Topological Vertex Research Articles

MacMahon KZ equation for Ding-Iohara-Miki algebra

We derive a generalized Knizhnik-Zamolodchikov equation for the correlation function of the intertwiners of the vector and the MacMahon representations of Ding-Iohara-Miki algebra. These intertwiners are cousins of the refined topological vertex which is regarded as the intertwining operator of the Fock representation. The shift of the spectral parameter of the intertwiners is generated by the operator which is constructed from the universal R matrix. The solutions to the generalized KZ equation are factorized into the ratio of two point functions which are identified with generalizations of the Nekrasov factor for supersymmetric quiver gauge theories.

More on topological vertex formalism for 5-brane webs with O5-plane

We propose a concrete form of a vertex function, which we call O-vertex, for the intersection between an O5-plane and a 5-brane in the topological vertex formalism, as an extension of the work of [1]. Using the O-vertex it is possible to compute the Nekrasov partition functions of 5d theories realized on any 5-brane web diagrams with O5-planes. We apply our proposal to 5-brane webs with an O5-plane and compute the partition functions of pure SO(N) gauge theories and the pure G2 gauge theory. The obtained results agree with the results known in the literature. We also compute the partition function of the pure SU(3) gauge theory with the Chern-Simons level 9. At the end we rewrite the O-vertex in a form of a vertex operator.

Topological vertex for 6d SCFTs with \u21242-twist

We compute the partition function for 6d mathcal{N} = 1 SO(2N) gauge theories compactified on a circle with ℤ2 outer automorphism twist. We perform the computation based on 5-brane webs with two O5-planes using topological vertex with two O5-planes. As representative examples, we consider 6d SO(8) and SU(3) gauge theories with ℤ2 twist. We confirm that these partition functions obtained from the topological vertex with O5-planes indeed agree with the elliptic genus computations.

6D strings and exceptional instantons

We propose new ADHM-like methods to compute the Coulomb branch instanton partition functions of 5d and 6d supersymmetric gauge theories, with certain exceptional gauge groups or exceptional matters. We study $G_2$ theories with $n_{\bf 7}\leq 3$ matters in ${\bf 7}$, and $SO(7)$ theories with $n_{\bf 8}\leq 4$ matters in the spinor representation ${\bf 8}$. We also study the elliptic genera of self-dual instanton strings of 6d SCFTs with exceptional gauge groups or matters, including all non-Higgsable atomic SCFTs with rank $2$ or $3$ tensor branches. Some of them are tested with topological vertex calculus. We also explore a D-brane-based method to study instanton particles of 5d $SO(7)$ and $SO(8)$ gauge theories with matters in spinor representations, which further tests our ADHM-like proposals.

Generalized Macdonald Functions on Fock Tensor Spaces and Duality Formula for Changing Preferred Direction

An explicit formula is obtained for the generalized Macdonald functions on the $N$-fold Fock tensor spaces, calculating a certain matrix element of a composition of several screened vertex operators. As an application, we prove the factorization property of the arbitrary matrix elements of the multi-valent intertwining operator (or refined topological vertex operator) associated with the Ding--Iohara--Miki algebra (DIM algebra) with respect to the generalized Macdonald functions, which was conjectured by Awata, Feigin, Hoshino, Kanai, Yanagida and one of the authors. Our proof is based on the combinatorial and analytic properties of the asymptotic eigenfunctions of the ordinary Macdonald operator of $A$-type, and the Euler transformation formula for Kajihara and Noumi's multiple basic hypergeometric series. That factorization formula provides us with a reasonable algebraic description of the 5D (K-theoretic) Alday-Gaiotto-Tachikawa (AGT) correspondence, and the interpretation of the invariance under the preferred direction from the point of view of the $SL(2,\mathbb{Z})$ duality of the DIM algebra.

Refined Topological Branes

We study the open refined topological string amplitudes using the refined topological vertex. We determine the refinement of holonomies necessary to describe the boundary conditions of open amplitudes (which in particular satisfy the required integrality properties). We also derive the refined holonomies using the refined Chern–Simons theory.

Hopf superpolynomial from topological vertices

Link/knot invariants are series with integer coefficients, and it is a long-standing problem to get them positive and possessing cohomological interpretation. Constructing positive “superpolynomials” is not straightforward, especially for colored invariants. A simpler alternative is a multi-parametric generalization of the character expansion, which leads to colored “hyperpolynomials”. The third construction involves branes on resolved conifolds, which gives rise to still another family of invariants associated with composite representations. We revisit this triality issue in the simple case of the Hopf link and discover a previously overlooked way to produce positive colored superpolynomials from the DIM-governed four-point functions, thus paving a way to a new relation between super- and hyperpolynomials.

Quantum periods and spectra in dimer models and Calabi-Yau geometries

We study a class of quantum integrable systems derived from dimer graphs and also described by local toric Calabi-Yau geometries with higher genus mirror curves, generalizing some previous works on genus one mirror curves. We compute the spectra of the quantum systems both by standard perturbation method and by Bohr-Sommerfeld method with quantum periods as the phase volumes. In this way, we obtain some exact analytic results for the classical and quantum periods of the Calabi-Yau geometries. We also determine the differential operators of the quantum periods and compute the topological string free energy in Nekrasov-Shatashvili (NS) limit. The results agree with calculations from other methods such as the topological vertex.

Trinion conformal blocks from topological strings

In this paper we investigate the relation between conformal blocks of Liouville CFT and the topological string partition functions of the rank one trinion theory T2. The partition functions exhibit jumps when passing from one chamber in the parameter space to another. Such jumps can be attributed to a change of the integration contour in the free field representation of Liouville conformal blocks. We compare the partition functions of the T2 theories representing trifundamental half hypermultiplets in N = 2, d = 4 field theories to the partition functions associated to bifundamental hypermultiplets. We find that both are related to the same Liouville conformal blocks up to inessential factors. In order to establish this picture we combine and compare results obtained using topological vertex techniques, matrix models and topological recursion. We furthermore check that the partition functions obtained by gluing two T2 vertices can be represented in terms of a four point Liouville conformal block. Our results indicate that the T2 vertex offers a useful starting point for developing an analog of the instanton calculus for SUSY gauge theories with trifundamental hypermultiplets.

New quantum toroidal algebras from 5D mathcal{N} = 1 instantons on orbifolds

Quantum toroidal algebras are obtained from quantum affine algebras by a further affinization, and, like the latter, can be used to construct integrable systems. These algebras also describe the symmetries of instanton partition functions for 5D mathcal{N} = 1 supersymmetric quiver gauge theories. We consider here the gauge theories defined on an orbifold S1× ℂ2/ℤp where the action of ℤp is determined by two integer parameters (ν1, ν2). The corresponding quantum toroidal algebra is introduced as a deformation of the quantum toroidal algebra of mathfrak{gl} (p). We show that it has the structure of a Hopf algebra, and present two representations, called vertical and horizontal, obtained by deforming respectively the Fock representation and Saito’s vertex representations of the quantum toroidal algebra of mathfrak{gl} (p). We construct the vertex operator intertwining between these two types of representations. This object is identified with a (ν1, ν2)-deformation of the refined topological vertex, allowing us to reconstruct the Nekrasov partition function and the qq-characters of the quiver gauge theories.

Topological vertex/anti-vertex and supergroup gauge theory

We propose a new vertex formalism, called anti-refined topological vertex (anti-vertex for short), to compute the generalized topological string amplitude, which gives rise to the supergroup gauge theory partition function. We show the one-to-many correspondence between the gauge theory and the Calabi-Yau geometry, which is peculiar to the supergroup theory, and the relation between the ordinary vertex formalism and the vertex/anti-vertex formalism through the analytic continuation.

Zeolite LTA structure generation by Coordination Sequence and Vertex Symbol

Abstract A graph theory based computational program CVdecoder is proposed that enable us to construct a zeolite LTA structure topology from Coordination Sequence and Vertex Symbol information. During the generation process, a critical size of cluster (graph) denoting as “Minimal Fingerprint Structure (MFS)” is found. The generation process after the MFS formation is no longer requiring more topological information but vertex transitive. The cluster is also smallest characteristic cluster that could only be found in LTA, that may provide a useful view of zeolite polymorph selective nucleation.

A note on the S-dual basis in the free fermion system

Abstract The free fermion system is the simplest quantum field theory which has the symmetry of the Ding–Iohara–Miki algebra (DIM). DIM has S-duality symmetry, known as Miki automorphism, which defines the transformation of generators. We introduce the second set of the fermionic basis (S-dual basis) which implements the duality transformation. It may be interpreted as the Fourier dual of the standard basis, and the inner product between the standard and the S-dual is proportional to the Hopf link invariant. We also rewrite the general topological vertex in the form of an Awata–Feigin–Shiraishi intertwiner and show that it becomes more symmetric for the duality transformation.

Kerov functions revisited

The Schur functions play a crucial role in the modern description of HOMFLY polynomials for knots and of topological vertices in DIM-based network theories, which could merge into a unified theory still to be developed. The Macdonald functions do the same for hyperpolynomials and refined vertices, but merging appears to be more problematic. For a detailed study of this problem, more knowledge is needed about the Macdonald polynomials than is usually available. As a preparation for the discussion of the knot/vertices relation, we summarize the relevant facts and open problems about the Macdonald and, more generally, Kerov functions. Like Macdonald polynomials, they are triangular combinations of Schur functions, but orthogonal in a more general scalar product. We explain that parameters of the measure can be considered as a set of new time variables, and the Kerov functions are actually expressed through the Schur functions of these variables as well. Despite they provide an infinite-parametric extension of the Schur and Macdonald polynomials, the Kerov functions, and even the skew Kerov functions continue to satisfy the most important relations, like Cauchy summation formula, the transposition identity for reflection of the Young diagram and expression of the skew functions through the generalized Littlewood–Richardson structure constants. Since these are the properties important in most applications, one can expect that the Kerov extension exists for most of them, from the superintegrable matrix and tensor models to knot theory.

Coloured refined topological vertices and parafermion conformal field theories**To Professor Mikio Sato on his 90th birthday.

We extend the definition of the refined topological vertex to an n-coloured refined topological vertex that depends on n free bosons, and compute the 5D strip partition function made of N pairs of vertices and conjugate vertices. Using geometric engineering and the AGT correspondence, the 4D limit of this strip partition function is identified with a (normalized) matrix element of a (primary state) vertex operator that intertwines two (arbitrary descendant) states in a (generically non-rational) 2D conformal field theory with parafermion primary states.

CHL Calabi–Yau threefolds: curve counting, Mathieu moonshine and Siegel modular forms

A CHL model is the quotient of $\mathrm{K3} \times E$ by an order $N$ automorphism which acts symplectically on the K3 surface and acts by shifting by an $N$-torsion point on the elliptic curve $E$. We conjecture that the primitive Donaldson-Thomas partition function of elliptic CHL models is a Siegel modular form, namely the Borcherds lift of the corresponding twisted-twined elliptic genera which appear in Mathieu moonshine. The conjecture matches predictions of string theory by David, Jatkar and Sen. We use the topological vertex to prove several base cases of the conjecture. Via a degeneration to $\mathrm{K3} \times \mathbb{P}^1$ we also express the DT partition functions as a twisted trace of an operator on Fock space. This yields further computational evidence. An extension of the conjecture to non-geometric CHL models is discussed. We consider CHL models of order $N=2$ in detail. We conjecture a formula for the Donaldson-Thomas invariants of all order two CHL models in all curve classes. The conjecture is formulated in terms of two Siegel modular forms. One of them, a Siegel form for the Iwahori subgroup, has to our knowledge not yet appeared in physics. This discrepancy is discussed in an appendix with Sheldon Katz.

Three-Partition Hodge Integrals and the Topological Vertex

A conjecture on the relation between the cubic Hodge integrals and the topological vertex in topological string theory is resolved. A central role is played by the notion of generalized shift symmetries in a fermionic realization of the two-dimensional quantum torus algebra. These algebraic relations of operators in the fermionic Fock space are used to convert generating functions of the cubic Hodge integrals and the topological vertex to each other. As a byproduct, the generating function of the cubic Hodge integrals at special values of the parameters therein is shown to be a tau function of the generalized KdV (aka Gelfand-Dickey) hierarchies.

Can tangle calculus be applicable to hyperpolynomials?

We make a new attempt at the recently suggested program to express knot polynomials through topological vertices, which can be considered as a possible approach to the tangle calculus: we discuss the Macdonald deformation of the relation between the convolution of two topological vertices and the HOMFLY-PT invariant of the 4-component link L8n8, which both depend on four arbitrary representations. The key point is that both of these are related to the Hopf polynomials in composite representations, which are in turn expressed through composite Schur functions. The latter are further expressed through the skew Schur polynomials via the remarkable Koike formula. It is this decomposition that breaks under the Macdonald deformation and gets restored only in the (large N) limit of A±1⟶0. Another problem is that the Hopf polynomials in the composite representations in the refined case are “chiral bilinears” of Macdonald polynomials, while convolutions of topological vertices involve “non-chiral combinations” with one of the Macdonald polynomials entering with permuted t and q. There are also other mismatches between the Hopf polynomials in the composite representation and the topological 4-point function in the refined case, which we discuss.

Web construction of ABCDEFG and affine quiver gauge theories

The topological vertex formalism for 5d mathcal{N} = 1 gauge theories is not only a convenient tool to compute the instanton partition function of these theories, but it is also accompanied by a nice algebraic structure that reveals various kinds of nice properties such as dualities and integrability of the underlying theories. The usual refined topological vertex ormalism is derived for gauge theories with A-type quiver structure (and A-type gauge groups). In this article, we propose a construction with a web of vertex operators for all ABCDEF G-type and affine quivers by introducing several new vertices into the formalism, based on the reproducing of known instanton partition functions and qq-characters for these theories.

Rank-3 antisymmetric matter on 5-brane webs

We discuss Type IIB 5-brane configurations for 5d mathcal{N}=1 gauge theories with hypermultiplets in the rank-3 antisymmetric representation and with various other hypermultiplets, which flow from a UV fixed point at the infinite coupling. We propose 5-brane web diagrams for the theories of SU(6) and Sp(3) gauge groups with rank-3 antisymmetric matter and check our proposed 5-brane webs against several consistency conditions implied from the one-loop corrected prepotential. Using the obtained 5-brane webs for rank-3 antisymmetric matter, we apply the topological vertex method to compute the partition function for one of these SU(6) gauge theories.