Fundamental vortices, wall-crossing, and particle-vortex duality
Received: March
Fundamental vortices, wall-crossing, and particle-vortex duality
Chiung Hwang 0 2
Piljin Yi 0 2
Yutaka Yoshida 0 1
Open Access 0
c The Authors. 0
0 3d/1d connection is used to
1 Research Institute for Mathematical Sciences, Kyoto University
2 School of Physics, Korea Institute for Advanced Study
We explore 1d vortex dynamics of 3d supersymmetric Yang-Mills theories, as inferred from factorization of exact partition functions. Under Seiberg-like dualities, the 3d partition function must remain invariant, yet it is not a priori clear what should happen to the vortex dynamics. We observe that the 1d quivers for the vortices remain the same, and the net e ect of the 3d duality map manifests as 1d Wall-Crossing phenomenon; although the vortex number can shift along such duality maps, the ranks of the 1d quiver theory are una ected, leading to a notion of fundamental vortices as basic building blocks for topological sectors. For Aharony-type duality, in particular, where one must supply extra chiral elds to couple with monopole operators on the dual side, 1d wall-crossings of an in nite number of vortex quiver theories are neatly and collectively encoded by 3d determinant of such extra chiral elds. As such, 1d wall-crossing of the vortex theory encodes the particle-vortex duality embedded in the 3d Seiberg-like duality. For N = 4, the D-brane picture is used to motivate this 3d/1d connection, while, for N = 2, this ne-tune otherwise ambiguous vortex dynamics.
particle-vortex; duality; prove some identities of 3d supersymmetric partition functions for the Aharony duality; Duality in Gauge Field Theories; Supersymmetry and Duality; Solitons
1 Introduction 2 3 4
Vortex quantum mechanics
Vortices and Seiberg-like dualities
T [SU(N )]
N = 2 linear quiver gauge theories
N = 2 SQCDs
N = 4 SQCDs
Linear quiver examples
T [SU(N )]
N = 2 linear quiver examples
Fundamental vortices and particle-vortex duality
Aharony duality for supersymmetric partition functions
Introduction
In recent years, a plethora of exact partition functions became available for supersymmetric
gauge theories. The localization method, responsible for these, is powerful and universal
but such universality comes with costs. Much of the dynamics is lost, as the end result
depends only on handful of UV information, such as eld contents and their representation
under the gauge and the global symmetries. This should be hardly surprising. When the
spacetime that admits a circle, for example, the supersymmetric partition function can
be regarded as a re ned index, well-known to be robust under continuous deformations.
Despite this ultraviolet nature of the computation, these partition functions proved to quite
useful, for example as a litmus test for various dualities. For dimensions less than three
also, where there is no notion of vacuum expectation value of moduli, a UV theory often
ows down to a unique theory in IR. As such, the partition functions in such low dimensions
contain more useful information than one may generally hope for. The Gromov-Witten
index [6] are more obvious ones.
A trick of convenience involved in the localization computations, which lifts at
directions as much as possible, is to introduce chemical potentials and other susy-preserving
symmetries, which generically simplify the vacuum structures to those of isolated ones.
Note that not all theories admit such computations. When the theory must involve
superpotentials, for example, the number of the available avor symmetries get reduced. Also,
given a reduced
avor symmetry that allows some superpotential, the computation will
tend to compute the partition function for theories with generic superpotential consistent
with the avor charge assignment. The usual mantra that the localization is insensitive to
the details of the superpotential must be taken with such genericity presumed. In this note,
we will be considering theories where all matter elds acquire real masses, independent of
one another, which means that superpotential is turned o by imposing global symmetries.
When the number of matter multiplets and accompanying
avor symmetry are su
ciently large, quantum vacua then tend to be isolated [7]. One type, which we refer to as
the Higgs vacua, is such that chiral elds are turned on to cancel FI constants with the
Coulombic vev's pinned at some of the real masses. Another type, more characteristic of
condition entirely along the Coulomb branch with all chiral elds turned o . Although the
total number of vacua is invariant under continuous deformations of the theory, this split
of vacua between the Higgs type and the topological type is not robust, and in particular
a ected by signs of 3d FI constants. One interesting class of theories is where one can tune
the 3d FI constant so that all vacua are of Higgs variety. In such theories, the exact
partition functions on S1 bred over S2 are known to admit the so-called factorization [8{17]
where the partiti (...truncated)