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Quantum mechanics and the geometry of the Weyl character formula
Nuclear Physics B337 (1990) 467—486 North-Holland
QUANTUM MECHANICS AND THE GEOMETRY OF THE WEYL CHARACTER FORMULA* Orlando ALVAREZ** LPTHE***, Université Pien-e et Marie Curie, Paris VI, F-75252 Paris Cedex 05, France
I.M. SINGER Department of Mathematics, MIT, Cambridge, MA 02139, USA
Paul WINDEY LPTHE***, Université Pierre et Marie Curie, Paris VI, F-75252 Paris Cedex 05, France Received 19 December 1989
General field theoretic methods are developed which will allow a path integral derivation of the character formula for loop groups. The methods are introduced in the classical Weyl character case. The irreducible representations of a compact semi-simple Lie group G are realized as the ground States of a supersymmetric quantum mechanical system. The Hilbert space for the quantum mechanical system is the space of sections of a holomorphic line bundle L over the complex manifold G/T, where T is the maximal torus of G. The Weyl character formula is derived by an explicit path integral computation of the index of the Dolbeault 3L operator
1. Introduction This paper provides a quantum mechanical proof of the celebrated Weyl character formula for compact semi-simple Lie group G [lii. The setting is conceptually quite simple. One first finds a quantum mechanical system whose ground state is an irreducible (highest-weight) representation of G and then computes the trace of an element g E G in this ground state by path integral techniques. The system will be shown to be supersymmetric and the projection operator on the *
**
This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-ACO3-76SF00098, the Division of Applied Mathematics of the U.S. Department of Energy under Contract DE-FGO2-88ER25066, and in part by the National Science Foundation under grant PHY85-15857, and in part by the Centre National de Ia Recherche Scientifique. On leave from Department of Physics, University of California at Berkeley. Laboratoire associé No. 280 au CNRS.
0550-3213/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
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ground state will be the usual fermion parity operator (~ 1)~.This physical realization of the geometrical proof of the Weyl character formula rests on two constructions. The first one is the construction by Borel and Weil (see ref. [21for a review) of the irreducible representations of G as holomorphic sections of line bundles associated to the principal bundle ir: G G/T, where T is the maximal torus of G. For the sake of brevity we will refer to this construction and the theorems as BWT. The second one is the observation of Atiyah and Bott [3,4] that the Weyl character formula is the fixed-point formula for the T-index of the Dolbeault operator acting on this line bundle. This paper is a warm-up exercise for the field theoretic derivation of the Weyl—Kac character formula [51 for loop groups which will be published elsewhere [61.We know of at least six different ways of using physical methods to derive the Weyl character formula. In particular, a clear derivation using hamiltonian methods may be found in an elegant paper by Stone [7],where the physical implications are discussed in detail. Another derivation based on the co-adjoint orbit method and path integrals may be found in ref. [8]. Unfortunately, not all these methods generalize easily to the loop group case. The key technical points are finding a good set of coordinates on the loop group modulo its Cartan torus, incorporating supersymmetry and implementing the Borel—Weil line bundle efficiently. The techniques developed in this paper generalize to the ioop group case, LG/T. The rest of the paper is organized as follows. In sect. 2 we review briefly BWT and the Atiyah—Bott constructions and show their connection to our quantum mechanical setting in the simple well-known example of SU(2). In sect. 3 we review the physics of supersymmetric particles in an external gauge field on a manifold M. The particular case we have to consider, namely M = G/T is treated in detail in sect. 4 where we give a new geometric construction of supersymmetry on coset manifolds by introducing the notion of supersymmetric horizontality. We also show there how to obtain the different representations of the group by introducing line bundles on G/T. Finally, sect. 5 is devoted to the detailed derivation of the character formula using path integral techniques. —~
2. Borel—Weil theory and the Atiyah—Bott theorem One learns early in quantum mechanics that the spherical harmonics Yim(O, ~) provide a representation of SU(2) on the square integrable functions on 52• BWT in the case of SU(2) is a generalization of the spherical harmonics construction; it exploits the complex structure of S2 to construct irreducible representations of SU(2) as holomorphic objects. Before embarking into a concrete description of the methods, let us examine how BWT constructs representations of any compact simple Lie group G as holomorphic sections of a holomorphic line bundle. One first notes that the homogeneous space G/T, where T is a maximal torus of G, can be made into a complex manifold. To every character e~ of the torus T one
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associates a holomorphic line bundle LA over G/T; LA carries a left G-action. This simply means that the transition functions of the bundle are determined by e~”~ in a way we will describe shortly, and that the sections of LA will transform under the action of G. A consequence of BWT is that the holomorphic sections of this bundle form the irreducible representation of G whose highest weight is A. It is clear that from a quantum mechanical point of view we simply want to find a system with a hamiltonian invariant under the action of G and with its Hilbert space being the space of square integrable sections of LA. We now develop these ideas in the case of SU(2) and later show how they generalize to all Lie groups. Our goal here is to write down a quantum mechanical system which corresponds to BWT which we exploit to derive the Weyl character formula. Let g E SU(2) and let U(g) be a unitary operator of a unitary representation U of SU(2) acting on a Hilbert space which contains each irreducible representation exactly once. The matrix elements of U are defined by
KJ~mIU(g)I j’, m’)
=
6jjD,~~(g)
(2.1)
,
where j, m) is the state with spin j and magnetic quantum number m. The angular momentum commutation relations are defined by hermitian operators* satisfying [Jj’Jk]
(2.2)
1Cjk!jI.
To make a connection with the spherical harmonics one chooses a Dirac state IN) on the square integrable function on S2 which is the 6-function normalized state at the north pole. Any other such state may be obtained from IN) by a rotation. The (0, q) point in polar coordinates is obtained by first rotating by an angle 0 about the y-axis and subsequently by ~ about the z-axis. Thus we define the state 0, ~) by 0,
~)
~
e1°.’21N).
(2.3)
Notice that IN) is invariant under rotations about the z-axis. Among other things we are saying that S2 SU(2)/U(1). The spherical harmonics are defined by =
=
Kj,mI0,~).
(2.4)
We need a formula for in terms of the D-functions above. Using the invariance of IN) with respect to z-rotations one sees that
*
We use the quantum mechanical convention of not distinguishing between the angular momentum operators in a specific representation and the abstract generators.
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tains the basic formula ~m(0,~)*
4~D,~
0(q,0,0),
(2.5)
where D(q,, 0, x) means D(g) with g e’~~ e~°~2 ~ having angles 2 Euler SU(2)/U(1), (~, 0 , x). Note that D~0(~, 0, x) D~0(q,0,0), reaffirming that S and that the ~1m are functions on the group which are invariant under right multiplication by U(1). We shall view them shortly as sections of the trivial line bundle over 52• Finally we remark that the peaked state IN) is an infinite linear superposition of extended states =
=
=
IN)=
~‘~JIi~°).
(2.6)
j~O
One can ask what role the other D/,,m play on S2. As we shall see, BWT constructs spherical harmonics like objects with D/~ 1instead of D/~0. coordinates 2 is a complex manifold by exhibiting complex we argue thatdone S onFirst it. This will be through a choice of local coordinates on SU(2). Let =
f 1 ±if~.The angular momentum algebra (2.2), i.e. the Lie algebra of SU(2),
complexified, may be rewritten as [f3,J~] ±J~ =
[J~,J_] =2J3.
(2.7), (2.8)
Notice that f~ and J3 form a closed subalgebra. Let z be a given complex number, then one can uniquely find functions a(z, 2) and b(z, 2) such that 2~t+ e~2~S).I3 E SU(2) , (2.9) e~’~ e~z b(z, 2) is real, and a(z,2)
=
—2+...,
b(z,2)
=
~
(2.10),(2.11)
where the ellipsis denotes higher order. Every group element near the identity can uniquely be written as
g(z, ~,b) ~ =
e’°~’ 2)J~eb~,2)J 3+z~~J3
(2.12)
where t/i is real. It then follows that z and ~1iare good local coordinates for SU(2). Note that multiplication on the right 2by SU(2)/U(1). U(1) does not change the value inofeq. z. What is exemplified Therefore z is a good coordinate on S (2.12) is that locally one can write an arbitrary group element g in terms of a section g(z) of the principal bundle ~: SU(2) SU(2)/U(1) and an element e U(1) in the form g g(z)t One can also show that z defines a complex structure on the coset*. =
—~
=
*
A clear exposition in the spirit of this paper may be found in ref. [91.
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The complex structure can also be exhibited by studying SL(2, C), the complexification of SU(2). One divides SL(2, C) by the Borel subgroup, the complex group generated by f~ and f3. It inherits a natural complex structure from SL(2, C) and the coset space is again ~2• We can now explain the BWT construction for SU(2). The analog of the north pole state IN) is the highest-weight state j,j). Although IN) was a linear superposition of states2 of the form If~0) we usetoa single state f, j). define a state z) here analogous 0, ~) extended by Given a point z on S Iz)
=
e_j~2,z)e~_ ea
,2)j~
e~’2~~lj,j).
(2.13)
The use of a different section of SU(2) such as (2.12) with a z-dependent i~(i immediately tells us that Iz) is a section of a line bundle over S2. That is, z) differs from
~
e_jb~e~t_ ~
=
e~~~’Iz)
(2.14)
by an arbitrary phase ~‘(z, 2). If we define spherical harmonic like objects by =
~j,mIz),
(2.15)
then one can easily see by Taylor series expansion of z) that the above is an analytic function of z since J~j, j) 0. Notice that it is the c-number prefactor of e_Jb in eq. (2.13) that guarantees that the sections will be holomorphic. By construction, automatically, the (~Jm}defines a representation of SU(2). In fact notice that =
~tn(z)
=
e_1~~z~S)D~ 1(g(z,0)),
(2.16)
where g(z,0) is defined by eq. (2.12). By using the orthogonality properties of the one can show that ~ ~ is a natural hermitian metric on the line bundle. Thus we see that BWT constructs holomorphic sections of a line bundle on which the group is represented. In fact, the generators may be written as
a
=
—z—
+j,
(2.17)
z
a
3z ~,
a +2jz, 2—
(2.18), (2.19)
f_= —z
and the sections we have derived are precisely /?‘m(Z) a ~J-~m The above operators are hermitian in the natural metric. For future purposes it is important to observe that, for example, J 3 has both an “orbital” angular momentum part —za/az and a
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“spin” angular momentum part j. Thus when one transforms the states one picks up a “phase” in addition to changing the argument. This phase will play a central role in the path integral discussion. More precisely we observe that the functions on SU(2) which satisfy f(gf’) t23f(g) where t E U(1) are sections of a certain line bundle over 52• We wish to study the quantum mechanical system whose wave functions correspond to the sections of this line bundle. Since this line bundle has a natural connection one can easily see that the quantum mechanical system describes the motion of a particle moving on the sphere in the presence of a magnetic monopole. A more detailed study of these line bundles can be found in sect. 4. One last observation; objects of the form D~ 1(g(z,0)),where j’ >1, are also sections of the line bundle in question. However these sections correspond to states =
e1b~S)ezJ_
ea
,2)j~
e~z, ~‘~1j~j),
(2.20)
which are not holomorphic since J~j’~j) ~‘O if j’ >1. Our goal is to out the 2, out of pick all possible holomorphic sections, i.e. the kernel of a a-operator on S sections. These holomorphic sections will be the ground-state wave functions of a supersymmetric quantum mechanical system on G/T. BWT generalizes these ideas to an arbitrary compact semi-simple Lie group G with maximal torus T. We first discuss how to make G/T into a complex manifold. Let (H 1} be a basis for a Cartan subalgebra of G and ~1the corresponding set of roots. For a positive root* a, let Ea be the associated “raising” generator and let E_a be the “lowering” generator. In analogy with the SU(2) case, given a complex number za for each a > 0, one can construct a group element exp( ~ zaE_~)exp( ~ aa(z,2)Ea)exp(~bI(z,2)Hj), a>O
(2.21)
a>O
where b’ is real. Near the identity, a general group element may be written as
g(z,~)~ (2.22) Because {H~Ea, a > 0) form a closed subalgebra one can show that {zi define a complex structure on G/T. If A is a highest weight of a representation then the * In general, G/T has many complex structures. The complex structure is determined by the choice
of “positive roots”.
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highest-weight state IA, A) can be used to construct a holomorphic object
Iz)=exP(— Ebl(z,2)Ai)exp(EzaE_a)
x exp( Laa(z, 2)Ea)exp(
Eb’(z,
2)H~)IA~ A).
(2.23)
The holomorphicity follows from the highest-weight property EaIA, A) 0 for a > 0. Since the representation is finite-dimensional we always get a polynomial in Zr’. The line bundle of which Iz) is a section is characterized by the functions on G which satisfy =
f(gft) =p(t)f(g),
(2.24)
where t E T and p is the representation of T with infinitesimal character A. Since the matrix elements of the representation matrices form a complete set for the functions on G, it follows that the sections of the line bundle are associated with matrix elements of the form D,~~(g) where A is the highest weight of the representation, and ~ and v are weights. In particular we have seen that e~bD~ 5(g(z,0)) is a holomorphic object. Just as in the SU(2) case, e~~~)D;~(g(z,0)) will not be holomorphic in general though it is a section of the
line bundle we are studying. We now provide an argument for the irreducibility of the representation (see ref. [2], p. 21). Suppose the representation R is not irreducible; then there would be at least two independent highest-weight vectors v1 and v2 corresponding to the decomposition R into R1 ~ R2. As we previously remarked, the complexification G~acts on G/T; in fact, G/T GC/S, where S is the maximal solvable subgroup (Borel subgroup) generated by H, and Ea~a > 0. Also R extends to a representation* R~of G~.Since v1 and v2 are highest-weight vectors, R~(E~)v, 0, for a > 0 and i 1,2 and consequently R~(expEa)v, v,, for a> 0. Notice that exp(E~> OCaEa) sends the identity coset to a dense set in G/T (certainly an open set around the identity coset eT, see eq. (2.22)). That is the nilpotent group N {exp ~a > ocaEa} sends eT into NT, dense in G/T. So in this open dense set, the holomorphic sections v1 and v2 are determined by their value at eT. Either one or the other is zero or the ratio v1/v2 (or v2/v1) is a holomorphic function constant in a neighborhood of eT, thus constant. Hence v1 and v2 are not linearly independent. =
=
=
=
=
* We denote the representation of the group and the algebra by the same symbol.
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We illustrate this for SU(2) and SL(2, C). SL(2, C) acts on S2 fractional transformations. The generator
(~~)
E~=
=
C U ~ by linear
(2.25)
is the only “raising operator”. One easily sees that exp cE~:z z + c: the orbit of z 0 is exactly C S2 ~, an open dense set. Since exp cE~will leave any highest vector fixed, any highest-weight vector is determined by its value at 0. So there is only one up to scale. In summary, the complex manifold G/T has a line bundle LA over it which is associated with each infinitesimal character A of T. LA has holomorphic sections which furnish an irreducible representation of the group G whose highest weight is A. These sections will be eigenstates of the hamiltonian for the motion of a particle on G/T which interacts with the vector potential associated with the line bundle LA. We will have to isolate the holomorphic eigenfunctions in order to construct the desired representation. Since we are only interested in finding the character, we will not require detailed information about the eigenstates. We just have to construct the operator 3, which we accomplish through supersymmetty. To compute the character we need only count the number of solutions of a~P 0 for sections 11’ of the line bundle with an appropriate weight. Atiyah and Bott [3,4] observed that this could be done by computing the character index of a, which they do by their holomorphic Lefshetz fixed-point formula. To be more precise, the 0 operator maps sections of the holomorphic line bundle LA into sections of ~4(0. ‘kG/T) ~ L tkG/T, LA), where fl(p,q) are the (p, q) forms. In fact the elliptic complex is5 fl(~ —*
=
=
—
-
=
OLA:
A(~*)(G/T,LA) ....~n(O*)(G/TLA),
(2.26)
where fl(~*)(G/T LA) are all the (0, q) forms twisted by the line bundle LA. Since the index computes the difference of the kernel of a and the cokernel of_a, one has to show that the only non-trivial contribution arises from the kernel of a acting on A~°’°~(G/T, L). One needs to prove a vanishing theorem showing that all the other kernels and cokernels are empty [2]. The required vanishing theorem may be proved along the lines of Lichnerowicz theorem on the existence of harmonic spinors [101;that is 8~lJ=DtD
+
(a positive potential),
where D is the covariant derivative. Moreover, since we know (see sect. 4 for details) that T acts on the sections of LA, the kernel of a decomposes into
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representations of T. This enables us to compute the character-valued index which will then give us the Weyl character formula. Since we will actually work with the Dirac operator on G/T, we now discuss the connection between the space of spinors and exterior forms. The complex manifold G/T has real dimension 2n. The spinors on SO(2n) belong to a 2~-dimensional representation. The exterior algebra A~°’ *) is also 2’~dimensional. It is well known that in the standard embedding of U(n) into SO(2n), the spinor representation of SO(2n) decomposes into the direct sum of the anti-symmetric tensor representations of SU(n) [11]. As far as dimension counting is concerned it appears as if the spinors S(G/T) are the same as fl(° kG/fl. Although true at a single point of G/T, this cannot be the whole story because the spin- 4 features are missing. If K A~°kG/T)is the canonical line bundle of G/T, then the correct statement is *
=
S(G/T) Thus
0L,
®
K~2
,4(O
is the same as the Dirac operator ~LA:
S(G/T)
®
K”2 ® LA
*)(G/T)
~LA
coupled to K”2 as well as to LA:
S(G/T)
—*
(2.27)
®
K”2
® LA.
(2.28)
Here the line bundle K’~2is associated with the infinitesimal character p of the torus given by (2.29)
p=4~a. a>O
Remember that p has projection 1 onto each of the simple roots and thus it is “pure” spin 4 in the two-planes that define the rotations on G/T. If I E T and if I~(l)is the character-valued index of the Dirac operator coupled to the line bundle defined by the infinitesimal character i.’, then the character x
5(i) of the representation with highest weight A may be computed via the relation XA(l)
(2.30)
‘A+P(l).
3. The method In this section we discuss the connection between the supersymmetric point particle and the Dirac operator [12, 13] and its relation to the Weyl character formula. The lagrangian for a supersymmetric free particle moving in l~20~(euclidean time) is given by L
=
—
~
where x~(t)is the trajectory of the particles and
(3.1) ~‘,~(t)
is an anti-commuting vector
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field defined along the trajectory. The supersymmetty transformations are defined by ~
6t~,=i.~,
(3.2),(3.3)
where c is an anti-commuting parameter. One can easily verify that the Noether charge associated with the supersymmetry is (3.4) where p,,,, i.~is the canonically conjugate momentum. The operator canonical commutation and anti-commutation relations are given by =
[x~,pj=i6~~,
=6~,.
~
(3.5), (3.6)
Quantization involves finding a Hilbert spaces where one can represent the above algebra. The x and p commutation relations may be implemented on the square integrable functions on ~ L2(l~2d).The operator x~,is multiplication by ~ and —~a/a~. The anti-commutation relations define a Clifford algebra which may be satisfied by choosing ~/i~ y~/ ~ where {y~) are the Dirac gamma matrices acting on the 2”-dimensional spinor space C 2’~• The Hilbert space associated with (3.1) is the space of spinorial wave functions L2(l~2”)0 C2d. One easily sees that the supersymmetry charge (3.4) becomes the Dirac operator under quantization =
=
ax~
(3.7)
All these ideas generalize to special riemannian manifolds called spin manifolds which allow the existence of spin structures. The heat kernel formula for the index of Q [14—16]can be interpreted in the supersymmetric theory as Tr(—1)~e~’~’,
(3.8)
as observed by Witten [17]. It corresponds to doing the path integral with periodic boundary conditions [181on the fermions. Further work [19—21]demonstrated that the path integral may be used to derive the index formula. The strategy is to exploit the I~independence of eq. (3.8) and do a path integral computation in the f3 0 limit. The calculation reduces to an effective gaussian path integral. These ideas can be extended to the situation where one has a symmetry group G whose action commutes with the Dirac operator. Now the path integral of interest —*
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is the character partition function
1(g) =TrU(g)(_1)Fe_~,
(3.9)
where U(g) is a unitary representation of G on the appropriate Hilbert space. The index of the operator is in this case the character index. We now examine how the presence of the group action affects the previous discussion, using a variant of the method of refs. [19,22—24]. We need to know how to couple a gauge field A,~to the supersymmetric particle of charge q in a gauge covariant way. One verifies that the following lagrangian is supersymmetric and changes by a total time derivative under a gauge transformation: L
=
—
4~i~+iq(A~(x)i.~+ 4F~~(x)~çlç),
(3.10)
where FIL,, ~ a~A~ is the field strength. Canonical quantization leads now to the supercharge =
—
V~Q =
The hamiltonian H
=
4Q2
~—i— a~
—
qA~. /
(3.11)
is of the Pauli form with a o B coupling:
H= ~(_t~
_qAM) + ~
(3.12)
It is important to notice the covariance of the hamiltonian under a gauge transformation A,~—*A,~+ a,~,,while the Schrödinger wave function must transform as IP(x) ~ One recognizes in the latter the transformation properties of a section of a line bundle of charge q associated to the vector potential A. In the lagrangian formulation these gauge transformations will affect the boundary conditions one has to use in the computation of the path integral _~
fx(P)=xf[~] x(O)=x,
_f~dtL 0
To obtain the Weyl character formula from a quantum mechanical approach we apply all the above machinery to the case of a supersymmetric particle moving on G/T. Let us go back to the bosonic SU(2)/U(1) example of sect. 2. In this case the above abelian U(1) gauge transformation corresponds to the right U(1) action of the group. It is important to note that there is also a left U(1) action which acts on the coset S2. Its generator acting on the holomorphic sections is given by eq. (2.17)
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and the character partition function is Tre’°~e~”. If Ix) denotes the 6-function normalized state on S2 with support at x, then e’°~~Ix)e~~°Ix , 0) =
(3.13)
where the point x8 is the image of x under the rotation by 0 around the z-axis. This phase enters in the following way into any path integral computation: 2~
Tre13e~””=f
(3.14)
S
=
=
f d2xe’°~(x0Ie~”Ix)
(3.15)
f d2xe_b0JfX~~[~x]exp(_fdtL). 2
x(0)=x
0
(3.16)
S
So besides generating twisted boundary conditions on the path integral, the operator e’°~~ introduces an overall phase, which plays a central role in our method. We will see that the phase is determined uniquely by the line bundle in question. 4. Supersymmetry and line bundles on G/ T One could now derive the Weyl character formula: first we write down the standard supersymmetric non-linear sigma model L
=
4g~~(x)~~ +
D~
where g~ is the metric on G/T and (x~,~) are local coordinates on G/T with their fermionic partners and T acts on G/T. Next add to the lagrangian L the connection and curvature terms of eq. (3.10), given by explicit formulas from differential geometry. The path integral with the new lagrangian gives the fixedpoint computation needed to compute the character-valued index of the appropriately twisted Dirac operator [23,24]. We follow this procedure except for a significant detour. We derive supersymmetiy on the coset space in a new and we believe interesting way by introducing the notion of horizontal supersymmetty. In any case, we need horizontal supersymmetry to achieve the Weyl—Kac character formula for loop groups with central extension*. So the following discussion which generalizes to the loop group case should be seen as a warm-up exercise for that more difficult task. *
For clarity of the argument the loop group case will be treated in a separate publication
[61.
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To construct the required supersymmetric theory for a spinning particle on the coset G/T we use the fact that G is a principal fiber bundle over M G/T with structure group T. For such a principal bundle we can lift curves in M to curves in G by choosing a horizontal projection, or equivalently, a connection: the tangent vectors to the lifted curve are horizontal. Similarly, we will see that a supersymmetry on G and a connection can be used to induce a natural supersymmetrv on M. We now describe the construction of the induced supersymmetry. Let (x~,r’) be local coordinates on the bundle where x~are coordinates on M and r’ on the fiber. Although our construction is coordinate independent, local coordinates are pedagogically useful to display the explicit supersymmetric transformations (4.2) and (4.3). We note that the construction works for any principal T-bundles over a manifold M; the group structure of G is not necessary. Given a vector potential A in the base, a vector field (zn, k’) is said to be horizontal if A’,~z’~k’ 0. In particular, a curve (x~(x),r’(x)) is the horizontal lift of a curve x(t) in M if =
—
=
A’~V~’ —
=
0.
(4.1)
The connection provides a natural way of identifying vector fields on the base with horizontal vector fields on the bundle. The supersymmetry transformations are given by 6xlL
ei~’,
=
~
6~/J~ =
=
E~7’,
6i~~ = Er’.
(4.2) (4.3)
If we interpret the supersymmetric deformation of X~L as a vector field then it has a horizontal lift. If we restrict ourselves to the supersymmetric deformations in G which are horizontal then their projection onto the base manifold can be regarded as induced supersymmetric transformations of M. On the bundle we have a globally defined connection one-form with values in the Lie algebra of T given by w’ =A~dx~ dr’. A horizontal vector field is one that annihilates the connection. Requiring the bosonic supersymmetry transformations to define a horizontal vector field, i.e. annihilate the connection, imposes conditions on the fermionic partners —
0
Consistency requires
=A~6x~
=
~
=
E(A’~IJ~—
—
n’),
I5T’
(4.4)
that the supersymmetric partner of the connection defined by
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6w also vanish. The supersymmetric partner of the above equation is ‘i
—A~~ ~F~/J~/J~ —
=
0,
(4.5)
where F dA is the field strength. This last equation is just the equation of motion for a spinning particle in a generalized electromagnetic field. It has a Pauli type magnetic dipole coupling a B. Notice that it is different from eq. (4.1), which applies to spinless particles. In conclusion, the supersymmetry transformation on the base (4.2) is equivalent to imposing the horizontality conditions (4.4) and (4.5) on the bundle. We will refer to these as supersymmetric horizontality. The local coordinates are only used to make contact with familiar formulas. The method itself is coordinate independent and global. Abstractly, given a supersymmetry and a connection on a principal bundle we can induce a supersymmetiy on the base and the associated bundles by imposing supersymmetric horizontality. Now consider the case of the bundle G being a compact semi-simple Lie group with fiber T over G/T. Instead of expressing our formulas in terms of coordinates on the base we will express everything in terms of local sections of the bundle. Let g t ~ m be the Lie algebra of G with t a Cartan subalgebra and m the orthogonal complement. Two local sections are related by a gauge transformation =
=
g
2(x) =.~1(x)4)12(x),
(4.6)
and one finds ~‘(x)
d~2(x)
=
4~_1(x)~_1(x) d~1(x)4,2(x) +
4~’(x)d~12(x). (4.7)
An element X of the Lie algebra can be decomposed into its vertical component X~E t and its horizontal component Xm E m. The vertical projection (~1 d~)~ A(x) along the torus transforms as an abelian T-connection. we write a generic 1 E T thereIf exists a global one element g of the group as g ,~(x)t ‘, with t e form on G given by =
=
w
=
(g’ dg),
=
=
tAt’
—
dtt’.
(4.8)
The one-form w is independent of the cross section. It is the projection on t at the identity of G and is invariant under left multiplication. Now define a superfield G ge°’associated with g ~ G. The fermionic partner y takes values in the Lie algebra of G. In this section, 0 is a Grassmann variable. The coordinate superfields on G/T and on the fibers are X” =x~+ ~ and =
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Oif. Given a section (~(x),~(y)) we have G=GTl=g(1+0y)=~(x)(1+0~(x,~,))eT+O~ =
(~(x) + oa(x)~p’~(x))T’.
(4.9) (4.10)
The last line relates the section to the local coordinates. The supersymmetry transformations along a curve (x(t), ~fr(t))are defined by eq. (4.2), or equivalently 6G = e(a0
—
oa,)G.
(4.11)
The horizontal supersymmetly deformations must then obey w’=A~6x~~—6r’=0and
6w’=A’=O.
(4.12)
These conditions will be more useful in the future if written in terms of the sections rather than in terms of the coordinates (x~,~ We observe that given (g’ a~(x)), we can write w = ~ 6g), = (g’6g)~= 0 except that 6 is now a supersymmetric variation. The supersymmetric horizontality conditions become =
—
w
=
0
=
~~(x)
6w
=
0
=
(~‘a~~)~(x) — (ymym)t.
—
i~,
(4.13) (4.14)
The term bilinear in the fermions is actually a commutator because the fermions anti-commute. One can easily verify by the use of the Maurer—Cartan equations for the Lie group and the Cartan structural equations that this fermionic bilinear is the curvature of the bundle. The supersymmetric lagrangian on G is given by
LG= fdoTr(G’DG)(G’a1G) =Tr[(g_1g)2+~y+g_1gy2~.
(4.15)
This is the standard lagrangian for a supersymmetric particle on G with riemannian connection. If we decompose the superfield into its vertical and horizontal parts and if we impose the supersymmetric horizontality conditions then LG
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reduces to the supersymmetric lagrangian on G/T: LG/T = Tr{(~1(x)~(x))~ + (~m(~) +
+
Tr[(~1(x)g(x))m~m(x)~m(x)j.
(4.16)
Notice that it only depends on the coordinates on M, is covariant and supersymmetric. We want to emphasize that the fermionic term is the standard riemannian covariant derivative on G/T. This follows from supersymmetiy and the fact that the bosonic kinetic energy term is the standard riemannian one. Naively it appears as if the fermions are coupled to a T-connection besides the riemannian one. The illusion is caused by the fact that the original y was left-invariant. Since we are studying cosets of the form gT, the ~m transforms when one changes section and must be reflected in (4.16). In any event, the lagrangian (4.16) is the one described at the very beginning of this section, but derived differently. We now turn to the construction of the line bundles over G/T and to the study of the group action on them. A line bundle L over G/T is determined by an appropriate one-dimensional representation of the group T. A section of L is the same as a function on the entire principal bundle ir: G G/T satisfying —~
(4.17)
t/i(gt’) =p(t)~t(g).
It is important to keep in mind the difference between right- and left-action of T. The right-action lets us move up and down the fiber and tells us that the function transforms under the representation given by p. A local section of L must be parametrized by the coordinates x of G/T and defined by ~/i
t~i
~(x) =çli(~(x)),
(4.18)
where ~(x) is a given local section of G. This determines the left-action of T on LjtIi(x)
=
~Ii(Ig(x)).
t/i:
(4.19)
Note that l~(x)is a new element of G in the fiber above the point I . x = 7r(l~(x)) in the base. We can use right multiplication to relate lg(x) to the section by introducing t(x, 1) E T as follows l,~(x)= ,~(lx)t t(x 1). We can write L1~(x) = ~(g(I .x)t’(x, 1)) =p(t(x, I)~(l ~x)) =p(t(x,l))~(lx).
(4.20)
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In accordance with our discussion in sect. 2 we see that the transformation law (4.20) for the sections has both an “orbital” part and a “spin” part. Because of the presence of the spin part there are some phases which will have to be accounted for in the path integral computation. The coupling of the line bundle defined by infinitesimal character r’ is implemented by the method described by eq. (3.10). Adding the appropriate gauge coupling to (4.16) gives the lagrangian =
LG/T
+
((~-‘~)~ (~m~m)t) —
(4.21)
Note that the terms in parentheses are the connection and curvature terms (3.10).
5. Index calculation We have to calculate the index of the Dirac operator on G/T coupled to the appropriate line bundle. It is given by 1(1) =TrU(/)(—1)”e~”.
(5.1)
Because of the presence of U(l), the path integral has boundary conditions x(0) =x and x(f3) = 1 ~x ~x1. The key to the path integral proof of the fixed-point formula is the observation [23, 24] that for small I~the action behaves as 2//3. distance between x(0) and x(/3)~ The dominant contributions of the path integral will come from the neighborhood of those points in G/T which are left fixed by the action of T, i.e. x 1=x. Our computation reduces to one in perturbation theory around the fixed points. Since 1 E T acts on G/T on the left by mapping the coset x =gT to x1 = IgT, the fixed points are cosets gT such that lgT = gT for all 1 ~ T. This means that g E the normalizer of T. Thus the fixed points of the T action on G/T may be identified with the Weyl group of T, W(T) = N(T)/T as expected from the fixed-point formula [3,4]. Let n E N(T); then w E N(T)/T defined by w = nT is a fixed point. Near this point a section of the super-group may be written as S(t) =ne~(t)eoist),
(5.2)
where ~ and ~ have components only in the m directions. The boundary conditions on S(t) (which respect the supersymmetty) are S(/3)T==IS(0)T.
(5.3)
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It follows that the boundary conditions on 2 and ~ are i(j3)
=l~2(0)I;’,
((13) =1(0)1~’,
(5.4),(5.5)
where l~ n 11n. Notice that l~only depends on the choice of coset w nT. For, if n 11n 1T n2T then nj 1 n~’ln2.Thus 1~is determined by wE W(T). To study the action to quadratic order near each fixed point, we write =
=
-
=
=
1= e’°, where 0
E
1~ e’°~,
(5.6), (5.7)
=
t and 0~ n’0~ is just the adjoint representation. If =
2(t) =i ~ 2’~(t)Ea
(5.8)
a~I
with 2~(t)
=
2~(t),then the boundary conditions become 2”(f3)
=
etao~*2(~(0).
(5.9)
because 1’Ealw e~~O~*Ea. Using a similar expansion for ~ we find the same boundary conditions. The transformation properties of the sections of the line bundle near the Weyl point w nT with coordinates 2 are easily computed using eq. (4.20): =
=
L,q(2) =p(l;’)ç~(l~2) =
e~’’°”ç&(e~’~~°”2”)
et 0~(e1~02~*) (5.10) 1. In the neighborhood of the Weyl points, the where v~1, to wvw’ and order a~ waw lagrangian quadratic is =
=
.
=
L
=
Tr~(i—4[r’,2])
+
Tr(~_~[v,(])(.
The Dirac index for the line bundle characterized by 4(1)
=
~ wEW(T)
e~det{_
—(— dt
dt
-
(5.11)
v,
H~,.])}det(_
-
~[Li,
.])~(5.12)
t
is obtained by performing the gaussian path integral defined by eq. (5.11) with twisted boundary conditions (5.9). Notice the inclusion of the end point phase factor contribution and the sum over the fixed points (minima of the action). A
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Weyl character formula
straightforward evaluation leads to the formula
~
I.~(l)
=
e~”~°~ fl . . 2zsin~a(0~) a>0
(5.13)
.
wEW(T)
The Weyl character formula is obtained by using observation (2.31) relating the character with the index:
X~(l)=IA+P(l).
(5.14)
Using path integrals we have derived the Weyl character formula:
XA(l)
=
~
e’~°~ w~W(T)
~ a>0
.
.
2isin~a(0~)
(5.15)
6. Interlude In this paper we derived the Weyl character formula from the path integral. The standard supersymmetric lagrangian had to be modified (3.10) to couple ~ to a line bundle. We also had to allow paths that were not closed in order to obtain the T-index as in refs. [23,24]. We introduced the notion of horizontal supersymmetry. Though not essential to our derivation of the Weyl character formula, it will be important in the construction of the supersymmetric model on LG/T. There, curves on LG/T have to be lifted to curves on LG (in a horizontal supersymmetric way) so that ultimately we can obtain a supersymmetric sigma model lagrangian based on G and derive the Weyl—Kac character formula. The details of this construction are technically complicated. In order not to obscure the principal ideas, we will publish them elsewhere. We would like to thank C. Itzykson for his friendly exhortations to complete this project and M. Picco for carefully reading the manuscript. The Aspen Center for Physics, le Laboratoire de Physique Théorique et des Hautes Energies, the Massachusetts Institute of Technology, Rutgers University, la Universidad Autónoma de Madrid and the University of California at Berkeley are warmly thanked for providing crucial brief two-body interactions in this long-range collaboration. References [1] H. Weyl, The classical groups (Princeton Univ. Press, Princeton, 1946) [2] A.M. Pressley and G.B. Segal, Loop groups (Oxford Univ. Press, Oxford, 1986) 13] M. Atiyah and R. Bott, Ann. Math. 86 (1967) 374
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14] M. Atiyah and R. Bott, Ann. Math. 88 (1968) 451 [51V. Kac, Infinite dimensional lie algebras (Cambridge
Univ. Press, Cambridge, 1985) [6] 0. Alvarez, I.M. Singer and P. Windey, The Dirac loop space index and character formulae. Talk given at Infinite Dimensional Symmetries in Mathematics and Physics, June 1989, Paris. In preparation [7] M. Stone, NucI. Phys. B314 (1989) 557 [8] A. Alekseev, L. Fadeev and S. Shatashvili, Quantisation of the symplectic orbits of the compact Lie groups by means of the functional integral. In the Gelfand Festschrift, J. Geom. Phys, to be published [9] B. Zumino, Geometry of the Virasoro group for physicists, in: Particle Physics Cargése 1987, ed. M. Levy et al., NATO-ASI Series B, Physics 173 (Plenum, New York, 1988) [10] A. Lichnerowicz, C.R. Acad. Sci. (Paris), Sér. 1 Math. 257 (1963) 7 [11]II. Georgi, Lie algebras in particle physics (Benjamin/Cummings, Menlo Park, 1982) [12] 5. Deser and B. Zumino, Phys. Lett. B65 (1926) 369 [13] L. Brink, P. Di Vecchia and P. Howe, Phys. Lett. B65 (1976) 471 [141H.P. McKean and I.M. Singer, J. Diff. Geom. 1 (1967) 43 [15] M. Atiyah, R. Bott and V.K. Patodi, Invent. Math. 19 (1973) 279 [161PB. Gilkey, Adv. in Math. 10 (1973) 344 [17] E. Witten, NucI. Phys. B202 (1982) 253 [181 S. Cecotti and L. Girardello, Phys. Lett. BilO (1982) 39 [19] L. Alvarez-Gaumé, Commun. Math. Phys. 90 (1983) 161 [20] D. Friedan and P. Windey, NucI. Phys. B235 (1984) 395 [21] E. Witten, unpublished, 1983, see M. Atiyah’s exposition in the Astérisque volume in honor of H. Weyl [22] E. Witten, J. Diff. Geom. 17 (1982) 661 [23] M. Goodman and F. Witten, NucI. Phys. B271 (1986) 21 [24] M. Goodman, Commun. Math. Phys. 107 (1986) 391
QUANTUM MECHANICS AND THE GEOMETRY OF THE WEYL CHARACTER FORMULA* Orlando ALVAREZ** LPTHE***, Université Pien-e et Marie Curie, Paris VI, F-75252 Paris Cedex 05, France
I.M. SINGER Department of Mathematics, MIT, Cambridge, MA 02139, USA
Paul WINDEY LPTHE***, Université Pierre et Marie Curie, Paris VI, F-75252 Paris Cedex 05, France Received 19 December 1989
General field theoretic methods are developed which will allow a path integral derivation of the character formula for loop groups. The methods are introduced in the classical Weyl character case. The irreducible representations of a compact semi-simple Lie group G are realized as the ground States of a supersymmetric quantum mechanical system. The Hilbert space for the quantum mechanical system is the space of sections of a holomorphic line bundle L over the complex manifold G/T, where T is the maximal torus of G. The Weyl character formula is derived by an explicit path integral computation of the index of the Dolbeault 3L operator
1. Introduction This paper provides a quantum mechanical proof of the celebrated Weyl character formula for compact semi-simple Lie group G [lii. The setting is conceptually quite simple. One first finds a quantum mechanical system whose ground state is an irreducible (highest-weight) representation of G and then computes the trace of an element g E G in this ground state by path integral techniques. The system will be shown to be supersymmetric and the projection operator on the *
**
This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-ACO3-76SF00098, the Division of Applied Mathematics of the U.S. Department of Energy under Contract DE-FGO2-88ER25066, and in part by the National Science Foundation under grant PHY85-15857, and in part by the Centre National de Ia Recherche Scientifique. On leave from Department of Physics, University of California at Berkeley. Laboratoire associé No. 280 au CNRS.
0550-3213/90/$03.50 © Elsevier Science Publishers B.V. (North-Holland)
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ground state will be the usual fermion parity operator (~ 1)~.This physical realization of the geometrical proof of the Weyl character formula rests on two constructions. The first one is the construction by Borel and Weil (see ref. [21for a review) of the irreducible representations of G as holomorphic sections of line bundles associated to the principal bundle ir: G G/T, where T is the maximal torus of G. For the sake of brevity we will refer to this construction and the theorems as BWT. The second one is the observation of Atiyah and Bott [3,4] that the Weyl character formula is the fixed-point formula for the T-index of the Dolbeault operator acting on this line bundle. This paper is a warm-up exercise for the field theoretic derivation of the Weyl—Kac character formula [51 for loop groups which will be published elsewhere [61.We know of at least six different ways of using physical methods to derive the Weyl character formula. In particular, a clear derivation using hamiltonian methods may be found in an elegant paper by Stone [7],where the physical implications are discussed in detail. Another derivation based on the co-adjoint orbit method and path integrals may be found in ref. [8]. Unfortunately, not all these methods generalize easily to the loop group case. The key technical points are finding a good set of coordinates on the loop group modulo its Cartan torus, incorporating supersymmetry and implementing the Borel—Weil line bundle efficiently. The techniques developed in this paper generalize to the ioop group case, LG/T. The rest of the paper is organized as follows. In sect. 2 we review briefly BWT and the Atiyah—Bott constructions and show their connection to our quantum mechanical setting in the simple well-known example of SU(2). In sect. 3 we review the physics of supersymmetric particles in an external gauge field on a manifold M. The particular case we have to consider, namely M = G/T is treated in detail in sect. 4 where we give a new geometric construction of supersymmetry on coset manifolds by introducing the notion of supersymmetric horizontality. We also show there how to obtain the different representations of the group by introducing line bundles on G/T. Finally, sect. 5 is devoted to the detailed derivation of the character formula using path integral techniques. —~
2. Borel—Weil theory and the Atiyah—Bott theorem One learns early in quantum mechanics that the spherical harmonics Yim(O, ~) provide a representation of SU(2) on the square integrable functions on 52• BWT in the case of SU(2) is a generalization of the spherical harmonics construction; it exploits the complex structure of S2 to construct irreducible representations of SU(2) as holomorphic objects. Before embarking into a concrete description of the methods, let us examine how BWT constructs representations of any compact simple Lie group G as holomorphic sections of a holomorphic line bundle. One first notes that the homogeneous space G/T, where T is a maximal torus of G, can be made into a complex manifold. To every character e~ of the torus T one
/
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Weyl character formula
469
associates a holomorphic line bundle LA over G/T; LA carries a left G-action. This simply means that the transition functions of the bundle are determined by e~”~ in a way we will describe shortly, and that the sections of LA will transform under the action of G. A consequence of BWT is that the holomorphic sections of this bundle form the irreducible representation of G whose highest weight is A. It is clear that from a quantum mechanical point of view we simply want to find a system with a hamiltonian invariant under the action of G and with its Hilbert space being the space of square integrable sections of LA. We now develop these ideas in the case of SU(2) and later show how they generalize to all Lie groups. Our goal here is to write down a quantum mechanical system which corresponds to BWT which we exploit to derive the Weyl character formula. Let g E SU(2) and let U(g) be a unitary operator of a unitary representation U of SU(2) acting on a Hilbert space which contains each irreducible representation exactly once. The matrix elements of U are defined by
KJ~mIU(g)I j’, m’)
=
6jjD,~~(g)
(2.1)
,
where j, m) is the state with spin j and magnetic quantum number m. The angular momentum commutation relations are defined by hermitian operators* satisfying [Jj’Jk]
(2.2)
1Cjk!jI.
To make a connection with the spherical harmonics one chooses a Dirac state IN) on the square integrable function on S2 which is the 6-function normalized state at the north pole. Any other such state may be obtained from IN) by a rotation. The (0, q) point in polar coordinates is obtained by first rotating by an angle 0 about the y-axis and subsequently by ~ about the z-axis. Thus we define the state 0, ~) by 0,
~)
~
e1°.’21N).
(2.3)
Notice that IN) is invariant under rotations about the z-axis. Among other things we are saying that S2 SU(2)/U(1). The spherical harmonics are defined by =
=
Kj,mI0,~).
(2.4)
We need a formula for in terms of the D-functions above. Using the invariance of IN) with respect to z-rotations one sees that
*
We use the quantum mechanical convention of not distinguishing between the angular momentum operators in a specific representation and the abstract generators.
470
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Wey! characterformula
tains the basic formula ~m(0,~)*
4~D,~
0(q,0,0),
(2.5)
where D(q,, 0, x) means D(g) with g e’~~ e~°~2 ~ having angles 2 Euler SU(2)/U(1), (~, 0 , x). Note that D~0(~, 0, x) D~0(q,0,0), reaffirming that S and that the ~1m are functions on the group which are invariant under right multiplication by U(1). We shall view them shortly as sections of the trivial line bundle over 52• Finally we remark that the peaked state IN) is an infinite linear superposition of extended states =
=
=
IN)=
~‘~JIi~°).
(2.6)
j~O
One can ask what role the other D/,,m play on S2. As we shall see, BWT constructs spherical harmonics like objects with D/~ 1instead of D/~0. coordinates 2 is a complex manifold by exhibiting complex we argue thatdone S onFirst it. This will be through a choice of local coordinates on SU(2). Let =
f 1 ±if~.The angular momentum algebra (2.2), i.e. the Lie algebra of SU(2),
complexified, may be rewritten as [f3,J~] ±J~ =
[J~,J_] =2J3.
(2.7), (2.8)
Notice that f~ and J3 form a closed subalgebra. Let z be a given complex number, then one can uniquely find functions a(z, 2) and b(z, 2) such that 2~t+ e~2~S).I3 E SU(2) , (2.9) e~’~ e~z b(z, 2) is real, and a(z,2)
=
—2+...,
b(z,2)
=
~
(2.10),(2.11)
where the ellipsis denotes higher order. Every group element near the identity can uniquely be written as
g(z, ~,b) ~ =
e’°~’ 2)J~eb~,2)J 3+z~~J3
(2.12)
where t/i is real. It then follows that z and ~1iare good local coordinates for SU(2). Note that multiplication on the right 2by SU(2)/U(1). U(1) does not change the value inofeq. z. What is exemplified Therefore z is a good coordinate on S (2.12) is that locally one can write an arbitrary group element g in terms of a section g(z) of the principal bundle ~: SU(2) SU(2)/U(1) and an element e U(1) in the form g g(z)t One can also show that z defines a complex structure on the coset*. =
—~
=
*
A clear exposition in the spirit of this paper may be found in ref. [91.
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The complex structure can also be exhibited by studying SL(2, C), the complexification of SU(2). One divides SL(2, C) by the Borel subgroup, the complex group generated by f~ and f3. It inherits a natural complex structure from SL(2, C) and the coset space is again ~2• We can now explain the BWT construction for SU(2). The analog of the north pole state IN) is the highest-weight state j,j). Although IN) was a linear superposition of states2 of the form If~0) we usetoa single state f, j). define a state z) here analogous 0, ~) extended by Given a point z on S Iz)
=
e_j~2,z)e~_ ea
,2)j~
e~’2~~lj,j).
(2.13)
The use of a different section of SU(2) such as (2.12) with a z-dependent i~(i immediately tells us that Iz) is a section of a line bundle over S2. That is, z) differs from
~
e_jb~e~t_ ~
=
e~~~’Iz)
(2.14)
by an arbitrary phase ~‘(z, 2). If we define spherical harmonic like objects by =
~j,mIz),
(2.15)
then one can easily see by Taylor series expansion of z) that the above is an analytic function of z since J~j, j) 0. Notice that it is the c-number prefactor of e_Jb in eq. (2.13) that guarantees that the sections will be holomorphic. By construction, automatically, the (~Jm}defines a representation of SU(2). In fact notice that =
~tn(z)
=
e_1~~z~S)D~ 1(g(z,0)),
(2.16)
where g(z,0) is defined by eq. (2.12). By using the orthogonality properties of the one can show that ~ ~ is a natural hermitian metric on the line bundle. Thus we see that BWT constructs holomorphic sections of a line bundle on which the group is represented. In fact, the generators may be written as
a
=
—z—
+j,
(2.17)
z
a
3z ~,
a +2jz, 2—
(2.18), (2.19)
f_= —z
and the sections we have derived are precisely /?‘m(Z) a ~J-~m The above operators are hermitian in the natural metric. For future purposes it is important to observe that, for example, J 3 has both an “orbital” angular momentum part —za/az and a
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Weyl character formula
“spin” angular momentum part j. Thus when one transforms the states one picks up a “phase” in addition to changing the argument. This phase will play a central role in the path integral discussion. More precisely we observe that the functions on SU(2) which satisfy f(gf’) t23f(g) where t E U(1) are sections of a certain line bundle over 52• We wish to study the quantum mechanical system whose wave functions correspond to the sections of this line bundle. Since this line bundle has a natural connection one can easily see that the quantum mechanical system describes the motion of a particle moving on the sphere in the presence of a magnetic monopole. A more detailed study of these line bundles can be found in sect. 4. One last observation; objects of the form D~ 1(g(z,0)),where j’ >1, are also sections of the line bundle in question. However these sections correspond to states =
e1b~S)ezJ_
ea
,2)j~
e~z, ~‘~1j~j),
(2.20)
which are not holomorphic since J~j’~j) ~‘O if j’ >1. Our goal is to out the 2, out of pick all possible holomorphic sections, i.e. the kernel of a a-operator on S sections. These holomorphic sections will be the ground-state wave functions of a supersymmetric quantum mechanical system on G/T. BWT generalizes these ideas to an arbitrary compact semi-simple Lie group G with maximal torus T. We first discuss how to make G/T into a complex manifold. Let (H 1} be a basis for a Cartan subalgebra of G and ~1the corresponding set of roots. For a positive root* a, let Ea be the associated “raising” generator and let E_a be the “lowering” generator. In analogy with the SU(2) case, given a complex number za for each a > 0, one can construct a group element exp( ~ zaE_~)exp( ~ aa(z,2)Ea)exp(~bI(z,2)Hj), a>O
(2.21)
a>O
where b’ is real. Near the identity, a general group element may be written as
g(z,~)~ (2.22) Because {H~Ea, a > 0) form a closed subalgebra one can show that {zi define a complex structure on G/T. If A is a highest weight of a representation then the * In general, G/T has many complex structures. The complex structure is determined by the choice
of “positive roots”.
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highest-weight state IA, A) can be used to construct a holomorphic object
Iz)=exP(— Ebl(z,2)Ai)exp(EzaE_a)
x exp( Laa(z, 2)Ea)exp(
Eb’(z,
2)H~)IA~ A).
(2.23)
The holomorphicity follows from the highest-weight property EaIA, A) 0 for a > 0. Since the representation is finite-dimensional we always get a polynomial in Zr’. The line bundle of which Iz) is a section is characterized by the functions on G which satisfy =
f(gft) =p(t)f(g),
(2.24)
where t E T and p is the representation of T with infinitesimal character A. Since the matrix elements of the representation matrices form a complete set for the functions on G, it follows that the sections of the line bundle are associated with matrix elements of the form D,~~(g) where A is the highest weight of the representation, and ~ and v are weights. In particular we have seen that e~bD~ 5(g(z,0)) is a holomorphic object. Just as in the SU(2) case, e~~~)D;~(g(z,0)) will not be holomorphic in general though it is a section of the
line bundle we are studying. We now provide an argument for the irreducibility of the representation (see ref. [2], p. 21). Suppose the representation R is not irreducible; then there would be at least two independent highest-weight vectors v1 and v2 corresponding to the decomposition R into R1 ~ R2. As we previously remarked, the complexification G~acts on G/T; in fact, G/T GC/S, where S is the maximal solvable subgroup (Borel subgroup) generated by H, and Ea~a > 0. Also R extends to a representation* R~of G~.Since v1 and v2 are highest-weight vectors, R~(E~)v, 0, for a > 0 and i 1,2 and consequently R~(expEa)v, v,, for a> 0. Notice that exp(E~> OCaEa) sends the identity coset to a dense set in G/T (certainly an open set around the identity coset eT, see eq. (2.22)). That is the nilpotent group N {exp ~a > ocaEa} sends eT into NT, dense in G/T. So in this open dense set, the holomorphic sections v1 and v2 are determined by their value at eT. Either one or the other is zero or the ratio v1/v2 (or v2/v1) is a holomorphic function constant in a neighborhood of eT, thus constant. Hence v1 and v2 are not linearly independent. =
=
=
=
=
* We denote the representation of the group and the algebra by the same symbol.
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We illustrate this for SU(2) and SL(2, C). SL(2, C) acts on S2 fractional transformations. The generator
(~~)
E~=
=
C U ~ by linear
(2.25)
is the only “raising operator”. One easily sees that exp cE~:z z + c: the orbit of z 0 is exactly C S2 ~, an open dense set. Since exp cE~will leave any highest vector fixed, any highest-weight vector is determined by its value at 0. So there is only one up to scale. In summary, the complex manifold G/T has a line bundle LA over it which is associated with each infinitesimal character A of T. LA has holomorphic sections which furnish an irreducible representation of the group G whose highest weight is A. These sections will be eigenstates of the hamiltonian for the motion of a particle on G/T which interacts with the vector potential associated with the line bundle LA. We will have to isolate the holomorphic eigenfunctions in order to construct the desired representation. Since we are only interested in finding the character, we will not require detailed information about the eigenstates. We just have to construct the operator 3, which we accomplish through supersymmetty. To compute the character we need only count the number of solutions of a~P 0 for sections 11’ of the line bundle with an appropriate weight. Atiyah and Bott [3,4] observed that this could be done by computing the character index of a, which they do by their holomorphic Lefshetz fixed-point formula. To be more precise, the 0 operator maps sections of the holomorphic line bundle LA into sections of ~4(0. ‘kG/T) ~ L tkG/T, LA), where fl(p,q) are the (p, q) forms. In fact the elliptic complex is5 fl(~ —*
=
=
—
-
=
OLA:
A(~*)(G/T,LA) ....~n(O*)(G/TLA),
(2.26)
where fl(~*)(G/T LA) are all the (0, q) forms twisted by the line bundle LA. Since the index computes the difference of the kernel of a and the cokernel of_a, one has to show that the only non-trivial contribution arises from the kernel of a acting on A~°’°~(G/T, L). One needs to prove a vanishing theorem showing that all the other kernels and cokernels are empty [2]. The required vanishing theorem may be proved along the lines of Lichnerowicz theorem on the existence of harmonic spinors [101;that is 8~lJ=DtD
+
(a positive potential),
where D is the covariant derivative. Moreover, since we know (see sect. 4 for details) that T acts on the sections of LA, the kernel of a decomposes into
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representations of T. This enables us to compute the character-valued index which will then give us the Weyl character formula. Since we will actually work with the Dirac operator on G/T, we now discuss the connection between the space of spinors and exterior forms. The complex manifold G/T has real dimension 2n. The spinors on SO(2n) belong to a 2~-dimensional representation. The exterior algebra A~°’ *) is also 2’~dimensional. It is well known that in the standard embedding of U(n) into SO(2n), the spinor representation of SO(2n) decomposes into the direct sum of the anti-symmetric tensor representations of SU(n) [11]. As far as dimension counting is concerned it appears as if the spinors S(G/T) are the same as fl(° kG/fl. Although true at a single point of G/T, this cannot be the whole story because the spin- 4 features are missing. If K A~°kG/T)is the canonical line bundle of G/T, then the correct statement is *
=
S(G/T) Thus
0L,
®
K~2
,4(O
is the same as the Dirac operator ~LA:
S(G/T)
®
K”2 ® LA
*)(G/T)
~LA
coupled to K”2 as well as to LA:
S(G/T)
—*
(2.27)
®
K”2
® LA.
(2.28)
Here the line bundle K’~2is associated with the infinitesimal character p of the torus given by (2.29)
p=4~a. a>O
Remember that p has projection 1 onto each of the simple roots and thus it is “pure” spin 4 in the two-planes that define the rotations on G/T. If I E T and if I~(l)is the character-valued index of the Dirac operator coupled to the line bundle defined by the infinitesimal character i.’, then the character x
5(i) of the representation with highest weight A may be computed via the relation XA(l)
(2.30)
‘A+P(l).
3. The method In this section we discuss the connection between the supersymmetric point particle and the Dirac operator [12, 13] and its relation to the Weyl character formula. The lagrangian for a supersymmetric free particle moving in l~20~(euclidean time) is given by L
=
—
~
where x~(t)is the trajectory of the particles and
(3.1) ~‘,~(t)
is an anti-commuting vector
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field defined along the trajectory. The supersymmetty transformations are defined by ~
6t~,=i.~,
(3.2),(3.3)
where c is an anti-commuting parameter. One can easily verify that the Noether charge associated with the supersymmetry is (3.4) where p,,,, i.~is the canonically conjugate momentum. The operator canonical commutation and anti-commutation relations are given by =
[x~,pj=i6~~,
=6~,.
~
(3.5), (3.6)
Quantization involves finding a Hilbert spaces where one can represent the above algebra. The x and p commutation relations may be implemented on the square integrable functions on ~ L2(l~2d).The operator x~,is multiplication by ~ and —~a/a~. The anti-commutation relations define a Clifford algebra which may be satisfied by choosing ~/i~ y~/ ~ where {y~) are the Dirac gamma matrices acting on the 2”-dimensional spinor space C 2’~• The Hilbert space associated with (3.1) is the space of spinorial wave functions L2(l~2”)0 C2d. One easily sees that the supersymmetry charge (3.4) becomes the Dirac operator under quantization =
=
ax~
(3.7)
All these ideas generalize to special riemannian manifolds called spin manifolds which allow the existence of spin structures. The heat kernel formula for the index of Q [14—16]can be interpreted in the supersymmetric theory as Tr(—1)~e~’~’,
(3.8)
as observed by Witten [17]. It corresponds to doing the path integral with periodic boundary conditions [181on the fermions. Further work [19—21]demonstrated that the path integral may be used to derive the index formula. The strategy is to exploit the I~independence of eq. (3.8) and do a path integral computation in the f3 0 limit. The calculation reduces to an effective gaussian path integral. These ideas can be extended to the situation where one has a symmetry group G whose action commutes with the Dirac operator. Now the path integral of interest —*
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is the character partition function
1(g) =TrU(g)(_1)Fe_~,
(3.9)
where U(g) is a unitary representation of G on the appropriate Hilbert space. The index of the operator is in this case the character index. We now examine how the presence of the group action affects the previous discussion, using a variant of the method of refs. [19,22—24]. We need to know how to couple a gauge field A,~to the supersymmetric particle of charge q in a gauge covariant way. One verifies that the following lagrangian is supersymmetric and changes by a total time derivative under a gauge transformation: L
=
—
4~i~+iq(A~(x)i.~+ 4F~~(x)~çlç),
(3.10)
where FIL,, ~ a~A~ is the field strength. Canonical quantization leads now to the supercharge =
—
V~Q =
The hamiltonian H
=
4Q2
~—i— a~
—
qA~. /
(3.11)
is of the Pauli form with a o B coupling:
H= ~(_t~
_qAM) + ~
(3.12)
It is important to notice the covariance of the hamiltonian under a gauge transformation A,~—*A,~+ a,~,,while the Schrödinger wave function must transform as IP(x) ~ One recognizes in the latter the transformation properties of a section of a line bundle of charge q associated to the vector potential A. In the lagrangian formulation these gauge transformations will affect the boundary conditions one has to use in the computation of the path integral _~
fx(P)=xf[~] x(O)=x,
_f~dtL 0
To obtain the Weyl character formula from a quantum mechanical approach we apply all the above machinery to the case of a supersymmetric particle moving on G/T. Let us go back to the bosonic SU(2)/U(1) example of sect. 2. In this case the above abelian U(1) gauge transformation corresponds to the right U(1) action of the group. It is important to note that there is also a left U(1) action which acts on the coset S2. Its generator acting on the holomorphic sections is given by eq. (2.17)
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and the character partition function is Tre’°~e~”. If Ix) denotes the 6-function normalized state on S2 with support at x, then e’°~~Ix)e~~°Ix , 0) =
(3.13)
where the point x8 is the image of x under the rotation by 0 around the z-axis. This phase enters in the following way into any path integral computation: 2~
Tre13e~””=f
(3.14)
S
=
=
f d2xe’°~(x0Ie~”Ix)
(3.15)
f d2xe_b0JfX~~[~x]exp(_fdtL). 2
x(0)=x
0
(3.16)
S
So besides generating twisted boundary conditions on the path integral, the operator e’°~~ introduces an overall phase, which plays a central role in our method. We will see that the phase is determined uniquely by the line bundle in question. 4. Supersymmetry and line bundles on G/ T One could now derive the Weyl character formula: first we write down the standard supersymmetric non-linear sigma model L
=
4g~~(x)~~ +
D~
where g~ is the metric on G/T and (x~,~) are local coordinates on G/T with their fermionic partners and T acts on G/T. Next add to the lagrangian L the connection and curvature terms of eq. (3.10), given by explicit formulas from differential geometry. The path integral with the new lagrangian gives the fixedpoint computation needed to compute the character-valued index of the appropriately twisted Dirac operator [23,24]. We follow this procedure except for a significant detour. We derive supersymmetiy on the coset space in a new and we believe interesting way by introducing the notion of horizontal supersymmetty. In any case, we need horizontal supersymmetry to achieve the Weyl—Kac character formula for loop groups with central extension*. So the following discussion which generalizes to the loop group case should be seen as a warm-up exercise for that more difficult task. *
For clarity of the argument the loop group case will be treated in a separate publication
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To construct the required supersymmetric theory for a spinning particle on the coset G/T we use the fact that G is a principal fiber bundle over M G/T with structure group T. For such a principal bundle we can lift curves in M to curves in G by choosing a horizontal projection, or equivalently, a connection: the tangent vectors to the lifted curve are horizontal. Similarly, we will see that a supersymmetry on G and a connection can be used to induce a natural supersymmetrv on M. We now describe the construction of the induced supersymmetry. Let (x~,r’) be local coordinates on the bundle where x~are coordinates on M and r’ on the fiber. Although our construction is coordinate independent, local coordinates are pedagogically useful to display the explicit supersymmetric transformations (4.2) and (4.3). We note that the construction works for any principal T-bundles over a manifold M; the group structure of G is not necessary. Given a vector potential A in the base, a vector field (zn, k’) is said to be horizontal if A’,~z’~k’ 0. In particular, a curve (x~(x),r’(x)) is the horizontal lift of a curve x(t) in M if =
—
=
A’~V~’ —
=
0.
(4.1)
The connection provides a natural way of identifying vector fields on the base with horizontal vector fields on the bundle. The supersymmetry transformations are given by 6xlL
ei~’,
=
~
6~/J~ =
=
E~7’,
6i~~ = Er’.
(4.2) (4.3)
If we interpret the supersymmetric deformation of X~L as a vector field then it has a horizontal lift. If we restrict ourselves to the supersymmetric deformations in G which are horizontal then their projection onto the base manifold can be regarded as induced supersymmetric transformations of M. On the bundle we have a globally defined connection one-form with values in the Lie algebra of T given by w’ =A~dx~ dr’. A horizontal vector field is one that annihilates the connection. Requiring the bosonic supersymmetry transformations to define a horizontal vector field, i.e. annihilate the connection, imposes conditions on the fermionic partners —
0
Consistency requires
=A~6x~
=
~
=
E(A’~IJ~—
—
n’),
I5T’
(4.4)
that the supersymmetric partner of the connection defined by
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Weyl character formula
6w also vanish. The supersymmetric partner of the above equation is ‘i
—A~~ ~F~/J~/J~ —
=
0,
(4.5)
where F dA is the field strength. This last equation is just the equation of motion for a spinning particle in a generalized electromagnetic field. It has a Pauli type magnetic dipole coupling a B. Notice that it is different from eq. (4.1), which applies to spinless particles. In conclusion, the supersymmetry transformation on the base (4.2) is equivalent to imposing the horizontality conditions (4.4) and (4.5) on the bundle. We will refer to these as supersymmetric horizontality. The local coordinates are only used to make contact with familiar formulas. The method itself is coordinate independent and global. Abstractly, given a supersymmetry and a connection on a principal bundle we can induce a supersymmetiy on the base and the associated bundles by imposing supersymmetric horizontality. Now consider the case of the bundle G being a compact semi-simple Lie group with fiber T over G/T. Instead of expressing our formulas in terms of coordinates on the base we will express everything in terms of local sections of the bundle. Let g t ~ m be the Lie algebra of G with t a Cartan subalgebra and m the orthogonal complement. Two local sections are related by a gauge transformation =
=
g
2(x) =.~1(x)4)12(x),
(4.6)
and one finds ~‘(x)
d~2(x)
=
4~_1(x)~_1(x) d~1(x)4,2(x) +
4~’(x)d~12(x). (4.7)
An element X of the Lie algebra can be decomposed into its vertical component X~E t and its horizontal component Xm E m. The vertical projection (~1 d~)~ A(x) along the torus transforms as an abelian T-connection. we write a generic 1 E T thereIf exists a global one element g of the group as g ,~(x)t ‘, with t e form on G given by =
=
w
=
(g’ dg),
=
=
tAt’
—
dtt’.
(4.8)
The one-form w is independent of the cross section. It is the projection on t at the identity of G and is invariant under left multiplication. Now define a superfield G ge°’associated with g ~ G. The fermionic partner y takes values in the Lie algebra of G. In this section, 0 is a Grassmann variable. The coordinate superfields on G/T and on the fibers are X” =x~+ ~ and =
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Weyl character formula
481
Oif. Given a section (~(x),~(y)) we have G=GTl=g(1+0y)=~(x)(1+0~(x,~,))eT+O~ =
(~(x) + oa(x)~p’~(x))T’.
(4.9) (4.10)
The last line relates the section to the local coordinates. The supersymmetry transformations along a curve (x(t), ~fr(t))are defined by eq. (4.2), or equivalently 6G = e(a0
—
oa,)G.
(4.11)
The horizontal supersymmetly deformations must then obey w’=A~6x~~—6r’=0and
6w’=A’=O.
(4.12)
These conditions will be more useful in the future if written in terms of the sections rather than in terms of the coordinates (x~,~ We observe that given (g’ a~(x)), we can write w = ~ 6g), = (g’6g)~= 0 except that 6 is now a supersymmetric variation. The supersymmetric horizontality conditions become =
—
w
=
0
=
~~(x)
6w
=
0
=
(~‘a~~)~(x) — (ymym)t.
—
i~,
(4.13) (4.14)
The term bilinear in the fermions is actually a commutator because the fermions anti-commute. One can easily verify by the use of the Maurer—Cartan equations for the Lie group and the Cartan structural equations that this fermionic bilinear is the curvature of the bundle. The supersymmetric lagrangian on G is given by
LG= fdoTr(G’DG)(G’a1G) =Tr[(g_1g)2+~y+g_1gy2~.
(4.15)
This is the standard lagrangian for a supersymmetric particle on G with riemannian connection. If we decompose the superfield into its vertical and horizontal parts and if we impose the supersymmetric horizontality conditions then LG
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Weyl character formula
reduces to the supersymmetric lagrangian on G/T: LG/T = Tr{(~1(x)~(x))~ + (~m(~) +
+
Tr[(~1(x)g(x))m~m(x)~m(x)j.
(4.16)
Notice that it only depends on the coordinates on M, is covariant and supersymmetric. We want to emphasize that the fermionic term is the standard riemannian covariant derivative on G/T. This follows from supersymmetiy and the fact that the bosonic kinetic energy term is the standard riemannian one. Naively it appears as if the fermions are coupled to a T-connection besides the riemannian one. The illusion is caused by the fact that the original y was left-invariant. Since we are studying cosets of the form gT, the ~m transforms when one changes section and must be reflected in (4.16). In any event, the lagrangian (4.16) is the one described at the very beginning of this section, but derived differently. We now turn to the construction of the line bundles over G/T and to the study of the group action on them. A line bundle L over G/T is determined by an appropriate one-dimensional representation of the group T. A section of L is the same as a function on the entire principal bundle ir: G G/T satisfying —~
(4.17)
t/i(gt’) =p(t)~t(g).
It is important to keep in mind the difference between right- and left-action of T. The right-action lets us move up and down the fiber and tells us that the function transforms under the representation given by p. A local section of L must be parametrized by the coordinates x of G/T and defined by ~/i
t~i
~(x) =çli(~(x)),
(4.18)
where ~(x) is a given local section of G. This determines the left-action of T on LjtIi(x)
=
~Ii(Ig(x)).
t/i:
(4.19)
Note that l~(x)is a new element of G in the fiber above the point I . x = 7r(l~(x)) in the base. We can use right multiplication to relate lg(x) to the section by introducing t(x, 1) E T as follows l,~(x)= ,~(lx)t t(x 1). We can write L1~(x) = ~(g(I .x)t’(x, 1)) =p(t(x, I)~(l ~x)) =p(t(x,l))~(lx).
(4.20)
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In accordance with our discussion in sect. 2 we see that the transformation law (4.20) for the sections has both an “orbital” part and a “spin” part. Because of the presence of the spin part there are some phases which will have to be accounted for in the path integral computation. The coupling of the line bundle defined by infinitesimal character r’ is implemented by the method described by eq. (3.10). Adding the appropriate gauge coupling to (4.16) gives the lagrangian =
LG/T
+
((~-‘~)~ (~m~m)t) —
(4.21)
Note that the terms in parentheses are the connection and curvature terms (3.10).
5. Index calculation We have to calculate the index of the Dirac operator on G/T coupled to the appropriate line bundle. It is given by 1(1) =TrU(/)(—1)”e~”.
(5.1)
Because of the presence of U(l), the path integral has boundary conditions x(0) =x and x(f3) = 1 ~x ~x1. The key to the path integral proof of the fixed-point formula is the observation [23, 24] that for small I~the action behaves as 2//3. distance between x(0) and x(/3)~ The dominant contributions of the path integral will come from the neighborhood of those points in G/T which are left fixed by the action of T, i.e. x 1=x. Our computation reduces to one in perturbation theory around the fixed points. Since 1 E T acts on G/T on the left by mapping the coset x =gT to x1 = IgT, the fixed points are cosets gT such that lgT = gT for all 1 ~ T. This means that g E the normalizer of T. Thus the fixed points of the T action on G/T may be identified with the Weyl group of T, W(T) = N(T)/T as expected from the fixed-point formula [3,4]. Let n E N(T); then w E N(T)/T defined by w = nT is a fixed point. Near this point a section of the super-group may be written as S(t) =ne~(t)eoist),
(5.2)
where ~ and ~ have components only in the m directions. The boundary conditions on S(t) (which respect the supersymmetty) are S(/3)T==IS(0)T.
(5.3)
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It follows that the boundary conditions on 2 and ~ are i(j3)
=l~2(0)I;’,
((13) =1(0)1~’,
(5.4),(5.5)
where l~ n 11n. Notice that l~only depends on the choice of coset w nT. For, if n 11n 1T n2T then nj 1 n~’ln2.Thus 1~is determined by wE W(T). To study the action to quadratic order near each fixed point, we write =
=
-
=
=
1= e’°, where 0
E
1~ e’°~,
(5.6), (5.7)
=
t and 0~ n’0~ is just the adjoint representation. If =
2(t) =i ~ 2’~(t)Ea
(5.8)
a~I
with 2~(t)
=
2~(t),then the boundary conditions become 2”(f3)
=
etao~*2(~(0).
(5.9)
because 1’Ealw e~~O~*Ea. Using a similar expansion for ~ we find the same boundary conditions. The transformation properties of the sections of the line bundle near the Weyl point w nT with coordinates 2 are easily computed using eq. (4.20): =
=
L,q(2) =p(l;’)ç~(l~2) =
e~’’°”ç&(e~’~~°”2”)
et 0~(e1~02~*) (5.10) 1. In the neighborhood of the Weyl points, the where v~1, to wvw’ and order a~ waw lagrangian quadratic is =
=
.
=
L
=
Tr~(i—4[r’,2])
+
Tr(~_~[v,(])(.
The Dirac index for the line bundle characterized by 4(1)
=
~ wEW(T)
e~det{_
—(— dt
dt
-
(5.11)
v,
H~,.])}det(_
-
~[Li,
.])~(5.12)
t
is obtained by performing the gaussian path integral defined by eq. (5.11) with twisted boundary conditions (5.9). Notice the inclusion of the end point phase factor contribution and the sum over the fixed points (minima of the action). A
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485
Weyl character formula
straightforward evaluation leads to the formula
~
I.~(l)
=
e~”~°~ fl . . 2zsin~a(0~) a>0
(5.13)
.
wEW(T)
The Weyl character formula is obtained by using observation (2.31) relating the character with the index:
X~(l)=IA+P(l).
(5.14)
Using path integrals we have derived the Weyl character formula:
XA(l)
=
~
e’~°~ w~W(T)
~ a>0
.
.
2isin~a(0~)
(5.15)
6. Interlude In this paper we derived the Weyl character formula from the path integral. The standard supersymmetric lagrangian had to be modified (3.10) to couple ~ to a line bundle. We also had to allow paths that were not closed in order to obtain the T-index as in refs. [23,24]. We introduced the notion of horizontal supersymmetry. Though not essential to our derivation of the Weyl character formula, it will be important in the construction of the supersymmetric model on LG/T. There, curves on LG/T have to be lifted to curves on LG (in a horizontal supersymmetric way) so that ultimately we can obtain a supersymmetric sigma model lagrangian based on G and derive the Weyl—Kac character formula. The details of this construction are technically complicated. In order not to obscure the principal ideas, we will publish them elsewhere. We would like to thank C. Itzykson for his friendly exhortations to complete this project and M. Picco for carefully reading the manuscript. The Aspen Center for Physics, le Laboratoire de Physique Théorique et des Hautes Energies, the Massachusetts Institute of Technology, Rutgers University, la Universidad Autónoma de Madrid and the University of California at Berkeley are warmly thanked for providing crucial brief two-body interactions in this long-range collaboration. References [1] H. Weyl, The classical groups (Princeton Univ. Press, Princeton, 1946) [2] A.M. Pressley and G.B. Segal, Loop groups (Oxford Univ. Press, Oxford, 1986) 13] M. Atiyah and R. Bott, Ann. Math. 86 (1967) 374
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Weyl character formula
14] M. Atiyah and R. Bott, Ann. Math. 88 (1968) 451 [51V. Kac, Infinite dimensional lie algebras (Cambridge
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