By Strömberg G.
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One can first compactify the IIa string and take a similar limit to obtain nϩ1-dimensional MSYM, with a similar interpretation. , 1997; Motl, 1997) and this is one of the main pieces of confirming evidence for the proposal. Another approach, spelled out by Taylor (1997), is to define toroidal compactification using the general theory of D-branes on quotient spaces discussed in Sec. C. Letting U i ϭ ␥ (g i ) for a set of generators of Zn , and taking AϭMatn (C), this leads to Eqs. : Noncommutative field theory j j j U Ϫ1 i X U i ϭX ϩ ␦ i 2 R i .
One way to make this claim precise has been proposed by Polychronakos (2001). One first observes that Eq. (126) (considered as a function of one-dimensional positions x ϭRe z ) is the ground state for a Calogero model, defined by the quantum-mechanical Hamiltonian Hϭ ͚ 1 2 m ͑ mϩ1 ͒ 1 1 p ϩ 2 x 2 ϩ . 2 2 2 Ͻ ͑ x Ϫx ͒ 2 ͚ (128) One can continue and identify all the excited Calogero states with excited Laughlin wave functions (Hellerman and Van Raamsdonk, 2001). It is furthermore known (Olshanetsky and Perelomov, 1976) that this model can be obtained from a matrixvector U(N) gauged quantum mechanics, with action (Polychronakos, 1991) Sϭ ͵ ͩ dt Tr ⑀ ij X i D 0 X j Ϫ 1 X 2 ϩ2mA 0 22 i ϩ †D 0 , ͪ (129) where X i are Hermitian matrices with iϭ1,2 and is a complex vector.
131) is bounded. A condition which does guarantee Tr͓ A,B ͔ ϭ0 is for A to be bounded and B to be trace class, roughly meaning that its eigenvalues form an absolutely convergent series. More precisely, A is trace class if ͉ A ͉ 1 is finite, where ͉ A ͉ p ϭ ͓ Tr(A † A) p/2͔ 1/p is the Schatten p norm. More generally, the p-summable operators are those for which ͉ A ͉ p is finite. This is more or less the direct analog of the conventional condition that a function (or some power of it) be integrable.