47 research outputs found
Vortex Counting and Lagrangian 3-manifolds
To every 3-manifold M one can associate a two-dimensional N=(2,2)
supersymmetric field theory by compactifying five-dimensional N=2
super-Yang-Mills theory on M. This system naturally appears in the study of
half-BPS surface operators in four-dimensional N=2 gauge theories on one hand,
and in the geometric approach to knot homologies, on the other. We study the
relation between vortex counting in such two-dimensional N=(2,2) supersymmetric
field theories and the refined BPS invariants of the dual geometries. In
certain cases, this counting can be also mapped to the computation of
degenerate conformal blocks in two-dimensional CFT's. Degenerate limits of
vertex operators in CFT receive a simple interpretation via geometric
transitions in BPS counting.Comment: 70 pages, 29 figure
Deformations of symplectic cohomology and exact Lagrangians in ALE spaces
We prove that the only exact Lagrangian submanifolds in an ALE space are
spheres. ALE spaces are the simply connected hyperkahler manifolds which at
infinity look like C^2/G for any finite subgroup G of SL(2,C). They can be
realized as the plumbing of copies of the cotangent bundle of a 2-sphere
according to ADE Dynkin diagrams. The proof relies on symplectic cohomology.Comment: 35 pages, 3 figures, minor changes and corrected typo
Holomorphic Anomaly in Gauge Theory on ALE space
We consider four-dimensional Omega-deformed N=2 supersymmetric SU(2) gauge
theory on A1 space and its lift to five dimensions. We find that the partition
functions can be reproduced via special geometry and the holomorphic anomaly
equation. Schwinger type integral expressions for the boundary conditions at
the monopole/dyon point in moduli space are inferred. The interpretation of the
five-dimensional partition function as the partition function of a refined
topological string on A1x(local P1xP1) is suggested.Comment: 28 page
HyperK\"ahler quotients and N=4 gauge theories in D=2
We consider certain N=4 supersymmetric gauge theories in D=2 coupled to
quaternionic matter multiplets in a minimal way. These theories admit as
effective theories sigma-models on non-trivial HyperK\"ahler manifolds obtained
as HyperK\"ahler quotients. The example of ALE manifolds is discussed. (Based
on a talk given by P. Fr\'e at the F. Gursey Memorial Conference, Istanbul,
June 1994).Comment: 22 pages, Latex, no figure
Monopoles and clusters
We define and study certain hyperkaehler manifolds which capture the
asymptotic behaviour of the SU(2)-monopole metric in regions where monopoles
break down into monopoles of lower charges. The rate at which these new metrics
approximate the monopole metric is exponential, as for the Gibbons-Manton
metric.Comment: v2.: relation to calorons mentioned; added explanation
Argyres-Seiberg duality and the Higgs branch
We demonstrate the agreement between the Higgs branches of two N=2 theories
proposed by Argyres and Seiberg to be S-dual, namely the SU(3) gauge theory
with six quarks, and the SU(2) gauge theory with one pair of quarks coupled to
the superconformal theory with E_6 flavor symmetry. In mathematical terms, we
demonstrate the equivalence between a hyperkaehler quotient of a linear space
and another hyperkaehler quotient involving the minimal nilpotent orbit of E_6,
modulo the identification of the twistor lines.Comment: 27 pages; v2: published versio
Topological Chern-Simons Sigma Model
We consider topological twisting of recently constructed Chern-Simons-matter
theories in three dimensions with N=4 or higher supersymmetry. We enumerate
physically inequivalent twistings for each N, and find two different twistings
for N=4, one for N=5,6, and four for N=8. We construct the two types of N=4
topological theories, which we call A/B-models, in full detail. The A-model has
been recently studied by Kapustin and Saulina. The B-model is new and it
consists solely of a Chern-Simons term of a complex gauge field up to
BRST-exact terms. We also compare the new theories with topological Yang-Mills
theories and find some interesting connections. In particular, the A-model
seems to offer a new perspective on Casson invariant and its relation to
Rozansky-Witten theory.Comment: 31 pages, no figure; v2. references adde
Hyperkahler Metrics from Periodic Monopoles
Relative moduli spaces of periodic monopoles provide novel examples of
Asymptotically Locally Flat hyperkahler manifolds. By considering the
interactions between well-separated periodic monopoles, we infer the asymptotic
behavior of their metrics. When the monopole moduli space is four-dimensional,
this construction yields interesting examples of metrics with self-dual
curvature (gravitational instantons). We discuss their topology and complex
geometry. An alternative construction of these gravitational instantons using
moduli spaces of Hitchin equations is also described.Comment: 23 pages, latex. v2: an erroneous formula is corrected, and its
derivation is given. v3 (published version): references adde
Branes, Instantons, And Taub-NUT Spaces
ALE and Taub-NUT (or ALF) hyper-Kahler four-manifolds can be naturally
constructed as hyper-Kahler quotients. In the ALE case, this construction has
long been understood in terms of D-branes; here we give a D-brane derivation in
the Taub-NUT case. Likewise, instantons on ALE spaces and on Taub-NUT spaces
have ADHM-like constructions related to hyper-Kahler quotients. Here we refine
the analysis in the Taub-NUT case by making use of a D-brane probe, and give an
application to M-theory.Comment: 63 p
Surface Operators in Abelian Gauge Theory
We consider arbitrary embeddings of surface operators in a pure,
non-supersymmetric abelian gauge theory on spin (non-spin) four-manifolds. For
any surface operator with a priori simultaneously non-vanishing parameters, we
explicitly show that the parameters transform naturally under an SL(2, Z) (or a
congruence subgroup thereof) duality of the theory. However, for
non-trivially-embedded surface operators, exact S-duality holds only if the
quantum parameter effectively vanishes, while the overall SL(2, Z) (or a
congruence subgroup thereof) duality holds up to a c-number at most,
regardless. Via the formalism of duality walls, we furnish an alternative
derivation of the transformation of parameters - found also to be consistent
with a switch from Wilson to 't Hooft loop operators under S-duality. With any
background embedding of surface operators, the partition function and the
correlation functions of non-singular, gauge-invariant local operators on any
curved four-manifold, are found to transform like modular forms under the
respective duality groups.Comment: 30 pages. Minor refinemen