620,171 research outputs found
Seiberg-Witten Monopole Equations on Noncommutative R^4
It is well known that, due to vanishing theorems, there are no nontrivial
finite action solutions to the Abelian Seiberg-Witten (SW) monopole equations
on Euclidean four-dimensional space R^4. We show that this is no longer true
for the noncommutative version of these equations, i.e., on a noncommutative
deformation R^4_\theta of R^4 there exist smooth solutions to the SW equations
having nonzero topological charge. We introduce action functionals for the
noncommutative SW equations and construct explicit regular solutions. All our
solutions have finite energy. We also suggest a possible interpretation of the
obtained solutions as codimension four vortex-like solitons representing
D(p-4)- and anti-D(p-4)-branes in a Dp-anti-Dp brane system in type II
superstring theory.Comment: 33 pages, v2: typos corrected, to appear in J.Math.Phy
Domain Wall Equations, Hessian of Superpotential, and Bogomol'nyi Bounds
An important question concerning the classical solutions of the equations of
motion arising in quantum field theories at the BPS critical coupling is
whether all finite-energy solutions are necessarily BPS. In this paper we
present a study of this basic question in the context of the domain wall
equations whose potential is induced from a superpotential so that the ground
states are the critical points of the superpotential. We prove that the
definiteness of the Hessian of the superpotential suffices to ensure that all
finite-energy domain-wall solutions are BPS. We give several examples to show
that such a BPS property may fail such that non-BPS solutions exist when the
Hessian of the superpotential is indefinite.Comment: 25 page
Polynomial solutions to the WDVV equations in four dimensions
All polynomial solutions of the WDVV equations for the case n = 4 are determined. We find all five solutions predicted by Dubrovin, namely those corresponding to Frobenius structures on orbit spaces of finite Coxeter groups. Moreover we find two additional series of polynomial solutions of which one series is of semi-simple type (massive). This result supports Dubrovin's conjecture if modified appropriately
The Role of Boundary Conditions in Solving Finite-Energy, Two-Body, Bound-State Bethe-Salpeter Equations
The difficulties that typically prevent numerical solutions from being
obtained to finite-energy, two-body, bound-state Bethe-Salpeter equations can
often be overcome by expanding solutions in terms of basis functions that obey
the boundary conditions. The method discussed here for solving the
Bethe-Salpeter equation requires only that the equation can be Wick rotated and
that the two angular variables associated with rotations in three-dimensional
space can be separated, properties that are possessed by many Bethe-Salpeter
equations including all two-body, bound-state Bethe-Salpeter equations in the
ladder approximation. The efficacy of the method is demonstrated by calculating
finite-energy solutions to the partially-separated Bethe-Salpeter equation
describing the Wick-Cutkosky model when the constituents do not have equal
masses.Comment: 18 page
Chern-Simons Solitons, Toda Theories and the Chiral Model
The two-dimensional self-dual Chern--Simons equations are equivalent to the
conditions for static, zero-energy solutions of the -dimensional gauged
nonlinear Schr\"odinger equation with Chern--Simons matter-gauge dynamics. In
this paper we classify all finite charge solutions by first
transforming the self-dual Chern--Simons equations into the two-dimensional
chiral model (or harmonic map) equations, and then using the Uhlenbeck--Wood
classification of harmonic maps into the unitary groups. This construction also
leads to a new relationship between the Toda and chiral model
solutions
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