Hiroaki Kanno

Special Quantum Field Theories¶in Eight and Other Dimensions

Abstract:We build nearly topological quantum field theories in various dimensions. We give special attention to the case of eight dimensions for which we first consider theories depending only on Yang–Mills fields. Two classes of gauge functions exist which correspond to the choices of two different holonomy groups in SO(8), namely SU(4) and Spin(7). The choice of SU(4) gives a quantum field theory for a Calabi–Yau fourfold. The expectation values for the observables are formally holomorphic Donaldson invariants. The choice of Spin(7) defines another eight dimensional theory for a Joyce manifold which could be of relevance in M- and F-theories. Relations to the eight dimensional supersymmetric Yang–Mills theory are presented. Then, by dimensional reduction, we obtain other theories, in particular a four dimensional one whose gauge conditions are identical to the non-abelian Seiberg–Witten equations. The latter are thus related to pure Yang–Mills self-duality equations in 8 dimensions as well as to the N=1, D=10 super Yang–Mills theory. We also exhibit a theory that couples 3-form gauge fields to the second Chern class in eight dimensions, and interesting theories in other dimensions.

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

We argue the connection of Nekrasovs partition function in the Ω background and the moduli space of D-branes, suggested by the idea of geometric engineering and Gopakumar-Vafa invariants. In the instanton expansion of = 2 SU(2) Yang-Mills theory the Nakrasovs partition function with equivariant parameters 1,2 of toric action on 2 factorizes correctly as the character of SU(2)L × 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) F0. 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 T2 action allows us to obtain the generating functions of equivariant χy and elliptic genera of the Hilbert scheme of n points on 2 by the method of topological vertex.

REFINED BPS STATE COUNTING FROM NEKRASOV'S FORMULA AND MACDONALD FUNCTIONS

It has been argued that Nekrasovs 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 previously proposed (arXiv:hep-th/0502061) is a building block of Nekrasovs partition function with two equivariant parameters. Compared with another refined topological vertex by Iqbal, Kozcaz and Vafa (arXiv: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 ℂ2. 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.

Topological strings and Nekrasov's formulas

We apply the method of geometric transition and compute all genus topological closed string amplitudes compactified on local F0 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 = 2 SU(2) gauge theory with two parameters β and . β is related to the size of the fiber of F0 and corresponds to the string coupling constant. Thus Nekrasovs formula encodes all the information of topological string amplitudes on local F0 including the number of holomorphic curves at arbitrary genus. By taking suitable limits β and/or →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 Nekrasovs formula of SU(2) gauge theory with a matter field in the vector representation.

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 = n1 + ⋯ + nM 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 nI 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 nI. By explicit calculations, we check that the partition function agrees with the norm of a coherent state in the corresponding Verma module.

Topological strings, flat coordinates and gravitational descendants

Abstract We discuss physical spectra and correlation functions of topological minimal models coupled to topological gravity. We first study the BRST formalism of these theories and show that their BRST operator Q = Q s + Q v can be brought to Q s by a certain homotopy operator U , UQU −1 = Q s ( Q s and Q v are the N = 2 and diffeomorphism BRST operators, respectively). The reparametrization (anti)-ghost b mixes with the supercharge operator G under this transformation. The existence of this transformation enables us to use matter fields to represent cohomology classes of the operator Q . We explicitly construct gravitational descendants and show that they generate the higher-order KdV flows. We also evaluate genus-zero correlation functions and rederive basic recursion relations of two-dimensional topological gravity.

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

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 �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 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.

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

Abstract 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.

Explicit examples of DIM constraints for network matrix models

A bstractDotsenko-Fateev and Chern-Simons matrix models, which describe Nekrasov functions for SYM theories in different dimensions, are all incorporated into network matrix models with the hidden Ding-Iohara-Miki (DIM) symmetry. This lifting is especially simple for what we call balanced networks. Then, the Ward identities (known under the names of Virasoro/W