# k=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula

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1 20 VASILY PESTUN 3. Lecture: Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer Index theorems 3.. Index for a holomorphic vector bundle. For a holomorphic vector bundle E over a complex variety of dim C = n the index ind(, E) is defined as n ind(, E) = ( ) k dim H k (, E) (3.) The localization theorem in K-theory gives the index formula of Grothendieck-Riemann- Roch-Hirzebruch-Atiyah-Singer relating the index to the Todd class ind(, E) = ( 2π td(t,0 ) n ) ch(e) (3.2) 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 ind( /D, E) = dim ker /D dim coker /D (3.3) ind( /D, E) = ( 2π ) n Notice that on a Kahler manifold the Dirac complex is isomorphic to the Dolbeault complex /D : S S Ω 0,p () Ω 0,p ()... twisted by the square root of the canonical bundle K = Λ n (T,0 ) Â(T,0 ) ch(e) (3.4) /D = K 2 (3.5) consistently with the relation (.75) and the Riemann-Roch-Hirzebruch-Atiyah-Singer index formula 3.2. Euler characteristic from an index formula. The vector bundle E in the index formula (3.2) can be promoted to a complex E E (3.6) In particular, the index of the complex E = Λ (T,0 ) of (, 0)-forms on a Kahler variety equals the Euler characteristic of n n e() = ind(, Λ (T,0 ) ) = ( ) pq dim H p,q () (3.7) We find ch Λ (T,0 ) = q=0 p=0 n ( e x i ) (3.8) i=

2 EQUIVARIANT COHOMOLOGY AND LOCALIZATIONWINTER SCHOOL IN LES DIABLERETSJANUARY where x i are Chern roots of the curvature of the n-dimensional complex bundle T,0. Hence, the Todd index formula (3.2) gives e() = ( 2πı) n The above agrees with the Euler characteristic (.82) c n (T,0 ) (3.9) 3.3. Equivariant index formula (Dolbeault and Dirac). Let G be a compact connected Lie group. Suppose that is a complex variety and E is a holomorphic G-equivariant vector bundle over. Then the cohomology groups H (, E) form representation of G. In this case the index of E (3.) can be refined to an equivariant index or character n ind G (, E) = ( ) k ch G H k (, E) (3.0) where ch G H i (, E) is the character of a representation of G in the vector space H i (, E). More concretely, the equivariant index can be thought of as a gadget that attaches to G- equivariant holomorphic bundle E a complex valued adjoint invariant function on the group G n ind G (, E)(g) = ( ) k tr H k (,E) g (3.) on elements g G. The sign alternating sum (3.) is also known as the supertrace ind G (, E)(g) = str H (,E) g (3.2) The index formula (3.2) 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 (.42) ind(, E)(e φa T a ) = ( 2πı) n td G (T ) ch G (E) = ch G E e G (T ) ch G Λ T (3.3) Here φ a T a is an element of Lie algebra of G and e φa T a is an element of G, and T denotes the holomorphic tangent bundle of the complex manifold. If the set G of G-fixed points is discrete, then applying the localization formula (2.9) to the equivariant index (3.3) we find the equivariant Lefshetz formula ind(, E)(g) = x G tr Ex(g) det T,0 x ( g ) The Euler character is cancelled against the numerator of the Todd character. (3.4) 3.4. Example of equivariant index on CP. Let be CP and let E = O(n) be a complex line bundle of degree n over CP, and let G = U() equivariantly act on E as follows. Let z be a local coordinate on CP, and let an element t U() C send the point with coordinate z to the point with coordinate tz so that ch T,0 0 = t ch T,0 = t (3.5)

3 22 VASILY PESTUN where T,0 0 denotes the fiber of the holomorphic tangent bundle at z = 0 and similarly T,0 the fiber at z =. Let the action of U() on the fiber of E at z = 0 be trivial. Then the action of U() on the fiber of E at z = is found from the gluing relation to be of weight n, so that Then ind(, O(n), CP )(t) = s = z n s 0 (3.6) ch E z=0 =, ch E z= = t n (3.7) t n t t = t n = t n t k, n 0 0, n =, t n 2 t k, n < (3.8) We can check against the direct computation. Assume n 0. The kernel of is spanned by n 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 n ind T (, O(n), CP ) = t k (3.9) Even more explicitly, for illustration, choose a connection -form A with constant curvature F A = 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) A (0) = 2 in( cos θ)dα A(π) = in( cos θ)dα (3.20) 2 The gauge transformation between the two patches A (0) = A (π) in dα (3.2) 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 s (0) = z n s (π) A (0) = A (π) z n dz n. (3.22) The equivariant curvature F T of the connection A in the bundle E is given by F T = in(ω ɛ( cos θ)) (3.23) 2 as can be verified against the definition (.60) F T = F ɛi v A. Notice that to verify the expression for the equivariant curvature (3.23) in the patch near θ = π one needs to take into account contributions from the vertical component g dg of the connection A on the total space of the principal U() bundle and from the T -action on the fiber at θ = π with weight n. Then ch(e) θ=0 = exp(f T ) θ=0 = ch(e) θ=π = exp(f T ) θ=π = exp( inɛ) = t n (3.24) for t = exp(iɛ) in agreement with (3.8).

4 EQUIVARIANT COHOMOLOGY AND LOCALIZATIONWINTER SCHOOL IN LES DIABLERETSJANUARY A similar exercise gives the index for the Dirac operator on S 2 twisted by a magnetic field of flux n ind( /D, O(n), S 2 ) = tn/2 t n/2 (3.25) t 2 t 2 where now we have chosen the lift of the T -action symmetrically to be of weight n/2 at θ = 0 and of weight n/2 at θ = π. Also notice that up to overall multiplication by a power of t related to the choice of lift of the T -action to the fibers of the bundle E, the relation (3.5) holds ind( /D, O(n), S 2 ) = ind(, O(n ), CP ) (3.26) because on CP the canonical bundle is K = O( 2) Example of equivariant index on CP m.. Let = CP m be defined by the projective coordinates (x 0 : x : : x m ) and L n be the line bundle L n = O(n). Let T = U() (m) act on by (x 0 : x :... x m ) (t 0 x 0 : t x : : t m x m ) (3.27) and by t n k on the fiber of the bundle L n in the patch around the k-th fixed point x k =, x i k = 0. We find the index as a sum of contributions from m fixed points ind T (D) = m t n k j k ( (t j/t k )) (3.28) 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 the O(n) bundle over CP m. s n (t 0,..., t m ), n 0 ind T (D) = 0, m n < 0 (3.29) ( ) m t 0 t... t m s n m (t 0,..., t m ), n m where s n (t 0,..., t m ) are complete homogeneous symmetric polynomials. This result can be quickly obtained from the contour integral representation of the sum (3.28) dz z n m 2πi z m j=0 ( t j/z) = t n k j k ( (t j/t k )), (3.30) C 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 (3.28). 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 (3.28) with the ( ) sign. The residue at z = 0 contributes by (3.29). Also notice that the last line of (3.29) relates to the first line by the reflection t i t i t n k j k ( t j/t k ) = ( )m (t j k Thanks to Bruno Le Floch for the comment k ) n m ( j t j ) ( t j /t k ) (3.3)

5 24 VASILY PESTUN which is the consequence of the Serre duality on CP m Equivariant index on general flag manifold and Weyl formula for the character. The CP in example (3.25) can be thought of as a flag manifold SU(2)/U(), and (3.8) (3.25) 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 (3.26) 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. The generalization of formula (3.25) for generic flag manifold appearing from a co-adjoint orbit in g is called Kirillov character formula [8], [9], [20]. Let G be a compact simple Lie group. The Kirillov character formula equates 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. 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 (3.32) 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 (3.33) 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 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 λρ π λ (3.34) This is known as the Kirillov correspondence in geometric representation theory. 2 For a short presentation see exposition by J. Lurie at bwb.pdf

6 EQUIVARIANT COHOMOLOGY AND LOCALIZATIONWINTER SCHOOL IN LES DIABLERETSJANUARY Namely, if λ g is a weight, the symplectic structure ω is integral and there exists a line bundle L λ with a unitary connection of curvature ω. The line bundle L 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 λρ on the co-adjoint orbit λρ with the character χ λ of the irreducible representation of G with highest weight λ ind T ( /D)( λρ ) = χ λ (3.35) This formula can be easily proven using the Atiyah-Singer equivariant index formula ind T ( /D)( λρ ) = ch ( 2πı) n T (L)ÂT (T ) (3.36) λρ and the Atiyah-Bott-Berline-Vergne formula to localize the integral over λρ to the set of fixed points λρ T. The localization to λρ 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 λρ is the full flag manifold. The T -fixed points are in the intersection λρ t, and hence, the set of the T -fixed points is the Weyl orbit of λ ρ T λρ = Weyl(λ ρ) (3.37) At each fixed point p λρ T the tangent space T λρ 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 ÂT gives the Weyl denominator, the numerator of ÂT cancels with the Euler class e T (T ) in the localization formula, and the restriction of ch T (L) = e ω is e w(λρ) ind T ( /D)( λρ ) = ch ( 2πı) n T (L)Â(T ) = λρ w W e iw(λρ)ɛ α>0 (e 2 iαɛ e 2 iαɛ ) (3.38) We conclude that the localization of the equivariant index of the Dirac operator on λρ twisted by the line bundle L to the set of fixed points λρ T is precisely the Weyl formula for the character Equivariant index for differential operators. See the book by Atiyah [2]. Let E k be a family of vector bundles over a manifold indexed by integer k Z. Let G be a compact Lie group acting equivariantly 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 x), x (3.39) Let D k : Γ(E k ) Γ(E k ) be linear differential operators compatible with the G action, and let E be the complex (that is D k D k = 0) E : Γ(E 0 ) D 0 Γ(E ) D Γ(E 2 )... (3.40) 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 ind G (D) : g C (3.4)

7 26 VASILY PESTUN defined by ind G (D)(g) = k ( ) k tr H k (E) g (3.42) If the set G of G-fixed points is discrete, the Atiyah-Singer equivariant index formula is ind G (D) = k ( )k ch G (E k ) x (3.43) det Tx( g ) x G For the Dolbeault complex E k = Ω 0,k and D k = : Ω 0,k Ω 0,k Ω 0, Ω 0, (3.44) the index (3.43) agrees with (3.4) because the numerator in (3.43) decomposes as ch G E ch G Λ T0, and the denominator as ch G Λ T0, ch G Λ T,0 and the factor ch G Λ T0, cancels out. For example, 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 ind T (C, ) t = ( t)( t) = t = t k (3.45) where the 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 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. 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 (3.9). Similarly, for the complex on C r we obtain [ r ] ind T (C r, ) = (3.46) ( t k ) k= 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 [ r ] [ r ] ind T (S 2r, D) = ( t k ) ( t k ) = [ r k= k= ( t k ) ] [ r k= k= ( ) r t... t r ( t k ) where [] and [] denotes the expansions in positive and negative powers of t k. ] (3.47)

8 EQUIVARIANT COHOMOLOGY AND LOCALIZATIONWINTER SCHOOL IN LES DIABLERETSJANUARY Atiyah-Singer index formula for a free action G-manifold. Suppose that a compact Lie group G acts freely on a manifold and let Y = /G be the quotient, and let π : Y (3.48) 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 as in (3.40). The G T -equivariance means that the complex E and the operator D are pullbacks by π of a T -equivariant complex Ẽ and operator D on the base Y E = π Ẽ, D = π D (3.49) We want to compute the G T -equivariant index ind G T (D; ) for the complex (E, D) on the total space 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 (3.48). Then, for each irrep α R G we consider a complex Ẽ W α on Y obtained by tensoring Ẽ with the vector bundle W α over Y. The Atiyah-Singer formula is ind G T (D; ) = α R G ind T ( D W α ; Y )χ α. (3.50) 3.9. Index of Dirac operator on odd-dimensional sphere S 2r. We consider an example relevant for localization on odd-dimensional spheres S 2r which are subject to the equivariant action of the maximal torus T r of the isometry group SO(2r). The sphere π : S 2r CP r is the total space of the S Hopf fibration over the complex projective space CP r. We will apply the equation (3.50) for a transversally elliptic operator D induced from the Dolbeault operator D = on CP r by the pullback π. To compute the index of operator D = π on π : S 2r CP r we apply (3.50) and use (3.29) and obtain [ ] [ ind(d, S 2r ) = ind T (, CP r ( ), O(n)) = r t ]... t r r n= k= ( t r k) k= ( t k ) (3.5) where [] and [] denotes the expansion in positive and negative powers of t k. See further review in Contribution [22] General Atiyah-Singer index formula. The Atiyah-Singer index formula for the Dolbeault and Dirac complexes and the equivariant index formula (3.43) can be generalized to a generic situation of an equivariant index of transversally elliptic complex (3.40). Let be a real manifold. Let π : T be the cotangent bundle. Let {E } be an indexed set of vector bundles on and π E be the vector bundles over T defined by the pullback. The symbol σ(d) of a differential operator D : Γ(E) Γ(F ) (3.40) is a linear operator σ(d) : π E π F which is defined by taking the highest degree part of the differential

9 28 VASILY PESTUN operator and replacing all derivatives by the conjugate coordinates p µ in the fibers of x µ T. For example, for the Laplacian : Ω 0 (, R) Ω 0 (, 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 given by σ( ) = g µν p µ p ν (3.52) where p µ are conjugate coordinates (momenta) on the fibers of T. 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 outside of the zero section T. 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. The Atiyah-Singer formula for the index of the complex (3.40) is ind G (D, ) = Â (2π) dim R G (π T ) ch G (π E ) (3.53) T Here T denotes the total space of the cotangent bundle of with canonical orientation such that dx dp dx 2 dp 2... is a positive element of Λ top (T ). Let n = dim R. Let π T denote the vector bundle of dimension n over the total T obtained as pullback of T to T. The ÂG-character of π T is ( ) Â G (π R G T ) = det π T (3.54) e R G/2 e R G/2 where R G denotes the G-equivariant curvature of the bundle π T. Notice that the argument of Â is n n matrix where n = dim R T (real dimension of ) while if general index formula is specialized to Dirac operator on Kahler manifold as in (3.4) the argument of the Â- character is an n n matrix where n = dim C T,0 (complex dimension of ). Even though the integration domain T is non-compact the integral (3.54) is well-defined because of the (G-transversal) ellipticity of the complex π E. For illustration take the complex to be E D 0 E. Since σ(d) : π E 0 π E is an isomorphism outside of the zero section we can pick a smooth connection on π E 0 and π E such that its curvature on E 0 is equal to the curvature on E away from a compact tubular neighborhood U ɛ of T. Then ch G (π E ) is explicitly vanishing away from U ɛ and the integration over T reduces to integration over the compact domain U ɛ. It is clear that under localization to the fixed points of the G-action on the general formula (3.54) reduces to the fixed point formula (3.43). This is due to the fact that the numerator in the Â-character det π T R G = Pf TT (R G ) is the Euler class of the tangent bundle T T to T which cancels with the denominator in (2.9), while the restriction of the denominator of (3.54) to fixed points is equal to (3.54) or (3.43), because det e R G =, since R G is a curvature of the tangent bundle T with orthogonal structure group.

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