arXiv subjects

Hiroaki Kanno

Publications and source records attributed to Hiroaki Kanno.

At least 37 records · Page 2Linked to original sources

arXiv2011

Macdonald operators and homological invariants of the colored Hopf link

Using a power sum (boson) realization for the Macdonald operators, we investigate the Gukov, Iqbal, Kozcaz and Vafa (GIKV) proposal for the homological invariants of the colored Hopf link, which include Khovanov-Rozansky homology as a special case. We prove the polynomiality of the invariants obtained by GIKV's proposal for arbitrary representations. We derive a closed formula of the invariants of the colored Hopf link for antisymmetric representations. We argue that a little amendment of GIKV's proposal is required to make all the coefficients of the polynomial non-negative integers.

math.QA↗

arXiv2011

Instanton counting with a surface operator and the chain-saw quiver

We describe the moduli space of SU(N) instantons in the presence of a general surface operator of type N=n_1+ ... +n_M in terms of the representations of the so-called chain-saw quiver, which allows us to write down the instanton partition function as a summation over the fixed point contributions labeled by Young diagrams. We find that the instanton partition function depends on the ordering of n_I which fixes a choice of the parabolic structure. This is in accord with the fact that the Verma module of the W-algebra also depends on the ordering of n_I. By explicit calculations, we check that the partition function agrees with the norm of a coherent state in the corresponding Verma module.

hep-th↗

arXiv2009

Quiver Matrix Model and Topological Partition Function in Six Dimensions

We consider a topological quiver matrix model which is expected to give a dual description of the instanton dynamics of topological U(N) gauge theory on D6 branes. The model is a higher dimensional analogue of the ADHM matrix model that leads to Nekrasov's partition function. The fixed points of the toric action on the moduli space are labeled by colored plane partitions. Assuming the localization theorem, we compute the partition function as an equivariant index. It turns out that the partition function does not depend on the vacuum expectation values of Higgs fields that break U(N) symmetry to U(1)^N at low energy. We conjecture a general formula of the partition function, which reduces to a power of the MacMahon function, if we impose the Calabi-Yau condition. For non Calabi-Yau case we prove the conjecture up to the third order in the instanton expansion.

hep-th↗

arXiv2009

Refined BPS state counting from Nekrasov's formula and Macdonald functions

It has been argued that the Nekrasov's partition function gives the generating function of refined BPS state counting in the compactification of M theory on local Calabi-Yau spaces. We show that a refined version of the topological vertex we proposed before (hep-th/0502061) is a building block of the Nekrasov's partition function with two equivariant parameters. Compared with another refined topological vertex by Iqbal-Kozcaz-Vafa (hep-th/0701156), our refined vertex is expressed entirely in terms of the specialization of the Macdonald symmetric functions which is related to the equivariant character of the Hilbert scheme of points on C^2. We provide diagrammatic rules for computing the partition function from the web diagrams appearing in geometric engineering of Yang-Mills theory with eight supercharges. Our refined vertex has a simple transformation law under the flop operation of the diagram, which suggests that homological invariants of the Hopf link are related to the Macdonald functions.

hep-th↗

arXiv2008

Instanton calculus and chiral one-point functions in supersymmetric gauge theories

We compute topological one-point functions of the chiral operator Tr ϕ^k in the maximally confining phase of U(N) supersymmetric gauge theory. These one-point functions are polynomials in the equivariant parameter \hbar and the parameter of instanton expansion q=Λ^{2N} and are of particular interest from gauge/string theory correspondence, since they are related to the Gromov-Witten theory of P^1. Based on a combinatorial identity that gives summation formula over Young diagrams of relevant functions, we find a relation among chiral one-point functions, which recursively determines the \hbar expansion of the generating function of one-point functions. Using a result from the operator formalism of the Gromov-Witten theory, we also present a vacuum expectation value of the loop operator Tr e^{itϕ}.

hep-th↗

arXiv2008

Instanton Calculus and Loop Operator in Supersymmetric Gauge Theory

We compute one-point function of the glueball loop operator in the maximally confining phase of supersymmetric gauge theory using instanton calculus. In the maximally confining phase the residual symmetry is the diagonal U(1) subgroup and the localization formula implies that the chiral correlation functions are the sum of the contributions from each fixed point labeled by the Young diagram. The summation can be performed exactly by operator formalism of free fermions, which also featured in the equivariant Gromov-Witten theory of P^1. By taking the Laplace transformation of the glueball loop operator, we find an exact agreement with the previous results for the generating function (resolvent) of the glueball one-point functions.

hep-th↗

arXiv2006

Universal Character and Large N Factorization in Topological Gauge/String Theory

We establish a formula of the large N factorization of the modular S-matrix for the coupled representations in U(N) Chern-Simons theory. The formula was proposed by Aganagic, Neitzke and Vafa, based on computations involving the conifold transition. We present a more rigorous proof that relies on the universal character for rational representations and an expression of the modular S-matrix in terms of the specialization of characters.

hep-th↗

arXiv2005

Instanton counting, Macdonald function and the moduli space of D-branes

We argue the connection of Nekrasov's partition function in the Ωbackground and the moduli space of D-branes, suggested by the idea of geometric engineering and Gopakumar-Vafa invariants. In the instanton expansion of N=2 SU(2) Yang-Mills theory the Nakrasov's partition function with equivariant parameters ε_1, ε_2 of toric action on C^2 factorizes correctly as the character of SU(2)_L \times SU(2)_R spin representation. We show that up to two instantons the spin contents are consistent with the Lefschetz action on the moduli space of D2-branes on (local) F_0. We also present an attempt at constructing a refined topological vertex in terms of the Macdonald function. The refined topological vertex with two parameters of T^2 action allows us to obtain the generating functions of equivariant χ_y and elliptic genera of the Hilbert scheme of n points on C^2 by the method of topological vertex.

hep-th↗

arXiv2004

Geometric transitions, Chern-Simons gauge theory and Veneziano type amplitudes

We consider the geometric transition and compute the all-genus topological string amplitudes expressed in terms of Hopf link invariants and topological vertices of Chern-Simons gauge theory. We introduce an operator technique of 2-dimensional CFT which greatly simplifies the computations. We in particular show that in the case of local Calabi-Yau manifolds described by toric geometry basic amplitudes are written as vacuum expectation values of a product vertex operators and thus appear quite similar to the Veneziano amplitudes of the old dual resonance models. Topological string amplitudes can be easily evaluated using vertex operator algebra.

hep-th↗

arXiv2003

Topological Strings and Nekrasov's formulas

We apply the method of geometric transition and compute all genus topological closed string amplitudes compactified on local {\bf F}_0 by making use of the Chern-Simons gauge theory. We find an exact agreement of the results of our computation with the formula proposed recently by Nekrasov for {\cal N}=2 SU(2) gauge theory with two parameters βand \hbar. βis related to the size of the fiber of {\bf F}_0 and \hbar corresponds to the string coupling constant. Thus Nekrasov's formula encodes all the information of topological string amplitudes on local {\bf F}_0 including the number of holomorphic curves at arbitrary genus. By taking suitable limits βand/or \hbar \to 0 one recovers the four-dimensional Seiberg-Witten theory and also its coupling to external graviphoton fields. We also compute topological string amplitude for the local 2nd del Pezzo surface and check the consistency with Nekrasov's formula of SU(2) gauge theory with a matter field in the vector representation.

hep-th↗

arXiv2002

Harmonic Forms and Deformation of ALC metrics with Spin(7) holonomy

Asymptotically locally conical (ALC) metric of exceptional holonomy has an asymptotic circle bundle structure that accommodates the M theory circle in type IIA reduction. Taking Spin(7) metrics of cohomogeneity one as explicit examples, we investigate deformations of ALC metrics, in particular that change the asymptotic S^1 radius related to the type IIA string coupling constant. When the canonical four form of Spin(7) holonomy is taken to be anti-self-dual, the deformations of Spin(7) metric are related to the harmonic self-dual four forms, which are given by solutions to a system of first order differential equations, due to the metric ansatz of cohomogeneity one. We identify the L^2-normalizable solution that deforms the asymptotic radius of the M theory circle.

hep-th↗

arXiv2001

On Spin(7) holonomy metric based on SU(3)/U(1)

We investigate the $Spin(7)$ holonomy metric of cohomogeneity one with the principal orbit $SU(3)/U(1)$. A choice of U(1) in the two dimensional Cartan subalgebra is left as free and this allows manifest $Σ_3=W(SU(3))$ (= the Weyl group) symmetric formulation. We find asymptotically locally conical (ALC) metrics as octonionic gravitational instantons. These ALC metrics have orbifold singularities in general, but a particular choice of the U(1) subgroup gives a new regular metric of $Spin(7)$ holonomy. Complex projective space ${\bf CP}(2)$ that is a supersymmetric four-cycle appears as a singular orbit. A perturbative analysis of the solution near the singular orbit shows an evidence of a more general family of ALC solutions. The global topology of the manifold depends on a choice of the U(1) subgroup. We also obtain an $L^2$-normalisable harmonic 4-form in the background of the ALC metric.

hep-th↗

arXiv2001

On Spin(7) holonomy metric based on SU(3)/U(1): II

We continue the investigation of Spin(7) holonomy metric of cohomogeneity one with the principal orbit SU(3)/U(1). A special choice of U(1) embedding in SU(3) allows more general metric ansatz with five metric functions. There are two possible singular orbits in the first order system of Spin(7) instanton equation. One is the flag manifold $SU(3)/T^2$ also known as the twister space of CP(2) and the other is CP(2) itself. Imposing a set of algebraic constraints, we find a two-parameter family of exact solutions which have SU(4) holonomy and are asymptotically conical. There are two types of asymptotically locally conical (ALC) metrics in our model, which are distingushed by the choice of $S^1$ circle whose radius stabilizes at infinity. We show that this choice of M theory circle selects one of possible singular orbits mentioned above. Numerical analyses of solutions near the singular orbit and in the asymptotic region support the existence of two families of ALC Spin(7) metrics: one family consists of deformations of the Calabi hyperKahler metric, the other is a new family of metrics on a line bundle over the twister space of CP(2).

hep-th↗

arXiv2000

Five-Dimensional Gauge Theories and Local Mirror Symmetry

We study the dynamics of 5-dimensional gauge theory on $M_4\times S^1$ by compactifying type II/M theory on degenerate Calabi-Yau manifolds. We use the local mirror symmetry and shall show that the prepotential of the 5-dimensional SU(2) gauge theory without matter is given exactly by that of the type II string theory compactified on the local ${\bf F}_2$, i.e. Hirzebruch surface ${\bf F}_2$ lying inside a non-compact Calabi-Yau manifold. It is shown that our result reproduces the Seiberg-Witten theory at the 4-dimensional limit $R\to 0$ ($R$ denotes the radius of $S^1$) and also the result of the uncompactified 5-dimensional theory at $R\to \infty$. We also discuss SU(2) gauge theory with $1\le N_f\le 4$ matter in vector representations and show that they are described by the geometry of the local ${\bf F}_2$ blown up at $N_f$ points.

hep-th↗

arXiv1998

Donaldson-Witten Functions of Massless N=2 Supersymmetric QCD

We study the Donaldson-Witten function in four-dimensional topological gauge theory which is constructed from N=2 supersymmetric SU(2) gauge theory with $N_f < 4$ massless fundamental hypermultiplets. When $N_f = 2,3$, the strong-coupling singularities with multiple massless monopoles appear in the moduli space (the u-plane) of the Coulomb branch. We show that the invariants made out of such singularities exhibit a property which is similar to the one expected for four-manifolds of generalized simple type.

hep-th↗

arXiv1998

Picard-Fuchs Equation and Prepotential of Five Dimensional SUSY Gauge Theory Compactified on a Circle

Five dimensional supersymmetric gauge theory compactified on a circle defines an effective N=2 supersymmetric theory for massless fields in four dimensions. Based on the relativistic Toda chain Hamiltonian proposed by Nekrasov, we derive the Picard-Fuchs equation on the moduli space of the Coulomb branch of SU(2) gauge theory. Our Picard-Fuchs equation agrees with those from other approaches; the spectral curve of XXZ spin chain and supersymmetric cycle in compactified M theory. By making use of a relation to the Picard-Fuchs equation of SU(2) Seiberg-Witten theory, we obtain the prepotential and the effective coupling constant that incorporate both a perturbative effect of Kaluza-Klein modes and a non-perturbative one of four dimensional instantons. In the weak coupling regime we check that the prepotential exhibits a consistent behavior in large and small radius limits of the circle.

hep-th↗

arXiv1997

Higher-dimensional WZW Model on Kähler Manifold and Toroidal Lie Algebra

We construct a generalization of the two-dimensional Wess-Zumino-Witten model on a $2n$-dimensional Kähler manifold as a group-valued non-linear sigma model with an anomaly term containing the Kähler form. The model is shown to have an infinite-dimensional symmetry which generates an $n$-toroidal Lie algebra. The classical equation of motion turns out to be the Donaldson-Uhlenbeck-Yau equation, which is a $2n$-dimensional generalization of the self-dual Yang-Mills equation.

hep-th↗

arXiv1997

Two-toroidal Lie Algebra as Current Algebra of Four-dimensional Kähler WZW Model

We investigate the structure of an infinite-dimensional symmetry of the four-dimensional Kähler WZW model, which is a possible extension of the two-dimensional WZW model. We consider the SL(2,R) group and, using the Gauss decomposition method, we derive a current algebra identified with a two-toroidal Lie algebra, a generalization of the affine Kac-Moody algebra. We also give an expression of the energy-momentum tensor in terms of currents and extra terms.

hep-th↗