81 research outputs found
3d analogs of Argyres-Douglas theories and knot homologies
We study singularities of algebraic curves associated with 3d N=2 theories
that have at least one global flavor symmetry. Of particular interest is a
class of theories T_K labeled by knots, whose partition functions package
Poincare polynomials of the S^r-colored HOMFLY homologies. We derive the
defining equation, called the super-A-polynomial, for algebraic curves
associated with many new examples of 3d N=2 theories T_K and study its
singularity structure. In particular, we catalog general types of singularities
that presumably exist for all knots and propose their physical interpretation.
A computation of super-A-polynomials is based on a derivation of corresponding
superpolynomials, which is interesting in its own right and relies solely on a
structure of differentials in S^r-colored HOMFLY homologies.Comment: 40 pages, 14 figure
Reconstructing GKZ via topological recursion
In this article, a novel description of the hypergeometric differential
equation found from Gel'fand-Kapranov-Zelevinsky's system (referred to GKZ
equation) for Givental's -function in the Gromov-Witten theory will be
proposed. The GKZ equation involves a parameter , and we will
reconstruct it as the WKB expansion from the classical limit via
the topological recursion. In this analysis, the spectral curve (referred to
GKZ curve) plays a central role, and it can be defined as the critical point
set of the mirror Landau-Ginzburg potential. Our novel description is derived
via the duality relations of the string theories, and various physical
interpretations suggest that the GKZ equation is identified with the quantum
curve for the brane partition function in the cohomological limit. As an
application of our novel picture for the GKZ equation, we will discuss the
Stokes matrix for the equivariant model and the
wall-crossing formula for the total Stokes matrix will be examined. And as a
byproduct of this analysis we will study Dubrovin's conjecture for this
equivariant model.Comment: 66 pages, 13 figures, 6 tables; v2: new subsections added, minor
revisions, typos corrected; v3: minor revisions, typos correcte
Super-A Polynomial
We review a construction of a new class of algebraic curves, called super-A-polynomials, and their quantum generalizations. The super-A-polynomial is a two-parameter deformation of the A-polynomial known from knot theory or Chern-Simons theory with SL(2,C) gauge group. The two parameters of the super-A-polynomial encode, respectively, the t-deformation which leads to the "refined A-polynomial", and the Q-deformation which leads to the augmentation polynomial of knot contact homology. For a given knot, the super-A-polynomial encodes the asymptotics of the corresponding S^r-colored HOMFLY homology for large r, while the quantum super-A-polynomial provides recursion relations for such homology theories for each r. The super-A-polynomial also admits a simple physical interpretation as the defining equation for the space of SUSY vacua in a circle compactification of the effective 3d N=2 theory associated to a given knot (complement). We discuss properties of super-A-polynomials and illustrate them in many examples
The boundary length and point spectrum enumeration of partial chord diagrams using cut and join recursion
We introduce the boundary length and point spectrum, as a joint
generalization of the boundary length spectrum and boundary point spectrum in
arXiv:1307.0967. We establish by cut-and-join methods that the number of
partial chord diagrams filtered by the boundary length and point spectrum
satisfies a recursion relation, which combined with an initial condition
determines these numbers uniquely. This recursion relation is equivalent to a
second order, non-linear, algebraic partial differential equation for the
generating function of the numbers of partial chord diagrams filtered by the
boundary length and point spectrum.Comment: 16 pages, 6 figure
Partial chord diagrams and matrix models
In this article, the enumeration of partial chord diagrams is discussed via
matrix model techniques. In addition to the basic data such as the number of
backbones and chords, we also consider the Euler characteristic, the backbone
spectrum, the boundary point spectrum, and the boundary length spectrum.
Furthermore, we consider the boundary length and point spectrum that unifies
the last two types of spectra. We introduce matrix models that encode
generating functions of partial chord diagrams filtered by each of these
spectra. Using these matrix models, we derive partial differential equations -
obtained independently by cut-and-join arguments in an earlier work - for the
corresponding generating functions.Comment: 42 pages, 14 figure
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