Vortices and monopoles in a harmonic trap
HJE
Vortices and monopoles in a harmonic trap
David Tong 0
Carl Turner 0
Cambridge 0
OWA U.K. 0
0 Department of Applied Mathematics and Theoretical Physics, University of Cambridge
The -deformation is a harmonic trap, penning certain excitations near the origin in a manner consistent with supersymmetry. Here we explore the dynamics of BPS monopoles and vortices in such a trap. We pay particular attention to monopoles in the Higgs phase, when they are con ned to a vortex string. Unusually for BPS solitons, the mass of these con ned monopoles is quadratic in the topological charges. We compute an index theorem to determine the number of collective coordinates of con ned monopoles. Despite being restricted to move on a line, we nd that they have a rich dynamics. As the strength of the trap increases, the number of collective coordinates can change, sometimes with constituent monopoles disappearing, sometimes with new ones emerging.
Supersymmetric gauge theory; Solitons Monopoles and Instantons
1 Introduction 2 Solitons in a harmonic trap
Monopoles as kinks on the worldsheet
A matrix model for vortex strings
Kinks on the worldsheet
Counting zero modes for con ned monopoles
4
Summary and discussion
A Dynamics on the vortex moduli space B
Dynamics of U(1) vortices in a trap C An index theorem for non-Abelian kinks 1 2
lational invariance. It has proven to be both a powerful tool for computation [1, 2], and
a useful device to highlight connections between di erent theories, most notably
fourdimensional gauge theories and two dimensional integrable systems [3, 4].
In this paper we take a more prosaic view of the
-background. We view it simply
as a harmonic trap, analogous to those which arise in condensed matter physics. Its role
is to restrict certain excitations to lie close to the origin. The excitations that we will be
interested in are solitons. In supersymmetric gauge theories, BPS solitons typically have a
number of nice properties, both physical and mathematical. The
-background provides
a harmonic trap which is consistent with supersymmetry and, correspondingly, preserves
many of these nice properties.
There are at least two motivations to study solitons in the -background. The rst
is purely classical. A harmonic trap squeezes solitons together. Yet this is often where
solitons are at their most interesting. They no longer appear as a point-like objects and
their extended, non-linear nature becomes apparent. They lose their individuality, merging
into each other to form something new, often with interesting structures and collective
excitations.
{ 1 {
The second motivation is more quantum in origin. Solitons provide a semi-classical
springboard to study some of the interesting dualities that are induced by the
-background. In particular, we have in mind the 4d/2d duality described in [5{7], relating the
Seiberg-Witten curve to twisted superpotentials of 2d sigma-models. This is an extension of
an earlier duality [8, 9] which found an explanation in the dynamics of vortex strings [10, 11].
Here we study the vortex strings relevant for the extended duality. More recently, there
have been studies of 3d gauge theories [12] and 5d gauge theories [13] in the presence of the
-background and we will describe the vortices and monopoles relevant for these theories.
What we do.
We study solitons in N
= 2 four-dimensional theories with an
and the index theorem (3.8) for these objects.
We nd that the presence of the harmonic trap endows these con ned monopoles with
a rich dynamics. A generic, higher-charge monopole can split into constituent monopoles,
each free to move up and down along the string. However, the mass of each of these
constituents has an extra term which, unusually for BPS solitons, is quadratic in the
magnetic ux charges. This can be thought of a binding energy between the monopole and
other ux tubes which also lie in the trap.
As one increases the strength of the harmonic trap, the number of collective coordinates
jumps. Sometimes this re ects the fact that some of the constituent monopoles become
massless and disappear; sometimes it re ects the fact the new constituent monopoles
appear. One of the surprising features is that monopoles with charges that one might naively
have thought of as anti-BPS can apparently become BPS in the presence of the trap. Much
of the paper is devoted to telling this story.
The paper also includes a number of other results. In particular, in two appendices we
study the dynamics of vortices in the presence of a harmonic trap. The e ect of the trap
is to induce a potential on the vortex moduli space, so that the ground state of vortices is
an incompressible disc lying at the origin of the plane. We show that, for U(
1
) vortices,
the collective excitations of this disc have a description as a eld theory living on the edge
of the disc.
2
Solitons in a harmonic trap
The theory of interest consists of a U(N ) gauge eld A , coupled to a real adjoint s (...truncated)