LECTURES ON EQUIVARIANT LOCALIZATION

VASILY PESTUN

Abstract. These are informal notes of the lectures on equivariantlocalization given at the program “Geometry of Strings and Fields”at The Galileo Galilei Institute in September, 2013. All remarksand corrections are welcomed from the participants.

Contents

0.1. References 11. Cartan model of equivariant cohomology 21.1. Cartan model 21.2. Equivariant characteristic classes in Cartan model 41.3. Frequently used characteristic classes 51.4. Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer

index formula 62. Equivariant integration 72.1. Thom isomorphism, the Euler class and the Atiyah-Bott-

Berline-Vergne integration formula 73. Equivariant index theory 113.1. Kirillov character formula 113.2. Index of a complex 143.3. Atiyah-Singer index formula 144. Equivariant cohomological field theories 174.1. Four-dimensional gauge theories 174.2. The Ω-background 18

0.1. References. The are multiple sources for these educational lec-tures. Some of them are to the original papers, some are pedagogicalexposition, or a combination of both. We assemble a list of referenceshere and excuse us for omitting references in the course

• Berline, Getzler, Vergne “Heat kernels and Dirac operators”(Springer, 1991)• Szabo “Equivariant Localization of Path Integrals” hep-th/9608068

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• Cordes, Moore, Ramgoolam “Lectures on 2D Yang-Mills The-ory, Equivariant Cohomology and Topological Field Theories”hep-th/9411210• Atiyah “Elliptic Operators and Compact Groups” (Lecture Notesin Math, 1974)• Bott, Tu “Equivariant characteristic classes in the Cartan model”,math/0102001• Vergne “Applications of equivariant cohomology” math/0607389• Atiyah-Bott “The moment map and equivariant cohqmology”,1982• Berline-Vergne “The equivariant Chern character and index ofG-invariant operators.” (Lectures at CIME, Venise 1992)• Witten “Introduction to cohomological field theories” Int. Jour.Mod. Phys. 1991• Witten “Topological Quantum Field Theory”, 1988• Kirillov, Lectures on the theory of group representations, 1971.Konstant, Quantization and Unitary representations, 1970. Kir-illov, Merits and Demerits of the orbit method, 1997• Mathai, Quillen “Superconnections, Thom classes and equivari-ant differential forms” 1986• Atiyah, Jeffrey “Topological Lagrangians and cohomology”, 1990• Nekrasov, Okounkov “Seiberg-Witten Theory and Random Par-titions” hep-th/0306238• Pestun “Localization of gauge theory on a four-sphere and su-persymmetric Wilson loops” 0712.2824• Kamnitzer “Lectures on geometric constructions of the irreduciblerepresentations of GLn” 0912.0569

1. Cartan model of equivariant cohomology

Let X be a manifold and T be a Lie group acting on X. Oftenwe are interested in the geometry of the quotient space X/T . Theequivariant cohomologyH•T (X) is a proper definition ofH•(X/T ) whenthe quotient X/T is not smooth.

1.1. Cartan model. The Cartan model of the equivariant cohomologyH•T (X) uses the algebra of functions on t valued in differential formson X, that is Ω•(X) ⊗ C(t). The action of group element t ∈ T on a

LECTURES ON EQUIVARIANT LOCALIZATION 3

form α ∈ Ω•(X)⊗ C(t) is defined by1

(t · α)(X) = t · (α(t−1Xt)) ∀X ∈ t (1.2)

Let Ta be the basis in t and εa be respective coordinate functions ong, so that an element of t be written as εaTa, and f cab be the structureconstants [Ta, Tb] = f cabTc.

Infinitesimal action by a Lie algebra generator Ta on an elementα ∈ Ω•(X)⊗ C(g) is then given by

Ta · α = −La ⊗ 1− 1⊗ La (1.3)

where La is the geometrical Lie derivative by the vector field associatedto the generator Ta on Ω•(X) and La is the coadjoint action on C(g):for α ∈ C(g) we have

Laα = f cabεb ∂α

∂εc(1.4)

The T -invariant subspace in Ω•(X) ⊗ C(g) is called the algebra ofequivariant differential forms

ΩT (X) = (Ω•(X)⊗ T •(t∗))T (1.5)

which means that

α ∈ ΩT (X) ⇔ (La ⊗ 1 + 1⊗ La)α = 0 (1.6) eq:Lie-geom-alg

Define the equivariant Cartan differential dT by the formula

dT = d− εaiva (1.7)

where iva is the contraction with the vector field va associated to theaction by the Ta generator on X. The square of dT is the geometricLie derivative

d2T = −εaLva (1.8)

On equivariant forms α ∈ ΩT (X) the action by geometric Lie derivative−La is equivalent to the coadjoint action by La by (1.6). Therefore, onα ∈ ΩT (X) the d2

T acts as

d2Tα = εa(1⊗ La)α = εaf cabε

b ∂α

∂εc= 0 (1.9)

by the antisymmetry of the structure constants f cab = −f cba. We con-clude that d2

T = 0 when acted on equivariant differential form ΩT (X).We can consistently define grading on ΩT (X) by assignment

deg d = 1 deg iva = −1 deg εa = 2 (1.10) eq:grading

1And we remind that the T -action on differential forms α ∈ Ω•(M) is definedby the pullback

t · α = ρ∗t−1α (1.1)where ρt : M →M is the map defining the T -action on M

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and in then deg dT = 1 so that dT : ΩpT (X) → Ωp+1

T (X) where Ωp(X)defines the subspace of the degree p in ΩT (X) according to the grading(1.10).

Since d2T = 0 on graded space Ω•T (X), and dT increases the grading

by 1, we have the differential complex

· · · dT→ ΩpT (X)

dT→ Ωp+1T (X)

dT→ . . .

and we defind the cohomology of the (ΩT (X), dT ) complex in the stan-dard fashion

H•T (X) ≡ Ker dT/Im dT (1.11)If X = pt is a point then dT = 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 (1.12)

1.2. Equivariant characteristic classes in Cartan model. LetE → X be a T -equivariant G-principal bundle and let DA = d + Abe the T -invariant connection and A is the connection g-valued 1-formon the total space of E (such connection always exists by the aver-aging procedure for compact T ). Then we define the T -equivariantconnection

DTA = DA − εaiva (1.13)

and the T -equivariant curvature

F TA = (Dt

A)2 + εa ⊗ Lva (1.14)

which is in fact is an element of Ω2T (X)

F TA = FA + εa ⊗ Lva − [εa ⊗ iva , 1⊗DA] = FA − εaivaA (1.15) eq:equivariant-curvature

The connection A is the g-valued 1-form on the total space of theprincipal bundle E. Let XT be the T -fixed point set in X. If theequivariant curvature F T is evaluated on XT , only vertical componentof iva contributes to the formula (1.15) and va pairs with the verticalcomponent of the connection A on the T -fiber of E given by g−1dg.The T -action on G-fiber induces the homomorphism

ρ : t→ g (1.16)

and let ρ(Ta) be the images of Ta basis elements.An differential form representing an ordinary characteristic class for

a G-bundle with a connection A can be taken to be P (FA) where P isan Ad-invariant polynomial on g and FA is the curvature.

In the same fashion a T -equivariant differential form representingT -equivariant characteristic class is an element in Ω2

T (X) defined asP (F T

A ) where F TA is the equivariant curvature.

LECTURES ON EQUIVARIANT LOCALIZATION 5

Above the T -fixed points with the T -action on G-fibers specified byρ : t→ g the characteristic class is

P (FA − εaρ(Ta)) (1.17)

1.3. Frequently used characteristic classes.

1.3.1. Chern character. For U(n) bundle with curvature two-form Fvalued in the Lie algebra u(n) define the Chern character by trace inthe fundamental representation

ch(F ) = tr eF =n∑i=1

exi (1.18)

where xi are eigenvalues of F .

Remark 1. Our conventions for characteristic classes differs from thefrequently used conventions where xi are eigenvalues of i

2πF by factor

of −2πi. In our conventions the characteristic class of degree 2n needsto be multiplied by 1/(−2πi)n to be integral.

1.3.2. Chern classes. The Chern classes cn are generated by the for-mula

det(1 + tF ) =n∑n=0

tncn (1.19)

1.3.3. Euler class. For principal SO(2n) bundle with curvature F theEuler class is

e = Pf(−iF ) (1.20)so that the integral Euler characteristic χ(X) is

χ(X) =1

(−2πi)n

∫X

e(TX) (1.21)

Example 1. Let X = R2 ' C〈z〉 and T = U(1) acting by z 7→ tz wheret is the fundamental character of T . The T -equivariant U(1) curvatureis

FT (TX) = −iε (1.22)and the T -equivariant Euler class is

eT (TX) = −iε (1.23)

Example 2. For a two-sphere X = S2 the curvature of the tangentSO(2) bundle in the local orthonormal frame (e1, e2) is R1212 = 1 andPf(−iR) = −i ωS2 where ωS2 is the standard volume form

χ(S2) =1

−2πi

∫(−i)ωS2 = 2 (1.24)

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Remark 2. If U(n) curvature is presented as SO(2n) curvature by em-bedding

i 7→(

0 −11 0

)(1.25)

then

e(Fso(n)) = Pf(−iFso(2n)

)= detFu(n) = cn =

n∏i=1

xi (1.26)

1.3.4. The Todd class and A-roof class. The Todd class is defined as

td =n∏i=1

xi1− e−xi

(1.27)

and A class is defined as

A = e−12

∑xi td =

n∏i=1

xiexi/2 − e−xi/2

(1.28)

Notice that on Kahler manifold the Dirac complex

D : S+ → S−

is isomorphic to the Dolbeault complex

· · · → Ω0,p(X)∂→ Ω0,p+1(X)→ . . .

twisted by the square root of the canonical bundle

D = ∂ ⊗K12 (1.29)

consistently with the relation between the A class, the Todd class andthe Riemann-Roch-Hirzebruch-Atiyah-Singer index formula

1.4. Grothendieck-Riemann-Roch-Hirzebruch-Atiyah-Singer in-dex formula. For a holomorphic vector bundle E over a holomorphicmanifold X of complex dimension n the index ind(∂, E) is defined asind(∂, E) =

∑nk=0(−1)k dimH i(X,O(E)) and can be computed by

ind(∂, E) =1

(−2πi)n

∫X

td(TX) ch(E) (1.30)

Similarly, the index of Dirac operatorD : S+ → S− from the positivechirality spinors S+ to the negative chirality spinors S−, twisted by avector bundle E, is defined as ind(D,E) = dim kerD − dim cokerDand is given by the Atiyah-Singer index formula

ind(D,E) =1

(−2πi)n

∫X

A(TX) ch(E) (1.31) eq:equivariant-Dirac

LECTURES ON EQUIVARIANT LOCALIZATION 7

2. Equivariant integration

2.1. Thom isomorphism, the Euler class and the Atiyah-Bott-Berline-Vergne integration formula. A map

f : F → X

of manifolds induces natural pushfoward map on the homology

f∗ : H•(F )→ H•(X)

and pullback on the cohomology

f ∗ : H•(X)→ H•(F )

In the situation when there is Poincare duality between homologyand cohomology we can construct pushforward operation on the coho-mology

f∗ : H•(F )→ H•(X) (2.1)We can display the pullback and pushforward maps on the diagram

H•(F )f∗−→←−f∗H•(X) (2.2)

For example, if F and X are compact manifolds and i : F → X is theinclusion, then for the pushforward map f∗ : H•(F )→ H•(X) we find

f∗1 = ΦF (2.3) eq:f1

where ΦF is the cohomology class in H•(X) which is Poincare dual tothe manifold F ⊂ X: for a form α on X we have∫

F

α =

∫X

ΦF ∧ α (2.4)

If X is the total space of the orthogonal vector bundle π : X → F overthe oriented manifold F then ΦF (X) is called the Thom class of thevector 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 importantproperty of the Thom class ΦF for submanifold F → X is

f ∗ΦF = e(νF ) (2.5)

where e(νF ) is the Euler class of the normal bundle to F in X. Com-bined with (2.3) the last equation gives

f ∗f∗1 = e(νF ) (2.6)

as a map H•(F )→ H•(F ).Now we consider T -equivariant cohomologies for a compact abelian

Lie group T acting on X. Let F = XT be the set of the T fixed points

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in X. Then the equivariant Euler class eT (νF ) is invertible, thereforethe identity map on H•T (X) can be presented as

1 = f∗1

eT (νF )f ∗ (2.7)

Let πX : X → pt be the map from a manifold X to a point pt. Thepushforward operator πX∗ : H•T (X) → H•T (pt) corresponds to the inte-gration of the cohomology class over X. The pushforward is functorial.For maps F f→ X

πX→ pt we have the composition πX∗ f∗ = πF∗ forF

πF→ pt. So we arrive to the Atiyah-Bott integration formula

πX∗ = πF∗f ∗

eT (νF )(2.8)

or more explicitly ∫X

α =

∫F

f ∗α

eT (νF )(2.9)

Example 3. Harmonic oscillator, Gaussian integral and Duistermaat-Heckman integration formula

Let X = (R2, ω) be the phase space of the one-dimensional harmonicoscillator with Hamiltonian

H =1

2(p2 + q2) (2.10)

and the symplectic structure

ω = dp ∧ dq = dH ∧ dφ (2.11)

where φ is the polar angle. The Hamiltonian equations generate theHamiltonian vector field vH = ∂φ = −q∂p + p∂q by the definition

dH = −ivHω (2.12)

The vector field ∂φ is the action by the basis element T of the Liealgebra t of the compact group T = U(1) action on manifold X.

The dT -equivariant differential on X is

dT = d− ε ivH (2.13)

where εa is the coordinate function on the Lie algebra u(1) ' R.The equivariant differential form

α = exp(t(ω − εH))

is dT -closeddT (ω − εH) = 0 (2.14)

LECTURES ON EQUIVARIANT LOCALIZATION 9

The elementary Gaussian integration gives

1

−2πi

∫X

α = − 1

2πi

∫t ωe−tεH =

1

−iε=

i∗Fα

e(νF )(2.15)

where F is the T -fixed point p = q = 0 and e(νF ) is the equivariantEuler class of the normal bundle to F ⊂ X.

The result is t-independent. The reason is the form α is Q-exact

α = (d− εiv)(1

2pdq − 1

2qdp) (2.16)

A constant function on X is Q-closed but not Q-exact, so that local-ization holds but the result is not t-independent.

Example 4. The SU(2)-spin and the co-adjoint orbit SU(2)/U(1) = S2

Let (X,ω) be the two-sphere S2 with coordinates (θ, φ) and sym-plectic structure

ω = sin θdθ ∧ dφ (2.17)

Let Hamiltonian function be

H = − cos θ (2.18)

so thatω = dH ∧ dφ (2.19)

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

ωT = ω − εH = sin θdθ ∧ dφ+ ε cos θ

is dT -closed. Locally there is degree 1 form V such that ωT = dTV , forexample

V = (− cos θ)dφ (2.20)

but globally such A is not defined. The dT -cohomology class [α] of theform α is non-zero. Let

α = etωT (2.21)

The integration shows

1

−2πi

∫X

α =1

−2πi

∫t ωe−tεH =

(e−tεH |θ=0

−iε− e−tεH |θ=π

−iε

)=∑F

i∗Fα

e(νF )

(2.22) eq:integrationThe result is the sum of the contributions of the T -fixed points θ = 0and θ = π.

Let Ln = O(n) be the complex line bundle over S2 = CP1 of the firstChern class n = c1 = 1

−2πi

∫XF . We choose connection 1-form A with

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constant curvature FA = −12inω, denoted in the patch around θ = 0

by A(0) and in the patch around θ = π by A(π)

A(0) = −1

2in(1− cos θ)dφ A(π) = −1

2in(−1− cos θ)dφ (2.23)

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 thesections s(0), s(π) in the patches around θ = 0 and θ = π

s(0) = zns(π) A(0) = A(π) + zndz−n. (2.25)

The equivariant curvature FT of the connection A in the Ln bundlewith a suitable lift of the T -action on the fibers is again given by

FT = −1

2inωT (2.26)

as can be verified agains the definition (1.15) FT = F − εivA, takinginto contribution from the vertical component g−1dg of the connectionA on the total space of the principal U(1) bundle and the lift2 of the Taction on the fiber at θ = 0 with weight 1

2n and on the fiber at θ = π

with weight −12n.

Therefore, for t = −12in we have α = eFT = ch(FT ) where FT is the

equivariant curvature of the O(n) bundle, and (2.22) implies

1

−2πi

∫X

eFT =2 sin n

2ε

ε(2.27)

Now let us compute the T -equivariant index of the Dirac operatoron S2 twisted by the Ln bundle using (1.31) and the localization tothe fixed points. We need to multiply the contributions from the northand south poles by the the T -equivariant A class of the tangent bundle,which for both fixed points is given by AT (TX)|θ=0 = AT (TX)|θ=π =

iεeiε/2−e−iε/2 = ε

2 sin ε2, and we find that the equivariant index

indT (D) =1

−2πi

∫X

AT (TX)eFT =sin n

2ε

sin 12ε

(2.28) eq:Kirillov

is the SU(2) character of irreducible representation with highest weightn− 1 in the conventions where 1 is the fundamental representation.

2It is easy to see that this is correct assignment by checking that for the tangentbundle n = 2 and the fiber at the north pole is acted with weight 1 while the fiberat the south pole is acted with weight −1 because of the opposite orientation.

LECTURES ON EQUIVARIANT LOCALIZATION 11

3. Equivariant index theory

3.1. Kirillov character formula. Let G be a compact simple Liegroup. The Kirillov character formula equates that the T -equivariantindex of the Dirac operator indT (D) on the G-coadjoint orbit of theelement λ+ρ ∈ g∗ with the character χλ of the G irreducible represen-tation with highest weight λ. Here ρ is the Weyl weight ρ =

∑α>0 α.

The character χλ is a function g→ C determined by the representa-tion of the Lie group G with highest weight λ as

χλ : X 7→ trλ eX , X ∈ g (3.1)

Let Xλ be an orbit of the co-adjoint action by G on g∗. Such orbitis specified by an element λ ∈ t∗/W where t is the Lie algebra ofthe maximal torus T ⊂ G and W is the Weyl group. The co-adjointorbit Xλ is a homogeneous symplectic G-manifold with the canonicalsymplectic structure ω defined at point x ∈ X ⊂ g∗ on tangent vectorsin g by the formula

ωx(•1, •2) = 〈x, [•1, •2]〉 •1, •2 ∈ g (3.2)

(The converse is also true: any homogeneous symplectic G-manifold islocally isomorphic to a coadjoint orbit of G or central extension of it).

The minimal possible stabilizer of λ is the maximal abelian sub-group T ⊂ G, and the maximal co-adjoint orbit is G/T . Such orbit iscalled full flag manifold. The real dimension of the full flag manifoldis 2n = dimG− rkG, and is equal to the number of roots of g. If thestabilizer of λ is larger group H, such that T ⊂ H ⊂ G, the orbit Xλ iscalled degenerate flag manifolds G/H. A degenerate flag manifold is aprojection from the full flag manifold with fibers isomorphic to H/T .

Flag manifolds are equipped with natural complex and Kahler struc-ture. There is expliclitly holomorphic realization of the flag mani-folds as a complex quotient GC/PC where GC is the complexificationof the compact group G and PC ⊂ GC is a parabolic subgroup. Letg = g−⊕ h⊕ g+ be the standard decomposition of g into the Cartan halgebra and the upper triangular g+ and lower triangular g− subspaces.

The minimal parabolic subgroup is known as Borel subgroup BC, itsLie algebra is conjugate to h⊕g+. The Lie algebra of generic parabolicsubgroup PC ⊃ BB is conjugate to the direct sum of h⊕g+ and a propersubspace of g−.

Full flag manifolds with integral symplectic structure are in bijectionwith irreducible G-representations πλ of highest weight λ

Xλ+ρ ↔ πλ (3.3)

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This is known as Kirillov correspondence of the geometric representa-tion theory.

Namely, if λ ∈ g∗ is a weight, the symplectic structure ω is integraland there exists a line bundle L→ Xλ with the unitary connection ofcurvature ω. The line bundle L → Xλ is acted by the maximal torusT ⊂ G and we can study the T -equivariant geometric objects. TheKirillov-Berline-Getzler-Vergne character formula equates the equivari-ant index of the Dirac operator D twisted by the line bundle L→ Xλ+ρ

on the co-adjoint orbit Xλ+ρ with the character χλ of irreducible rep-resentation of G with the highest weight λ

indT (D)(Xλ+ρ) = χλ (3.4)

This formula can be easily proved using the Atiyah-Singer equivariantindex formula

indT (D)(Xλ+ρ) =1

(−2πi)n

∫Xλ+ρ

chT (L)AT (TX) (3.5)

and Atiyah-Bott formula to localize the integral over Xλ+ρ to the setof fixed points XT

λ+ρ.The localization to XT

λ+ρ yields the Weyl formula for the character.Indeed, the stabilizer of λ + ρ, where λ is a dominant weight, is theCartan torus T ⊂ G. The co-adjoint orbitXλ+ρ is the full flag manifold.The T -fixed points are in the intersection Xλ+ρ ∩ t, and hence, the setof the T -fixed points is the Weyl orbit of λ+ ρ

XTλ+ρ = Weyl(λ+ ρ) (3.6)

At each fixed point p ∈ XTλ+ρ the tangent space TXλ+ρ|p is generated

by the root system of g. The tangent space is a complex T -module⊕α>0Cα with weights α given by the positive roots of g. Consequently,the denominator of the AT gives the Weyl denominator, the numera-tor of the AT cancels with the Euler class eT (TX) in the localizationformula, and the restriction of the chT (L) = eω is ew(λ+ρ)

1

(−2πi)n

∫Xλ+ρ

chT (L)A(TX) =∑w∈W

eiw(λ+ρ)ε∏α>0(e

12iαε − e− 1

2iαε)

(3.7)

We conclude the localization of the equivariant index of the Dirac op-erator on Xλ+ρ twisted by the line bundle L to the set of the fixedpoints XT

λ+ρ is precisely the Weyl formula for the character.The Kirillov correspondence between the index of the Dirac operator

of L→ Xλ+ρ is closedly related to the Borel-Weyl-Bott theorem.

LECTURES ON EQUIVARIANT LOCALIZATION 13

Let BC be a Borel subgroup of GC, TC be the maximal torus, λ anintegral weight of TC. A weight λ defines a one-dimensional represen-tation of BC by pulling back the representation on TC = BC/UC whereUC is the unipotent radical of BC.3 Let Lλ → GC/BC be the associatedline bundle, and O(Lλ) be the sheaf of regular local sections of Lλ. Forw ∈WeylG define action of w on a weight λ by w ∗ λ := w(λ+ ρ)− ρ.The Borel-Weyl-Bott theorem is

H l(w)(GC/BC,O(Lλ)) =

Rλ, w ∗ λ is dominant0, w ∗ λ is not dominant

(3.8)

where Rλ is the irreducible G-module with the highest weight λ. Weremark that if there exists w ∈ WeylG such that w ∗ λ is dominantweight, then w is unique. There is no w ∈ WeylG such that w ∗ λ isdominant if in the basis of the fundamental weights Λi some of thecoordinates of λ+ ρ vanish.

Example 5. For G = SU(2) the GC/BC = CP1, integral weight of TCis an integer n ∈ BZ, and the line bundle Ln is the O(n) bundle overCP1. The Weyl weight is ρ = 1.

The weight n ≥ 0 is dominant and theH0(CP1,O(n)) is the SL(2,C)module of highest weight n (in the basis of fundamental weights ofSL(2)).

For weight n = −1 the H i(CP1,O(−1)) is empty for all i as there isno Weyl transformation w such that w ∗ n is dominant (equivalently,because ρ+ n = 0).

For weight n ≤ −2 the w is the Z2 reflection and w ∗n = −(n+ 1)−1 = −n− 2 is dominant and H1(CP1,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 theDirac complex on G/B is that Dirac operator is precisely the Dolbeaultoperator shifted by the square root of the canonical bundle

S+(X) S−(X) = K12

∑(−1)pΩ0,p(X) (3.9)

and consequentlyind(Xλ+ρ, D ⊗ Lλ+ρ) = ind(GC/BC, ∂ ⊗ Lλ) (3.10)

Borel-Bott-Weyl theorem has generalization for incomplete flag man-ifolds. Let PC be a parabolic subgroup of GC with BC ⊂ PC and letπ : GC/BC → GC/PC denote the canonical projection. Let E → GC/PCbe a vector bundle associated to an irreducible finite dimensional PC

3The unipotent radical UC is generated by g+

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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 aone-dimensional BC-module L and

Hk(GC/PC,O(E)) = Hk(GC/BC,O(L))

3.2. Index of a complex. Let Ek be vector bundles over a manifoldX. Let T be a compact Lie group acting on X and the bundles Ek.The action of T on a bundle E induces canonically linear action on thespace of sections Γ(E). For t ∈ T and section φ ∈ Γ(E) the action is

(tφ)(x) = tφ(t−1x), x ∈ X (3.11) eq:section-action

Let Di be linear differential operators compatible with the T action,and let E be the complex

E : Γ(E0)D0→ Γ(E1)

D1→ Γ(E2)→ . . . (3.12)

Since Di are T -equivariant operators, the T -action on Γ(Ei) inducesthe T -action on the cohomology H i(E). The equivariant index of thecomplex E is the virtual character

indT (D) : t→ C (3.13)

defined byindT (D) =

∑k

(−1)k trHk(E) t (3.14)

3.3. Atiyah-Singer index formula. If the set XT of T -fixed pointsis discrete, the Atiyah-Singer equivariant index formula is

indT (D) =∑x∈XT

∑k(−1)k chT (Ek)|xdetνx(1− t−1)

(3.15)

Example 6. The equivariant index of ∂ : Ω0,0(X) → Ω0,1(X) on X =C〈x〉 under the T = U(1) action x 7→ t−1x where t ∈ T is the funda-mental character is contributed by the fixed point x = 0 as

indT (∂) =1− t

(1− t)(1− t)=

1

1− t=∞∑k=0

tk (3.16)

where denominator is the determinant of the operator 1 − t over thetwo-dimensional normal bundle to 0 ∈ C spanned by the vectors ∂xand ∂x with T eigenvalues t and t. In the numerator, 1 comes from theequivariant Chern character on the fiber of the trivial line bundle atx = 0 and −t comes from the equivariant Chern character on the fiberof the bundle of (0, 1) forms dx.

LECTURES ON EQUIVARIANT LOCALIZATION 15

We can compare the expansion in power series in tk of the index withthe direct computation. The terms tk for k ∈ Z≥0 come from the localT -equivariant holomorphic functions xk which span the kernel of ∂ onC〈x〉. The cokernel is empty by the Poincare lemma.

Example 7. Now let X = CP1 and Ln be the holomorphic line bundlewith first Chern class c1 = n. Let ∂ : Ω0,0(Ln) → Ω0,1(Ln). Let x bethe local coordinate in the patch around x = 0 and x = x−1 be thelocal coordinate in the patch around x = ∞. Let group T = U(1) actequivariantly on Ln → X by

x 7→ t−1x (3.17)

with a certain lift on Ln: define the action of T on the fiber of Ln atx = 0 to be trivial, and consequently T acts on sections in Ω0,1(Ln)by sending the section φx(x, x)dx to tφx(tx, tx)dx. Therefore, on thefibers of Ln and Ln ⊗ T ∗X at x = 0 the T acts with characters 1 and t.

There are two fixed points x = 0 and x =∞. The x = 0 contributesas in the previous example by

indT (D,Ln)|x=0 =1

1− t(3.18)

At x = ∞ we need to know the action of T on the fibers of Ln andLn ⊗ T ∗X . Let x = x−1. A section φ ∈ Ω0,0(Ln) in the x coordinatepatch is related to φ in the x coordinate patch by

φ(x) = xnφ(x) (3.19)

A constant section 1 in the patch x is acted in the same way as sectionxn in the patch x, that is with weight tn. Therefore, the fiber of Ln atx = 0 is acted with tn. Consequently, the contribution to the index ofthe point x = 0 is

indT (D,Ln)|x=0 =tn

1− t−1(3.20)

The total index is

indT (D,Ln)|x=0+indT (D,Ln)|x=0 =1

1− t+

tn

1− t−1=

∑n

k=0 tk, n ≥ 0

0, n = −1,

−t−1∑−n−2

k=0 t−k, n < −1

(3.21)We can check agains the direct computation. Assume n ≥ 0. The

kernel of D is spanned by n + 1 holomorphic sections of O(n) of theform xk for k = 0, . . . , n, the cokernel is empty by Riemann-Roch. The

16 VASILY PESTUN

section xk is acted by t ∈ T with weight tk. Therefore

indT (D,Ln) =n∑k=0

tk (3.22)

Example 8. Let X = CPm be defined by the projective coordinates(x0 : x1 : · · · : xm) and Ln be the line bundle O(L) = O(n). LetT = U(1)(m+1) act on the bundle X by

(x0 : x1 : . . . xm) 7→ (t−10 x0 : t−1

1 x1 : · · · : t−1m xm) (3.23)

and by tnk on the fiber of the bundle E in the patch around k-th fixedpoint xk = 1, xi 6=k = 0. We find the index as a sum of contributionsfrom m+ 1 fixed points

indT (D) =m∑k=0

tnk∏j 6=k(1− (tj/tk))

(3.24) eq:fixed

For n > 0 the index is a homogeneous polynomial in C[t−10 , . . . , t−1

m ]of degree n representing the character on the space of holomorphicsections of O(n) bundle over CPm.

indT (D) =

sn(t0, . . . , tm), n ≥ 0

0, −m ≤ n < 0

(−1)mt−10 t−1

1 . . . t−1m s−n−m−1(t−1

0 , . . . , t−1m ), n ≤ −m− 1

(3.25) eq:cp-answerwhere sn(t0, . . . , tm) are complete homegeneous symmetric polynomi-als. This result can be quickly obtained from the contour integralrepresentation of the sum (3.24)

1

2πi

∮C

dz

z

zn∏mj=0(1− tj/z)

=m∑k=0

tnk∏j 6=k(1− (tj/tk))

, (3.26)

If n ≥ −m we pick the contour of integration C to enclose all residuesz = tj. 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 residueat z = ∞ which is computed by expanding all fractions in inversepowers of z, and is given by complete homogeneous polynomial in ti ofdegree n.

If n < −m we assume that the contour of integration is a smallcircle around the z = 0 and does not include any of the residues z = tj.Summing the residues outside of the contour, and taking that z = ∞does not contribute, we get (3.24) with the (−) sign . The residue atz = 0 contributes by (3.25).

LECTURES ON EQUIVARIANT LOCALIZATION 17

4. Equivariant cohomological field theories

Often the path integral for supersymmetric field theories can be rep-resented in the form

Z =

∫Xα (4.1)

where X is the (infinite-dimensional) superspace of fields of the theoryover which we integrate in the path integral the measure α closed withrespect to a supercharge operator Q

Qα = 0 (4.2)

4.1. Four-dimensional gauge theories. The Donaldson-Witten twistof the N = 2 supersymmetric off-shell vector multiplet for a theory ona four-manifold M produces the topological gauge multipletQAµ = ψµQψµ = DAφ

Qχ+

µν = H+µν

QHµν = −[φ, χµν ]

Qφ = η

Qη = −[φ, φ]Qφ = 0

(4.3)The operator Q can be given the geometric interpretation of the

equivariant Cartan differential for the action of group G of gauge trans-formations on the space A(M, g)×ΠΩ2+(M, g)×Ω0(M, g) where A isthe affine space of connections Aµ on a principal G-bundle over M , theΠΩ2+(M, g) is the space of the g-valued fermionic self-dual forms χ+

µν

onM and Ω0(M, g) is the space of g-valued functions φ onM . The Liealgebra of G is parameterized by the g-valued function φ on M . Thefields ψµ, H+

µν , η represent respectively the Q-differential of Aµ, χ+µν , φ

In the gauge theory path integral we integrate over φ in Lie(G) andover the fields of the gauge-fixing multiplet, the fermionic g-valuedfields c, c and the bosonic g-valued field b. It turns out that the field φis naturally interpreted as the differential of the of the fermionic ghostc, 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 overΠΩ0(M, g)× ΠΩ0(M, g). Finally, the gauge theory path integral is

Z =∑

k∈H4(M,Z)⊕r

qk

∫X=Ak(M,g)×ΠΩ2+(M,g)×Ω0(M,g)×ΠΩ0(M,g)×ΠΩ0(M,g)

α

(4.4)where we sum over topologically equivalence classes of the G-bundlesonM labelled by an r-tuple of integers k = (k1, . . . , kr) with generatingparameters q = (q1, q2, . . . , qr) if G is a product of compact simple Liegroups G = G1 × · · · ×Gr.

18 VASILY PESTUN

In addition to the action by the gauge group G the fields mighttransform under an additional global symmetry T and we can makeall geometrical objects to be T -equivariant. The integration over X byAtiyah-Bott localization formula (here we take its super and infinite-dimensional version) reduces to the sum over the set of fixed points X T

that we assume to be discrete

Z =∑k

qk∑p∈XT

i∗pα

eT (TA(M, g)p × ΠΩ2+(M, g)× ΠΩ0(M, g))(4.5)

The T -equivariant Euler classes of Ω0(M, g)× ΠΩ0(M, g) cancel.The Euler class eT (TA(M, g)p × ΠΩ2+(M, g) × ΠΩ0(M, g)) is most

conveniently computed from the Chern character chT (TA(M, g)p ×ΠΩ2+(M, g) × ΠΩ0(M, g)) for which we can use Atiyah-Singer indextheorem.

Namely we consider the T -equivariant character for the complex ofthe self-dual equations

E : Ω0(M, g)d→ Ω1(M, g)

d+→ Ω2+(M, g) (4.6)

and notice that because of the reversed parity we have

eT (E) =1

eT (TA(M, g)p × ΠΩ2+(M, g)× ΠΩ0(M, g))(4.7)

4.2. The Ω-background. Now we consider the concrete example ofthe gauge theory in so called Ω-background on M = R4 = C2

〈z1,z2〉.Since M is non-compact, it is convenient to consider to consider the

topological gauge theory complex build from the principal G-bundle onM with a fixed framing at ∞ ∈ M . The gauge transformation of thefiber at ∞ ∈M generate the equivariant action by the group G whichwe can restrict to its maximal abelian subgroup TG. Therefore, weconsider the group T〈ε1,ε2,a〉 = U(1)2× TG〈a〉 acting by z1 → t−1

1 z1, z2 →t−12 z2 where t1 = eiε1 , t2 = eiε2 and by ad(g) transformations on the gvalued fields.

First we compute the contribution from the trivial fixed point: theprincipal G-bundle has trivial topology, k = 0, the gauge connection istrivial and all other fields vanish.

The equivariant Euler class of E is conveniently found from the equi-variant Chern class of E , or the index of the self-dual complex.

LECTURES ON EQUIVARIANT LOCALIZATION 19

We find the index

indT (E) ≡ chT (E) =∑α>0

ω=eiα·a

ω + ω + t1t2ω + t1t2ω − (t1ω + t1ω + t2ω + t2ω)

(1− t1)(1− t1)(1− t2)(1− t2)

=∑α>0

ω=eiα·a

ω1

(1− t1)(1− t2)+ ω

1

(1− t1)(1− t2)(4.8)

The above line does not include contribution from the subspace of gwith the zero adjoint weights, that is from the Cartan of g as such con-tribution is a-independent and is often not-interesting in the physicalapplications.

Then Euler class iseT (E) =

∏α>0

Gε1,ε2(α · ai)Gε1,ε2(ε1 + ε2 − α · ai) (4.9)

whereGε1,ε2(x)

reg=

∏n1≥0,n2≥0

(x+ n1ε1 + n2ε2) (4.10)

is a properly regularized Weierstrass infinite product formula for theinverse of the double-gamma function of Barnes.

Institute for Advanced Study, PrincetonE-mail address: pestun@ias.edu