Changing the preferred direction of the refined topological vertex

Abstract

We consider the issue of the slice invariance of refined topological string amplitudes, which means that they are independent of the choice of the preferred direction of the refined topological vertex. We work out two examples. The first example is a geometric engineering of five-dimensional U(1) gauge theory with a matter. The slice invariance follows from a highly non-trivial combinatorial identity which equates two known ways of computing the chi_y genus of the Hilbert scheme of points on C^2. The second example is concerned with the proposal that the superpolynomials of the colored Hopf link are obtained from a refinement of topological open string amplitudes. We provide a closed formula for the superpolynomial, which confirms the slice invariance when the Hopf link is colored with totally anti-symmetric representations. However, we observe a breakdown of the slice invariance for other representations.

35 pages, 3 figures; (v3) a few improvements, references updated

References in corpus (11)

• The Refined Topological Vertex
• Refined BPS state counting from Nekrasov's formula and Macdonald functions
• Refined Topological Vertex and Instanton Counting
• Link Homologies and the Refined Topological Vertex
• Refined Topological Vertex, Cylindric Partitions and the U(1) Adjoint Theory
• Matrix Factorizations and Kauffman Homology
• Quantum (sl_n, \land V_n) link invariant and matrix factorizations
• Macdonald operators and homological invariants of the colored Hopf link
• Flop Invariance of Refined Topological Vertex and Link Homologies
• Topological String Partition Functions as Equivariant Indices
• Instantons and the 5D U(1) gauge theory with extra adjoint

Cited by in corpus (32)

• Superpolynomials for toric knots from evolution induced by cut-and-join operators
• Topological strings and 5d T_N partition functions
• Explicit examples of DIM constraints for network matrix models
• Toric Calabi-Yau threefolds as quantum integrable systems. R-matrix and RTT relations
• Exact partition functions of Higgsed 5d $T_N$ theories
• Quiver Matrix Model and Topological Partition Function in Six Dimensions
• (q,t)-KZ equation for Ding-Iohara-Miki algebra
• Macdonald operators and homological invariants of the colored Hopf link
• New quantum toroidal algebras from 5D $\mathcal{N}=1$ instantons on orbifolds
• The MacMahon R-matrix
• Fiber-base duality from the algebraic perspective
• On Factorization of Generalized Macdonald Polynomials
• Challenges of beta-deformation
• K-theoretic Donaldson-Thomas theory and the Hilbert scheme of points on a surface
• Kerov functions for composite representations and Macdonald ideal
• A non-torus link from topological vertex
• Elliptic lift of the Shiraishi function as a non-stationary double-elliptic function
• Can tangle calculus be applicable to hyperpolynomials?
• Trinion Conformal Blocks from Topological strings
• Check-Operators and Quantum Spectral Curves
• The Condensate from Torus Knots
• Factorization of the 3d superconformal index
• Geometric transition in Non-perturbative Topological string
• Non-Stationary Ruijsenaars Functions for $\kappa =t^{-1/N}$ and Intertwining Operators of Ding-Iohara-Miki Algebra
• ADHM Wilson line defect indices
• Quasi-Hopf twist and elliptic Nekrasov factor
• Hopf superpolynomial from topological vertices
• Matrix model and dimensions at hypercube vertices
• Quantum $W_{1+\infty }$ subalgebras of BCD type and symmetric polynomials
• A Representation-Theoretic Approach to $qq$-Characters
• Intertwining operator and integrable hierarchies from topological strings
• Quiver matrix model of ADHM type and BPS state counting in diverse dimensions