30 research outputs found
Spectral networks and Fenchel-Nielsen coordinates
We explain that spectral networks are a unifying framework that incorporates
both shear (Fock-Goncharov) and length-twist (Fenchel-Nielsen) coordinate
systems on moduli spaces of flat SL(2,C) connections, in the following sense.
Given a spectral network W on a punctured Riemann surface C, we explain the
process of "abelianization" which relates flat SL(2)-connections (with an
additional structure called "W-framing") to flat C*-connections on a covering.
For any W, abelianization gives a construction of a local Darboux coordinate
system on the moduli space of W-framed flat connections. There are two special
types of spectral network, combinatorially dual to ideal triangulations and
pants decompositions; these two types of network lead to Fock-Goncharov and
Fenchel-Nielsen coordinates respectively.Comment: 63 pages; v2: expository improvements, journal versio
Topological Strings and Quantum Curves
This thesis presents several new insights on the interface between
mathematics and theoretical physics, with a central role for fermions on
Riemann surfaces. First of all, the duality between Vafa-Witten theory and WZW
models is embedded into string theory. Secondly, this model is generalized to a
web of dualities connecting topological string theory and N=2 supersymmetric
gauge theories to a configuration of D-branes that intersect over a Riemann
surface. This description yields a new perspective on topological string theory
in terms of a KP integrable system based on a quantum curve. Thirdly, this
thesis describes a geometric analysis of wall-crossing in N=4 string theory.
And lastly, it offers a novel approach to construct metastable vacua in type
IIB string theory.Comment: PhD thesis, July 2009, 308 pages, 65 figure
From SO/Sp instantons to W-algebra blocks
We study instanton partition functions for N=2 superconformal Sp(1) and SO(4)
gauge theories. We find that they agree with the corresponding U(2) instanton
partitions functions only after a non-trivial mapping of the microscopic gauge
couplings, since the instanton counting involves different renormalization
schemes. Geometrically, this mapping relates the Gaiotto curves of the
different realizations as double coverings. We then formulate an AGT-type
correspondence between Sp(1)/SO(4) instanton partition functions and chiral
blocks with an underlying W(2,2)-algebra symmetry. This form of the
correspondence eliminates the need to divide out extra U(1) factors. Finally,
to check this correspondence for linear quivers, we compute expressions for the
Sp(1)-SO(4) half-bifundamental.Comment: 83 pages, 29 figures; minor change
Nonsupersymmetric Flux Vacua and Perturbed N=2 Systems
We geometrically engineer N=2 theories perturbed by a superpotential by
adding 3-form flux with support at infinity to local Calabi-Yau geometries in
type IIB. This allows us to apply the formalism of Ooguri, Ookouchi, and Park
[arXiv:0704.3613] to demonstrate that, by tuning the flux at infinity, we can
stabilize the dynamical complex structure moduli in a metastable,
supersymmetry-breaking configuration. Moreover, we argue that this setup can
arise naturally as a limit of a larger Calabi-Yau which separates into two
weakly interacting regions; the flux in one region leaks into the other, where
it appears to be supported at infinity and induces the desired superpotential.
In our endeavor to confirm this picture in cases with many 3-cycles, we also
compute the CIV-DV prepotential for arbitrary number of cuts up to fifth order
in the glueball fields.Comment: 70 pages (47 pages + 4 appendices), 10 figure
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
Towards a 4d/2d correspondence for Sicilian quivers
We study the 4d/2d AGT correspondence between four-dimensional instanton
counting and two-dimensional conformal blocks for generalized SU(2) quiver
gauge theories coming from punctured Gaiotto curves of arbitrary genus. We
propose a conformal block description that corresponds to the elementary SU(2)
trifundamental half-hypermultiplet, and check it against Sp(1)-SO(4) instanton
counting.Comment: 39 pages, 11 figure
Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions
We show that various holomorphic quantities in supersymmetric gauge theories
can be conveniently computed by configurations of D4-branes and D6-branes.
These D-branes intersect along a Riemann surface that is described by a
holomorphic curve in a complex surface. The resulting I-brane carries
two-dimensional chiral fermions on its world-volume. This system can be mapped
directly to the topological string on a large class of non-compact Calabi-Yau
manifolds. Inclusion of the string coupling constant corresponds to turning on
a constant B-field on the complex surface, which makes this space
non-commutative. Including all string loop corrections the free fermion theory
is elegantly formulated in terms of holonomic D-modules that replace the
classical holomorphic curve in the quantum case.Comment: 67 pages, 6 figure