Linking numbers in higher dimensions and their generalization including gauge fields are studied in the context of BF theories. The linking numbers associated to n-manifolds with smooth flows generated by divergence-free pvector fields, endowed with an invariant flow measure are computed in different cases. They constitute invariants of smooth dynamical systems (for non-singular flows) and generalizes previous results for the 3-dimensional case. In particular, they generalizes to higher dimensions the Arnold's asymptotic Hopf invariant for the three-dimensional case. This invariant is generalized by a twisting with a non-abelian gauge connection. The computation of the asymptotic Jones-Witten invariants for flows is naturally extended to dimension n = 2p + 1. Finally we give a possible interpretation and implementation of these issues in the context of string theory.
Abelian T-duality in Gauged Linear Sigma Models (GLSM) forms the basis of the physical understanding of Mirror Symmetry as presented by Hori and Vafa. We consider an alternative formulation of Abelian T-duality on GLSM's as a gauging of a global U(1) symmetry with the addition of appropriate Lagrange multipliers. For GLSMs with Abelian gauge groups and without superpotential we reproduce the dual models introduced by Hori and Vafa. We extend the construction to formulate non-Abelian T-duality on GLSMs with global non-Abelian symmetries. The equations of motion that lead to the dual model are obtained for a general group, they depend in general on semi-chiral superfields; for cases such as SU(2) they depend on twisted chiral superfields. We solve the equations of motion for an SU(2) gauged group with a choice of a particular Lie algebra direction of the vector superfield. This direction covers a non-Abelian sector that can be described by a family of Abelian dualities. The dual model Lagrangian depends on twisted chiral superfields and a twisted superpotential is generated. We explore some non-perturbative aspects by making an Ansatz for the instanton corrections in the dual theories. We verify that the effective potential for the U(1) field strength in a fixed configuration on the original theory matches the one of the dual theory. Imposing restrictions on the vector superfield, more general non-Abelian dual models are obtained. We analyze the dual models via the geometry of their susy vacua.
By wrapping D3-branes over 3-cycles on a half-flat manifold, we construct an effective supersymmetric black hole in the N ¼ 2 low-energy theory in four dimensions. Specifically, we find that the torsion cycles present in a half-flat compactification, corresponding to the mirror symmetric image of electric Neveu-Schwarz flux on a Calabi-Yau manifold, manifest in the half-flat black hole as quantum hair. We compute the electric and magnetic charges related to the quantum hair and also the mass contribution to the effective black hole. We find that by wrapping a number of D3-branes equal to the order of the discrete group associated to the torsional part of the half-flat homology, the effective charge and mass terms vanish. We compute the variation of entropy and the corresponding temperature associated with the loss of quantum hair. We also comment on the equivalence between canceling Freed-Witten anomaly and the assumption of self-duality for the 5-form field strength. Finally from a K-theoretical perspective, we compute the presence of discrete Ramond-Ramond charge of D-branes wrapping torsional cycles in a half-flat manifold.
In the present work some geometric and topological implications of noncommutative Wilson loops are explored via the Seiberg-Witten map. In the abelian Chern-Simons theory on a three dimensional manifold, it is shown that the effect of noncommutativity is the appearance of 6 n new knots at the n-th order of the Seiberg-Witten expansion. These knots are trivial homology cycles which are Poincaré dual to the high-order Seiberg-Witten potentials. Moreover the linking number of a standard 1-cycle with the Poincaré dual of the gauge field is shown to be written as an expansion of the linking number of this 1-cycle with the Poincaré dual of the Seiberg-Witten gauge fields. In the process we explicitly compute the noncommutative 'Jones-Witten' invariants up to first order in the noncommutative parameter. Finally in order to exhibit a physical example, we apply these ideas explicitly to the Aharonov-Bohm effect. It is explicitly displayed at first order in the noncommutative parameter, we also show the relation to the noncommutative Landau levels.
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