aa r X i v : . [ h e p - t h ] J a n On the r–matrix of M(embrane)–theory
Jens Hoppe
Abstract.
Supersymmetrizable theories, such as M(em)branesand associated matrix–models related to Yang–Mills theory, pos-sess r–matrices
While the Lax–pairs found in [1, 2], in contrast to standard integrablesystems (see e.g. [3]), naively do not seem to provide any non–trivialconserved quantity, it is also unlikely that they will not be useful. Asa start I would like to point out that Lax–pairs arising from supersym-metrizability generically do have an r–matrix associated with them(which in principle is not even particularly difficult to explicitly calcu-late), including the infinite–dimensional case of relativistic higher di-mensional extended objects (see e.g. [4] for a review) such as Membranetheory, whose discretized version, a SU(N)–invariant matrix–model [5],is known to be subtle in several ways (e.g. possessing classical solu-tions extending to infinity, but quantum–mechanically purely discretespectrum [6, 7], while when supersymmetrized [8] changing “again” tocontinuous [9, 10]); in this sense making the existence of a rather spe-cial Lax–pair for them not too surprising.Let me first illustrate the idea by considering ˙ L = [ L , M ] , ˙ L = [ L , M ] L = N X a =1 ( γ a p a − γ a + N ∂ a w ) , L = X a ( γ a ∂ a w + γ a + N p a ) M = − N X a,b =1 γ a γ b + N ∂ ab w (1)where w = w ( x , x , . . . , x N ) and the hermitean Clifford matrices γ i =1 ... N, satisfying(2) γ i γ j + γ j γ i = 2 δ ij · , by many first interpreted negatively, then [12, 13] positively cp.[1](missing in eq.(12), dWHN: 305/1988). JENS HOPPE canonically realized as 2 N × N dimensional tensorproducts of Pauli–matrices. While in that canonical representation L and L anticom-mute and square to a multiple of the unit matrix, β, β ′ = 1 , L β L β ′ + L β ′ L β = 2 δ ββ ′ (2 H := ~p + ( ∇ w ) ) (1), due to the polynomials of degree ≤ γ i closing under com-mutation, forming a (spinor–) representation of so (2 N + 1), may alsobe considered in the defining , ‘vector’ representation of so (2 N + 1), inwhich ˜ L ( λ ) := ( ˜ L + λ ˜ L ) and ˜ M , instead of having non–zero elementsdistributed over many of the 2 N × N entries, take the simple form˜ L ( λ ) = i (cid:18) v T − v N × N (cid:19) =: √ H √ λ + 1 K ( λ ) v = (cid:18) ~p + λ ~ ∇ wλ~p − ~ ∇ w (cid:19) =: √ H √ λ + 1 e ( λ )˜ M = (cid:18) − A (cid:19) , A = (cid:18) w ab − w ab (cid:19) N × N , (4)as when representing γ ij = ( γ i γ j − γ j γ i ) by M ij := E ij − E ji (gen-erating so (2 N ) ⊂ so (2 N + 1)) iγ k will correspond to the generators M k = E k − E k of so (2 N + 1),(5) [ M µν , M ρλ ] = δ νρ M µλ ± µ, ν, ρ, λ = 0 , , . . . , N .It is trivial to check that(6) ˙˜ L ( λ ) = [ ˜ L ( λ ) , ˜ M ] ⇔ ˙ e ( λ ) = Ae, e ∈ S N − are equivalent to the equations of motion ˙ x a = p a , ˙ p a = − w ab ∂ b w ,( w ab := ∂ ab w ) being the Hessian of the ‘superpotential’ w . However,in contrast with L and L in the spinor representation each having N eigenvalues + √ H and N eigenvalues −√ H , the Lax–matrix ˜ L ( λ ),as given in (4), will have only two non–zero eigenvalues (which, divingby √ H √ λ , i.e. considering the normalized matrix K ( λ ), may betaken to be ± e ± , i.e. K ( λ ) = U ( λ ) − U † ( λ ) ,U ( λ ) = 1 √ (cid:18) + i − i . . . e e √ n . . . √ n N − (cid:19) = ( u u . . . u N − )(7) n the r–matrix of M(embrane)–theory 3 with ( e, n , . . . , n N − ) forming an orthonormal basis of R N .As the eigenvalues of K ( λ ) are numerical constants (= +1 , − , . . . { X , Y } := { X ⊗ , ⊗ Y } := { X ij , Y kl } E ij ⊗ E kl that K := K ⊗ and K := × K satisfy { K ( λ ) , K ( λ ) } = (cid:2) [ U ( λ ) , K ] , K (cid:3) = (cid:2) [ U ( λ ) , K ] , K (cid:3) = [˚ r , K ] − [˚ r , K ] U := { U , U } U − U − ˚ r = 12 [ U , K ] , ˚ r = 12 [ U , K ] = −
12 [ U , K ];(9)and J ( λ ) := √ HK ( λ ) will therefore satisfy { J , J } = H { K , K } + 12 ( ˙ K K − ˙ K K )= [ H ˚ r −
12 ˜ M K , K ] − (1 ↔ r –matrix for the Lax–pair ( J ( λ ) , ˜ M = (cid:0) A (cid:1) ) with equa-tions of motion ˙ e = Ae , e ( λ ) ∈ S N − is(11) r ( λ ) = √ H ˚ r ( λ ) −
12 ˜ M K ( λ ) 1 √ H .
Having gone through this simple example it is almost correct to saythat one has understood all supersymmetrizable systems (from thisperspective) whose supercharges are linear in the Clifford generators,resp. ‘fermions’ (and at this stage it would be tempting to see a re-lation to other, old and new - see e.g. [11], and references therein -statements about supersymmetric systems); one ‘only’ gets different(for field–theories: infinite dimensional) unit vectors e and different(‘more complicated’) antihermitean operators A (cp. (6)) satisfying˙ e ( λ ) = Ae ( λ ).Consider now the membrane–matrix model (cp.(2) J a =0 of [1]), i.e.(12) v = (cid:18)P β λ β P βαa P β λ β Q βαa (cid:19) = (cid:18)P β λ β ~p β P β λ β ~q β (cid:19) = (cid:18) ~p ( λ ) ~q ( λ ) (cid:19) = (cid:18) ~p~q (cid:19) P βαa = d X t =1 p ta γ tβα , Q βαa = 12 ( γ st ) βα f abc x sb x tc , where the f abc are totally antisymmetric (real) structure constants of su ( N ), a, b, c = 1 . . . N −
1, the x sb and p tc are canonically conjugate JENS HOPPE variables, the γ t are real symmetric σ × σ matrices satisfying γ s γ t + γ t γ s = δ st , the time–evolution is given by(13) H = 12 ( P βαa P βαa + Q βαa Q βαa ) = 12 ( ~p β ~p β + ~q β ~q β ) , which is independent of β , sum over ( αa ) = (11) . . . ( σ, N −
1) and(14) J a = f abc x sb p sc ! = 0 , which also implies v T v = (2 H )( ~λ ); the equations of motion can bewritten in the form (cp.(6) of [1])(15) ˙ ~q = Ω ~p, ˙ ~p = Ω ~q, and the Lax–pair [1], when going to the defining vector representationof so (2 n + 1), n = σ ( N − ∈ N , becomes –as explained above– J ( λ ) = i (cid:18) v T − v (cid:19) p ~λ ˜ M = (cid:18) − A (cid:19) , A n × n = (cid:18) (cid:19) = − A T Ω αa,α ′ a ′ = f aa ′ c x tc γ tαα ′ (16)with J = U √ H −√ H U † U = i √ − i √ . . . e √ e √ n . . . n ∗ ! (17)and the 2 n unit vectors ( e, n . . . n ∗ ) being orthonormal, and(18) ˙ J ( λ ) = [ J ( λ ) , ˜ M ] ⇔ ˙ e = Ae being equivalent to the matrix–model equations of motion.As shown above, the su ( N )–invariant membrane matrix model of [5]therefore possesses an r–matrix, { J ( λ ) , J ( λ ) } = [ r ( λ ) , J ] − (1 ↔ r = (cid:0)
12 [ U , J ] −
12 ˜ M J H (cid:1) ;(19) n the r–matrix of M(embrane)–theory 5 note that the normalisation of J is chosen such that tr J = H (as aconsistency check, one can calculate − Tr ( r J ) = ˜ M tr J H + tr [ U , J ]which indeed gives ˜ M ; that ˚ r does not give any contribution meansthat it is in some sense ‘trivial’, i.e. not influencing the time–evolution;dimensionally [ x ] ∼ E , [ p ] ∼ E so [ ∂∂x ∂∂p ] = E − = [ ˜ MH ] = E E ).What about the infinite–dimensional case of membrane–theory? v βαa = Z Y a ( ϕ ) (cid:0) p i ρ γ iβα + 12 { x i , x j } γ ijβα (cid:1) ρ d ϕ a ∈ N { x i , x j } ( ϕ ) := ε rs ρ ∂ r x i ∂ s x j Z Y a Y b ρ d ϕ = δ ab , ∞ X a =0 Y a ( ϕ ) Y a ( ˜ ϕ ) = δ ( ϕ, ˜ ϕ ) ρ , (20) ˙ J ( λ ) = [ J ( λ ) , ˜ M ] , Ω αa,α ′ a ′ = g aa ′ c x ic γ iαα ′ , g abc = Z Y a { Y b , Y c } ρ d ϕJ = i (cid:18) v † − v (cid:19) √ λ † λ , ˜ M = (cid:18) − A (cid:19) , A = (cid:18) (cid:19) . (21)While in principle having to worry about potentially diverging infinitesums, and Lie–algebraically one would have to identify a well–definedalgebra, I think that (21), and the infinite–dimensional analogue of(19), should be fine, for various reasons: v † v = 2 λ † λH implies thatfor fixed (trivially conserved) energy all components of v are finite,and v √ λ † λH =: e will be a unit vector; the norm of each row of Ω (orcolumn, given by ( αa )) is R { Y a , x i } ρ d ϕ , which is clearly finite, asone integrates over a compact manifold, and if { Y a , x i } was infinite forsome a , the potential term of H could not be finite; the arising scalarproducts of infinite–dimensional vectors (corresponding to R ρ d ϕ ofthe product of corresponding square–integrable functions) therefore in-volve only vectors of finite norm. One may also write (21), resp. theequations of motion, in the following compact suggestive forms:˙ q βα = γ iαα ′ { p βα ′ , x i } , ˙ p βα = γ iαα ′ { q βα ′ , x i } , or as˙ V = { V, X } := { V βα , X δε } E βα E δε = −{ X, V T } T JENS HOPPE resp. ˙ Q βα = { P βα ′ , X α ′ α } = −{ X αα ′ , P α ′ β } = − ˙ Q αβ ˙ P βα = { Q βα ′ , X α ′ α } = + { X αα ′ , Q α ′ β } = + ˙ P αβ (22)(using that, as finite matrices, P is symmetric, Q = { X, X } antisym-metric and X αα ′ := γ iαα ′ x i = X α ′ α , ˙ X = P , ¨ X = { X, { X, X }} ).Finally, note [15] and that L = P + iQ , respectively ˙ L = i { L ∗ , X } could be considered a generalization to arbitrary d of (39) in [14], turn-ing into a real Lax–pair for the Wick–rotated/Euclidean equation ofmotion (though care is needed for the definition of a Lie–algebra in-volving matrix–valued functions on the membrane). Acknowledgement.
I would like to thank J. Fr¨ohlich and I. Kostovfor discussion, T. Ratiu for having raised the question on which Lie–algebra the Lax–pair (12) of [1] is formulated, and O. Lechtenfeld forsending me a copy of his Ph.D. thesis.
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