84 research outputs found
Wall Crossing As Seen By Matrix Models
The number of BPS bound states of D-branes on a Calabi-Yau manifold depends
on two sets of data, the BPS charges and the stability conditions. For D0 and
D2-branes bound to a single D6-brane wrapping a Calabi-Yau 3-fold X, both are
naturally related to the Kahler moduli space M(X). We construct unitary
one-matrix models which count such BPS states for a class of toric Calabi-Yau
manifolds at infinite 't Hooft coupling. The matrix model for the BPS counting
on X turns out to give the topological string partition function for another
Calabi-Yau manifold Y, whose Kahler moduli space M(Y) contains two copies of
M(X), one related to the BPS charges and another to the stability conditions.
The two sets of data are unified in M(Y). The matrix models have a number of
other interesting features. They compute spectral curves and mirror maps
relevant to the remodeling conjecture. For finite 't Hooft coupling they give
rise to yet more general geometry \widetilde{Y} containing Y.Comment: 44 pages, 9 figures, published versio
Wall-crossing, free fermions and crystal melting
We describe wall-crossing for local, toric Calabi-Yau manifolds without
compact four-cycles, in terms of free fermions, vertex operators, and crystal
melting. Firstly, to each such manifold we associate two states in the free
fermion Hilbert space. The overlap of these states reproduces the BPS partition
function corresponding to the non-commutative Donaldson-Thomas invariants,
given by the modulus square of the topological string partition function.
Secondly, we introduce the wall-crossing operators which represent crossing the
walls of marginal stability associated to changes of the B-field through each
two-cycle in the manifold. BPS partition functions in non-trivial chambers are
given by the expectation values of these operators. Thirdly, we discuss crystal
interpretation of such correlators for this whole class of manifolds. We
describe evolution of these crystals upon a change of the moduli, and find
crystal interpretation of the flop transition and the DT/PT transition. The
crystals which we find generalize and unify various other Calabi-Yau crystal
models which appeared in literature in recent years.Comment: 61 pages, 14 figures, published versio
Topological recursion for chord diagrams, RNA complexes, and cells in moduli spaces
We introduce and study the Hermitian matrix model with potential
V(x)=x^2/2-stx/(1-tx), which enumerates the number of linear chord diagrams of
fixed genus with specified numbers of backbones generated by s and chords
generated by t. For the one-cut solution, the partition function, correlators
and free energies are convergent for small t and all s as a perturbation of the
Gaussian potential, which arises for st=0. This perturbation is computed using
the formalism of the topological recursion. The corresponding enumeration of
chord diagrams gives at once the number of RNA complexes of a given topology as
well as the number of cells in Riemann's moduli spaces for bordered surfaces.
The free energies are computed here in principle for all genera and explicitly
for genera less than four.Comment: 34 pages, 2 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
Quantum Curves and D-Modules
In this article we continue our study of chiral fermions on a quantum curve.
This system is embedded in string theory as an I-brane configuration, which
consists of D4 and D6-branes intersecting along a holomorphic curve in a
complex surface, together with a B-field. Mathematically, it is described by a
holonomic D-module. Here we focus on spectral curves, which play a prominent
role in the theory of (quantum) integrable hierarchies. We show how to
associate a quantum state to the I-brane system, and subsequently how to
compute quantum invariants. As a first example, this yields an insightful
formulation of (double scaled as well as general Hermitian) matrix models.
Secondly, we formulate c=1 string theory in this language. Finally, our
formalism elegantly reconstructs the complete dual Nekrasov-Okounkov partition
function from a quantum Seiberg-Witten curve.Comment: 63 pages, 9 figures; revised published versio
Crystal Melting and Wall Crossing Phenomena
This paper summarizes recent developments in the theory of
Bogomol'nyi-Prasad-Sommerfield (BPS) state counting and the wall crossing
phenomena, emphasizing in particular the role of the statistical mechanical
model of crystal melting. This paper is divided into two parts, which are
closely related to each other. In the first part, we discuss the statistical
mechanical model of crystal melting counting BPS states. Each of the BPS state
contributing to the BPS index is in one-to-one correspondence with a
configuration of a molten crystal, and the statistical partition function of
the melting crystal gives the BPS partition function. We also show that smooth
geometry of the Calabi-Yau manifold emerges in the thermodynamic limit of the
crystal. This suggests a remarkable interpretation that an atom in the crystal
is a discretization of the classical geometry, giving an important clue as to
the geometry at the Planck scale.In the second part we discuss the wall
crossing phenomena. Wall crossing phenomena states that the BPS index depends
on the value of the moduli of the Calabi-Yau manifold, and jumps along real
codimension one subspaces in the moduli space. We show that by using type
IIA/M-theory duality, we can provide a simple and an intuitive derivation of
the wall crossing phenomena, furthermore clarifying the connection with the
topological string theory. This derivation is consistent with another
derivation from the wall crossing formula, motivated by multi-centered BPS
extremal black holes. We also explain the representation of the wall crossing
phenomena in terms of crystal melting, and the generalization of the counting
problem and the wall crossing to the open BPS invariants.Comment: PhD thesis, 129 pages, 39 figures, comments welcome; v2: typos
corrected, references added, now in IJMPA format; v3: figures correcte
3d-3d Correspondence Revisited
In fivebrane compactifications on 3-manifolds, we point out the importance of
all flat connections in the proper definition of the effective 3d N=2 theory.
The Lagrangians of some theories with the desired properties can be constructed
with the help of homological knot invariants that categorify colored Jones
polynomials. Higgsing the full 3d theories constructed this way recovers
theories found previously by Dimofte-Gaiotto-Gukov. We also consider the
cutting and gluing of 3-manifolds along smooth boundaries and the role played
by all flat connections in this operation.Comment: 43 pages + 1 appendix, 6 figures Version 2: new appendix on flat
connections in the 3d-3d correspondenc
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