ESI The Erwin Schr�odinger International Boltzmanngasse 9Institute for Mathematical Physics A-1090 Wien, AustriaNotes on Equivariant LocalizationAnton Alekseev

Vienna, Preprint ESI 744 (1999) August 30, 1999Supported by Federal Ministry of Science and Transport, AustriaAvailable via http://www.esi.ac.at

Notes on Equivariant LocalizationAntonAlekseevInstitutionen f�or Teoretisk Fysik, Uppsala Universitet, Box 803, S-751 08, Uppsala,SwedenAbstract. We review the localization formula due to Berline-Vergne and Atiyah-Bott,with applications to the exact stationary phase phenomenon discovered by Duistermaat-Heckman. We explain the Weil model of equivariant cohomology and recall its relationto BRST. We show how to quantize the Weil model, and obtain new localization for-mulas which, in particular, apply to Hamiltonian spaces with group valued momentmaps.1 IntroductionThe purpose of these lecture notes is to present the localization formulas forequivariant cocycles. The localization phenomenon was �rst discovered by Duis-termaat and Heckman in [DH], and then explained in the works of Berline-Vergne[BV] and Atiyah-Bott [AB]. The main idea of the localization formulas is simi-lar to the residue formula: a multi-dimensional integral is evaluated exactly bysumming up a number of the �xed point contributions.In Section 2 we review the localization formula of [BV] and [AB]. We usean elementary example of the sphere S2 as an illustration. Then, we outlinethe relation between the localization formulas and Hamiltonian Mechanics, andrecover the Duistermaat-Heckman formula [DH].In Section 3 we discuss the relations between the localization formulas and thegroup actions. In the case of the Duistermaat-Heckman formula, localization isintimately related to the symmetry group of the underlying Hamiltonian system.In particular, we compare the equivariant di�erential to the BRST di�erential.In Section 4 we explain how to quantize the equivariant cohomology. ThisSection is based on the papers [AMM], [AM], [AMW1] and [AMW2]. We end upby presenting the new localization formula which is derived in [AMW2]. Somesimple applications of this new formula can be found in [P]. Section 4 is basedon the joint works with A.Malkin, E.Meinrenken and C.Woodward.These notes do not touch upon various applications of localization formulasin Physics. Usually, one proceeds by extrapolating the localization phenomenonto path integrals. Some of the most exciting examples of this approach can befound in [MNP], [W2], [G], [BT], [MNS]. In fact, [W2] was the original motivationfor the formulas of Section 3.I am grateful to the organizers and participants of the 38th SchladmingWinter School for the inspiring atmosphere!

2 AntonAlekseev2 Localization FormulasIn this Section we review the localization formula due to Berline-Vergne [BV] andAtiyah-Bott [AB]. It is then used to derive the exact stationary phase formuladue to Duistermaat and Heckman [DH]. The presentation is illustrated at theelementary example of sphere S2.2.1 Stationary phase methodIn this section we recall the stationary phase method. It applies when one isinterested in the asymptotic behavior at large s of the integralI(s) = Z 1�1 dxeisf(x)g(x): (1)Here we assume that functions f(x) and g(x) are real, and su�ciently smooth.At large s > 0 the leading contribution into the integral (1) is given by theneighborhood of the critical points of f(x), where its derivative in x vanishes.Let x0 be such a critical point. Then, one can approximate f(x) near x0 by the�rst two terms of the Taylor series,f(x) = f(x0) + 12f 00(x0)(x� x0)2 + : : : ;where : : : stand for the higher order terms.The leading contribution of the critical point x0 into the integral I(s) is givenby a simpler integralI0(s) = g(x0)eisf(x0) Z 1�1 dxe i2 sf 00(x0)(x�x0)2 :This integral is Gaussian, and can be computed explicitly,I0(s) = g(x0)ei(sf(x0)+"�4 )� 2�sjf 00(x0)j� 12Here " is the sign on the second derivative f 00(x0).A similar formula holds for multi-dimensional integrals,I(s) = Z dnxg(x)eisf(x): (2)Again, the leading contrinution into the asymptotics at large s is given by thecritical points of f(x), where its gradient vanishes rf = 0. At the critical pointx0 one can expand f(x) into the Taylor series,f(x) = f(x0) + 12 nXi;j=1 f 00ij(x0)(x � x0)i(x� x0)j + : : : ; (3)

Notes on Equivariant Localization 3where f 00ij = @2f@xi@xj :We assume that the critical point is non-degenerate, that is, the matrix f 00ij isinvertible. Then, the leading contribution of x0 into the integral I(s) is of theform, I0(s) = g(x0)eisf(x0)�2�s �n2 ei� �4j det(f 00(x0))j 12 : (4)Here � = �+ � �� is the signature of the matrix f 00ij, �+ and �� are numbers ofpositive and negative eigenvalues, respectively.In general, one can have several critical points. Then, on can add the leadingcontributions (4) to obtain the approximate answer for I(s),I(s) � �2�s �n2 Xi g(xi)eisf(xi) ei�i �4j det(f 00(xi))j 12 (5)Of course, there is no reason for the right hand side to be the exact answer forI(s). But sometimes this is the case! Such a situation is called exact stationaryphase, and will be studied in these notes.Example: sphere S2 The simplest example of the exact stationary phasephenomenon is the computation of the following integral. Consider the unitsphere S2 de�ned by equation x2 + y2 + z2 = 1. We choose g(x; y; z) = 1 andf(x; y; z) = z. Then, the integral I(s) is of the form,I(s) = ZS2 dA eisz; (6)where dA is the area element normalized in the standard way, RS2 dA = 4�.The critical points of the function f(x; y; z) = z are the North and the Southpoles of the sphere. At both points one can use x and y as local coordinates toobtain, z � 1� 12(x2 + y2)near the North pole, and z � �1 + 12(x2 + y2)near the South pole. Thus, for the stationary phase approximation one obtains,I(s) � 2�s (�ieis + ie�is) = 4� sin(s)s : (7)Here we have used that at both North and South poles det(fij) = 1, and that�N = �2 and �S = 2.

4 AntonAlekseevNow we can compare the `approximate' result (7) with the exact calculation.It is convenient to use polar angles 0 < � < �; 0 < � < 2�. Then, the coordinatefunction z = cos(�), and the area element is dA = d cos(�)d�. The simplecalcuation gives,I(s) = Z d cos(�)d�eis cos(�) = 2� eis � e�isis = 4� sin(s)s : (8)This expression coincides with the stationary phase result (7).In the following sections we shall see that the equality of the exact andapproximate results (7) amd (8) is not a coincidence. In fact, this is the symplectexample of equivariant localization.2.2 Equivariant cohomologyStokes's theorem and residue formula The main tool in proving the lo-calization formula will be the generalization of the Stokes's integration formula.The latter states that given an exact di�erential form �, � = d�, its integralover the domainD can be expressed as an integral of � over the boundary of D,ZD d� = Z@D �: (9)As a warm up excercise we prove the standard residue formula using theStokes's formula (9). Given a function f(z) analytic on the complex plane withthe exception of some �nite number of poles, its integral over a closed contourC is given by the sum of residues at the poles inside C,12�i ZC f(z)dz =Xi reszi f: (10)We naturally choose � in the form,� = 12�i f(z)dz;the domain D is the interior of C, and its boundary is C. The form � is givenby equation, � = d� = 12�id(f(z)dz) = 12�i (�@f)d�z ^ dz;where �@f is the partical derivative in �z. The function f(z) is analytic. Hence, �vanishes everywhere except for the poles. We conclude, that � is a distributionsupported at some number of points. Such a distribution is a sum of �-functionsand its derivatives. The only terms which contribute into the integral of � overD are �-functions at the poles. They give rise to the residues,�@ reszi f2�i(z � zi) = (reszi f)�(z � zi):

Notes on Equivariant Localization 5Now we use the Stokes's formula,12�i ZC f(z)dz = ZDXi (reszi f)�(z � zi) =Xi reszi f;and recover the residue formula.S1-action and �xed points The derivation of the localization formula requiresmore structure on the integration domain. In particular, the notion of symmetryplays an important role. We assume that our symmetry is continuous. In partic-ular, this may be the action of the circle group S1, which is our main example inthis Section. For instance, in the case of S2 there is an action of S1 by rotationsaround the z-axis.We always choose the integration domain to be a compact manifoldM with-out boundary (as in the case of S2). The S1-action de�nes a vector �eld v = @=@�on M . Zeroes of v correspond to �xed points of the circle action. For simplicity,we assume that all the �xed points are isolated. This is only possible if the di-mension of the manifold is even, n = 2m. Given such a �xed point x0 one canwrite the action near this point asxi(�) = xi0 +Rij(�)(x� x0)j + : : : ;where : : : stand for higher order terms in x � x0. It is easy to see that onecan lenearize the action (drop higher order terms). The matrix R gives a linearrepresentation of S1, R(�1)R(�2) = R(�1 + �2);and satis�es condition R(2�) = id. By the appropriate choice of the basis sucha matrix can always be represented as a direct sum of 2� 2 blocks, each blockof the form � cos(��) sin(��)� sin(��) cos(��)� ;where � is an integer. We can denote the corresponding local coordinates byxi; yi where i = 1 : : :m. In these local coordinates the vector �eld v has theform, v = mXi=1 �i �xi @@yi � yi @@xi� :The integers �i are called indices of the vector �eld v at the point x0.In fact the indices �i are de�ned up to a sign: the ip of coordinates xi andyi changes the sign of the corresponding index. In what follows we shall need aproduct of all indices corresponding to the given �xed point, �1 : : : �m. It is wellde�ned is the tangent space at the �xed point is oriented: one should choose thecoordinate system (x1; y1; : : : ; xm; ym) with positive orinetation. This conditiondetermines the product of indices in a unique way. In particular, if the manifoldis oriented, the products of indices at �xed points are well de�ned.

6 AntonAlekseevIn the following we shall use that one can always choose an S1-invariantmetric on M . Indeed, given any metric, one can always average it over the S1-action. In particular, this metric at the �xed point x0 can always be chosen inthe form, g = mXi=1(dx2i + dy2i ): (11)Equivariant di�erential Now we are ready to de�ne the equivariant di�eren-tial, and equivariant cohomology.We de�ne the space of equivariant forms onMas S1 invariant di�erential fomrs with values in functions of one variable, whichwe denote by �. Typically, such a di�erential form is a polynomial,�(�) = NXs=0�s�s;where �s are S1-invariant di�erential forms. We shall also need equivariant formswith more complicated �-dependence.Sometimes it is convenient to decompose equivariant forms according to thedegree, �(�) = nXj=0�j(�);where �j(�) is a form of degree j which takes values in functions of �.The di�erential on the space of equivariant forms is de�ned by formula,dS1 = d + i��v; (12)where �v is the contraction with respect to the vector �eld v. One can assign toparameter � degree 2 in order to make the equivariant di�erential homogeneous.Unfortunately, this arrangement is only meaningful for equivariant di�erentialforms polynomial in �.It is the basic property of the di�erential (12) that it squares to zero on thespace of equivariant forms. Indeed,d2S1 = (d + i��v)2 = i�(d�v + �vd) = i�Lv;where we have used Cartan's formula for Lv. The Lie derivative Lv vanishes onequivariant forms, and so does d2S1 .Using the di�erential (12) one can de�ne equivariantly closed forms, dS1� =0, and equivariantly exact forms � = dS1�. Because d2S1 = 0, equivariantly exctforms are automatically equivariantly closed, and one can de�ne equivariantcohomologyHS1(M ) as the quotient of the space of (equivariantly) closed formsby the space of (equivariantly) exact forms. If �(�) is an equivariant cocycle, itsatis�es the closedness condition,(d + i��v)�(�) = 0;

Notes on Equivariant Localization 7which implies a number of equations for the forms �k(�),d�k�2(�) + i��v�k(�) = 0: (13)Note that this recurrence relation has step 2. Hence, odd and even degree partsof an equivarient cocycle are also equivariant cocycles. If the manifoldM is evendimensional, the closedness condition relates the top degree component �n(�)and the function �0(�). We shall see that exactly this relation is used in thelocalization formula.The Stokes' integration formula generalizes to equivariantly exact forms. In-deed, ZD(d + i��v)� = Z@D �;where RD �v� = 0 because the integrand has a vanishing top degree component(at least one degree is eaten up by �v). In particular, if the integration domainhas no boundary, the integral of any equivariantly exact form vanishes, and theintegration map descends to equivariant cohomology. That is, given a class [�] 2HS1(M ) one can choose any representative � in integrate it over M . The resultis a function of � which is independent of the representative: the representativesdi�er by an exact form, and the integral of an exact form vanishes.The localization formula is a tool of computing integral of equivariant co-cycles in terms of �xed points. This formula was discovered by Belrine-Vergne[BV] and by Atiyah-Bott [AB]. For an equivariant cocycle �(�), its integral overM is given by, ZM �(�) = �2�i� �n2 Xp (�0(�))(xp)�p1 : : : �pm ; (14)where the index p labels �xed points of the circle action (we assume that all ofthem are isolated), and �p1 ; : : : ; �pm are indices of the p's �xed point.Note that the integral on the left hand side of (14) depends only on the topdegree component �n(�) of the cocycle �(�). At the same time the right handside is expressed in terms of the zero degree component �0(�). This is possiblebecause �n(�) and �0(�) are related by the recurrence relations (13) expressingclosedness of �(�).Also note that even if the equivariant form �(�) is smooth at � = 0, the righthand side of (14) contains the divergent factor ��n=2. The singulariy at � = 0 iscanceled by the sum of contributions of �xed points, which has a zero of degreen=2 at � = 0. This idea leads to reside formulas [JK].2.3 Proof of localization formulaIn this section we give a proof of the localization formula (14). This proof wassuggested by Witten in [W1]. The idea is as follows. Choose an S1-invariantmetric g on M , with behavior at �xed points given by (11), and de�ne a 1-form = g(v; �);

8 AntonAlekseevsuch that �u = g(v; u) for any vector �eld u on M . The form is S1-invariant(because the metric is S1-invariant). De�ne an equivariantly exact form�(�) = dS1 = d + i��v = d + i�g(v; v):Note that near a �xed point is of the form � �12 mXk=1�k(xkdyk � ykdxk);and the form �(�) is given by�(�) � � mXk=1 �kdxk ^ dyk + i�2 mXk=1�2k(x2k + y2k):Let us consider the equivariant formeis�(�) � 1 = 1Xk=1 (is)kk! (dS1 )k = dS1 1Xk=1 (is)kk! (dS1 )k�1! :It is equivariantly exact, and hence,ZM �(�) = ZM �(�)eis�(�) (15)for any value of the parameter s. In particular, if � > 0, one van consider the limits ! +1. On one hand, the asymptotics of the integral (15) can be computedusing the stationary phase method. But on the other hand, the answer does notdepend on s. Hence, it is su�cient to extract the term in the asymptotics whichdoes not depend on s, and this will be the exact answer for the integral!At large s the form eis�(�) = eisd �s�g(v;v)is exponentially small everywhere except for the small neighborhoods of the �xedpoints where v = 0 and g(v; v) = 0. So, the �xed points are at the same time thecritical points which give contributions into the stationary phase asymptotics.The leading contribution in s of the critical point xp is given by(�0(�))(xp) mYk=1�is�pk Z dxkdyke� s�2 P(�ik)2(x2k+y2k)� (16)Here we have used the fact that the 2-form isd which enters the integrand isproportional to s, and, hence, it gives the leading contribution into the integra-tion measure. The integral in (16) is Gaussian. It yields�2�i� �m (�0(�))(xp)�p1 : : : �pmwhich is indeed independent of s. Summing up these contributions for all �xedpoints xp we obtain the localization formula (14).

Notes on Equivariant Localization 92.4 Duistermaat-Heckman formulaPerhaps, the most well-known application of the localization formulas is theDuistermaat-Heckman formula in symplectic geometry [DH]. In fact, it was dis-covered before the general localization principle was formulated.The framework is as follows: we have a closed 2-form !,d! = 0;which satis�es the nodegeneracy condition. That is,�u! = 0for some vector �eld u onM implies u = 0. The pair (M;!) is called a symplecticmanifold. A standard example is R2m with coordinates pi and qi and the 2-form! = mXi=1 dpi ^ dqi:A vector �eld v is called Hamiltonian if there exists a function H such that�v! + dH = 0:The function H is called the Hamiltonian of v. A symplectic manifold is alwayseven dimensional (otherwise ! is necessarily degenerate), and has the volumeform L = !mm! = [e! ]topcalled the Liouville form. The volume form L is invariant with respect to allHamiltonian vector �elds.Now assume that M is compact, and carries a circle action. In addition, letthe corresponding vector �eld v be Hamiltonian, with HamltonianH. Then, onecan de�ne the following integral,I(�) = ZM L ei�H ; (17)which is called the Duistermaat-Heckman integral, and can be evaluated usinglocalization theorm.First, we de�ne the equivariant extension of the symplectic form !,!(�) = ! + i�H:It is an equivariantly closed form,dS1!(�) = (d + i��v)(! + i�H) = d! + i�(�v! + dH) = 0:Here we have used closedness of ! and the de�nition of the Hamiltonian vector�eld.

10 AntonAlekseevNext, we de�ne an equivarinat Liouville form,L(�) = e!(�) = ei�H mXk=1 !kk! ;where the sum terminates because higher powers of ! vanish. The form L(�) isalso equivariantly closed, and, moreover,ZM L(�) = Z L ei�H = I(�):Now we apply the localization formula (14) to the left hand side to obtain theDuistermaat-Heckman formula,I(�) = �2�i� �m Xp ei�H(xp)�p1 : : : �pm : (18)Example: sphere S2. Getting back to the example of the sphere S2, weshow that our observation on exact stationary phase is a particular case of theDuistermaat-Heckman formula.Let us choose the area form on the sphere as the symplectic form. It is clearlyclosed, and non-degenerate. In the polar angles �; � the vector �eld generatingrotations around z-axis is of the form v = @=@�. Then, the Hamiltonian of v isdetermined by equation, �� @@�� d cos(�)d� + dH = 0;which imples H = cos(�) = z (up to a shift by a constant). Thus, the integralwhich we would like to compute,I(�) = ZS2 dA ei�zis the Duistermaat-Heckman integral, and is given by the Duistermaat-Heckmanformula (18).There are two �xed points of the S1-action on S2, the North pole and theSouth pole. The values of the Hamiltonian are given by zN = 1; zS = �1, andthe indices of the S1-action are �N = 1 and �S = �1. Then, formula (18) yieldsI(�) = 2�i� �ei�zN�N � e�i�zS�S � = 4� sin(�)�con�rming the results we obtained before.

Notes on Equivariant Localization 113 Weil model of equivariant cohomologyIn this Section we develop technical tools for dealing with equivariant cohomol-ogy for any compact group G. We begin by introducing the Weil algebra, andthe Weil model of equivariant cohomology. Then we establish the equivalenceto the Cartan model which we used in the case of G = S1. Finally, we give anexpression for the equivariant Liouville form in the Weil model, and introduceequivariant cohomology with generalized coe�cients.For another physicist-oriented review of the subject see [CMR].3.1 Weil algebra and Weil di�erentialGroup actions on manifolds. In general, we shall study the situation whenthe group acting on M is not necessarily a circle S1. Let G be a compact con-nected Lie group, and G be its Lie algebra. In many situations it will be conve-nient to choose a basis feag � G, and the dual basis feag in the space G�. Wedenote by fcab the structure constants in this basis,[ea; eb] = fcabec:If the group G acts on the manifold M , one can associate to each elemente 2 G a fundamental vector �eld on M which we denote by eM . For instance,the vector �elds corresponding to the basis elements ea are (ea)M . The Liederivatives and contractions corresponding to these fundamental vector �eldsact on the space of di�erential forms (M ). We denote them by La and �a,respectively. They satisfy the following relations,[La; �b] = fcab�c; (19)[La; Lb] = fcabLc;[d; �a] = La:Here d is the de Rham di�erential, and [; ] stands for the super-commutator. Forinstance, [d; �a] = d�a + �ad.In a more abstract setting we can say that equations (19) de�ne a super-algebra G with generators La; �a; d. If M is a G-manifold, the space of forms(M ) carries a representation of G, where La are represented by Lie derivatives,�a by contractions, and d by the de Rham di�erential.Weil algebra. In this section we construct a special representation of the al-gebra G called the Weil algebra. It was suggested by H.Cartan in C1 as an`algebraic model' of the space of forms on the classifying space EG.By de�nition, the Weil algebra WG is the product of the symmetric andexterior algebras of the dual space to the Lie algebra of G,WG := SG� ^G�: (20)

12 AntonAlekseevThe algebra SG� is the algebra of polynomials on G. It is convenient to introducegenerators va of SG� corresponding to the basis elements ea 2 G�. The generatorsva correspond to linear functions on G, and naturally commute with each other,vavb � vbva = 0:We denote the generators of the exterior algebra ^G� by ya. They satisfy theanti-commutation relations, yayb + ybya = 0:One can introduce a grading on WG by assigning degree 2 to va and degree 1 toya, W lG = �j+2k=lSkG� ^jG�:Following H.Cartan, one can view WG as a model of the space of forms on EG,such that each ya corresponds to a 1-form, and each va corresponds to a 2-form.There is an action of G on WG de�ned as follows. Operators La are de�nedon generators, La(vc) = �fcabvb; La(yb) = �fcabyc;and extended by the Leibniz rule. In a similar fashion, one de�nes contractions�a, �a(vb) = 0; �a(yb) = �ba:Finally, the Weil di�erential d is de�ned byd(ya) = va � 12fabcybyc; d(va) = �fabcybvc:These formulas have a simple geometric meaning: if one interprets ya as compo-nents of a connection on a principal G-bundle, then the �rst formula,va = dya + 12fabcybyc;is the standard de�nition of the curvature. The second formula,dva + fabcybvc = 0gives the Bianchi identity.

Notes on Equivariant Localization 13Relation to BRST. The di�erential on WG can be written in the form,d = ya(La 1) + (va � 12fabcybyc)�a;where (La 1) is the Lie derivative acting only on the elements of SG�.It is often compared to the BRST di�erential which is de�ned as follows. LetV be a representation of the group G. Then, the BRST di�erential acts on thespace V ^G�, and is given by formula,dBRST = ya(La 1)� 12fabcybyc�a:In the physical interpretation, ya are called ghosts, and denoted by ca. Thedual contractions �a are called anti-ghosts, and denoted by ba. The ghosts andanti-ghosts (generators of ^G� and contractions) satisfy the anti-commutationrelation, ckbl + blck = �kl :If we introduce a special notation for generators of the G-action on V , Ta :=(La 1), we get the standard formula for the BRST di�erential,dBRST = caTa � 12fabccbccba:The main di�erence between the Weil di�erential (and the equivariant di�er-ential) and the BRST di�erential is the extra term va�a in the Weil di�erential.One can interpret it as a BRST di�erential for the abelian Lie algebra G� withgenerators va and ghosts ba := �a. One can say that the Weil di�erential is asum of two BRST di�erentials,d = dGBRST + dG�BRST :3.2 Weil model of equivariant cohomologyIn this section we de�ne the Weil model of equivariant cohomology, and provethat it is equivalent to the Cartan model introduced before. Then, we extendour consideration to equivariant cohomology with generalized coe�cients.De�nition of the Weil model. LetM be a G-manifold. It is our goal to de�nethe space of equivariant forms in the Weil model, and the equivariant di�erentialon this space.Consider the product (M )WG of the space of di�erential forms onM andof the Weil algebra WG. If one interprets WG as the space of di�erential formson EG, the product is naturally interpreted as (M � EG). Both (M ) andWG carry representations of G. Hence, one can de�ne the diagonal action on the

14 AntonAlekseevtensor product. That is, La; �a and d are de�ned as operators on ((M ) SG�)by formulas, La = La 1 + 1 La;�a = �a 1 + 1 �a;d = d 1 + 1 d:We de�ne the space of equivariant forms on M as the basic part of (M )WG, G(M ) := f� 2 (M )WGjLa� = 0; �a� = 0g:In more geometric terms we are looking at the principal G-bundleM �EG! (M �EG)=G;and de�ne G(M ) as the space of basic forms. These are forms which can beobtained as pull-backs of di�erential forms on the quotient space (M �EG)=G.The space of equivariant forms G(M ) carries the action of the combineddi�erential (d 1 + 1 d). One de�nes the equivariant cohomology of M asHG(M ) := H(G(M ); d 1 + 1 d):In the next section we show that this de�nition is equivalent to the de�nition inthe Cartan model which we used in the case of G = S1.Equivalence to the Cartan model. In Section 2 we used a simpler model forS1-equivariant cohomology. This model does not use anti-commuting variablesya, and is called the Cartan model [C2]. A simple transformation which estab-lishes the relation between Weil and Cartan models was suggested by Kalkman[K].Let us de�ne an operator � on the space (M )WG by formula,� := exp(��a ya):The key property of � is�(�a 1 + 1 �a)��1 = 1 �a: (21)In order to prove this equality we use the formula,�X��1 = 1Xj=0 1j! adj(��a ya)X: (22)The calculation gives,ad(��a ya)(�a 1 + 1 �a) = ��a 1;12! ad2(��a ya)(�a 1 + 1 �a) = 0:

Notes on Equivariant Localization 15After substitution to (22) we obtain equation (21).The action of � maps the forms annihilated by (�a 1+ 1 �a) to the formsannihilated by 1 �a. That is, it is mapped to (M )SG� . Taking into accountthat � commutes with the diagonal action of La, we conclude that the space ofequivariant forms G(M ) is mapped to� : G(M )! ((M ) SG�)G:This is the new model of the space of equivariant forms called Cartan model.We already worked with it in the case of G = S1.Next, we compute the equivariant di�erential in the Cartan model. We applyformula (22) to the equivariant di�erential,ad(��a ya)(d 1 + 1 d) = La ya � �a (va � 12fabcybyc);12!ad2(��a ya)(d 1 + 1 d) = �12fcab�c yayb;13!ad2(��a ya)(d 1 + 1 d) = 0:We add all the terms and take into account that (La 1 + 1 La) and 1 �aannihilate the image of the space of equivariant forms. The �nal result for thedi�erential in the Cartan model is quite simple,dG := d 1� �a va:In the case of G = S1 one should put v = �i� to recover the expression (12) fordS1.Equivariant Liouville form. As before, examples of equivariant classes areprovided by symplectic geometry. Let (M;!) be a symplectic manifold, andassume that the G-action onM is Hamiltonian. That is, there is a set of Hamil-tonians, Ha such that 1 �a! = dHa; (23)and LaHb = fcabHc: (24)The collection of functions Ha can be viewed as the moment map H :M ! G�with Ha = hH; eai. The property (24) expresses equivariance of the map H withrespect to the G-action on M and the co-adjoint action on G�.Again, one can de�ne the equivariant extension of the Liouville form in Car-tan model, !(v) := ! +Hava:This form is equivariantly closed,dG!(v) = (d� va�a)(! + vbHb) = d! + va(dHa � �a!) = 0:1 Note that we have changed the sign convention in comparison to the previous Section.

16 AntonAlekseevThe same form in the Weil model is expressed by!W := ��1!(v) = ! � yadHa +Ha(va � 12fabcybyc);where we have used the property �adHb = fcabHc implied by (24).Finally, we introduce the equivariant Liouville forms in Cartan and Weilmodels, L(v) := e!(v) = e! evaHa ;andLW := e!W = exp(!) exp(�yadHa) exp��12Hafabcybyc� exp(vaHa): (25)In the next Section we shall use the expression for LW to intoduce the theoryof `group-valued' Hamltonians.Generalized coe�cients. The last technical ingredient needed in the nextSection is the notion of equivariant cohomology with generalized coe�cients. Itis sometimes convenient to make a Fourier-Laplace transform in the variables vasuch that they become distributions on the space G� supported at the origin,va = � @@�a �0;where �a are linear coordinates on G�, and �0 is the �-function supported at theorigin.Then, it is natural to replace the space of polynomials SG� in the de�nitionof WG by the space of all compactly supported distributions E 0(G�). We denotethe extended Weil algebra byWG := E 0(G�) ^G�;and the corresponding equivariant cohomology by HG(M ).For instance, in the new notations the equivariant Liouville form containsthe �-function supported at the value H of the moment map,LW = exp(!) exp(�yadHa) exp��12Hafabcybyc� �H :The distribution part of LW is not supported at the origin, and de�nes a classin HG(M ).4 Group-valued equivariant localizationIn this Section we explain how to quantize the Weil algebra [AM], and de�ne thegroup valued equivariant cohomology. This leads to the new localization theorem[AMW2], and new moment map theory [AMM].In contrast to the previous sections we only sketch the results. At this stagethe proofs are too involved for these notes. So, refer the reader to the originalpapers.

Notes on Equivariant Localization 174.1 Non-commutative Weil algebraIn this section we introduce the non-commutative counterpart of the Weil lage-bra. In the exposition we follow [AM].Invariant inner product on G. As before, we assume that G is a compactconnected Lie group. In addition, we choose an invariant innder product on theLie algebra G, and suppose that G is a direct product of a compact simply-connected Lie group and a torus.We denote the innder product on G by (�; �). It induces a number of newstructures. First, one can identify G with its dual space G�. The basis feag canbe chosen orthonormal, (ea; eb) = �ab. The corresponding structure constants[ea; eb] = fabcec are anti-symmetric with respect to the permutation of any twoindices. Thus, one can de�ne an element � 2 (^3G)G by formula,� := 16fabcea eb ec:We de�ne the left- and right-invariant vector �elds eLa and eRa on the groupG, and the dual left- and right-invariant 1-forms, �La and �Ra . They satisfy theMaurer-Cartan structure equations,d�La = �12fabc�Lb �Lc ; d�Ra = 12fabc�Rb �Rc :Using the identi�cation ^3G �= ^3G�, one can de�ne a bi-invariant 3-form on G,� := 112fabc�La �Lb �Lc = 112fabc�Ra �Rb �Rc :Finally, we introduce the distributions on G with support at the group unitcorresponding to the vector �elds eLa and eRa ,uLa := �eLa �e; uRa := �eRa �e:Here �e is the �-function supported at the group unit.De�nition of non-commutative Weil algebra. We recall that the Weil al-gebraWG is a tensor product of symmetric and exterior algebras of the space G�.The non-commutative Weil algebra is a tensor product of the non-commutativecounterparts of these algebras. The symmetric algebra is replaced by the uni-versal enveloping algebra U (G) with generators ua and relationsuaub � ubua = fabcuc:The exterior algebra is replaced by the Cli�ord algebra Cl(G) with generatorsxa and relations xaxb + xbxa = �ab:

18 AntonAlekseevHere we use the fact that the basis feag is orthonormal. We denote the non-commutative Weil algebra by WG,WG := U (G) Cl(G):Similar to WG the algebra WG carries a representation of G. The action ofthe LIe derivatives La is de�ned on generators,La(ub) = fabcuc; La(xb) = fabcxc:The contractions �a are given by formulas,�a(ub) = 0; �a(xb) = �ab:Finally, the Weil di�erential has its analog on WG,d(xa) = ua � 12fabcxbxc; d(ua) = �fabcxbuc:These formulas are very simlar to those which de�ne the G-action on WG. Theimportant di�erence is that in the non-commutative algebra WG, the operatorsLa; �a; d are inner derivations,La = ad(ua � 12fabcxbxc); �a = ad(xa); d = ad(xaua � 16fabcxaxbxc):As in the case of WG, one can introduce the non-commutative Weil algebrawith generalized coe�cients,WG := E 0(G) Cl(G):For any G-manifold M one can now de�ne the space of `group-valued' equiv-ariant forms, ((M )WG)basic and ((M ) WG)basic and the `group-valued'equivariant cohomology,HG(M ) := H(((M )WG)basic; d 1 + 1 d);HG(M ) := H(((M ) WG)basic; d 1 + 1 d)It is our main goal to present the localization formulas for classes in HG(M ) andin HG(M ).

Notes on Equivariant Localization 194.2 Group-valued moment mapsIn this section we give examples of equivariant cocycles which give rise to classesin HG(M ). We follow the paper [AMW1]. The idea is to �nd the counterpartof formula (25) for the equivariant Liouville form. We shall see that it naturallyleads to moment maps with values in the Lie group rather than in the dual ofthe Lie algebra.The right hand side of (25) is a product of four factors,exp(!) exp(�yadHa) exp��12Hafabcyayb� �H 2 (M ) E 0(G�) ^G�where ! is a 2-form on M and H is the moment map H : M ! G�. In thegroup-valued case we still need a 2-form !, but the moment map should takevalues in the group G, � : M ! G. The reason is that instead of the space ofdistrinutions on the dual of the Lie algebra E 0(G�) we now have the space ofdistributions on the group E 0(G). Then, the �rst and the last terms in (25) havetheir counterparts, exp(!) and ��.The third term is related to the spinor representation of G. In more detail,choose H = Haea 2 g and de�ne a map � : G! Cl(G) by formula,� (eH ) = exp��12Hafabcxbxc� :If G is a product of a compact simply-connected Lie group and a torus, the map� is well-de�ned, and de�nes the representation of G (see e.g. [BGV]),� (g1g2) = � (g1)� (g2):There are two possible candidates for the role of the second term, exp(�xa���La )and exp(�xa���Ra ). We notice thatexp(�xa���Ra )� (�) = � (�) exp(�xa���La ):Thus, we can choose either left- or right-invariant Maurer-Cartan forms, but weshould position them on the di�erent sides of � (�).Finally, our candidate for an group-valued equivariant Liouville form is,LW = exp(!) exp(�xa���Ra )� (�) ��: (26)The question is: under what conditions on ! and �, the form LW is an equivari-antly closed? According to [AMW1], there are 2 conditions to be satis�ed: �rst,the di�erential of the 2-form ! is a pull-back of the bi-invariant 3-form � on G,d! = ���: (27)Second, there is an analog of the moment map condition,�a! = 12��(�La + �Ra ): (28)We call a triple (M;!;�) which satis�es these conditions a group-valued Hamil-tonian space.

20 AntonAlekseevExamples of group-valued Hamiltonian spaces Our �rst example of agroup-valued Hamiltonian G-space is the torus T 2 = S1 � S1. We parametrizeby it two angles, (�; ), choose the S1 action� : (�; ) 7! (�; + �)and the two form, ! = d ^ d�:Then, �� @@ �! = d�;and one can de�ne the moment map � : (�; ) 7! � with values in the group S1.The 3-form � vanishes on S1 for dimensional reasons, which is consistent withclosedness of !.Our second example is a bit more complicated. We consider the group G =SU (2), choose any element f 2 G, and consider the corresponding conjugacyclass, Cf := fgfg�1j g 2 Gg:In other words, these are all unitary 2 by 2 matrices with the same eigenvaluesas f . If f is e or �e, the corresponding conjugacy class f is a point. Otherwise, itis a 2-sphere. We de�ne the moment map on Cf as its embedding into G. Then,the 2-form ! is uniquely determined by the conditions (27) and (28). Up to ascalar factor, ! coincides with the area form dA induced by the identi�cationwith the 2-sphere. If the eigenvalues of f are exp(i�) and exp(�i�), one gets[AMW1], ! = sin(�) dA:Our last example is the product of two copies of SU (2), D := SU (2)�SU (2).One can view it as a nonabelian counterpart of the torus T 2. We view D as anSU (2) � SU (2)-manifold, with the action,(g; h) : (a; b) 7! (gah�1; gbh�1);and the moment map, � : (a; b) 7! (ab; a�1b�1):The corresponding 2-form which satis�es conditions (27) and (28) is [AMW1],! = 12(a��La b��Ra + a��Ra b��La ):4.3 Group-valued localizationIn this section we explain how the localization principle works for the classes inHG(M ). We begin by recalling some standard fact from the theory of Lie groups.We assume that G is a direct product of a compact semi-simple Lie group anda compact torus.

Notes on Equivariant Localization 21Some facts about compact Lie groups. Let G be a product of a compactsemi-simple Lie group and a torus. Let T be a maximal torus in G and T be itsLie algebra. The Lie algebra T contains the integral lattice � � T ,� = fx 2 T j exp(x) = eg:The dual space T � contains the dual lattice ��. By choosing the set of positiveroots R+ we de�ne the positive Weyl chamber T �+ � T �, and the set of dominantweights ��\T �+ . A dominant weight � de�nes a unique irreducible highest weightrepresentation V� of G. The representation V� contains the highest weight vectorv� which satis�es the following conditions,e� � v� = 0;where e� are the generators corresponding to positive roots, andh� � v� = (�; �)v�;where h� are the elements of T corresponding to the roots. All irreducible rep-resentations V� possess a Hermitian invariant scalar product. We normalize v�such that (v�; v�) = 1. Then, for each dominant weight � one can de�ne twofunctions on G, the character, ��(g) = TrV�g;and the `spherical harmonics',��(g) = (v�; g � v�):A special weight is given by the half-sum of positive roots,� = 12 X�2R+ �:For example, for G = SU (2) the representations are parametrized by thespin j = 0; 1=2; 1 : : :. The weight � corresponds to j = 1=2. The correspondingrepresentation is two-dimentional, and we obtain,� 12 �a bc d� = a+ d; � 12 �a bc d� = a:Usually, we identify T and T � using the scalar product. Then, one can viewthe dominant weights as the elements of T . Because the weights belong to a spe-cial lattice, the corresponding one-parameter subgroups T� = fexp(s�)g (exceptfor � = 0) are circle subgroups of T . Note that a typical one-paraemter subgroupof T is dense in T . So, the subgroups T� are very special, and this will play animportant role in the localization theorem for HG(M ).

22 AntonAlekseevFinally, we recall that the classical r-matrix is an element in ^2G de�ned byformula, r = X�2R+ e� ^ e��:It is convenient to represent r as r = rabtatb. The r-matrix satis�es the calssicalYang-Baxter equation, Cyclabc(rasfsbtrtc) = 14fabc;where Cyclabc stands for summation over the cyclic permutations of the indecesa; b; c.The localization formula. Now we are ready to formulate the new localizationformula. Let M be a compact G-manifold, and � 2 ((M ) WG)basic be anequivariant cocycle, (dM + dW)� = 0:Then, one can de�ne an integral of � over M ,ZM � 2 (WG)basic:The elements of the space (WG)basic are annihilated by contractions, and, hence,belong to E 0(G)1. By G-invariance, these distributions should be conjugation-invariant, (WG)basic �= E 0(G)G. A conjugation-invariant distribution is com-pletely characterized by its pairings with charatcers of irreducible representationsof G, �� = hZM �; ��i:It is easy to show that the numbers �� do not depend on the representative inthe cohomology class.The localization formula [AMW2] gives expressions for �� in terms of the�xed points of the action of T�+� (note the shift by � ). As usual, we assumethat all these �xed points are isolated. Then, one obtains,�� = �2�i �m dimV� Xp hexp(12�(r))�;��i(p)�p1 : : : �pm : (29)Here the dimension ofM is 2m, the dimension of the representation V� is dimV�,�(r) is de�ned as rab�a�b, and �pi are the indices of the circle action of T�+� atthe point p.Formula (29) simpli�es if M is a Hamiltonian space with group valued mo-ment map, and the cocycle is the equivariant Liouville form on M . In this casethe localization formula reads [AMW2],L� = �2�i �m dimV� Xp ��+�(H(p))�p1 : : : �pm : (30)

Notes on Equivariant Localization 23Note that if p is a �xed point for some T�+�, the value of the moment mapH(p) belongs to the maximal torus T . The spherical harmonics ��+� de�nes acharacter of T which generalizes the expression exp(i�H(p)) in the Duistermaat-Heckman formula. (18).Some simple application of the formula (30) can be found in [P]. In particular,certain integrals over the sphere S4, and the space SU (2) � SU (2) �= S3 � S3can be computed using this technique. A more ambitious task is to show thatformula (18) gives precise meaning to the path integrals of [W2].References[AB] Atiyah, M., Bott, R. (1984): The moment map and equivariant cohomology.Topology 23 no. 1, 1-28[AMM] Alekseev, A., Malkin, A., Meinrenken E. (1998): Lie group valued momentmaps. J. Di�erential Geom. 48 no. 3, 445-495[AM] Alekseev, A., Meinrenken, E. (1999): The non-commutative Weil algebra.Preprint math.DG/990352, to be published in Inv. Math.[AMW1] Alekseev, A., Meinrenken, E., Woodward, C. (1999): Duistermaat-Heckmandistributions for group-valued moment maps. Preprint math.DG/9903087[AMW2] Alekseev, A., Meinrenken, E., Woodward, C. (1999): Group-valued equiv-ariant localization. Preprint math.DG/9905130[BV] Berline, N., Vergne, M. (1983): Z�ero d'un champ de vecteurs et classes car-act�eristiques �equivariantes. Duke Math. J. 50, 539-549[BGV] Berline, N., Getzler, E., Vergne, M. (1992): Heat kernels and Dirac operators.Grundlehren der mathematischen Wissenschaften, vol. 298, Springer-Verlag, Berlin-Heidelberg-New York[BT] Blau, M., Thompson, G. (1995): Equivariant K�ahler Geometry and Localizationin G=G Model. Nucl. Phys. B439, 367-394[C1] Cartan, H. (1950): La transgression dans un groupe de Lie et dans un �br�eprincipal. In Colloque de topologie (espaces �br�es), Bruxelles, 73-81[C2] Cartan, H. (1950): Notions d'alg�ebre di��erentielle; application aux groupes de Lieet aux vari�et�es o�u op�re un groupe de Lie. In Colloque de topologie (espaces �br�es),Bruxelles[CMR] Cordes, S., Moore, G., Ramgoolam, S. (1995): Lectures on 2D Yang-MillsTheory, Equivariant Cohomology and Topological Field Theories. Nucl. Phys. Proc.Suppl. 41, 184-244[DH] Duistermaat, J.J., Heckman, G.J. (1982): One the variation in the cohomologyof the symplectic form on the reduced phase space. Inv. Math. 69, 259-268[G] Gerasimov, A. (1993): Localization in GWZW and Verlinde formula. Preprinthep-th/9305090[MNP] A., Morozov, A., Niemi, A., Palo, K. (1991): Supersymmetry and loop spacegeometry. Phys. Lett. B 271 365-371[JK] Je�rey, L., Kirwan, F. (1995): Localization for Nonabelian Group Actions. Topol-ogy 34, 291-327[K] Kalkman, J. (1993): A BRST model applied to symplectic geometry. Ph.D. thesis,Universiteit Utrecht.[MNS] Moore, G., Nekrasov, N., Shatashvili, S. (1997): Integrating over Higgsbranches. Preprint hep-th/9712241

24 AntonAlekseev[P] Plamenevskaya, O. (1999): A Residue Formula for SU(2)-valued Moment Maps.Preprint math.DG/9906093[W1] Witten, E. (1982): Supersymmetry and Morse theory. J. Di�erential Geom. 17,661-692[W2] Witten, E. (1992): Two-dimensional gauge theory revisited. J. Geom. Phys. 9,303-368