Dijkgraaf–Vafa theory as large-N reduction

Abstract

We construct a large-N twisted reduced model of the four-dimensional super YangMills theory coupled to one adjoint matter. We first consider a non-commutative version of the four-dimensional superspace, and then give the mapping rule between matrices and functions on this space explicitly. The supersymmetry is realized as a part of the internal U (∞) gauge symmetry in this reduced model. Our reduced model can be compared with the Dijkgraaf-Vafa theory that claims the low-energy glueball superpotential of the original gauge theory is governed by a simple one-matrix model. We show that their claim can be regarded as the large-N reduction in the sense that the one-matrix model they proposed can be identified with our reduced model. The map between matrices and functions enables us to make direct identities between the free energies and correlators of the gauge theory and the matrix model. As a byproduct, we can give a natural explanation for the unconventional treatment of the one-matrix model in the Dijkgraaf-Vafa theory where eigenvalues lie around the top of the potential.

The large-Nreduction in matrix quantum mechanics — a bridge between BFSS and IKKT —

“…We fit the results to O 1 lat = C 1 + C 2 a + C 3 a 2 , from which we obtain the continuum limit. 3 Figure 2 shows that the results obtained in the continuum limit for various τ can be nicely fitted to lim a→0 O 1 lat = C 4 + C 5 τ −1 , from which we obtain the τ → ∞ limit. We redo this analysis for N = 20, 32.…”

“…This motivated us 2 Here we have taken an average over the lattice site n to increase the statistics, but one should obtain the same result for O1(n) = 1 N tr(Xi(n) 2 ) in the τ → ∞ limit as far as the site n is well separated from the boundaries (1 n T ). 3 The coefficient C2 of the linear term turns out to be tiny as one can see also from figure 1, but this is not the case for the other observables listed in table 1.…”

“…This implies that the IKKT model with a small mass term in 9 directions, which should be removed after taking the large-N limit, is equivalent (in the sense of Eguchi-Kawai) to the BFSS model. In particular if the IKKT model has 4 extended directions, the BFSS model should have 3 out of 9 transverse directions extended, thus breaking the SO(9) symmetry down to SO (3) spontaneously.…”

“…Recent developments in the exact calculation of the low-energy superpotential in N = 1 supersymmetric gauge theories [2] may also be regarded as a particular application of the large-N reduction [3]. The ultimate aim of our work is to apply the original idea to supersymmetric gauge theories, but in a more conventional way.…”

D=2,N=2, Supersymmetric theories on Non(anti)commutative Superspace

“…The complete kinetic action for the non(anti)commutative N = 2 the action, to include the most general superpotentials, as shown: 20) where apart from the Kähler potential K(S,S), we also have superpotentials W (S) andW (S). Like the case for K(S,S) in eqn.…”