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Figure from:Collective field theory of a singular supersymmetric matrix model

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The square root factors are treated by expanding about the large N background configuration, which clearly generates an infinite perturbative expansion. I IEE EIDE IIE Sg EEE ESE Now, we turn our attention to the N dependence of the Hamiltonian and the supercharges. It is well known that he Feynman diagrams of a matrix theory can be topologically classified according to their genus [28]. Writing the superpotential 0 in terms of the supermatrix ® [15,17]

Figure 3 The square root factors are treated by expanding about the large N background configuration, which clearly generates an infinite perturbative expansion. I IEE EIDE IIE Sg EEE ESE Now, we turn our attention to the N dependence of the Hamiltonian and the supercharges. It is well known that he Feynman diagrams of a matrix theory can be topologically classified according to their genus [28]. Writing the superpotential 0 in terms of the supermatrix ® [15,17]

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The above commutation relations (13) - (16) are unusual in that the bosonic momentum does not commute with the fermionic fields. This is not however, hard to understand. Classically, the above fields correspond to a density description of a super Calogero model [10,14]:
The above commutation relations (13) - (16) are unusual in that the bosonic momentum does not commute with the fermionic fields. This is not however, hard to understand. Classically, the above fields correspond to a density description of a super Calogero model [10,14]:
and the corresponding supercharges are The hamiltonian becomes
and the corresponding supercharges are The hamiltonian becomes
Requiring that supersymmetry is preserved to leading order, we obtain an integral equation for the vacuum config- uration of the collective field: it then follows that a diagram of genus I carries a factor N?~2". It is clear that for 0 of the above form, the expressions Wa and v'(VNz) are both independent of N. Thus rescaling « > /Na,v > Ty and ¢ > VN¢, our constraint on ¢ becomes N independent (f dxd(x) = 1) and the N dependence of the Hamiltonian is made explicit:
Requiring that supersymmetry is preserved to leading order, we obtain an integral equation for the vacuum config- uration of the collective field: it then follows that a diagram of genus I carries a factor N?~2". It is clear that for 0 of the above form, the expressions Wa and v'(VNz) are both independent of N. Thus rescaling « > /Na,v > Ty and ¢ > VN¢, our constraint on ¢ becomes N independent (f dxd(x) = 1) and the N dependence of the Hamiltonian is made explicit:
we find that the zeroes of L"* (wa?) obey (46). Thus we see that the collective field do in fact describes the density 2 of zeroes of the Laguerre polynomials in the limit that these polynomials have an infinite number of nodes. In th case of the harmonic oscillator (7 = 0) Jevicki and Levine [10] proved explicitly that the collective field describes thi density of zeroes of the Hermite polynomials which are of course special cases of the Laguerre polynomials. To solve this equation, we use the results of Calegero’s work [23], based on the classical results of Stieltjies [30]. ley show that the zeroes of the Laguerre polynomial L% (a) obey:
we find that the zeroes of L"* (wa?) obey (46). Thus we see that the collective field do in fact describes the density 2 of zeroes of the Laguerre polynomials in the limit that these polynomials have an infinite number of nodes. In th case of the harmonic oscillator (7 = 0) Jevicki and Levine [10] proved explicitly that the collective field describes thi density of zeroes of the Hermite polynomials which are of course special cases of the Laguerre polynomials. To solve this equation, we use the results of Calegero’s work [23], based on the classical results of Stieltjies [30]. ley show that the zeroes of the Laguerre polynomial L% (a) obey:
The change of co-ordinates (52) is easily integrated to yield an explicit expression for q: This term can be made to vanish if we systematically restrict ourselves to odd fluctuations 7, which we assume from now on. Notice that from (33) this simply implies that our density field ¢(2) is always even, which is natural in view of the ground state of the previous section. Thus, under this restriction, the bosonic spectrum is that of a massless scalar. In order to exhibit this explicitly, one changes to the time of flight variable gq [3] we have just got the terms associated in the bosonic string theory to a massless scalar particle [3]. This term could, in principle, modify the spectrum in an unexpected way. For the potential we consider here, using the fact that f{ dxO,n = 0,this term may be simplified to:
The change of co-ordinates (52) is easily integrated to yield an explicit expression for q: This term can be made to vanish if we systematically restrict ourselves to odd fluctuations 7, which we assume from now on. Notice that from (33) this simply implies that our density field ¢(2) is always even, which is natural in view of the ground state of the previous section. Thus, under this restriction, the bosonic spectrum is that of a massless scalar. In order to exhibit this explicitly, one changes to the time of flight variable gq [3] we have just got the terms associated in the bosonic string theory to a massless scalar particle [3]. This term could, in principle, modify the spectrum in an unexpected way. For the potential we consider here, using the fact that f{ dxO,n = 0,this term may be simplified to:
where the prime on the sum indicates a sum over even integers only, and h.c. denotes the hermitean conjugate. VI. DISCUSSION AND CONCLUSIONS
where the prime on the sum indicates a sum over even integers only, and h.c. denotes the hermitean conjugate. VI. DISCUSSION AND CONCLUSIONS