96 research outputs found
Combinatorial structure of colored HOMFLY-PT polynomials for torus knots
We rewrite the (extended) Ooguri-Vafa partition function for colored
HOMFLY-PT polynomials for torus knots in terms of the free-fermion
(semi-infinite wedge) formalism, making it very similar to the generating
function for double Hurwitz numbers. This allows us to conjecture the
combinatorial meaning of full expansion of the correlation differentials
obtained via the topological recursion on the Brini-Eynard-Mari\~no spectral
curve for the colored HOMFLY-PT polynomials of torus knots.
This correspondence suggests a structural combinatorial result for the
extended Ooguri-Vafa partition function. Namely, its coefficients should have a
quasi-polynomial behavior, where non-polynomial factors are given by the Jacobi
polynomials. We prove this quasi-polynomiality in a purely combinatorial way.
In addition to that, we show that the (0,1)- and (0,2)-functions on the
corresponding spectral curve are in agreement with the extension of the colored
HOMFLY-PT polynomials data.Comment: 40 pages; section 10 addressing the quantum curve was added, as well
as some remarks regarding Meixner polynomials thanks to T.Koornwinde
Topological open string amplitudes on local toric del Pezzo surfaces via remodeling the B-model
We study topological strings on local toric del Pezzo surfaces by a method
called remodeling the B-model which was recently proposed by Bouchard, Klemm,
Marino and Pasquetti. For a large class of local toric del Pezzo surfaces we
prove a functional formula of the Bergman kernel which is the basic constituent
of the topological string amplitudes by the topological recursion relation of
Eynard and Orantin. Because this formula is written as a functional of the
period, we can obtain the topological string amplitudes at any point of the
moduli space by a simple change of variables of the Picard-Fuchs equations for
the period. By this formula and mirror symmetry we compute the A-model
amplitudes on K_{F_2}, and predict the open orbifold Gromov-Witten invariants
of C^3/Z_4.Comment: 31 pages, 4 figures. v2: an example in Subsection 4.3 added, a
footnote in Subsection 4.4 added, minor errors in Appendix E corrected, a
reference added. v3: typos correcte
Torus Knots and Mirror Symmetry
We propose a spectral curve describing torus knots and links in the B-model. In particular, the application of the topological recursion to this curve generates all their colored HOMFLY invariants. The curve is obtained by exploiting the full symmetry of the spectral curve of the resolved conifold, and should be regarded as the mirror of the topological D-brane associated with torus knots in the large N Gopakumar-Vafa duality. Moreover, we derive the curve as the large N limit of the matrix model computing torus knot invariant
The SYZ mirror symmetry and the BKMP remodeling conjecture
The Remodeling Conjecture proposed by Bouchard-Klemm-Mari\~{n}o-Pasquetti
(BKMP) relates the A-model open and closed topological string amplitudes (open
and closed Gromov-Witten invariants) of a symplectic toric Calabi-Yau 3-fold to
Eynard-Orantin invariants of its mirror curve. The Remodeling Conjecture can be
viewed as a version of all genus open-closed mirror symmetry. The SYZ
conjecture explains mirror symmetry as -duality. After a brief review on SYZ
mirror symmetry and mirrors of symplectic toric Calabi-Yau 3-orbifolds, we give
a non-technical exposition of our results on the Remodeling Conjecture for
symplectic toric Calabi-Yau 3-orbifolds. In the end, we apply SYZ mirror
symmetry to obtain the descendent version of the all genus mirror symmetry for
toric Calabi-Yau 3-orbifolds.Comment: 18 pages. Exposition of arXiv:1604.0712
Open orbifold Gromov-Witten invariants of [C^3/Z_n]: localization and mirror symmetry
We develop a mathematical framework for the computation of open orbifold
Gromov-Witten invariants of [C^3/Z_n], and provide extensive checks with
predictions from open string mirror symmetry. To this aim we set up a
computation of open string invariants in the spirit of Katz-Liu, defining them
by localization. The orbifold is viewed as an open chart of a global quotient
of the resolved conifold, and the Lagrangian as the fixed locus of an
appropriate anti-holomorphic involution. We consider two main applications of
the formalism. After warming up with the simpler example of [C^3/Z_3], where we
verify physical predictions of Bouchard, Klemm, Marino and Pasquetti, the main
object of our study is the richer case of [C^3/Z_4], where two different
choices are allowed for the Lagrangian. For one choice, we make numerical
checks to confirm the B-model predictions; for the other, we prove a mirror
theorem for orbifold disc invariants, match a large number of annulus
invariants, and give mirror symmetry predictions for open string invariants of
genus \leq 2.Comment: 44 pages + appendices; v2: exposition improved, misprints corrected,
version to appear on Selecta Mathematica; v3: last minute mistake found and
fixed for the symmetric brane setup of [C^3/Z_4]; in pres
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