We study nonabelian vortices (flux tubes) in SU(N) gauge theories, which are responsible for the confinement of (nonabelian) magnetic monopoles. In particular a detailed analysis is given of N = 2 SQCD with gauge group SU(3) deformed by a small adjoint chiral multiplet mass. Tuning the bare quark masses (which we take to be large) to a common value m, we consider a particular vacuum of this theory in which an SU(2) subgroup of the gauge group remains unbroken. We consider 5 ≥ N f ≥ 4 flavors so that the SU(2) sub-sector remains non asymptotically free: the vortices carrying nonabelian fluxes may be reliably studied in a semi-classical regime. We show that the vortices indeed acquire exact zero modes which generate global rotations of the flux in an SU(2) C+F group. We study an effective world sheet theory of these orientational zero modes which reduces to an N = 2 O(3) sigma model in (1+1) dimensions. Mirror symmetry then teaches us that the dual SU(2) group is not dynamically broken.

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We find the phase and flavor symmetry breaking pattern of each N = 1 supersymmetric vacuum of SU (n c ) and U Sp(2n c ) gauge theories, constructed from the exactly solvable N = 2 theories by perturbing them with small adjoint and generic bare hypermultiplet (quark) masses. In SU (n c ) theories with n f ≤ n c the vacua are labelled by an integer r, in which the flavor U (n f ) symmetry is dynamically broken to U (r) × U (n f − r) in the limit of vanishing bare hyperquark masses. In the r = 1 vacua the dynamical symmetry breaking is caused by the condensation of magnetic monopoles in the n f representation. For general r, however, the monopoles in the n f C r representation, whose condensation could explain the flavor symmetry breaking but would produce too-many Nambu-Goldstone multiplets, actually "break up" into "magnetic quarks": the latter with nonabelian interactions condense and induce confinement and dynamical symmetry breaking. In U Sp(2n c ) theories with n f ≤ n c + 1, the flavor SO(2n f ) symmetry is dynamically broken to U (n f ), but with no description in terms of a weakly coupled local field theory. In both SU (n c ) and U Sp(2n c ) theories, with larger numbers of quark flavors, besides the vacua with these properties, there exist also vacua in free magnetic phase, with unbroken global symmetry.

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ABSTRACT:We prove that the linear delta expansion for energy eigenvalues of the quantum mechanical anharmonic oscillator converges to the exact answer if the order dependent trial frequency Ω is chosen to scale with the order as Ω = CN γ ; 1/3 < γ < 1/2, C > 0 as N → ∞. It converges also for γ = 1/3, if C ≥ α c g 1/3 , α c ≃ 0.570875, where g is the coupling constant in front of the operator q 4 /4. The extreme case with γ = 1/3, C = α c g 1/3 corresponds to the choice discussed earlier by Seznec and Zinn-Justin and, more recently, by Duncan and Jones.