Supersymmetric vortex defects in two dimensions
JHE
Supersymmetric vortex defects in two dimensions
Kazuo Hosomichi 0 1 4
Sungjay Lee 0 1 2
Takuya Okuda 0 1 3
0 85 Hoegi-ro , Dongdaemun-gu, Seoul 02455 , South Korea
1 Meguro-ku, Tokyo 153-8902 , Japan
2 Korea Institute for Advanced Study
3 University of Tokyo , Komaba
4 Department of Applied Physics, National Defense Academy
We study codimension-two BPS defects in 2d N gauge theories, focusing especially on those characterized by vortex-like singularities in the dynamical or non-dynamical gauge eld. We classify possible SUSY-preserving boundary conditions on charged matter elds around the vortex defects, and derive a formula for defect correlators on the squashed sphere. We also prove an equivalence relation between vortex defects and 0d-2d coupled systems. Our defect correlators are shown to be consistent with the mirror symmetry duality between Abelian gauged linear sigma models and Landau-Ginzburg models, as well as that between the minimal model and its orbifold. We also study the vortex defects inserted at conical singularities.
Conformal Field Theory; Field Theories in Lower Dimensions; Supersymmet-
-
HJEP01(28)3
1 Introduction and summary
2 SUSY gauge theories and vortex defects in 2d
Construction of SUSY theories
BPS vortex defects
Smeared vortex defect con gurations
2.4 0d-2d coupled systems
3 Vortex defect correlators on the squashed sphere
Computation of the partition function (review)
3.2 Introduction of vortex defects
Comparison with smearing
4 Relations between defect operators
4.1
Relations between wave functions
4.2 0d multiplets from localized modes
4.3 The relations
5 Defect correlators in Abelian GLSMs
6
Mirror symmetry for vortex defects
Hori-Vafa mirror symmetry
N = 2 minimal model and its orbifold
7 Vortex defects at conical singularities
Orbifold projection
Normal and ipped boundary conditions for conical singularities
Resolution of the conical singularities and the localized modes
2.1
2.2
2.3
3.1
3.3
6.1
6.2
7.1
7.2
7.3
8 Vortex defects in non-Abelian theories
9 Discussion
A The
! 0 limit of the bulk modes
B A review of mirror symmetry
C Interpretation of ( )
{ 1 {
Introduction and summary
In gauge theories in di erent dimensions, one can introduce defects of codimension 2 in a
number of ways. One standard way is the following. Let (r; ') be the polar coordinates
for the two dimensions transverse to the defect, with r = 0 the position of the defect. One
requires the gauge eld A to behave near the defect like
A
d';
(1.1)
where is a Lie algebra valued constant called vorticity. According to their dimensionality,
they are called surface operators in 4d [1], vortex loops in 3d and vortex defects in 2d.
equally important roles as order parameters in gauge theories [4, 5].
Another standard way to de ne a defect is to introduce dynamical variables or elds
localized on it and let them interact with the elds in the bulk. For 4d supersymmetric
gauge theories, another de nition has been proposed based on embedding the theory into
a larger theory and \Higgsing" by a position-dependent scalar vev [6]. Moreover, there are
examples where the defects based on di erent de nitions are believed to be equivalent [7, 8].
Studying the relations among di erent de nitions of defects will therefore be a key for their
better understanding.
BPS defects in supersymmetric theories are especially interesting, since their protected
sector can often be determined precisely. In particular, for systems with Lagrangian
descriptions, SUSY localization allows us to evaluate the protected observables by explicit
path integration. For example, exact path integrals have been worked out for the coupled
1d-3d systems on S3 in [9], and for the (0d-)2d-4d systems on S4 in [10{12], see also [13, 14].
On the other hand, we have not yet reached a fully satisfactory understanding for the
defects de ned by the boundary condition (1.1), though exact vortex loop observables in
Abelian 3d gauge theories on S3 and S2
S1 were worked out in [15, 16], and a proposal
for surface operator vev on S4 were given in [17]. A major di culty here is in
nding the
right de nition of the path integral measure for the gauge elds as well as charged matter
elds under the singular boundary condition.
The aim of this paper is to propose a fully precise de nition of the defects of the
type (1.1) in the path integral formalism. As the simplest setting to study this problem,
we take 2d N = (2; 2) supersymmetric gauge theories of vector and chiral multiplets, and
focus on vortex defects preserving half of the supersymmetry. Throughout the paper we
work in Euclidean signature. In the standard N = (2; 2) terminology, the defects are either
in the twisted chiral or anti-twisted chiral rings. Based on our de nition we study various
aspects of 2d vortex defects, which include their relation to other type of defects or the
transformation property under mirror symmetry.
Boundary conditions on charged matter.
De ning path integration und (...truncated)