the Dolbeault complex Ω 0,p () Ω 0,p+ ()... twisted by the square root of the canonical bundle i= x i e x i/2 e x i/2 (.28) D = K 2 (.29) consistently with the relation between the  class, the Todd class and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula.4. Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer index formula. For a holomorphic vector bundle E over a holomorphic manifold of complex dimension n the index ind(, E) is defined as ind(, E) = n k=0 ( )k dim H i (, O(E)) and can be computed by ind(, E) = td(t ( 2πi) n ) ch(e) (.30) Similarly, the index of Dirac operator D : S + S from the positive chirality spinors S + to the negative chirality spinors S, twisted by a vector bundle E, is defined as ind(d, E) = dim ker D dim coker D and is given by the Atiyah-Singer index formula ind(d, E) = Â(T ( 2πi) n ) ch(e) (.3) {eq:equivariant-dirac
LECTURES ON EQUIVARIANT LOCALIZATION 7 2. Equivariant integration 2.. Thom isomorphism, the Euler class and the Atiyah-Bott- Berline-Vergne integration formula. A map f : F of manifolds induces natural pushfoward map on the homology and pullback on the cohomology f : H (F ) H () f : H () H (F ) In the situation when there is Poincare duality between homology and cohomology we can construct pushforward operation on the cohomology f : H (F ) H () (2.) We can display the pullback and pushforward maps on the diagram H (F ) f f H () (2.2) For example, if F and are compact manifolds and i : F is the inclusion, then for the pushforward map f : H (F ) H () we find f = Φ F (2.3) {eq:f} where Φ F is the cohomology class in H () which is Poincare dual to the manifold F : for a form α on we have α = Φ F α (2.4) F If is the total space of the orthogonal vector bundle π : F over the oriented manifold F then Φ F () is called the Thom class of the vector bundle and f : H (F ) H () is the Thom isomorphism: to a form α on F we associate a form Φ π α on. The important property of the Thom class Φ F for submanifold F is f Φ F = e(ν F ) (2.5) where e(ν F ) is the Euler class of the normal bundle to F in. Combined with (2.3) the last equation gives f f = e(ν F ) (2.6) as a map H (F ) H (F ). Now we consider T -equivariant cohomologies for a compact abelian Lie group T acting on. Let F = T be the set of the T fixed points
8 VASILY PESTUN in. Then the equivariant Euler class e T (ν F ) is invertible, therefore the identity map on H T () can be presented as = f e T (ν F ) f (2.7) Let π : pt be the map from a manifold to a point pt. The pushforward operator π : H T () H T (pt) corresponds to the integration of the cohomology class over. The pushforward is functorial. For maps F f π pt we have the composition π f = π F for F πf pt. So we arrive to the Atiyah-Bott integration formula or more explicitly π = π F α = f e T (ν F ) F f α e T (ν F ) (2.8) (2.9) Example 3. Harmonic oscillator, Gaussian integral and Duistermaat- Heckman integration formula Let = (R 2, ω) be the phase space of the one-dimensional harmonic oscillator with Hamiltonian and the symplectic structure H = 2 (p2 + q 2 ) (2.0) ω = dp dq = dh dφ (2.) where φ is the polar angle. The Hamiltonian equations generate the Hamiltonian vector field v H = φ = q p + p q by the definition dh = i vh ω (2.2) The vector field φ is the action by the basis element T of the Lie algebra t of the compact group T = U() action on manifold. The d T -equivariant differential on is d T = d ɛ i vh (2.3) where ɛ a is the coordinate function on the Lie algebra u() R. The equivariant differential form is d T -closed α = exp(t(ω ɛh)) d T (ω ɛh) = 0 (2.4)
LECTURES ON EQUIVARIANT LOCALIZATION 9 The elementary Gaussian integration gives α = t ωe tɛh = 2πi 2πi iɛ = i F α e(ν F ) (2.5) where F is the T -fixed point p = q = 0 and e(ν F ) is the equivariant Euler class of the normal bundle to F. The result is t-independent. The reason is the form α is Q-exact α = (d ɛi v )( 2 pdq qdp) (2.6) 2 A constant function on is Q-closed but not Q-exact, so that localization holds but the result is not t-independent. Example 4. The SU(2)-spin and the co-adjoint orbit SU(2)/U() = S 2 Let (, ω) be the two-sphere S 2 with coordinates (θ, φ) and symplectic structure ω = sin θdθ dφ (2.7) Let Hamiltonian function be so that H = cos θ (2.8) ω = dh dφ (2.9) and the Hamiltonian vector field v H = φ. The differential form ω T = ω ɛh = sin θdθ dφ + ɛ cos θ is d T -closed. Locally there is degree form V such that ω T = d T V, for example V = ( cos θ)dφ (2.20) but globally such A is not defined. The d T -cohomology class [α] of the form α is non-zero. Let α = e tω T (2.2) The integration shows α = ( e t ωe tɛh tɛh θ=0 = 2πi 2πi iɛ ) e tɛh θ=π = iɛ F The result is the sum of the contributions of the T -fixed points θ = 0 and θ = π. Let L n = O(n) be the complex line bundle over S 2 = CP of the first Chern class n = c = 2πi F. We choose connection -form A with i F α e(ν F ) (2.22) {eq:integration}
0 VASILY PESTUN constant curvature F A = inω, denoted in the patch around θ = 0 2 by A (0) and in the patch around θ = π by A (π) A (0) = 2 in( cos θ)dφ A(π) = in( cos θ)dφ (2.23) 2 The gauge transformation between the two patches A (0) = A (π) in dφ (2.24) is consistent with the defining O(n) bundle transformation rule for the sections s (0), s (π) in the patches around θ = 0 and θ = π s (0) = z n s (π) A (0) = A (π) + z n dz n. (2.25) The equivariant curvature F T of the connection A in the L n bundle with a suitable lift of the T -action on the fibers is again given by F T = 2 inω T (2.26) as can be verified agains the definition (.5) F T = F ɛi v A, taking into contribution from the vertical component g dg of the connection A on the total space of the principal U() bundle and the lift 2 of the T action on the fiber at θ = 0 with weight n and on the fiber at θ = π 2 with weight n. 2 Therefore, for t = in we have α = 2 ef T = ch(f T ) where F T is the equivariant curvature of the O(n) bundle, and (2.22) implies e F T = 2 sin nɛ 2 (2.27) 2πi ɛ Now let us compute the T -equivariant index of the Dirac operator on S 2 twisted by the L n bundle using (.3) and the localization to the fixed points. We need to multiply the contributions from the north and south poles by the the T -equivariant  class of the tangent bundle, which for both fixed points is given by ÂT (T ) θ=0 = ÂT (T ) θ=π = iɛ = ɛ, and we find that the equivariant index e iɛ/2 e iɛ/2 2 sin ɛ 2 ind T (D) =  T (T )e F T 2πi = sin n 2 ɛ sin 2 ɛ (2.28) {eq:kirillov} is the SU(2) character of irreducible representation with highest weight n in the conventions where is the fundamental representation. 2 It is easy to see that this is correct assignment by checking that for the tangent bundle n = 2 and the fiber at the north pole is acted with weight while the fiber at the south pole is acted with weight because of the opposite orientation.
LECTURES ON EQUIVARIANT LOCALIZATION 3. Equivariant index theory 3.. Kirillov character formula. Let G be a compact simple Lie group. The Kirillov character formula equates that the T -equivariant 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. Here ρ is the Weyl weight ρ = α>0 α. 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 : tr λ e, g (3.) Let 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 λ is a homogeneous symplectic G-manifold with the canonical symplectic structure ω defined at point x g on tangent vectors in g by the formula ω x (, 2 ) = x, [, 2 ], 2 g (3.2) (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 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 larger group H, such that T H G, the orbit λ is called degenerate flag manifolds 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 expliclitly 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 B 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 λ+ρ π λ (3.3)
2 VASILY PESTUN This is known as Kirillov correspondence of the geometric representation theory. Namely, if λ g is a weight, the symplectic structure ω is integral and there exists a line bundle L λ with the unitary connection of curvature ω. The line bundle L λ is acted 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 λ+ρ on the co-adjoint orbit λ+ρ with the character χ λ of irreducible representation of G with the highest weight λ ind T (D)( λ+ρ ) = χ λ (3.4) This formula can be easily proved using the Atiyah-Singer equivariant index formula ind T (D)( λ+ρ ) = ch ( 2πi) n T (L)ÂT (T ) (3.5) λ+ρ and Atiyah-Bott formula to localize the integral over λ+ρ to the set of fixed points λ+ρ T. The localization to λ+ρ T 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 λ+ρ is the full flag manifold. The T -fixed points are in the intersection λ+ρ t, and hence, the set of the T -fixed points is the Weyl orbit of λ + ρ T λ+ρ = Weyl(λ + ρ) (3.6) At each fixed point p λ+ρ T the tangent space T 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 of the ÂT gives the Weyl denominator, the numerator of the ÂT cancels with the Euler class e T (T ) in the localization formula, and the restriction of the ch T (L) = e ω is e w(λ+ρ) ( 2πi) n λ+ρ ch T (L)Â(T ) = w W e iw(λ+ρ)ɛ α>0 (e 2 iαɛ e 2 iαɛ ) (3.7) We conclude the localization of the equivariant index of the Dirac operator on λ+ρ twisted by the line bundle L to the set of the fixed points λ+ρ T is precisely the Weyl formula for the character. The Kirillov correspondence between the index of the Dirac operator of L λ+ρ is closedly related to the Borel-Weyl-Bott theorem.
LECTURES ON EQUIVARIANT LOCALIZATION 3 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. 3 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 action of w on a weight λ by w λ := w(λ + ρ) ρ. The Borel-Weyl-Bott theorem is H l(w) (G C /B C, O(L λ )) = { R λ, w λ is dominant 0, w λ is not dominant (3.8) where R λ is the irreducible G-module with the highest weight λ. 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. Example 5. For G = SU(2) the G C /B C = CP, integral weight of T C is an integer n BZ, and the line bundle L n is the O(n) bundle over CP. The Weyl weight is ρ =. The weight n 0 is dominant and the H 0 (CP, O(n)) is the SL(2, C) module of highest weight n (in the basis of fundamental weights of SL(2)). For weight n = the H i (CP, O( )) is empty for all i as there is no Weyl transformation w such that w n is dominant (equivalently, because ρ + n = 0). For weight n 2 the w is the Z 2 reflection and w n = (n + ) = n 2 is dominant and H (CP, O(n)) is an irreducible SL(2, C) module of highest weight n 2. The relation between Borel-Weil-Bott theorem for G/B and the Dirac complex on G/B is that Dirac operator is precisely the Dolbeault operator shifted by the square root of the canonical bundle and consequently S + () S () = K 2 ( ) p Ω 0,p () (3.9) ind( λ+ρ, D L λ+ρ ) = ind(g C /B C, L λ ) (3.0) Borel-Bott-Weyl theorem has generalization for incomplete 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 3 The unipotent radical U C is generated by g +
4 VASILY PESTUN module, and let O(E) the the sheaf of local regular sections of E. Then O(E) is isomorphic to the the direct image sheaf π O(L) for a one-dimensional B C -module L and H k (G C /P C, O(E)) = H k (G C /B C, O(L)) 3.2. Index of a complex. Let E k be vector bundles over a manifold. Let T be a compact Lie group acting on and the bundles E k. The action of T on a bundle E induces canonically linear action on the space of sections Γ(E). For t T and section φ Γ(E) the action is (tφ)(x) = tφ(t x), x (3.) {eq:section-action} Let D i be linear differential operators compatible with the T action, and let E be the complex E : Γ(E 0 ) D 0 Γ(E ) D Γ(E 2 )... (3.2) Since D i are T -equivariant operators, the T -action on Γ(E i ) induces the T -action on the cohomology H i (E). The equivariant index of the complex E is the virtual character defined by ind T (D) : t C (3.3) ind T (D) = k ( ) k tr H k (E) t (3.4) 3.3. Atiyah-Singer index formula. If the set T of T -fixed points is discrete, the Atiyah-Singer equivariant index formula is ind T (D) = k ( )k ch T (E k ) x (3.5) det νx ( t ) x T Example 6. The equivariant index of : Ω 0,0 () Ω 0, () on = C x under the T = U() action x t x where t T is the fundamental character is contributed by the fixed point x = 0 as t ind T ( ) = ( t)( t) = t = t k (3.6) where denominator is the determinant of the operator t over the two-dimensional normal bundle to 0 C spanned by the vectors x and x with T eigenvalues t and t. In the numerator, 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, ) forms d x. k=0
LECTURES ON EQUIVARIANT LOCALIZATION 5 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 Poincare lemma. Example 7. Now let = CP and L n be the holomorphic line bundle with first Chern class c = n. Let : Ω 0,0 (L n ) Ω 0, (L n ). Let x be the local coordinate in the patch around x = 0 and x = x be the local coordinate in the patch around x =. Let group T = U() act equivariantly on L n by x t x (3.7) with a certain lift on L n : define the action of T on the fiber of L n at x = 0 to be trivial, and consequently T acts on sections in Ω 0, (L n ) by sending the section φ x (x, x)d x to tφ x (tx, t x)d x. Therefore, on the fibers of L n and L n T at x = 0 the T acts with characters and t. There are two fixed points x = 0 and x =. The x = 0 contributes as in the previous example by ind T (D, L n ) x=0 = t (3.8) At x = we need to know the action of T on the fibers of L n and L n T. Let x = x. A section φ Ω 0,0 (L n ) in the x coordinate patch is related to φ in the x coordinate patch by φ(x) = x n φ( x) (3.9) A constant section in the patch x is acted in the same way as section x n in the patch x, that is with weight t n. Therefore, the fiber of L n at x = 0 is acted with t n. Consequently, the contribution to the index of the point x = 0 is The total index is ind T (D, L n ) x=0 = ind T (D, L n ) x=0 +ind T (D, L n ) x=0 = t + tn t = tn t (3.20) n k=0 tk, n 0 0, n =, t n 2 k=0 t k, n < (3.2) We can check agains the direct computation. Assume n 0. The kernel of D is spanned by n + holomorphic sections of O(n) of the form x k for k = 0,..., n, the cokernel is empty by Riemann-Roch. The
6 VASILY PESTUN section x k is acted by t T with weight t k. Therefore n ind T (D, L n ) = t k (3.22) Example 8. Let = CP m be defined by the projective coordinates (x 0 : x : : x m ) and L n be the line bundle O(L) = O(n). Let T = U() (m+) act on the bundle by k=0 (x 0 : x :... x m ) (t 0 x 0 : t x : : t m x m ) (3.23) and by t n k on the fiber of the bundle E in the patch around k-th fixed point x k =, x i k = 0. We find the index as a sum of contributions from m + fixed points m t n k ind T (D) = j k ( (t (3.24) {eq:fixed} j/t k )) k=0 For n > 0 the index is a homogeneous polynomial in C[t 0,..., t m ] of degree n representing the character on the space of holomorphic sections of O(n) bundle over CP m. s n (t 0,..., t m ), n 0 ind T (D) = 0, m n < 0 ( ) m t 0 t... t m s n m (t 0,..., t m ), n m (3.25) {eq:cp-answer} where s n (t 0,..., t m ) are complete homegeneous symmetric polynomials. This result can be quickly obtained from the contour integral representation of the sum (3.24) 2πi C dz z z n m m j=0 ( t j/z) = k=0 t n k j k ( (t j/t k )), (3.26) 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 (3.24). 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 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 (3.24) with the ( ) sign. The residue at z = 0 contributes by (3.25).
LECTURES ON EQUIVARIANT LOCALIZATION 7 4. Equivariant cohomological field theories Often the path integral for supersymmetric field theories can be represented in the form Z = α (4.) where is the (infinite-dimensional) superspace of fields of the theory over which we integrate in the path integral the measure α closed with respect to a supercharge operator Q Qα = 0 (4.2) 4.. Four-dimensional gauge theories. The Donaldson-Witten twist of the N = 2 supersymmetric off-shell vector multiplet for a theory on a four-manifold M produces the topological gauge multiplet { QA µ = ψ µ Qψ µ = D A φ { Qχ + µν = H + µν QH µν = [φ, χ µν ] { Q φ = η Qη = [φ, φ] (4.3) The operator Q can be given the geometric interpretation of the equivariant Cartan differential for the action of group G of gauge transformations on the space A(M, g) ΠΩ 2+ (M, g) Ω 0 (M, g) where A is the affine space of connections A µ on a principal G-bundle over M, the ΠΩ 2+ (M, g) is the space of the g-valued fermionic self-dual forms χ + µν on M and Ω 0 (M, g) is the space of g-valued functions φ on M. The Lie algebra of G is parameterized by the g-valued function φ on M. The fields ψ µ, H µν, + η represent respectively the Q-differential of A µ, χ + µν, φ In the gauge theory path integral we integrate over φ in Lie(G) and over the fields of the gauge-fixing multiplet, the fermionic g-valued fields c, c and the bosonic g-valued field b. It turns out that the field φ is naturally interpreted as the differential of the of the fermionic ghost c, and the bosonic field b is the differential of the fermionic field c. Then integration over (φ, c) and ( c, b) is interpreted as integration over ΠΩ 0 (M, g) ΠΩ 0 (M, g). Finally, the gauge theory path integral is Z = α k H 4 (M,Z) r q k =A k (M,g) ΠΩ 2+ (M,g) Ω 0 (M,g) ΠΩ 0 (M,g) ΠΩ 0 (M,g) (4.4) where we sum over topologically equivalence classes of the G-bundles on M labelled by an r-tuple of integers k = (k,..., k r ) with generating parameters q = (q, q 2,..., q r ) if G is a product of compact simple Lie groups G = G G r. Qφ = 0
8 VASILY PESTUN In addition to the action by the gauge group G the fields might transform under an additional global symmetry T and we can make all geometrical objects to be T -equivariant. The integration over by Atiyah-Bott localization formula (here we take its super and infinitedimensional version) reduces to the sum over the set of fixed points T that we assume to be discrete Z = k q k p T i pα e T (T A(M, g) p ΠΩ 2+ (M, g) ΠΩ 0 (M, g)) (4.5) The T -equivariant Euler classes of Ω 0 (M, g) ΠΩ 0 (M, g) cancel. The Euler class e T (T A(M, g) p ΠΩ 2+ (M, g) ΠΩ 0 (M, g)) is most conveniently computed from the Chern character ch T (T A(M, g) p ΠΩ 2+ (M, g) ΠΩ 0 (M, g)) for which we can use Atiyah-Singer index theorem. Namely we consider the T -equivariant character for the complex of the self-dual equations E : Ω 0 (M, g) d Ω (M, g) d+ Ω 2+ (M, g) (4.6) and notice that because of the reversed parity we have e T (E) = e T (T A(M, g) p ΠΩ 2+ (M, g) ΠΩ 0 (M, g)) (4.7) 4.2. The Ω-background. Now we consider the concrete example of the gauge theory in so called Ω-background on M = R 4 = C 2 z,z 2. Since M is non-compact, it is convenient to consider to consider the topological gauge theory complex build from the principal G-bundle on M with a fixed framing at M. The gauge transformation of the fiber at M generate the equivariant action by the group G which we can restrict to its maximal abelian subgroup T G. Therefore, we consider the group T ɛ,ɛ 2,a = U() 2 T G a acting by z t z, z 2 t 2 z 2 where t = e iɛ, t 2 = e iɛ 2 and by ad(g) transformations on the g valued fields. First we compute the contribution from the trivial fixed point: the principal G-bundle has trivial topology, k = 0, the gauge connection is trivial and all other fields vanish. The equivariant Euler class of E is conveniently found from the equivariant Chern class of E, or the index of the self-dual complex.
We find the index LECTURES ON EQUIVARIANT LOCALIZATION 9 ind T (E) ch T (E) = α>0 ω=e iα a = α>0 ω=e iα a ω ω + ω + t t 2 ω + t t 2 ω (t ω + t ω + t 2 ω + t 2 ω) ( t )( t )( t 2 )( t 2 ) ( t )( t 2 ) + ω ( t )( t 2 ) (4.8) The above line does not include contribution from the subspace of g with the zero adjoint weights, that is from the Cartan of g as such contribution is a-independent and is often not-interesting in the physical applications. Then Euler class is where e T (E) = α>0 G ɛ,ɛ 2 (α a i )G ɛ,ɛ 2 (ɛ + ɛ 2 α a i ) (4.9) G ɛ,ɛ 2 (x) reg = n 0,n 2 0 (x + n ɛ + n 2 ɛ 2 ) (4.0) is a properly regularized Weierstrass infinite product formula for the inverse of the double-gamma function of Barnes. Institute for Advanced Study, Princeton E-mail address: pestun@ias.edu