aa r X i v : . [ h e p - t h ] F e b Representation spaces for the membrane matrix model
Jens Hoppe
Abstract.
The SU ( N )–invariant matrix model potential is writ-ten as a sum of squares with only four frequencies (whose multi-plicities and simple N –dependence are calculated). Difficult problems, unless one is willing to give up on them, should beviewed from different perspectives. For the membrane matrix model a Lax–pair was recently given in [1], a set of ‘dual’ variables, in whichthe Hamiltonian is of the form ( ~p + ~q ), introduced in [2], and anr–matrix derived in [3]. Here I would like to point out that if oneintroduces real symmetric matrices Y = ( Y ab ) a,b =1 ...N − n = ( ~x a · ~x b )as variables, the potential − d X i,j =1 Tr[ X i , X j ] = f abc f ade x ib x jc x id x je = Tr (cid:0) Y F ( Y ) (cid:1) = W ( Y )(1)becomes a diagonalizable quadratic form in Y , as the map Y → F ( Y ) = − P a F a Y F a (cid:0) F ( Y ) (cid:1) ab = Tr( F a F b Y )( F a ) bc = − f abc (2)is symmetric. When trying to calculate the eigenvalues of F , I noticedthat for each pair ( ab ) the 4–dimensional subspace spanned by thesymmetric n × n matrices (for the definition of the invariant d –tensorand many useful identities, see [6]) E ab := δ ab n × n (∆ ab ) cd := 2 δ ab δ cd − δ ac δ bd − δ ad δ bc H ab := d abc D e , ( D e ) cg := d ecg Z ab := D a D b + D b D a (3) [4](see also [5]) JENS HOPPE is left invariant by F , and diagonalization of the simple 4 × N − N − N N − N − N
00 2 N N Z ab ∆ ab H ab E ab gives, apart from the immediate(5) F ( E ab ) = N E ab , F ( H ab ) = N H ab the eigenmatrices (with eigenvalue ∓ K ab := Z ab − N + 2 N ∆ ab − N + 4 N + 2 H ab + 2 N + 2 N + 1 E ab M ( N ≥ ab := Z ab + N − N ∆ ab − N − N − H ab − N − N − E ab , (6)with the understanding that Z ab and H ab are (put to) zero when N = 2.It is also not difficult to see that(7) M ( N =3) ab ≡ , X a M aa = 0 = X a K aa , d abc M ab = 0 = d abc K ab , and to determine for small N the number of independent matricesof ( E, H, M, K )–type, namely (1 , , ,
0) for N = 2, (1 , , ,
0) for N = 3, and (1 , , ,
20) for N = 4 (in each of these cases togetherspanning the n ( n +1)2 –dimensional space of symmetric n × n matrices).As the eigenspaces with different eigenvalues can not mix (as the mapcommutes with the group/algebra–action) it is immediate that theycorrespond to representation spaces under the action of SU ( N ), andas there are (for N ≥
3) precisely 4 (for N = 3 only 3) such irre-ducible spaces occurring in the symmetric part of the tensor productof 2 adjoint–representations (1 , . . . ,
1) of A l ∼ = SU ( l + 1), for l > (cid:0) (1 0 . . . × (1 0 . . . (cid:1) s = (0 . . . ⊕ (1 0 . . . ⊕ (2 , , . . . , , ⊕ (0 , , . . . , , E, H, K, M )–type to which we will, apart fromusing the above–mentioned Dynkin–labels, refer to as the
E, H, K, resp. M –representations. While the dimension of the H (= adjoint)–representation is of course n = N − E trivially = 1, thedimension of the K (= (2 , , . . . , , M (cid:0) ∧ = (0 , , . . . , , (cid:1) rep-resentations is slightly less trivial (though of course known; elementary epresentation spaces for the membrane matrix model 3 derivations are given in the appendix):dim( K ) = N ( N − N + 3)4dim( M ) = N ( N + 1)( N − N ( N − for the total dimension of the space of sym-metric n × n matrices). Y = y W + ~y H ~W H + ~y K ~W K + ~y M ~W M , with the W ’s orthonormal basesfor the respective irreducible representation–spaces, then gives(9) W ( Y ) = N y + N ~y H − ~y K + ~y M . The at first surprising − sign ( W ( Y ) = − Tr[ X i , X j ] ≥ N × N matrices X i ) brings one to the important issue thatthe Y ’s in (1) are not arbitrary (symmetric) matrices; they are (as= QQ T ) positive–semidefinite, and in fact, if n > d , necessarily ofsmaller than general rank.As an example, consider the d = 2, N = 3 matrix model; then thesingular value decomposition gives(10) Y = λ ~u ~u T + λ ~u ~u T where ~u and ~u are orthonormal vectors in R , and λ ≥ λ ≥ Y ; so containing only 7 + 6 + 2 = 15 parameters .Nevertheless (9), which is of a tantalizing simple form, should be useful.What about N → ∞ ? Despite of the, simple N dependence of thefrequencies (and multiplicities; naively one should think that it is easyto see which modes are the most important ones as N → ∞ ; note thatwhen summing their products the leading power of N cancels), and(6) converging to well–defined expressions, the N → ∞ limit seems difficult , for a variety of reasons. As indicated already by (7), and clearfrom general considerations, the 4 invariant subspaces E , H , K and M should most conveniently be discussed by corresponding projectors P α =1 , , , (resp. α = E, H, K, M ), forming a partition of the identity,with e.g.(11) P H = NN − d abe d a ′ b ′ e = P H , F ( P H Y ) = N P H Y ) . Using various SU ( N )–identities, in particular (cp. [7])(12) f abe f cde = 2 N ( δ ac δ bd − δ ad δ bc ) + ( d ace d bde − d ade d bce ) many thanks to R. Suter for a related discussion JENS HOPPE the corresponding projectors P K and P M are not difficult to work out,and in fact (I noticed that after finding (6)) have been worked out, inthe context of QCD [8]. The projectors however do not converge as N → ∞ . Moreover, the following general problem exists: while theredo exist bases of SU ( N ) in which the structure constants f abc con-verge (to those, g abc , of the Lie–algebra of area preserving diffeomor-phisms; (the fuzzy sphere [5] was invented in precisely this context),and d ( N ) abc → d ∞ abc too, as well as the central object f abc f ade (sum over a , cp. (1)) converging to g abc g ade , similarly (cp. (11)) d abe d a ′ b ′ e (thesum over e is finite for fixed ab , a ′ b ′ ) their action on the 4 subspaces,resp. projectors, involves multiple sums where the range of the indicesis not finite. Another aspect of the arising subtleties, and difficulties,can be demonstrated by looking at (12). As explained e.g. in [9], thenormalisation for the f ’s and d ’s suitable to take the limit is such that(12) becomes(13) 1 N ˜ f abc ˜ f cde = 2( δ ac δ bd − δ ad δ bc ) + ( ˜ d ace ˜ d bde − ˜ d ade ˜ d bce ) . Indeed, with ˜ d ∞ ace = R Y a ( ϕ ) Y c ( ϕ ) Y e ( ϕ ) ρ d ϕ , the Y a ( ϕ ) being orthonor-mal eigenfunctions of the Laplacian on the parameter–surface, the rhs. is zero for N = ∞ . Vice versa this however shows that if decomposingthe relevant operator, ˜ f abc ˜ f cde in the nomalisation where ˜ f ∞ abc is finite(and the sum over e as well) decomposing it with respect to f ’s and˜ d ’s, which effectively is done in [8] (for finite N ), involves (for infi-nite N ) a finite part of ∞ ·
0. Let me at this point go back to howI came to consider the matrices given in (3). The adjoint action (ofthe a –th generator of SU ( N )) on the symmetric n × n matrix Y (thesymmetric part of the tensor–product of two copies of the Lie–algebra)is, possibly up to an overall sign, commutation with the matrix F a (cp.(2)), i.e. [ F a , Y ] (and the map F commutes with the SU ( N ) action: − [ F a , F ( Y )] = [ F a , F c Y F c ] = [ F a , F c ] Y F c + F c Y [ F a , F c ] + F c [ F a , Y ] F c = ± f abc ( F b Y F c + F c Y F b ) + F c [ F a , Y ] F c = − F ([ F a , Y ])). Due to [ F a , D b ]being (again, not worrying about the signs in this qualitative argu-ment) f abc D c , the n dimensional subspace consisting of linear combina-tions of the D ’s is clearly invariant (giving the (1 0 . . . 0 1), resp. H –space). This being so easy, the obvious next step was to consider D a D b (+ D b D a , to get symmetric matrices), i.e. Z ab . Calculating F ( Z ab ),which involves ∆ ab then led to (3), resp. (4-7). If on the other handone wants (‘only’) to understand the representation theory, it is natu-ral to look for identities involving the Z ab (which can not be linearlyindependent when taken together with the first order polynomials in epresentation spaces for the membrane matrix model 5 the D ’s, as too many), and there one finds that(14) d abc Z bc = 2 d abc D b D c = D a N − N (which then is already most of the final answer). Unfortunately thescaling, ˜ d ( N ) abc = √ N d abc , (cp.e.g.[9]), that is known to converge to thetotally symmetric tensor,(15) ˜ d ∞ abc = ˜ d abc = Z Y a ( ϕ ) Y b ( ϕ ) Y c ( ϕ ) ρ d ϕ := h abc for functions on P (compact orientable, surface of genus g ) does notcancel (actually: enhances) the diverging factor on the rhs. of (14),and while the naive analogue of the Z ab ,(16) ( ˜ Z ab ) cd = ( ˜ D a ˜ D b + ˜ D b ˜ D a ) cd = ( ˜ d ace ˜ d bde + a ↔ b )is well–defined,(17) ( ˜ d abc ˜ Z bc ) fg = ˜ d abc ( ˜ d bfe ˜ d cge + b ↔ c )is not (seen by inserting (15) resp. indicated by the triple sum over bce in (17), involving truly infinite sums).Similarly, concerning the decomposition of adjoint ⊗ adjoint for sdiff Σ:defining infinite matrices G α and H β ( α, β = 1 . . . ∞ ) by(18) ( G α ) βγ := − g αβγ ( H α ) βγ := h αβγ satisfying (note: no convergence–problems, as each row and column ofthe matrices G α and H β has only a finite number of non–zero entries)(19) [ G α , G β ] = g αβγ G γ , [ G α , H β ] = − g αβγ H γ ,G ( X ) := − G α XG α , resp. ( G ( X )) αβ := Tr G α G β X , is formally sdiff–invariant, − [ G α , G ( x )] = [ G α , G ε XG ε ] = − G ([ G α , X ]) and, due to(19), the subspace consisting of linear combinations of the H γ certainlycorresponds to an adjoint representation, G ( H ε ) = . . . = + g αβε g αβγ H γ gives a diverging eigenvalue on that H –space (‘consistent’ with havinggotten N for finite N ). My reason for, still, being fairly optimistic aboutunderstanding the N → ∞ limit this way are twofold: firstly, puremathematics (understanding sdiff, whose structure strongly dependson the genus, hence must be reflected by the representation theory);secondly: as for classical motions of given energy the potential is bydefault finite/for a (regular) minimal surface (without singularities)all local quantities are finite/the apparent divergencies one gets abovemay actually tell one how to proceed, i.e. which collective degrees offreedom the system chooses. JENS HOPPE
Acknowledgement.
I am grateful to M. Bordemann for valuablediscussions.
Appendix
The K – and M – representations The easiest way to calculate the dimensions, and see that for SU ( N )the symmetric part of the tensor product of 2 adjoints contains only 4irreducible representations is( A ij A kl ) s = ( A ikjl ) s = 12 ( A ikjl + A kilj )= ˆ A ikjl + ˜ A ikjl = ˆ A ( ik )( jl ) + ˆ A [ ik ][ jl ] + ˜ A ikjl (A1)where both ˆ A and ˜ A are symmetric under (cid:0) ij (cid:1) ↔ (cid:0) kl (cid:1) , ˆ A is traceless withrespect to any upper and lower index (while A ikjl is traceless only withrespect to (cid:0) ij (cid:1) and (cid:0) kl (cid:1) ), ˜ A ikjl is a linear combination of the N quanti-ties A pnnq (= A npqn ; p, q = 1 . . . N ), ˆ A ( ik )( jl ) is symmetric in the upper, andlower indices (i.e. taking into account the traceless–condition, givingrise to a (cid:0) N ( N +1)2 (cid:1) − N = N ( N + 3)( N −
1) dimensional space, the(2 0 . . . K ) while ˆ A [ ik ][ jl ] is antisymmetric in the upperand lower indices (and traceless) giving rise to a (cid:0) N ( N − (cid:1) − N = N ( N + 1)( N −
3) dimensional space, the (0 1 0 . . . M , which is part of the tensor product (0 1 0 . . . × (0 . . . ω = (0 1 0 . . .
0) (known to berealized on the exterior product of two defining representations, corre-sponding to A [ ij ] ’s) and ω N − = (0 . . . A [ kl ] space), and the traceless-nessconditions making it irreducible, i.e. (0 1 0 . . . N di-mensional space of ˜ A ikjl ’s (traceless with respect to (cid:0) ij (cid:1) and (cid:0) kl (cid:1) , but not (cid:0) il (cid:1) and (cid:0) kj (cid:1) ) gives an N − . . . N = 3 (and N = 4) are slightly special, as for N = 3 (cp. [10])ˆ A [ ik ][ jl ] = ε ikp ε jlq ˜ A pq , while the traceless–ness condition then says that ˜ A pq must be = 0; for N = 4, the antisymmetric part of ˆ A gives the (0 2 0)representation, lying in (0 1 0) × (0 1 0), the first (0 1 0) viewed as A [ ik ] ’s,the exterior square of (1 0 0), the second (0 1 0) as A [ jl ] ’s, the exteriorsquare of the (0 0 1) representation–space.Apart from these simple considerations, one may also calculate the di-mensions of the 2 non–trivial representations ( K and M ) as follows:Weyl’s dimension formula (see e.g. [11]) says that if all the roots of a epresentation spaces for the membrane matrix model 7 (semi–)simple Lie–algebra have the same length (which is the case for SU ( N ) ∼ = A N − l ),(A2) dim V ~n = Y α = P lj =1 k j α j ∈ φ + l P i =1 k i ( n i + 1) l P i =1 k i = Y α d α , where ~n = ( n , n , . . . , n l ) ∈ N l classifies the finite dimensional irre-ducible representations, α , . . . α l are the simple roots ( α i = ε i − ε i +1 =(0 . . . − . . . ~k = ( k , . . . , k l ) ∈ N l characterizes the differentpositive roots – which for A l are all of the form ε p − ε q = (0 1 0 . . . − ≤ p < q ≤ l + 1.For ~n = (2 , , . . . , ,
2) the numerator of d α will be equal to the denom-inator, k α = P k i , resp. k α + 2 or k α + 4, depending on whether α contains neither α nor α l , contains α (but not α l ), or α l (but not α ),resp.containing both α and α l . As all positive roots are of the form(A3) ε p − ε q = α p + α p +1 + . . . + α q − , the 4 factors (corresponding to the just mentioned 4 cases) arecase 1: 1case 2 ( ε − ε q ): l Q q =2 q +1 q − = · · ... · l +11 · · ... · l − = l ( l +12 ) = N ( N − case 3 ( ε p − ε l +1 ): l Q p =2 p +1 p − = N ( N − case 4 ( ε − ε l +1 ): l +4 l = N +3 N − , hence(A4) dim V (2 , ,..., , = N ( N − N + 3 N − N N − N + 3) . For ~n = (0 , , . . . , ,
0) there are, apart from α ’s containing neither α = ε − ε nor α l − = ε l − − ε l (trivially contributing factors d α = 1),the following cases: JENS HOPPE ε − ε q> = α + α + . . . + α q − : l − Y q =3 qq − · · . . . · l − · · . . . · l − l − ε − ε q> = α + . . . + α q − : l − Y q =3 q − q − · · . . . · l − · · . . . · l − l − ε − ε l = α + α + . . . + α l − : l + 1 l − ε − ε l +1 = α + . . . + α l : l + 2 lε − ε l = α + . . . + α l − : ll − ε − ε l +1 = α + . . . + α l : l + 1 l − ε p> − ε l = α p + . . . + α l − : ( l − (cid:0) = l − Y p =3 l − p + 1 l − p (cid:1) ε p> − ε l +1 = α p + . . . + α l : ( l − (cid:0) = l − Y p =3 l − p + 2 l − p + 1 (cid:1) , hence(A5)dim V (0 , , ,..., , , = (cid:0)
12 ( l − l − (cid:1) (cid:0) l + 1 l − (cid:1) l + 2 l ll −
2= 14 ( l − l + 1) ( l + 2) = N N + 1)( N − . (A6) References [1] J.Hoppe, arXiv:2101.01803[2] J.Hoppe, arXiv:2101.04495[3] J.Hoppe, arXiv:2101.11510[4] J.Hoppe, Ph.D. thesis, MIT 1982 http://dspace.mit.edu/handle/1721.1/15717[5] T.Banks, W.Fischler, D.Shenker, L.Susskind, Phys.Rev.D 55, 1997[6] H.E.Haber, arXiv:1912.13302[7] A.J.Macfarlane, A.Sudbery, P.H.Weisz, Com.Math.Phys.11, 1968[8] P.Arnold, arXiv:1904.04264[9] J.Hoppe, M.Trzetrzelewski, arXiv:1101.4403[10] S.Coleman,
Fun with SU (3), Seminar held in Trieste, 1965[11] R.Carter, Lie Algebras of Finite and Affine Type, Cambridge studies inadvanced mathematics 96, Cambridge University Press, 2005 Braunschweig University, Germany
Email address ::