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Generalized prequantization on principal bundles over homogeneous spaces and its applications
Journal of Geometry and Physics 13(1994)124—138 North-Holland
JOURNAL OF
G~METRYA~D
PHYSICS
Generalized prequantization on principal bundles over homogeneous spaces and its applications *
a
Maozheng Guoa, Mm Qiana and Zhao-hui Qian’~’1
Department of Mathematics, Peking University, Beijing 100 871, China b CCAST (World Laboratory), P.O. Box 8730, Beijing 100 080, China Received 14 June 1992 (Revised 13 April 1993)
In this paper, we set up a framework of generalized prequantization by starting from a single structure on the principal bundle associated with the Lie group pair (G, H). This procedure could be regarded as a kind of “postulated rules” taking the place of ordinary prequantization. Illustrative examples which show the usefulness of our scheme are, among others, compact semisimple Lie groups and loop groups. The model spaces of loop groups are also discussed. Keywords: geometric quantization 1991 MSC: 58 F 06
Introduction Geometric quantization in its well established form starts from a symplectic manifold (M, w), and a complex line bundle over M (cf. refs. [13, 17]). In case of a co-adjoint orbit M = G/H, due to Kirillov’s coorbit theory [6], what one gets after quantization is a certain representation of G, which is usually only one sector of the quantum mechanical Hilbert space. And it is always non-trivial to find a global form of the connection which meets the need of prequantization [13, 17]. In this paper, we set up a framework of generalized prequantization by starting from a single structure on the principal bundle P associated with the Lie group pair (G, H), which takes advantage of the Ehresmann connection induced from the G-actions on P. This procedure could be regarded as a kind of “postulated rules” taking the place of ordinary prequantization [13, 17]. *
Research partially supported by NNSFC Grant (1991—1995). Permanent address: do Prof. Pei-Xin Qian, Institute of Microelectronics, Tsinghua University, Beijing 100 084, China.
0393-0440/1994/$7.00 © 1994 Elsevier Science B.V. All rights reserved SSD10393-0440(93)E0030-S
M. Guo et al. / Generalized prequantization on principal bundles
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The conditions under which our suggested procedure can really be carried out are given in this paper. Illustrating examples which show the usefulness of our scheme are, among others, 1. a short and transparent treatment of geometric quantization and the Borel— Weil theorem for compact groups (other finite dimensional cases could easily be found also satisfying our conditions) (cf. refs. [1, 3, 12]) 2. the Borel—Weil theorem for 1oop groups LG (M = LG/T) as treated in ref. [10], 3. geometric quantization ofthe Wess—Zumino—Witten (WZW) model, whose correct phase space should be LG/G instead of LG/T as most people seem to take for granted [4, 9—11, 15, 16], and 4. the model spaces of ioop groups and the model space of the WZW model [11]. Actually we believe that all kinds of induced representations are susceptible to such explanations.
1. Generalized prequantization on principal bundles over homogeneous spaces Let G be a Lie group, and H be a closed Lie subgroup of G. Then G can be identified with a principal bundle on the homogeneous space G/H of G (the left coset space of H) taking H as both its fiber and its structure group. The projection mapping ~r : G G/H is just the natural quotient mapping induced by H. The left actions of G on G induce the actions of G on G/H, and they commute with it. The right actions of H on G are just the actions of the structure group H on the principal bundle iv: G G/H. Thus at each point g of G, we identify the Lie algebra 7~Iof H with the subspace of the tangent space Tg G as follows: —*
—~
ag:7(—~TgG,
(1.1)
~F_+rig(~),
where rig (~)is the tangent vector of TgG corresponding to the one-parameter current g exp ( t~)of G for t ~ R’ [1, 141. Let g, 7~Ibe the Lie algebras ofthe Lie groups G, H respectively. Suppose that there exists a direct sum decomposition of ~ which satisfies —
g=h~Ei~M,
[7-i,M]cM.
(1.2)
We define a 1i~-valuedone-form c~on G as ct(L~)(g)~
—~
for all ~ E g, all g E G,
(1.3)
where L~is the left-invariant vector field on G with L~(e) = and ~ is the 7~component of~under the decomposition of g in (1.2). Then c~is an Ehresmann ~,
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connection in the principal bundle it: G
—f
G/H [141, such that
a(a(~)) =
(1.4)
cs(R~~X)(g)= Adh.1cE(X)(g), for all ~ ~ 7-1, all h E H, all g E G, and all tangent vectors X E TgG, where the mapping a is defined in (1.1), Rh denotes the right action of h on G, and Rh is the tangent mapping of Rh. Due to ref. [141, the connection c~determines a horizontal space decomposition of TgG for each g e G as follows: *
TgGHgeVg,
(1.5)
where the vertical space Vg ~ {ag(c~) ~ ~ 7-1}, and the horizontal space Hg ~ kerag. According to ref. [141, the curvature of the connection ct is defined as a 7-1valued two-form Q on G such that Q(X,Y)(g)
(1.6)
~da(ivHg(X),mHg(Y))(g)
for all g E G, and all tangent vectors X, Y e TgG, where HHg: TgG projection. Then we have Q(L~,L~)=
[‘~M,
Hg is the (1.7)
1Ml7t,
Q(Rh~X,R~~Y)(g)= Adh.’Q(X, Y)(g),
for all E all h E H, all g ~ G, and all tangent vectors X, Y E TgG, where crM denotes the M-component of~. ~,
~
~,
Generally speaking, the curvature Q on the principal bundle it: G G/H is not the pullbackof a 7-1-valued two-form w on G/H by the mapping iv. However, in the case discussed below, the curvature Q is really the pullback of a 7-1-valued two-form w on G/H, which could be regarded as the basic structure of generalized prequantization of the principal bundle iv: G G/H (the counterpart of ordinary symplectic structures of classical phase spaces). By the left actions of G on G, and the induced actions of G on G/H, we define the infinitesimal actions of ~ on C°° (G) and C°° (G/H), respectively, as —*
—~
~G(F)(g) ~ ~F(exp(-t~)g)~o, def
= ~_fojv(exp(_tçe)g)~~.. 0,
(1.8)
for all ~ e all g E G, all F E C°°(G),and all f e C°°(G/H). In fact, we can relate ~G/H to ~ with the connection c~and details are summarized in the following proposition of generalized Kostant mappings of the principal bundle iv: G G/H. ~,
—p
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Proposition 1.1 (Generalized Kostant mappings). Suppose that the Lie algebras g, 7-1 of the Lie groups G, H satisfy (1.2), and that [M, MI cM~Z,
(1.9)
[7-1,y]cy,
7-1——Zey,
where Z is the center ofl-1. We define two linear mappings ô0 : ~ C°°(G/H,Z), respectively, as ôG(~)(g)~
(1.10)
C°° (G, 7-1) and
—#
:
ÔG/H
-(Adg.’~,
~
for all~E g, and alig e G. Then: (1) The curvature Q of the connection cv defined in (1.6) is the pullback of a Z-valued two-form w on G/H, which satisfies w(iv~(L~(g)), iv~(L~(g)))(iv(g)) =
(1.12)
[c~M,~)M]z
for all çe, ~j E g, and all g E G. (2) For all~E g, and allg E G, ~G(g)
+
= ~G/H(g) =
Lo0(~)(g) (1.13) (1.14)
~G/H(g)—a(~G(~))(g),
dôG/H(~) = i(çe~,~)w,
where ‘~G/Hdenotes the horizontal lift ofthe vectorfield ‘~G/Hon G/H determined by the horizontal decomposition of TgG in (1.5) corresponding to the Ehresmann connection cv. Due to the identity (1.14) it makes sense to call the vector field ‘~G/H on G/H a generalized Hamiltonian vector field of öG/H (~) corresponding to w. (cf refs. [13, 17]). Due Kostant to the identities (1. 13) andprincipal (1.14), we call the ö~, 5G/H generalized mappings of the bundle iv:mappings G G/H and ~ (cf refs. [6, 13, 17]). —f
The proof is by direct calculation, we omit it. Corollary 1.2. Suppose that the Lie algebras g, 7-1 satisfy the conditions in (1.2), and H is abelian. Then the
mappings öG, öG/H
defined in (1.12) satisfy
öG(c~)=ôG/H(~~)oivforall~e~, ~G(g)
= ~G/H(g)
+ Lo
0/~(~)olr(g)(g) =
(1.15) (116)
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M. Guo et a!. / Generalized prequantization on principal bundles
Remark. If the Lie algebras N of the Lie groups G, H satisfy the conditions supposed in proposition 1.1, then we can associate the left infinitesimal actions of g on C°°(G)with their induced actions of ~ on C°°(G/H)through the Ehresmann connection cv in proposition 1.1. This procedure is similar to ordinary prequantization of a complex line bundle on a symplectic manifold [13, 17]. However, since the Z-valued two-form w on G/H (corresponding to the curvature Q of the connection cv) in (1.12) may be degenerate, and the dimension of the fiber H of the principal bundle iv: G G/H may be larger than 1, we would like to call this procedure generalized prequantization of the principal bundle G on its homogeneous space G/H. ~,
—f
Definition 1.3. (Poisson structure of a subspace of C°° (G/H,Z), and generalized prequantization operators) Assume that the Lie algebras 7-1 satisfy the conditions (1.2), (1.9) and (1.10). We define a subspace C~°(G/H,Z)of C°°(G/H,Z)as ~,
(1.17) Then (1) For all ,~iE g, we define the Poisson bracket of two Z-valued smooth functions ÔG/H (~)and öG/H (ij) as ~,
def {öG/H(’~),ÔG/H(i7)} = W(17G/H, ‘~G/H),
(1.18) where w is a Z-valued two-form on G/H defined in (1.12), and ~G/H defined in (1.8) is the generalized Hamiltonian vector field of ÔG/H (~)which satisfies (1.14). (2) For all prequantization operators ~G/H (~)on ~ (G) as ÔG/H(~) ~~G/H
+
LoG(~)=
(1.19)
~G,
where c~is defined in (1.8), óG(~)in (1.11), and ~G/H is the horizontal lift of ~G/H.
Theorem 1.4. Suppose that the Lie algebras g, N of the Lie groups G, H satisfy the conditions (1.2), (1.9) and (1.10). Then: (1) for all ~ E G,
n]);
(1.20)
{ôG/H(c~),öG/HO7)} = ~GIH([’~,
the Lie algebra (C~° (G/H, 2), { }) is Lie-isomorphic to ~ (2) the generalized prequantization mapping Q : öG/H (~) ~G/H (~)is a Lie homomorphism from (C~’° (G/H, Z), { }) to the space of linear differential operators on C°° (G). moreover,
,
~
,
M.
Guo et al. / Generalized prequantization on principal bundles
129
Proof (1) By (1. 10), it is implied that [N, NI cy.
(1.21)
Then it follows from (1.2) and (1.21) that the identity [~, ‘lIz = [~M,
(1.22)
?1MIZ
holds for all ~, ,~e ç. Since {öG/H(~), ~G/H(’l)}(it(g)) = W(’lG/H, ~G/H)(7r(g)) =
=
w(—iv~(L(Adg.~1fl)~(g)),
[(Adg.”l)~,
it~(L(Adg._I~)~(g)))
(Adg.’~)M]z
[Adg.’,~, Adg.~Iz = (—Adg.’[~, ‘l])z =
= ôG/JJ([~~, p1),
it follows that the identity (1.20) holds for all ~~‘ie Moreover, (C~(G/H, 2), { , }) is a Lie algebra, and the mapping y, : ‘—p ôG/H(~) is a Lie isomorphism from g to C~(G/H,Z). (2) Since ~ is the representation of ~ on C°° (G), it follows from (1 .19) and (1.20) that the mapping Q is a Lie homomorphism. E ~.
Remark. Our generalized prequantization is just the representation of g on COC (G) by (1.19). Definition 1.3 and theorem 1.4 show that our generalized prequantization scheme is similar to the ordinary prequantization scheme [13, 171 in the definitions and properties of the Poisson bracket and prequantization operators. Since the 2-valued two-form w on G/H may be degenerate, the Poisson bracket and generalized prequantization operators in our scheme are only defined in a subspace C00°(G/H, 2) of C~(G/H, 2), which is not an algebra. An interesting question is how to extend the definitions of the Poisson bracket and generalized prequantization operators to a linear space which is larger than C~(G/H, 2), is a Poisson algebra, and guarantees a result similar to theorem 1.4(2). The following section is motivated by the question whether there exist some illustrative examples of the Lie group pairs (G, H), whose Lie algebras g, N satisfy the conditions (1.2), (1.9) and (1.10) supposed in proposition 1.1. The answer to this question is actually positive. In fact, the dimensions ofthe interesting examples range from finite dimensions to infinite dimensions. In particular,
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applying this method to loop groups [9, 101, we make contributions to the geometric quantization of the Wess—Zumino—Witten model in two-dimensional conformal field theory (cf. refs. [11, 161). 2. Applications of generalized prequantization 2.1. COMPACT SEMISIMPLE LIE GROUPS
Let G be a compact semisimple Lie group, and T be the maximal torus of G. It is easy to see that the principal bundle iv: G G/H satisfies the conditions supposed in theorem 1.2. In fact, due to Adams [11,the Lie algebras ~, Y of G, T satisfy —f
=
[T, TI
=
[‘T, MI
=
[M, MI
c
{0},
(2.1)
M~T,
where M
‘~
span{(efl e_~),i(ep + e_p) I /3 is a positive root of cc, and ep, e_p are the root vectors of Gc corresponding to the roots /3 and /3}. —
—
We see that the connection (1.2), (1.9) and (1 10) are satisfied, thus the connection cv and generalized Kostant mappings of this principal bundle can be obtained through proposition 1.1. Moreover, since (2.1) make the conditions in corollary 1.3 satisfied, generalized Kostant mappings are only determined by the linear mapping öG/T : ~ C°°(G/T,T), i.e., .
—~
=
—(Adg.’~)j
for all g e G, and all ~ e g. If ,~is a weight of G, then we obtain the induced complex line bundle C LA G/T equipped with the induced connection V(A) and induced Kostant mapping. It implies that the line bundle L2 is a “generalized” prequantization —.--~
line bundle on G/ T, where the word “generalized” means that the curvature w1 of the induced connection may not be a symplectic form on G/ T. We choose the Kãhlerian polarization P on G/ T [13, 17] as P(it(g))~{it~(L~(g)) I~~V+}, (2.2) 1÷is the positive root vector subspace of cc. The polarization space where A I~f(LA)~ {F E F(L
2) I V~~(g))F(g)= 0 for all tangent vectors X(iv(g)) E P(iv(g))},
(2.3)
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131
where F (LA) is the smooth section space of the complex line bundle LA, which can be identified with the following subspace ofthe smooth function space of G: F(LA) ~ {F E Cc~o(G)I F(gt) = )~‘(t)F(g)fora1lte T, allg e G}. (2.4) In fact, the polarization space I~”(LA) is just the holomorphic section space I~. of the holomorphic line bundle iv: LA Gc/B +, where B + denotes the Borel subgroup of G~corresponding to the positive roots of cc. The properties of the space I~are elucidated by the most famous Borel—Weil theorem for compact groups. —~
Theorem 2.1 (Borel—Weil theorem of compact groups [1, 3, 91). (1) The holomorphic line bundle LA has nonzero holomorphic sections if and only if the weight ofG is the antidominant weight ofG. (2) If)L is the antidominant weight of G, then the holomorphic section space I~ of the holomorphic line bundle LA is the irreducible unitary representation space of G with the lowest weight )~. ~.
Remark. The arguments above show that the irreducible unitary representation of G can be obtained through our generalized geometric quantization ofthe complex homogeneous space G/ T of G. Proposition 1.1 and corollary 1.2 imply that the induced Kostant mapping makes the induced representation of G coincide with the geometric quantization of G/ T. For more details of the calculation in the case of compact groups, please refer to refs. [6, 121. 2.2. LOOP GROUPS AND THEIR CENTRAL EXTENSIONS
Suppose that Gis a connected, simply connected and compact semisimple Lie group. Let LG be the 1oop group of G, and LG be the central extension of LG determined by the basic two-cocycle w of LG in ref. [91. LG can be regarded as the different principal bundles on its different homogeneous spaces. In this paper, we would like to discuss the following two fundamental homogeneous spaces [101: 1. LG/T: the principal bundle T x T LG LG/T, where T, T denote the center of LG, and the maximal torus of G, respectively. 2. LG/G: the principal bundle T x G LG .QG, where .QG is the subgroup of LG whose element y satisfies that y (1) = e where 1 e S1, and e is the unit element of G. In fact, QG ~ LG/G. —f
—~
—*
—~
LG/T. By the structure of the Lie algebra Lc of LG, it follows directly that the principal bundle Tx T LG LG/T satisfies the conditions supposed in proposition 1.1 and corollary 1.2. 2.2.1. The principal bundle iv :
—~
—f
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M. Guo et a!. / Generalized prequantization on principal bundles
Then this principal bundle iv : LG LG/ T can be equipped with the structures of the connection cv, and generalized Kostant mapping öLG/T. If A is a weight of LG [91,then we can induce a complex line bundle C LA LG/T on LG/T from this principal bundle, and impose the induced connection and induced Kostant mapping on the complex line bundle. Thus we regard LA as a generalized prequantization line bundle on LG/T. Moreover, by the general theory of Pressley and Segal [101, the fundamental homogeneous space LG/ T can be identified with a homogeneous complex manifold LGc/B + Ge, where B + G~consists of the boundary values of holomorphic maps —~
—f
y : {z : IzI < 1}
Ge
such that y(0) belongs to the Borel subgroup B~of Ge corresponding to the positive roots of cc. As in the case of compact groups, we choose the polarization P determined by the Lie subalgebra N~Gcof Lcc, which consists of the boundary values of holomorphic maps ~: {z: IzI < l} cc such that ~(0) belongs to the nilpotent Lie subalgebra .jV + of cc corresponding to the positive roots of cc. By the complex structure of LG/T, it follows that the polarization space ~ {F E F(LA) I V~(7))F(y) = 0 for ally E LG, and all tangent vectors X (iv (y)) E P (iv (y) ) } is just the holomorphic section space lj of the holomorphic line bundle LA on the homogeneous complex manifold LG/T (~ LGc/B~Ge). Being similar to compact groups, the Borel—Weil theorem for ioop groups [101 completely describes the space Ij. Theorem 2.2 (Borel—Weil theorem for loop groups [101). (1) The holomorphic line bundle LA on the homogeneous complex manifold LG/ T possesses non-vanishing holomorphic sections if and only if the weight A of LG is antidominant. (2) If the representation Ij is nonzero, then the representation is: (i) ofpositive energy, (ii) offinite type, (iii) essentially unitary, and (iv) irreducible, with lowest weight A. Remark. The Borel—Weil theorem of loop groups and induced Kostant mapping also identify the induced representation of LG on 1j with the generalized geometric quantization of its infinite dimensional homogeneous complex manifold LG/T ~ LGc/B~Gc.
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133
2.2.2. The principal bundle iv : JLG LG/G( ~ QG). The Wess—Zumino— Witten model with the gauge group Gplays an important role in two dimensional conformal field theory [2, 4, 16]. Following from the discussions in refs. [11] and [161, the classical phase space of the WZW model is Q Gx Q G. Since the left copy and the right copy of Q G in this classical phase space are homeomorphic, it is enough to investigate the geometric quantization of QG. Moreover, the geometric quantization of the classical phase space Q G x Q G is just the direct sum of the geometric quantizations of the two copies of QG [161. With the above arguments, it is natural to take the principal bundle T x G LG —p QG as the starting point of the geometric quantization ofQG. It follows that this principal bundle satisfies the conditions (1.2), (1.9), and (1. 10). By proposition 1.1, we obtain an Ehresmann connection and generalized Kostant mappings. If A = (0, 2, h) is an antidominant weight of JiG [10], where the level h (a positive integer) is the character ofthe centerT of LG, then 2 is an antidominant weight of G. By theorem 2.1 (Borel—Weil theorem for compact groups), there exists one and only one irreducible unitary representation (VA, PA) with the lowest weight 2, where the representation space VA is the holomorphic section space I~ofthe induced holomorphic line bundle LA on the homogeneous complex manifold G/T. With A, and the irreducible representation (1’~, PA) of G, the complex vector bundle VA ~ QG can be induced from the principal bundle T x G LG QG. Then we can also impose the structures of the induced connection and induced generalized Kostant mappings on this complex vector bundle. Thus we regard the complex vector bundle Ef) equipped with the structures stated above as the prequantization vector bundle of Q G. In order to obtain the polarization of the prequantization vector bundle of QG, we first briefly investigate the structure of the homogeneous space Q G ~ LG/G. As we know from ref. [10], QG can be identified with a homogeneous complex manifold LGc/L+ G~,where L+ Ge consists of the boundary values of the holomorphic maps —+
—~
—b
—i
—~
y: {z: IzI < l}
Ge.
Moreover, QG is a Kähler manifold whose Kähler form is just the basic twococycle w of LG [9]. Now we choose the polarization P of Q G as P(,v(y))~f{iv~(L~)) ~EL~cc}
(2.5)
for all y e LG, where the Lie subalgebra L~cc of Lcc consists of the boundary values of holomorphic maps ~: {z: IzI < l}
cc
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M. Guo et a!. / Generalized prequantization on principal bundles
such that ~(0) p
0P (EA(r))
=
0. The polarization space is defined as
¶~{ FE F(E~”~) I Vx(fl(A))F(y)
=
0 for all
(2.6)
yE LG, and all tangent vectors X(iv(y)) e where F (Er) denotes the smooth section space of the vector bundle E.~T),which can be identified with the following subspace of C~(LG, VA): F(E~T))~
{ FE
C~(LG,VA) I F(yt) = A(t)’F(y), F(yg) = p~(g).’F(y) for all y E LG, all t E T, and all g e G}.
(2.7)
Lemma 2.3 (The explicit form ofF0”(E~)).SupposethatA(= (0, 2, h)) isan antidominant weight oftG. Let (VA, PA) be the irreducible unitary representation ofGe induced by the antidominant weight 1 of G in theorem 2.1. We extend PA as the homomorphism PA from L~Gcx C’< to Hom(VA, VA), which satisfies PAIL~+GC =
1,
PAle>’ = h,
PAIGc
=
PA,
(2.8)
where c>< denotes the complexification of the center T of LG,and the subgroup L~Ge ofLGc consists of the boundary of the holomorphic maps y: {z: IzI < l}
‘Gc
such thaty(0) = e~G. We define EA as the induced holomorphic vector bundlefrom the complex principal bundle L~Gcx CX
LGc
—*
QG(~ LGc/L~Gc)
by the extended homomorphism PA: L~Gex CX Hom(VA, 13,). Then the polarization space F0” (Er) oftheprequantization vector bundleEj~ isjust the holomorphic section space FhOl (EA) of the holomorphic vector bundle VA_-~EA_*QGonQG. Proof (1) Since for all F E 17f(E~), all ~ E L~ce,and ally E LG, by (2.6), 0
=
Vr(L(y))(F)(Y)
=
L~(F)(y)= ~F(yexpt~)~~o,
it follows that F(yb) = F(y) for all b E ~ This implies that F E Fh01(EA), i.e., FO”(EA~”~) ç FhOJ(EA). (2) By the complex structure of QG, and the definition of the polarization space 17!(E~T))in (2.6), it follows directly that FhOJ(EA) c It follows from (1) and (2) above that the identity Fr(E~) = FhOI(EA) holds. E
M. Guo et al. / Generalized prequantization on principal bundles
1 35
Remark. Lemma 2.3 shows that generalized geometric quantization space ofQG is the holomorphic section space Fh01 (EA) of the holomorphic vector bundle EA of Q G. The question whether Fh01 (EA) is non-zero or not needs to be answered. The positive answer to this question is similar to the Borel—Weil theorem for loop groups (theorem 2.2). Lemma 2.4 (The model space of G [5, 7, 11]). Suppose that G is a compact
semisimple Lie group. We define the model space A of G as the homogeneous complex manifold Ge/Nt where N+ is the maximal nilpotent subgroup of G corresponding to the positive root vectors ofthe complexified Lie algebra cc of G. Then the holomorphicfunction space Hol (A) of A as the representation space of G can be decomposed into the direct sum ofthe irreducible representation spaces ofG in which each irreducible representation appears once and only once. Proof Let T be the maximal torus of G. Since TN~ = N~T, T has the right actions on the model space A of G. It follows that the right actions of T on A commute with the left actions of G on A. Since T is abelian, Hol(A) can be decomposed into the direct sum ~A HA, where HA denotes the weight space corresponding to the character 2 of T. In particular, HA is invariant under the induced actions of G on Hol(A). In fact, HA is just the holomorphic section space I~,of the induced holomorphic line bundle LA of G/T. By theorem 2.1 (Borel—Weil theorem for compact groups), HA is the irreducible representation space of G with 2 as the lowest weight. It is obvious that I~ c Hol(A) for each antidominant weight 2 of G. Thus each irreducible representation appears once and only once in the direct sum decomposition of Hol (A). Remark.
(1) Lemma 2.4 states that all irreducible representations of G can be naturally combined into the holomorphic function space Hol (A) of the model space A. This is just the reason whythe homogeneous complex manifold A( def Gc/N+) is called the model space of G. (2) This lemma is quite different from the Peter—Weyl theorem.space The Peter— 2 (G) (the square integrable function of G) Weyl theorem shows that L as the representation space of G can be decomposed into the direct sum of the irreducible representation spaces of G, and each irreducible representation VA appears (dim VA) times. The Peter—Weyl theorem for compact groups cannot be generalized to the case of infinite dimensional Lie groups. Theorem 2.5.IfA(= (0, 1, h)) is the antidominant weight of LG, then there
exists a linear isomorphism ~‘:Jj—~
Fh 01(EA),
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where Ij denotes the holomorphic section space of the holomorphic line bundle LA on LG/T in theorem 2.2. Moreover, FhOI(EA) is the irreducible representation ofLG with the lowest weight A. Proof In the proof of lemma 2.2, the space HA
{fEHol(Gc/N+)If([gt])_2(tY~f([g]) for all g E Gc, and all t E Tc}
(29)
is just the holomorphic section space I~, of the holomorphic line bundle LA on G/ T in theorem 2.1. Thus we take HA as VA in lemma 2.3. It also follows directly that the following identities hold: (2 10) FA ~ { f E Hol(LGc/N~Ge)I f([yt]) = A(tY1f([yI) forallyELG, andalltE Te xC><}, FhoI(EA) ~
~
{ f E HOl(LGe/Lj~Gc,VA) I f([yg]) = p~(g).’f([y]) for ally E LGe, and all g E Ge X CX} { f ~ Hol(LGe/L~Gc,Hol(Gc/N~)) I f([yg], [g’])= f([y], [gg’Il), f( [yt], [g]) = A(t)’f([y], [g]) forally ELGc, allg,g’e Ge, andalltE Te xC><}.
(2.11) We define the linear map ç~:1j—4F~ 0l(EA), ~(f)([y],[g]) ~
f([yg])
f~-~c’(f), forall~ELGe, allgEGc.
(2.12)
By the identities (2.10) and (2. 11), it implies that q is a linear isomorphism. Since A is an antidominant weight of LG, it follows from theorem 2.2 that TA is the irreducible representation of LG with the lowest weight A. Thus Fh01 (EA) is also the irreducible representation of .LG with the lowest weight A of LG. ~ Remark. The induced irreducible representation of LG on FhOl (EA) coincides with the generalized geometric quantization ofQG by theorem 2.5, and induced Kostant mappings. Theorem 2.6 (The model spaces of LG [11]). We define two spaces A1 and A2 ofLG as A1 ~LGe/N~Ge, A2~LGc/L~Ge, (2.13) respectively. Then: (1) the holomorphic function space Hol (A1) of A1 can be decomposed into the direct sum of the irreducible representations ofLG in which each irreducible representation appears once and only once.
M. Guo et al. / Generalizedprequantization on principal bundles
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(2) We define a linear space as NA
2
~ {Hol(LGe/L~Ge,Ho1(Ge/N~))I f([yg], for all y E LGe, all g, g’ E Ge}.
[g’])
f([y],
{gg’])
(2.14) Then there exists a linear isomorphism ~,: Hol (Ai) N~,2,which is defined in (2.12). Moreover, NA2 can be decomposed into the direct sum of the irreducible representations ofLG in which each irreducible representation appears only once. —~
Proof (1) Similar to the proof of lemma 2.4, it follows directly from theorem 2.2 (Borel—Weil theorem for loop groups) that the statement about Hol(A1) is true. (2) By lemma 2.4, Hol(Ge/N~) = ~A VA, where VA is the irreducible representation of G with the lowest weight 2 of G, and 2 runs over the set of antidominant weights of G. Since the following identity f([yg]) = p(g).”f([y])
(2.15)
holds for all f ~ N~2,all y E LGe, and all g E Ge, where p is the representation of G~on Hol(Ge/N~)induced by the left actions of Ge on Ge/N~,the space NA2 can be decomposed into the direct sum ~A7-~A2, where N~ = {fEHol(LGc/Lj~Ge, VA)If([yg]) for all y E LGe, and all g E Ge}
=p~(g).~’f([yI)
(216)
and 2 runs over all antidominant weights of G. By the fact that the center T of LG commutes with L~Ge, the center T has right actions on the model space A2 = LGc/L~Ge. Thus the space N~ under the right actions of T can be decomposed into the direct sum ~hN~ h) where the positive integer h is the character of T, and ~ h) is the corresponding characteristic space of T. In fact, the space N~h) can be identified with the holomorphic section space FhOl (EA) ofthe holomorphic vector bundle EA defined in lemma 2.3 by the weight A of LG. By theorem 2.5, ~ h) is nonzero if and only if A is an antidominant weight. And on the other hand, if A( = (0, 2, h)) is an antidominant weight, then it is easy to see that FhOl (EA) =
N~h) C ~
C
NA2.
Thus N42 can be decomposed into the direct sum of the irreducible representations of LG in which each irreducible representation appears only once. By the definition of ~ in (2.15), it follows that q is a linear isomorphism. Moreover, the statement about N42 in theorem 2.6(2) can also be proved by theorem 2.6(1) through the linear isomorphism ço. Li
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Remark. Theorem 2.6 shows that the model spaces A1 and A2 both can combine the irreducible representations of LG into a global structure such as Hol(A1) and NA2.
We would like to thank Prof. Robert J. Blattner for explaining some ofthe ideas involved in geometric quantization and quantum groups in his lectures given at Peking University in September, 1990. We also wish to thank Prof. K. Wu, Mr. F. Xu, and Mr. B.L. Wang for their useful discussions and explanations concerning the loop group theory, Ehresmann connections on principal bundles, and the WZW model in two-dimensional conformal field theory. The third author is especially grateful to Ms. M.Q. Guo and Ms. T. Xu for their help in preparing this manuscript. References [1] J.F. Adams, Lectures on Lie Groups (Benjamin, New York, 1969). [2] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333—380. [3] R. Bott, Homogeneous vector bundles, Ann. Math. 66 (1957) 203—248. [4] K. Gawcdzki, Wess—Zumino—Witten conformal field theory, in: Constructive Field Theory II (Erice, Italy, 1988). [5] TM. Gelfand, in: Proc. Intern. Congr. Math., Vol. 1 (1970) p. 95. [6] A.A. Kirillov, Elements of the theory of representations, in: Grundlehren der mathematischen Wissenschaften, Vol. 220 (Springer, Berlin, 1976). [7] H.S. La, P. Nelson and A.S. Schwarz, Virasoro model space, Commun. Math. Phys. 134 (1990) 539—554. [8] A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. Math. Ser. II, 65 (1957) 391—404. [9] AN. Pressley and GB. Segal, Loop Groups, Oxford Mathematical Monographs, Vol. 12 (Clarendon Press, Oxford, 1986) chs. 1—5. [10] AN. Pressley and G.B. Sega!, Loop Groups, Oxford Mathematical Monographs, Vol. 12 (Clarendon Press, Oxford, 1986) chs. 7—11. [11] Z.H. Qian, Fundamental mathematical theory of Wess—Zumino—Witten model, Master’s thesis, Peking University, Beijing (1991). [12] Z.H. Qian, M. Qian and M.Z. Guo, Geometric quantization and Borel—Weil theorem for compact groups, preprint CCAST 91-01 (1991). [13] J. Sniatycki, Geometric Quantization and Quantum Mechanics, Applied Mathematical Sciences, Vol. 30 (Springer, New York, 1980). [14] M. Spivak, A Comprehensive Introduction to Differential Geometry, Second Ed., Vol. II (Publish or Perish, Berkeley, 1979) ch. 8. [15] E. Witten, Non-abelian bosonization in two dimensions, Commun. Math. Phys. 92 (1984) 45 5—472. [16] E. Woodhouse, Geometric Quantization, Oxford Mathematical Monographs, Vol. 4 (Clarendon Press, Oxford).
JOURNAL OF
G~METRYA~D
PHYSICS
Generalized prequantization on principal bundles over homogeneous spaces and its applications *
a
Maozheng Guoa, Mm Qiana and Zhao-hui Qian’~’1
Department of Mathematics, Peking University, Beijing 100 871, China b CCAST (World Laboratory), P.O. Box 8730, Beijing 100 080, China Received 14 June 1992 (Revised 13 April 1993)
In this paper, we set up a framework of generalized prequantization by starting from a single structure on the principal bundle associated with the Lie group pair (G, H). This procedure could be regarded as a kind of “postulated rules” taking the place of ordinary prequantization. Illustrative examples which show the usefulness of our scheme are, among others, compact semisimple Lie groups and loop groups. The model spaces of loop groups are also discussed. Keywords: geometric quantization 1991 MSC: 58 F 06
Introduction Geometric quantization in its well established form starts from a symplectic manifold (M, w), and a complex line bundle over M (cf. refs. [13, 17]). In case of a co-adjoint orbit M = G/H, due to Kirillov’s coorbit theory [6], what one gets after quantization is a certain representation of G, which is usually only one sector of the quantum mechanical Hilbert space. And it is always non-trivial to find a global form of the connection which meets the need of prequantization [13, 17]. In this paper, we set up a framework of generalized prequantization by starting from a single structure on the principal bundle P associated with the Lie group pair (G, H), which takes advantage of the Ehresmann connection induced from the G-actions on P. This procedure could be regarded as a kind of “postulated rules” taking the place of ordinary prequantization [13, 17]. *
Research partially supported by NNSFC Grant (1991—1995). Permanent address: do Prof. Pei-Xin Qian, Institute of Microelectronics, Tsinghua University, Beijing 100 084, China.
0393-0440/1994/$7.00 © 1994 Elsevier Science B.V. All rights reserved SSD10393-0440(93)E0030-S
M. Guo et al. / Generalized prequantization on principal bundles
125
The conditions under which our suggested procedure can really be carried out are given in this paper. Illustrating examples which show the usefulness of our scheme are, among others, 1. a short and transparent treatment of geometric quantization and the Borel— Weil theorem for compact groups (other finite dimensional cases could easily be found also satisfying our conditions) (cf. refs. [1, 3, 12]) 2. the Borel—Weil theorem for 1oop groups LG (M = LG/T) as treated in ref. [10], 3. geometric quantization ofthe Wess—Zumino—Witten (WZW) model, whose correct phase space should be LG/G instead of LG/T as most people seem to take for granted [4, 9—11, 15, 16], and 4. the model spaces of ioop groups and the model space of the WZW model [11]. Actually we believe that all kinds of induced representations are susceptible to such explanations.
1. Generalized prequantization on principal bundles over homogeneous spaces Let G be a Lie group, and H be a closed Lie subgroup of G. Then G can be identified with a principal bundle on the homogeneous space G/H of G (the left coset space of H) taking H as both its fiber and its structure group. The projection mapping ~r : G G/H is just the natural quotient mapping induced by H. The left actions of G on G induce the actions of G on G/H, and they commute with it. The right actions of H on G are just the actions of the structure group H on the principal bundle iv: G G/H. Thus at each point g of G, we identify the Lie algebra 7~Iof H with the subspace of the tangent space Tg G as follows: —*
—~
ag:7(—~TgG,
(1.1)
~F_+rig(~),
where rig (~)is the tangent vector of TgG corresponding to the one-parameter current g exp ( t~)of G for t ~ R’ [1, 141. Let g, 7~Ibe the Lie algebras ofthe Lie groups G, H respectively. Suppose that there exists a direct sum decomposition of ~ which satisfies —
g=h~Ei~M,
[7-i,M]cM.
(1.2)
We define a 1i~-valuedone-form c~on G as ct(L~)(g)~
—~
for all ~ E g, all g E G,
(1.3)
where L~is the left-invariant vector field on G with L~(e) = and ~ is the 7~component of~under the decomposition of g in (1.2). Then c~is an Ehresmann ~,
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M. Guo et al. / Generalized prequantization on principal bundles
connection in the principal bundle it: G
—f
G/H [141, such that
a(a(~)) =
(1.4)
cs(R~~X)(g)= Adh.1cE(X)(g), for all ~ ~ 7-1, all h E H, all g E G, and all tangent vectors X E TgG, where the mapping a is defined in (1.1), Rh denotes the right action of h on G, and Rh is the tangent mapping of Rh. Due to ref. [141, the connection c~determines a horizontal space decomposition of TgG for each g e G as follows: *
TgGHgeVg,
(1.5)
where the vertical space Vg ~ {ag(c~) ~ ~ 7-1}, and the horizontal space Hg ~ kerag. According to ref. [141, the curvature of the connection ct is defined as a 7-1valued two-form Q on G such that Q(X,Y)(g)
(1.6)
~da(ivHg(X),mHg(Y))(g)
for all g E G, and all tangent vectors X, Y e TgG, where HHg: TgG projection. Then we have Q(L~,L~)=
[‘~M,
Hg is the (1.7)
1Ml7t,
Q(Rh~X,R~~Y)(g)= Adh.’Q(X, Y)(g),
for all E all h E H, all g ~ G, and all tangent vectors X, Y E TgG, where crM denotes the M-component of~. ~,
~
~,
Generally speaking, the curvature Q on the principal bundle it: G G/H is not the pullbackof a 7-1-valued two-form w on G/H by the mapping iv. However, in the case discussed below, the curvature Q is really the pullback of a 7-1-valued two-form w on G/H, which could be regarded as the basic structure of generalized prequantization of the principal bundle iv: G G/H (the counterpart of ordinary symplectic structures of classical phase spaces). By the left actions of G on G, and the induced actions of G on G/H, we define the infinitesimal actions of ~ on C°° (G) and C°° (G/H), respectively, as —*
—~
~G(F)(g) ~ ~F(exp(-t~)g)~o, def
= ~_fojv(exp(_tçe)g)~~.. 0,
(1.8)
for all ~ e all g E G, all F E C°°(G),and all f e C°°(G/H). In fact, we can relate ~G/H to ~ with the connection c~and details are summarized in the following proposition of generalized Kostant mappings of the principal bundle iv: G G/H. ~,
—p
M. Guo et al. / Generalized prequantization on principal bundles
127
Proposition 1.1 (Generalized Kostant mappings). Suppose that the Lie algebras g, 7-1 of the Lie groups G, H satisfy (1.2), and that [M, MI cM~Z,
(1.9)
[7-1,y]cy,
7-1——Zey,
where Z is the center ofl-1. We define two linear mappings ô0 : ~ C°°(G/H,Z), respectively, as ôG(~)(g)~
(1.10)
C°° (G, 7-1) and
—#
:
ÔG/H
-(Adg.’~,
~
for all~E g, and alig e G. Then: (1) The curvature Q of the connection cv defined in (1.6) is the pullback of a Z-valued two-form w on G/H, which satisfies w(iv~(L~(g)), iv~(L~(g)))(iv(g)) =
(1.12)
[c~M,~)M]z
for all çe, ~j E g, and all g E G. (2) For all~E g, and allg E G, ~G(g)
+
= ~G/H(g) =
Lo0(~)(g) (1.13) (1.14)
~G/H(g)—a(~G(~))(g),
dôG/H(~) = i(çe~,~)w,
where ‘~G/Hdenotes the horizontal lift ofthe vectorfield ‘~G/Hon G/H determined by the horizontal decomposition of TgG in (1.5) corresponding to the Ehresmann connection cv. Due to the identity (1.14) it makes sense to call the vector field ‘~G/H on G/H a generalized Hamiltonian vector field of öG/H (~) corresponding to w. (cf refs. [13, 17]). Due Kostant to the identities (1. 13) andprincipal (1.14), we call the ö~, 5G/H generalized mappings of the bundle iv:mappings G G/H and ~ (cf refs. [6, 13, 17]). —f
The proof is by direct calculation, we omit it. Corollary 1.2. Suppose that the Lie algebras g, 7-1 satisfy the conditions in (1.2), and H is abelian. Then the
mappings öG, öG/H
defined in (1.12) satisfy
öG(c~)=ôG/H(~~)oivforall~e~, ~G(g)
= ~G/H(g)
+ Lo
0/~(~)olr(g)(g) =
(1.15) (116)
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M. Guo et a!. / Generalized prequantization on principal bundles
Remark. If the Lie algebras N of the Lie groups G, H satisfy the conditions supposed in proposition 1.1, then we can associate the left infinitesimal actions of g on C°°(G)with their induced actions of ~ on C°°(G/H)through the Ehresmann connection cv in proposition 1.1. This procedure is similar to ordinary prequantization of a complex line bundle on a symplectic manifold [13, 17]. However, since the Z-valued two-form w on G/H (corresponding to the curvature Q of the connection cv) in (1.12) may be degenerate, and the dimension of the fiber H of the principal bundle iv: G G/H may be larger than 1, we would like to call this procedure generalized prequantization of the principal bundle G on its homogeneous space G/H. ~,
—f
Definition 1.3. (Poisson structure of a subspace of C°° (G/H,Z), and generalized prequantization operators) Assume that the Lie algebras 7-1 satisfy the conditions (1.2), (1.9) and (1.10). We define a subspace C~°(G/H,Z)of C°°(G/H,Z)as ~,
(1.17) Then (1) For all ,~iE g, we define the Poisson bracket of two Z-valued smooth functions ÔG/H (~)and öG/H (ij) as ~,
def {öG/H(’~),ÔG/H(i7)} = W(17G/H, ‘~G/H),
(1.18) where w is a Z-valued two-form on G/H defined in (1.12), and ~G/H defined in (1.8) is the generalized Hamiltonian vector field of ÔG/H (~)which satisfies (1.14). (2) For all prequantization operators ~G/H (~)on ~ (G) as ÔG/H(~) ~~G/H
+
LoG(~)=
(1.19)
~G,
where c~is defined in (1.8), óG(~)in (1.11), and ~G/H is the horizontal lift of ~G/H.
Theorem 1.4. Suppose that the Lie algebras g, N of the Lie groups G, H satisfy the conditions (1.2), (1.9) and (1.10). Then: (1) for all ~ E G,
n]);
(1.20)
{ôG/H(c~),öG/HO7)} = ~GIH([’~,
the Lie algebra (C~° (G/H, 2), { }) is Lie-isomorphic to ~ (2) the generalized prequantization mapping Q : öG/H (~) ~G/H (~)is a Lie homomorphism from (C~’° (G/H, Z), { }) to the space of linear differential operators on C°° (G). moreover,
,
~
,
M.
Guo et al. / Generalized prequantization on principal bundles
129
Proof (1) By (1. 10), it is implied that [N, NI cy.
(1.21)
Then it follows from (1.2) and (1.21) that the identity [~, ‘lIz = [~M,
(1.22)
?1MIZ
holds for all ~, ,~e ç. Since {öG/H(~), ~G/H(’l)}(it(g)) = W(’lG/H, ~G/H)(7r(g)) =
=
w(—iv~(L(Adg.~1fl)~(g)),
[(Adg.”l)~,
it~(L(Adg._I~)~(g)))
(Adg.’~)M]z
[Adg.’,~, Adg.~Iz = (—Adg.’[~, ‘l])z =
= ôG/JJ([~~, p1),
it follows that the identity (1.20) holds for all ~~‘ie Moreover, (C~(G/H, 2), { , }) is a Lie algebra, and the mapping y, : ‘—p ôG/H(~) is a Lie isomorphism from g to C~(G/H,Z). (2) Since ~ is the representation of ~ on C°° (G), it follows from (1 .19) and (1.20) that the mapping Q is a Lie homomorphism. E ~.
Remark. Our generalized prequantization is just the representation of g on COC (G) by (1.19). Definition 1.3 and theorem 1.4 show that our generalized prequantization scheme is similar to the ordinary prequantization scheme [13, 171 in the definitions and properties of the Poisson bracket and prequantization operators. Since the 2-valued two-form w on G/H may be degenerate, the Poisson bracket and generalized prequantization operators in our scheme are only defined in a subspace C00°(G/H, 2) of C~(G/H, 2), which is not an algebra. An interesting question is how to extend the definitions of the Poisson bracket and generalized prequantization operators to a linear space which is larger than C~(G/H, 2), is a Poisson algebra, and guarantees a result similar to theorem 1.4(2). The following section is motivated by the question whether there exist some illustrative examples of the Lie group pairs (G, H), whose Lie algebras g, N satisfy the conditions (1.2), (1.9) and (1.10) supposed in proposition 1.1. The answer to this question is actually positive. In fact, the dimensions ofthe interesting examples range from finite dimensions to infinite dimensions. In particular,
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M Guo et a!. / Generalized prequantization on principal bundles
applying this method to loop groups [9, 101, we make contributions to the geometric quantization of the Wess—Zumino—Witten model in two-dimensional conformal field theory (cf. refs. [11, 161). 2. Applications of generalized prequantization 2.1. COMPACT SEMISIMPLE LIE GROUPS
Let G be a compact semisimple Lie group, and T be the maximal torus of G. It is easy to see that the principal bundle iv: G G/H satisfies the conditions supposed in theorem 1.2. In fact, due to Adams [11,the Lie algebras ~, Y of G, T satisfy —f
=
[T, TI
=
[‘T, MI
=
[M, MI
c
{0},
(2.1)
M~T,
where M
‘~
span{(efl e_~),i(ep + e_p) I /3 is a positive root of cc, and ep, e_p are the root vectors of Gc corresponding to the roots /3 and /3}. —
—
We see that the connection (1.2), (1.9) and (1 10) are satisfied, thus the connection cv and generalized Kostant mappings of this principal bundle can be obtained through proposition 1.1. Moreover, since (2.1) make the conditions in corollary 1.3 satisfied, generalized Kostant mappings are only determined by the linear mapping öG/T : ~ C°°(G/T,T), i.e., .
—~
=
—(Adg.’~)j
for all g e G, and all ~ e g. If ,~is a weight of G, then we obtain the induced complex line bundle C LA G/T equipped with the induced connection V(A) and induced Kostant mapping. It implies that the line bundle L2 is a “generalized” prequantization —.--~
line bundle on G/ T, where the word “generalized” means that the curvature w1 of the induced connection may not be a symplectic form on G/ T. We choose the Kãhlerian polarization P on G/ T [13, 17] as P(it(g))~{it~(L~(g)) I~~V+}, (2.2) 1÷is the positive root vector subspace of cc. The polarization space where A I~f(LA)~ {F E F(L
2) I V~~(g))F(g)= 0 for all tangent vectors X(iv(g)) E P(iv(g))},
(2.3)
M. Guo et al. / Generalized prequantization on principal bundles
131
where F (LA) is the smooth section space of the complex line bundle LA, which can be identified with the following subspace ofthe smooth function space of G: F(LA) ~ {F E Cc~o(G)I F(gt) = )~‘(t)F(g)fora1lte T, allg e G}. (2.4) In fact, the polarization space I~”(LA) is just the holomorphic section space I~. of the holomorphic line bundle iv: LA Gc/B +, where B + denotes the Borel subgroup of G~corresponding to the positive roots of cc. The properties of the space I~are elucidated by the most famous Borel—Weil theorem for compact groups. —~
Theorem 2.1 (Borel—Weil theorem of compact groups [1, 3, 91). (1) The holomorphic line bundle LA has nonzero holomorphic sections if and only if the weight ofG is the antidominant weight ofG. (2) If)L is the antidominant weight of G, then the holomorphic section space I~ of the holomorphic line bundle LA is the irreducible unitary representation space of G with the lowest weight )~. ~.
Remark. The arguments above show that the irreducible unitary representation of G can be obtained through our generalized geometric quantization ofthe complex homogeneous space G/ T of G. Proposition 1.1 and corollary 1.2 imply that the induced Kostant mapping makes the induced representation of G coincide with the geometric quantization of G/ T. For more details of the calculation in the case of compact groups, please refer to refs. [6, 121. 2.2. LOOP GROUPS AND THEIR CENTRAL EXTENSIONS
Suppose that Gis a connected, simply connected and compact semisimple Lie group. Let LG be the 1oop group of G, and LG be the central extension of LG determined by the basic two-cocycle w of LG in ref. [91. LG can be regarded as the different principal bundles on its different homogeneous spaces. In this paper, we would like to discuss the following two fundamental homogeneous spaces [101: 1. LG/T: the principal bundle T x T LG LG/T, where T, T denote the center of LG, and the maximal torus of G, respectively. 2. LG/G: the principal bundle T x G LG .QG, where .QG is the subgroup of LG whose element y satisfies that y (1) = e where 1 e S1, and e is the unit element of G. In fact, QG ~ LG/G. —f
—~
—*
—~
LG/T. By the structure of the Lie algebra Lc of LG, it follows directly that the principal bundle Tx T LG LG/T satisfies the conditions supposed in proposition 1.1 and corollary 1.2. 2.2.1. The principal bundle iv :
—~
—f
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M. Guo et a!. / Generalized prequantization on principal bundles
Then this principal bundle iv : LG LG/ T can be equipped with the structures of the connection cv, and generalized Kostant mapping öLG/T. If A is a weight of LG [91,then we can induce a complex line bundle C LA LG/T on LG/T from this principal bundle, and impose the induced connection and induced Kostant mapping on the complex line bundle. Thus we regard LA as a generalized prequantization line bundle on LG/T. Moreover, by the general theory of Pressley and Segal [101, the fundamental homogeneous space LG/ T can be identified with a homogeneous complex manifold LGc/B + Ge, where B + G~consists of the boundary values of holomorphic maps —~
—f
y : {z : IzI < 1}
Ge
such that y(0) belongs to the Borel subgroup B~of Ge corresponding to the positive roots of cc. As in the case of compact groups, we choose the polarization P determined by the Lie subalgebra N~Gcof Lcc, which consists of the boundary values of holomorphic maps ~: {z: IzI < l} cc such that ~(0) belongs to the nilpotent Lie subalgebra .jV + of cc corresponding to the positive roots of cc. By the complex structure of LG/T, it follows that the polarization space ~ {F E F(LA) I V~(7))F(y) = 0 for ally E LG, and all tangent vectors X (iv (y)) E P (iv (y) ) } is just the holomorphic section space lj of the holomorphic line bundle LA on the homogeneous complex manifold LG/T (~ LGc/B~Ge). Being similar to compact groups, the Borel—Weil theorem for ioop groups [101 completely describes the space Ij. Theorem 2.2 (Borel—Weil theorem for loop groups [101). (1) The holomorphic line bundle LA on the homogeneous complex manifold LG/ T possesses non-vanishing holomorphic sections if and only if the weight A of LG is antidominant. (2) If the representation Ij is nonzero, then the representation is: (i) ofpositive energy, (ii) offinite type, (iii) essentially unitary, and (iv) irreducible, with lowest weight A. Remark. The Borel—Weil theorem of loop groups and induced Kostant mapping also identify the induced representation of LG on 1j with the generalized geometric quantization of its infinite dimensional homogeneous complex manifold LG/T ~ LGc/B~Gc.
M. Guo et al. / Generalizedprequantization on principal bundles
133
2.2.2. The principal bundle iv : JLG LG/G( ~ QG). The Wess—Zumino— Witten model with the gauge group Gplays an important role in two dimensional conformal field theory [2, 4, 16]. Following from the discussions in refs. [11] and [161, the classical phase space of the WZW model is Q Gx Q G. Since the left copy and the right copy of Q G in this classical phase space are homeomorphic, it is enough to investigate the geometric quantization of QG. Moreover, the geometric quantization of the classical phase space Q G x Q G is just the direct sum of the geometric quantizations of the two copies of QG [161. With the above arguments, it is natural to take the principal bundle T x G LG —p QG as the starting point of the geometric quantization ofQG. It follows that this principal bundle satisfies the conditions (1.2), (1.9), and (1. 10). By proposition 1.1, we obtain an Ehresmann connection and generalized Kostant mappings. If A = (0, 2, h) is an antidominant weight of JiG [10], where the level h (a positive integer) is the character ofthe centerT of LG, then 2 is an antidominant weight of G. By theorem 2.1 (Borel—Weil theorem for compact groups), there exists one and only one irreducible unitary representation (VA, PA) with the lowest weight 2, where the representation space VA is the holomorphic section space I~ofthe induced holomorphic line bundle LA on the homogeneous complex manifold G/T. With A, and the irreducible representation (1’~, PA) of G, the complex vector bundle VA ~ QG can be induced from the principal bundle T x G LG QG. Then we can also impose the structures of the induced connection and induced generalized Kostant mappings on this complex vector bundle. Thus we regard the complex vector bundle Ef) equipped with the structures stated above as the prequantization vector bundle of Q G. In order to obtain the polarization of the prequantization vector bundle of QG, we first briefly investigate the structure of the homogeneous space Q G ~ LG/G. As we know from ref. [10], QG can be identified with a homogeneous complex manifold LGc/L+ G~,where L+ Ge consists of the boundary values of the holomorphic maps —+
—~
—b
—i
—~
y: {z: IzI < l}
Ge.
Moreover, QG is a Kähler manifold whose Kähler form is just the basic twococycle w of LG [9]. Now we choose the polarization P of Q G as P(,v(y))~f{iv~(L~)) ~EL~cc}
(2.5)
for all y e LG, where the Lie subalgebra L~cc of Lcc consists of the boundary values of holomorphic maps ~: {z: IzI < l}
cc
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M. Guo et a!. / Generalized prequantization on principal bundles
such that ~(0) p
0P (EA(r))
=
0. The polarization space is defined as
¶~{ FE F(E~”~) I Vx(fl(A))F(y)
=
0 for all
(2.6)
yE LG, and all tangent vectors X(iv(y)) e where F (Er) denotes the smooth section space of the vector bundle E.~T),which can be identified with the following subspace of C~(LG, VA): F(E~T))~
{ FE
C~(LG,VA) I F(yt) = A(t)’F(y), F(yg) = p~(g).’F(y) for all y E LG, all t E T, and all g e G}.
(2.7)
Lemma 2.3 (The explicit form ofF0”(E~)).SupposethatA(= (0, 2, h)) isan antidominant weight oftG. Let (VA, PA) be the irreducible unitary representation ofGe induced by the antidominant weight 1 of G in theorem 2.1. We extend PA as the homomorphism PA from L~Gcx C’< to Hom(VA, VA), which satisfies PAIL~+GC =
1,
PAle>’ = h,
PAIGc
=
PA,
(2.8)
where c>< denotes the complexification of the center T of LG,and the subgroup L~Ge ofLGc consists of the boundary of the holomorphic maps y: {z: IzI < l}
‘Gc
such thaty(0) = e~G. We define EA as the induced holomorphic vector bundlefrom the complex principal bundle L~Gcx CX
LGc
—*
QG(~ LGc/L~Gc)
by the extended homomorphism PA: L~Gex CX Hom(VA, 13,). Then the polarization space F0” (Er) oftheprequantization vector bundleEj~ isjust the holomorphic section space FhOl (EA) of the holomorphic vector bundle VA_-~EA_*QGonQG. Proof (1) Since for all F E 17f(E~), all ~ E L~ce,and ally E LG, by (2.6), 0
=
Vr(L(y))(F)(Y)
=
L~(F)(y)= ~F(yexpt~)~~o,
it follows that F(yb) = F(y) for all b E ~ This implies that F E Fh01(EA), i.e., FO”(EA~”~) ç FhOJ(EA). (2) By the complex structure of QG, and the definition of the polarization space 17!(E~T))in (2.6), it follows directly that FhOJ(EA) c It follows from (1) and (2) above that the identity Fr(E~) = FhOI(EA) holds. E
M. Guo et al. / Generalized prequantization on principal bundles
1 35
Remark. Lemma 2.3 shows that generalized geometric quantization space ofQG is the holomorphic section space Fh01 (EA) of the holomorphic vector bundle EA of Q G. The question whether Fh01 (EA) is non-zero or not needs to be answered. The positive answer to this question is similar to the Borel—Weil theorem for loop groups (theorem 2.2). Lemma 2.4 (The model space of G [5, 7, 11]). Suppose that G is a compact
semisimple Lie group. We define the model space A of G as the homogeneous complex manifold Ge/Nt where N+ is the maximal nilpotent subgroup of G corresponding to the positive root vectors ofthe complexified Lie algebra cc of G. Then the holomorphicfunction space Hol (A) of A as the representation space of G can be decomposed into the direct sum ofthe irreducible representation spaces ofG in which each irreducible representation appears once and only once. Proof Let T be the maximal torus of G. Since TN~ = N~T, T has the right actions on the model space A of G. It follows that the right actions of T on A commute with the left actions of G on A. Since T is abelian, Hol(A) can be decomposed into the direct sum ~A HA, where HA denotes the weight space corresponding to the character 2 of T. In particular, HA is invariant under the induced actions of G on Hol(A). In fact, HA is just the holomorphic section space I~,of the induced holomorphic line bundle LA of G/T. By theorem 2.1 (Borel—Weil theorem for compact groups), HA is the irreducible representation space of G with 2 as the lowest weight. It is obvious that I~ c Hol(A) for each antidominant weight 2 of G. Thus each irreducible representation appears once and only once in the direct sum decomposition of Hol (A). Remark.
(1) Lemma 2.4 states that all irreducible representations of G can be naturally combined into the holomorphic function space Hol (A) of the model space A. This is just the reason whythe homogeneous complex manifold A( def Gc/N+) is called the model space of G. (2) This lemma is quite different from the Peter—Weyl theorem.space The Peter— 2 (G) (the square integrable function of G) Weyl theorem shows that L as the representation space of G can be decomposed into the direct sum of the irreducible representation spaces of G, and each irreducible representation VA appears (dim VA) times. The Peter—Weyl theorem for compact groups cannot be generalized to the case of infinite dimensional Lie groups. Theorem 2.5.IfA(= (0, 1, h)) is the antidominant weight of LG, then there
exists a linear isomorphism ~‘:Jj—~
Fh 01(EA),
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M Guo et a!. / Generalized prequantization on principal bundles
where Ij denotes the holomorphic section space of the holomorphic line bundle LA on LG/T in theorem 2.2. Moreover, FhOI(EA) is the irreducible representation ofLG with the lowest weight A. Proof In the proof of lemma 2.2, the space HA
{fEHol(Gc/N+)If([gt])_2(tY~f([g]) for all g E Gc, and all t E Tc}
(29)
is just the holomorphic section space I~, of the holomorphic line bundle LA on G/ T in theorem 2.1. Thus we take HA as VA in lemma 2.3. It also follows directly that the following identities hold: (2 10) FA ~ { f E Hol(LGc/N~Ge)I f([yt]) = A(tY1f([yI) forallyELG, andalltE Te xC><}, FhoI(EA) ~
~
{ f E HOl(LGe/Lj~Gc,VA) I f([yg]) = p~(g).’f([y]) for ally E LGe, and all g E Ge X CX} { f ~ Hol(LGe/L~Gc,Hol(Gc/N~)) I f([yg], [g’])= f([y], [gg’Il), f( [yt], [g]) = A(t)’f([y], [g]) forally ELGc, allg,g’e Ge, andalltE Te xC><}.
(2.11) We define the linear map ç~:1j—4F~ 0l(EA), ~(f)([y],[g]) ~
f([yg])
f~-~c’(f), forall~ELGe, allgEGc.
(2.12)
By the identities (2.10) and (2. 11), it implies that q is a linear isomorphism. Since A is an antidominant weight of LG, it follows from theorem 2.2 that TA is the irreducible representation of LG with the lowest weight A. Thus Fh01 (EA) is also the irreducible representation of .LG with the lowest weight A of LG. ~ Remark. The induced irreducible representation of LG on FhOl (EA) coincides with the generalized geometric quantization ofQG by theorem 2.5, and induced Kostant mappings. Theorem 2.6 (The model spaces of LG [11]). We define two spaces A1 and A2 ofLG as A1 ~LGe/N~Ge, A2~LGc/L~Ge, (2.13) respectively. Then: (1) the holomorphic function space Hol (A1) of A1 can be decomposed into the direct sum of the irreducible representations ofLG in which each irreducible representation appears once and only once.
M. Guo et al. / Generalizedprequantization on principal bundles
137
(2) We define a linear space as NA
2
~ {Hol(LGe/L~Ge,Ho1(Ge/N~))I f([yg], for all y E LGe, all g, g’ E Ge}.
[g’])
f([y],
{gg’])
(2.14) Then there exists a linear isomorphism ~,: Hol (Ai) N~,2,which is defined in (2.12). Moreover, NA2 can be decomposed into the direct sum of the irreducible representations ofLG in which each irreducible representation appears only once. —~
Proof (1) Similar to the proof of lemma 2.4, it follows directly from theorem 2.2 (Borel—Weil theorem for loop groups) that the statement about Hol(A1) is true. (2) By lemma 2.4, Hol(Ge/N~) = ~A VA, where VA is the irreducible representation of G with the lowest weight 2 of G, and 2 runs over the set of antidominant weights of G. Since the following identity f([yg]) = p(g).”f([y])
(2.15)
holds for all f ~ N~2,all y E LGe, and all g E Ge, where p is the representation of G~on Hol(Ge/N~)induced by the left actions of Ge on Ge/N~,the space NA2 can be decomposed into the direct sum ~A7-~A2, where N~ = {fEHol(LGc/Lj~Ge, VA)If([yg]) for all y E LGe, and all g E Ge}
=p~(g).~’f([yI)
(216)
and 2 runs over all antidominant weights of G. By the fact that the center T of LG commutes with L~Ge, the center T has right actions on the model space A2 = LGc/L~Ge. Thus the space N~ under the right actions of T can be decomposed into the direct sum ~hN~ h) where the positive integer h is the character of T, and ~ h) is the corresponding characteristic space of T. In fact, the space N~h) can be identified with the holomorphic section space FhOl (EA) ofthe holomorphic vector bundle EA defined in lemma 2.3 by the weight A of LG. By theorem 2.5, ~ h) is nonzero if and only if A is an antidominant weight. And on the other hand, if A( = (0, 2, h)) is an antidominant weight, then it is easy to see that FhOl (EA) =
N~h) C ~
C
NA2.
Thus N42 can be decomposed into the direct sum of the irreducible representations of LG in which each irreducible representation appears only once. By the definition of ~ in (2.15), it follows that q is a linear isomorphism. Moreover, the statement about N42 in theorem 2.6(2) can also be proved by theorem 2.6(1) through the linear isomorphism ço. Li
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M. Guo et a!. / Generalized prequantization on principal bundles
Remark. Theorem 2.6 shows that the model spaces A1 and A2 both can combine the irreducible representations of LG into a global structure such as Hol(A1) and NA2.
We would like to thank Prof. Robert J. Blattner for explaining some ofthe ideas involved in geometric quantization and quantum groups in his lectures given at Peking University in September, 1990. We also wish to thank Prof. K. Wu, Mr. F. Xu, and Mr. B.L. Wang for their useful discussions and explanations concerning the loop group theory, Ehresmann connections on principal bundles, and the WZW model in two-dimensional conformal field theory. The third author is especially grateful to Ms. M.Q. Guo and Ms. T. Xu for their help in preparing this manuscript. References [1] J.F. Adams, Lectures on Lie Groups (Benjamin, New York, 1969). [2] A.A. Belavin, A.M. Polyakov and A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B 241 (1984) 333—380. [3] R. Bott, Homogeneous vector bundles, Ann. Math. 66 (1957) 203—248. [4] K. Gawcdzki, Wess—Zumino—Witten conformal field theory, in: Constructive Field Theory II (Erice, Italy, 1988). [5] TM. Gelfand, in: Proc. Intern. Congr. Math., Vol. 1 (1970) p. 95. [6] A.A. Kirillov, Elements of the theory of representations, in: Grundlehren der mathematischen Wissenschaften, Vol. 220 (Springer, Berlin, 1976). [7] H.S. La, P. Nelson and A.S. Schwarz, Virasoro model space, Commun. Math. Phys. 134 (1990) 539—554. [8] A. Newlander and L. Nirenberg, Complex analytic coordinates in almost complex manifolds, Ann. Math. Ser. II, 65 (1957) 391—404. [9] AN. Pressley and GB. Segal, Loop Groups, Oxford Mathematical Monographs, Vol. 12 (Clarendon Press, Oxford, 1986) chs. 1—5. [10] AN. Pressley and G.B. Sega!, Loop Groups, Oxford Mathematical Monographs, Vol. 12 (Clarendon Press, Oxford, 1986) chs. 7—11. [11] Z.H. Qian, Fundamental mathematical theory of Wess—Zumino—Witten model, Master’s thesis, Peking University, Beijing (1991). [12] Z.H. Qian, M. Qian and M.Z. Guo, Geometric quantization and Borel—Weil theorem for compact groups, preprint CCAST 91-01 (1991). [13] J. Sniatycki, Geometric Quantization and Quantum Mechanics, Applied Mathematical Sciences, Vol. 30 (Springer, New York, 1980). [14] M. Spivak, A Comprehensive Introduction to Differential Geometry, Second Ed., Vol. II (Publish or Perish, Berkeley, 1979) ch. 8. [15] E. Witten, Non-abelian bosonization in two dimensions, Commun. Math. Phys. 92 (1984) 45 5—472. [16] E. Woodhouse, Geometric Quantization, Oxford Mathematical Monographs, Vol. 4 (Clarendon Press, Oxford).