20 research outputs found
Local Mirror Symmetry for One-Legged Topological Vertex
We prove the Bouchard-Mari\~no Conjecture for the framed one-legged
topological vertex by deriving the Eynard-Orantin type recursion relations from
the cut-and-join equation satisfied by the relevant triple Hodge integrals.
This establishes a version of local mirror symmetry for the local
geometry with one -brane.Comment: First revised versio
On ELSV-type formulae, Hurwitz numbers and topological recursion
We present several recent developments on ELSV-type formulae and topological
recursion concerning Chiodo classes and several kind of Hurwitz numbers. The
main results appeared in D. Lewanski, A. Popolitov, S. Shadrin, D. Zvonkine,
"Chiodo formulas for the r-th roots and topological recursion", Lett. Math.
Phys. (2016).Comment: 18 pages, comments are welcom
Chiodo formulas for the r-th roots and topological recursion
We analyze Chiodo's formulas for the Chern classes related to the r-th roots
of the suitably twisted integer powers of the canonical class on the moduli
space of curves. The intersection numbers of these classes with psi-classes are
reproduced via the Chekhov-Eynard-Orantin topological recursion. As an
application, we prove that the Johnson-Pandharipande-Tseng formula for the
orbifold Hurwitz numbers is equivalent to the topological recursion for the
orbifold Hurwitz numbers. In particular, this gives a new proof of the
topological recursion for the orbifold Hurwitz numbers.Comment: 19 pages, some correction
A matrix model for simple Hurwitz numbers, and topological recursion
We introduce a new matrix model representation for the generating function of
simple Hurwitz numbers. We calculate the spectral curve of the model and the
associated symplectic invariants developed in [Eynard-Orantin]. As an
application, we prove the conjecture proposed by Bouchard and Marino, relating
Hurwitz numbers to the spectral invariants of the Lambert curve exp(x)=y
exp(-y).Comment: 24 pages, 3 figure
Open String Invariants and Mirror Curve of the Resolved Conifold
For the resolved conifold with one outer D-brane in arbitrary framing, we
present some results for the open string partition functions obtained by some
operator manipulations. We prove some conjectures by Aganagic-Vafa and
Aganagic-Klemm-Vafa that relates such invariants to the mirror curve of the
resolved conifold. This establishes local mirror symmetry for the resolved
confolds for holomorphic disc invariants. We also verify an integrality
conjecture of such invariants by Ooguri-Vafa in this case and present closed
formulas for some Ooguri-Vafa type invariants in genus 0 and arbitrary genera
Holomorphic Anomaly in Gauge Theories and Matrix Models
We use the holomorphic anomaly equation to solve the gravitational
corrections to Seiberg-Witten theory and a two-cut matrix model, which is
related by the Dijkgraaf-Vafa conjecture to the topological B-model on a local
Calabi-Yau manifold. In both cases we construct propagators that give a
recursive solution in the genus modulo a holomorphic ambiguity. In the case of
Seiberg-Witten theory the gravitational corrections can be expressed in closed
form as quasimodular functions of Gamma(2). In the matrix model we fix the
holomorphic ambiguity up to genus two. The latter result establishes the
Dijkgraaf-Vafa conjecture at that genus and yields a new method for solving the
matrix model at fixed genus in closed form in terms of generalized
hypergeometric functions.Comment: 34 pages, 2 eps figures, expansion at the monopole point corrected
and interpreted, and references adde
A matrix model for the topological string II: The spectral curve and mirror geometry
In a previous paper, we presented a matrix model reproducing the topological
string partition function on an arbitrary given toric Calabi-Yau manifold.
Here, we study the spectral curve of our matrix model and thus derive, upon
imposing certain minimality assumptions on the spectral curve, the large volume
limit of the BKMP "remodeling the B-model" conjecture, the claim that
Gromov-Witten invariants of any toric Calabi-Yau 3-fold coincide with the
spectral invariants of its mirror curve.Comment: 1+37 page