We consider four-dimensional N = 2 supersymmetric gauge theories with gauge group U (N ) on R 3 ×S 1 , in the presence of a classical superpotential. The low-energy quantum superpotential is obtained by simply replacing the adjoint scalar superfield in the classical superpotential by the Lax matrix of the integrable system that underlies the 4d field theory. We verify in a number of examples that the vacuum structure obtained in this way matches precisely that in 4d, although the degrees of freedom that appear are quite distinct. Several features of 4d field theories, such as the possibility of lifting vacua from U (N ) to U (tN ), become particularly simple in this framework. It turns out that supersymmetric vacua give rise to a reduction of the integrable system which contains information about the field theory but also about the Dijkgraaf-Vafa matrix model. The relation between the matrix model and the quantum superpotential on R 3 × S 1 appears to involve a novel kind of mirror symmetry.
We perform three tests on our proposal to implement diffeomorphism invariance in the nonabelian D0-brane DBI action as a basepoint independence constraint between matrix Riemann normal coordinate systems. First we show that T-duality along an isometry correctly interchanges the potential and kinetic terms in the action. Second, we show that the method to impose basepoint independence using an auxiliary dN 2 -dimensional non-linear sigma model also works for metrics which are curved along the brane, provided a physical gauge choice is made at the end. Third, we show that without alteration this method is applicable to higher order in velocities. Testing specifically to order four, we elucidate the range of validity of the symmetrized trace approximation to the non-abelian DBI action.
We continue our study of nonperturbative superpotentials of four-dimensional N = 2 supersymmetric gauge theories with gauge group U (N ) on R 3 × S 1 , broken to N = 1 due to a classical superpotential. In a previous paper [1] we discussed how the low-energy quantum superpotential can be obtained by substituting the Lax matrix of the underlying integrable system directly into the classical superpotential. In this paper we prove algebraically that this recipe yields the correct factorization of the Seiberg-Witten curves, which is an important check of the conjecture. We will also give an independent proof using the algebraic-geometrical interpretation of the underlying integrable system.
An indication for the existence of a collective Myers solution in the non-abelian D0-brane BornInfeld action is the presence of a tachyonic mode in fluctuations around the standard diagonal background. We show that this computation for non-abelian D0-branes in curved space has the geometric interpretation of computing the eigenvalues of the geodesic deviation operator for U(N)-valued coordinates. On general grounds one therefore expects a geometric Myers effect in regions of sufficiently negative curvature. We confirm this by explicit computations for nonabelian D0-branes on a sphere and a hyperboloid. For the former the diagonal solution is stable, but not so for the latter. We conclude by showing that near the horizon of a Schwarzschild black hole one also finds a tachyonic mode in the fluctuation spectrum, signaling the possibility of a near-horizon gravitationally induced Myers effect.
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