August, SLAC-PUB-5067 1989
(9
Field Identifications in Coset Conformal Theories from Projection Matrices
C. AHN* Stanford Linear Accelerator Center Stanford University, Stanford, California
94909
and
M. A. WALTONt Physics Dept., McGill University Montreal, Quebec, Canada H9A 2T8
Abstract We demonstrate the usefulness of projection matrices for finite subalgebras h c 3 and their affine counterparts h c i in finding field identifications (and selection rules) in coset conformal field theories.
Submitted to Phys. Lett. B.
* Work supported by the U.S. Department of Energy under Contract DE-AC0376SF00515. Address after September 1, 1989: Newman Laboratory of Nuclear Studies, Cornell University, Ithaca, NY 14853. t Work supported in part by NSERC of Canada. Address after September 1, 1989: Departement de physique, Universitk Laval, Laval, Quebec, Canada GlK 7P4.
Coset conformal field theories[ 11 may include all two-dimensional rational conformal field theories[2]. For every finite Lie subalgebra h C 3, one can
construct a two-dimensional conformal field theoryi. If G, H are the (covering) Lie groups whose algebras are S, h, the embedding h c s will quite generally specify a relation between the centres B(G), B(H) of the two groups. We will explain how these relations may be identified. One of their consequences in the coset conformal field theory is a selection rule saying that certain primary fields do not occur[3,4]. Let i, k denote the Kac-Moody algebras that are the central extensions of the loop algebras of 3, h, respectively. Then the finite subalgebra h c s with index of embedding e induces an affine subalgebra kek c i, where the superscripts are the levels (see, for example, Reference [5]3. There exist automorphisms of G (k) which are not themselves elements of 4 (L) and are therefore called outer automorphisms[6]. The outer automorphisms of 4 permute the fundamental weights & = 0, 1, . . . , R; R = rank(g)) (p in -such a way as to leave the Dynkin diagram of ij invariant. Similarly, outer automorphisms of & permute the fundamental weights wO( cy = O,l, . . . ,r; r = rank(h) ) of A. The group of outer automorphisms of i, O(G), is isomorphic to the centre B(G). Relations between the centres B(G) and B(H) are therefore accompanied
1 We assume h is a maximal subalgebra of ij, otherwise the coset theory factors into (a/i) @ (k/h) theories, where k C ij is maximal.
.
1
by relations between the outer automorphism groups O(i) and O(k). One consequence of these outer automorphism relations is that certain fields in the coset conformal theory built from h C 3 must be identified[7,4]. Let A = C~zohP~P (X = C&,X, way> with 0 5 A,, E Z (0 5 X, E Z) be an affine weight of i(i). Also let A = C~=,A,w (x = C~,,X,w be the g(h) ) weight that is the finite restriction of A (X). Then the isomorphism O(i) E B(G) -- _ may be described in the following manner. If we denote an outer automorphism
by A E O(i), th ere exists a corresponding element of the centre a E B(G) whose
]A)]. The eigenvalue on a s representation with highest weight A is2 exp[2ri(& element cr E B(G) a so acts diagonally on representations of ij with the same 1 eigenvalues. For a representation with highest weight A, we have (symbolically) g - A = A . exp[2ri(& ]A)] where we have used (&oI&) = (& ]A). Similarly, for A c O(h), there exists Q! E B(H) such that a - X = X . exp[27ri(Aw0 ]A)] for all highest-weight representations X of k. Because of the form of the eigenvalues (1,3), to get relations between the centres of H and G we examine the relation between weights of s and 6. The
(AlA and (&,A are dot products of weights A, A and ;i, A determined by the Killing ) ) forms of B and ij, respectively, and normalised so that a long simple root satisfies (alcx) E lo!12 = 2.
: (1) ._
(2)
(3)
2
embedding 6 c s is specified by the projection that takes weights of s onto weights of h. Denoting a weight A of s by a column vector A = (AIR2 e . . AR)~, we can construct a so-called projection matrix F[8] such that .A is projected onto the weight FA of h. F is a r x R matrix with integer entries greater than whose r entries are the coefficients of or equal to zero. FA is a column vector the -- _ fundamental weights tia of 6. If we let F act on all the weights {A in a representation with highest } } weight A, the weights { FA of h will fill out several representations with highest weights xi. This can be denoted symbolically by A+ c xi i
-
.
and is known as a branching rule. Two embeddings with distinct projection __ matrices F are said to be equivalent when their branching rules are identical. Thus there are, in general, more than one valid projection matrices for the same embedding. This will be useful later on. Because of (1,2,3), B E B(G) and o E B(H) are identified if and only if (Aw IR) = (AwOlF&) mod 1
(5)
for all A. So with a projection matrix F, it is straightforward to find relations between the centres of G and H. To find the consequences of the centre relations (5), we study characters. Let x*(r) (x (r)) denote th e ( s p ecialised) character of the 4 (i) representation with highest weight A (X). Th e ch aracters of the coset theory are the branching 3
functions b?(r) [3] of the subalgebra &? > i, defined by
x^w = xY+m *
matrix notation (6) is x=xb Note that (Aw JX + ,B) = (Aw JX) mod 1 .
(6)
The corresponding coset fields are labelled by two highest weights (A, X). In
(7)
(8)
(& ]A + p) = (& ]A) mod 1
for any roots ,L?, - of h, g. Suppose A is the finite restriction of a weight A in the .* ,B 6 representation with highest weight A and FA is the restriction of a weight in , the k representation with highest weight X. Then (8) means the centre relation (5) implies (ho ]A = (Au mod 1 ) IX) . (9)
The representation with highest weight Awill branch only to those representations of & with highest weights X obeying (9) . This means only those primary fields (A X) obeying (9) app ear in the coset conformal theory. These , selection rules have been discussed previously (at least for particular cases) in references[3,4]. The selection rule can be expressed using the characters in the following
4
way: exp[27ri(Aw0 IX)]bt exp[27ri(b ]A)] = bf or in matrix notation, aba= b . (11)
(10)
The phases introduced in (11) by a E B(G) and o E B(H) must cancel, or else . the-element bi of b must vanish, implying that the corresponding primary field does not appear. Equation (11) a so requires that certain fields in the coset theory be iden1 tified. To see this, consider how the characters transform under the modular transformation S(r + -l/r). If
1.. (12)
X(-l/T) = xws
then from (7) we have b(-l/r) = S+b(T)S . (13)
Now in the space of characters of a Kac-Moody algebra i, it is the modular transformation S which diagonalises an outer automorphism A [9]: S+AS=a , w
thereby manifesting the isomorphism O(i) 2 B(G). A similar relation holds for
R:
S+AS=a 5 ,
(15)
where A E O(i), CE E B(H). Applying (13,14,15) to (11) then yields AbA=b .
(16)
The characters of the fields (AA, AX) and (A, X) are identical, and so they must be identified: -_ @A,AX) z (A,X) . (17)
.
Thus field identifications are a consequence of relations between the centres of G and H that may be easily found via (5) using a projection matrix F. These field identifications in coset conformal field theories have become of interest lately[4],esp ecially in connection with the N = 2 superconformal coset models[lO,ll] of Kazama and Suzuki[l2]. Of course, the field identifications (17) are simply consequences. of the rela- 1m tions between outer automorphisms of i and i c 4. One should not have to introduce characters to find them. In the following we will discuss how they may be discovered in a manner as direct as relations between centres are found. To do this we study projection matrices fi for the affine subalgebra Lek c G[13,14]. Since affine KaE-Moody algebras i, ?L have the fundamental weights _ g ,wo as well as those of the finite algebras S, h, the matrix $ is a (r+l) x (R+l) dimensional matrix3. An affine weight A (X) is written as a column vector
[Aoh - - .hlT ([X0&. . . &IT). Then the weight A of 4 is projected onto the weight @A of i.
3Here we assume both s and E are simple. Generalisation is straightforward.
6
One way to construct a projection matrix & for i > iek is to demand that the finite parts of affine weights be projected according to a valid matrix F for s > h. Denoting the elements of 8 by &, that is
this specifies all elements o # 0. The remaining elements are determined by -- _ requiring that a level k weight A of i be projected onto a level ek weight of ?L. The level of a i (?L) weight A(X) is A& (X,k ) where ~vp(kva) are the co-marks of i (A). So we demand
k @A P = e k ff if APkVp = k. Taking k = k and AP = 6; gives - k fiE = ekVV ,
(19)
(20)
-
or in matrix notation (k )TE = e(b )T , completing the determination of @ from F. Note in particular that (21)
An affine projection matrix manifests a relation between A E O(G) and A E O(k) if the following is true
A&=$ ,
(23)
7
where @ is another valid projection matrix. In (23) A and A are the matrices I which permute the rows and columns, respectively, of F in the manner pre- scribed by the corresponding outer automorphisms. (Note that-these matrices are in general of dimension smaller than those of Eqs. (14,15,16).) the type (23) with F= 5 were found in Refs [13,14]. . Unfortunately, we have no general test for a valid affine projection matrix. We can only check those that are built from a finite matrix F in the manner just described. The test is then simply the requirements of the F matrix that is a submatrix of F. A sufficient requirement[8] is that the matrix F produce the correct branching rule for the second smallest (i.e. not the scalar) irreducible representation of 3 into representations of 7~. This means we must restrict the 3 in (23) to those satisfying (22). This .* restricts us to a subset among the pairs A, 4 satisfying (23) in the general sense. Our ignorance concerning affine projection matrices therefore makes the centre relations (5) easier to verify. However, quite often there are matrices $ which manifest outer automorphism relations in an obvious way (see Refs. [13,14]). Furthermore, in all cases _ we have checked, there is a sufficient number of different F such that a coms plete set of relations may be derived from (23). At the very least, even with the technical restriction (22) imposed on F the relations (23) provide checks , on the centre relations. There is even a case when the relations (23) are the only ones that may Relations of
8
be simply verified. Suppose we drop for the moment the restriction that 6 is a maximal subalgebra of s (see footnote page l), and suppose h $ h c s is maximal. Suppose further there is a centre relation for this maximal subalgebra of the form -(bO]A) = (Aw ]FA) + (A IFii) mod 1 w , (24
where A w are an outer automorphism and the Oth fundamental weight of A . , Then if we consider the non-maximal embedding h c s , we do not have @& ]A) = (Au IFR) mod 1
(25)
even though A and A should be identified. On the other hand, a relation of the type (23) will exist, at least subject to the restrictions discussed above. The following examples should clarify our general discussion. . ..
Example 1
G = SO(7), H = SU(4).
Our first example is the subalgebra SO(~) > su(4), with index of embedding _ e = 1. This is an example of a regular maximal subalgebra, i.e. it can be understood by deleting a node from the extended Dynkin diagram of SO(~) (see, _ for example, Reference [ 151). The node omitted is the one representing the short simple root of so(7), so that the long roots and the negative of the highest root are projected onto the simple roots of su(4). So the finite subalgebra projection
matrix is
(26)
The affine matrix built from (26) by the method discussed above is
(27)
A sufficient check of the validity of F is that it reproduce the branching rule
c
( loo)T + (010)T + (000)T .L
.
(28)
.e
We can check (28) by letting 2 act on the states having the minimum La eigenvalue in the G(7) representation with highest weight [O1OO]T, since these states transform under SO( 7) as the representation with highest weight ( 100)T. - They are 0 -1 1 0 -L 1 0 -1 2 1 0 1 -2 2 1 -1 0 2 -1 0 0
(29)
and (28) is easily verified. Now s%(7) has outer automorphism group Z2, generated by CC, acting in the
10
following way on a weight A:
. -
The 24 outer automorphism group of G?(4) is generated by a, with action
a[XoXJ2X,]T = [XJlJ~A2]T .
It is simple to verify
(31)
(~ lh) = (&IA) = i As mod 1
(32)
(a2uo IFA) = (w IFA) = f As mod 1 ,
--
implying the following relation, of the form (5), between the centres of SO(7) __ and SU(4): (= [A) = (a2w01FA) mod 1
--
,
(33)
for all A. This relation implies that any field (A X) appearing in the coset , conformal theory, labelled by highest weights A X of representations of 4, L , , respectively, must satisfy the selection rule
(olc~ ) = (u2~ lA lX) mod 1 .
(34)
The corresponding relation of the type (23) between the outer automorphism
11
groups is also easily verified. We have 1 0 0 0 0 0 1 0 0 1 0 1 0 0 0 1 1 = j? (35)
and this matrix acting on the weights (29) reproduces the correct branching rule (28). Fields of the coset ~0(7)/su(4) theorymust therefore be identified as follows: (oh a2X) E (A X) , , Example 2 G=H@H. . (36)
Our second example is the diagonal embedding h c h $ h. A weight of i @ k .may be denoted [X, X/IT, where X is a weight of the first i and X of the second. If we demand that a pure k weight [X, OIT or [0, XIT is projected onto the same weight [XIT of the diagonal subalgebra, we get
(37)
where c = 1,2 specify the two summands of k $ k and (Y, QI denote the fundamental weights of A. So a weight [X, X of i $ k is projected onto the weight lT [A + X of L ]T Now consider any outer automorphism A of fi. The corresponding automorphism of i = i $ i is A = A @ A. Since (& \A) = (Au (X) + (Ati 12
(38)
and (Aw ]FA) = (Aw + X lX ) (39)
we have a centre relation of the type (5) for all A. If A= [p, 01~ and X = [clT are highest weights of ?L $ ?L and & representations, respectively, th.en only those fields obeying the selection rule (9) may occur in the diagonal coset theory. In . -this example, it means FA- 1 = ,!i + 5 - 5 must lie in the root lattice of h[3,4]. so s Equation (23) a 1 ho Id ob viously, with 2 = g: A@A @I A) = ? .
(40)
Therefore the field ([p, (T]~, [<IT) is identified with ([Ap, AnIT, [A<lT). Example 3 G = SU(6) , H = SU(2) @I W(3).
The last example illustrates that quite nontrivial relations exist-between the .centres of G and H. It also shows the limitations imposed by the technical restriction (22) on the relations (23) that can be found. The embedding %(p) x Z( Q )L c %(pq) with k = 1 was studied in _ Reference [13]. In this example we will not restrict k, but set p = 2 and q = 3, just for the sake of simplicity. The following is a valid projection matrix[l3]: 3 0 2 0 2 1 1 1 3 0 0 2 0 2 1 0 1 1 3 0 0 0 2 2 0 1 0 1
(41)
0.0
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Let A6, AZ, A3 be the generators of the outer automorphism groups 0(&I(6)), mw), O@(3)), respectively, so that (Ai)i = 1, i = 6,2,3. Then this projection matrix immediately gives (I@ As)F(A~)~ = g .
(42)
On the other hand, with F the finite projection matrix contained in (41), we have the following centre relation (AGui\h) = (Api + (A3)2ui1FA) ...