20 research outputs found
Topological Strings with Scaling Violation and Toda Lattice Hierarchy
We show that there is a series of topological string theories whose
integrable structure is described by the Toda lattice hierarchy. The monodromy
group of the Frobenius manifold for the matter sector is an extension of the
affine Weyl group introduced by Dubrovin. These
models are generalizations of the topological string theory with scaling
violation. The logarithmic Hamiltonians generate flows for the puncture
operator and its descendants. We derive the string equation from the
constraints on the Lax and the Orlov operators. The constraints are of
different type from those for the string theory. Higher genus expansion
is obtained by considering the Lax operator in matrix form.Comment: 27 pages, latex, no figure
Picard-Fuchs Ordinary Differential Systems in N=2 Supersymmetric Yang-Mills Theories
In general, Picard-Fuchs systems in N=2 supersymmetric Yang-Mills theories
are realized as a set of simultaneous partial differential equations. However,
if the QCD scale parameter is used as unique independent variable instead of
moduli, the resulting Picard-Fuchs systems are represented by a single ordinary
differential equation (ODE) whose order coincides with the total number of
independent periods. This paper discusses some properties of these Picard-Fuchs
ODEs. In contrast with the usual Picard-Fuchs systems written in terms of
moduli derivatives, there exists a Wronskian for this ordinary differential
system and this Wronskian produces a new relation among periods, moduli and QCD
scale parameter, which in the case of SU(2) is reminiscent of scaling relation
of prepotential. On the other hand, in the case of the SU(3) theory, there are
two kinds of ordinary differential equations, one of which is the equation
directly constructed from periods and the other is derived from the SU(3)
Picard-Fuchs equations in moduli derivatives identified with Appell's
hypergeometric system, i.e., Burchnall's fifth order ordinary differential
equation published in 1942. It is shown that four of the five independent
solutions to the latter equation actually correspond to the four periods in the
SU(3) gauge theory and the closed form of the remaining one is established by
the SU(3) Picard-Fuchs ODE. The formula for this fifth solution is a new one.Comment: \documentstyle[12pt,preprint,aps,prb]{revtex}, to be published in J.
Math. Phy
Picard-Fuchs Equations and Whitham Hierarchy in N=2 Supersymmetric SU(r+1) Yang-Mills Theory
In general, Whitham dynamics involves infinitely many parameters called
Whitham times, but in the context of N=2 supersymmetric Yang-Mills theory it
can be regarded as a finite system by restricting the number of Whitham times
appropriately. For example, in the case of SU(r+1) gauge theory without
hypermultiplets, there are r Whitham times and they play an essential role in
the theory. In this situation, the generating meromorphic 1-form of the Whitham
hierarchy on Seiberg-Witten curve is represented by a finite linear combination
of meromorphic 1-forms associated with these Whitham times, but it turns out
that there are various differential relations among these differentials. Since
these relations can be written only in terms of the Seiberg-Witten 1-form,
their consistency conditions are found to give the Picard-Fuchs equations for
the Seiberg-Witten periods.Comment: to be published in J. Math. Phys, revtex, 14 page
Instanton Correction of Prepotential in Ruijsenaars Model Associated with N=2 SU(2) Seiberg-Witten Theory
Instanton correction of prepotential of one-dimensional SL(2) Ruijsenaars
model is presented with the help of Picard-Fuchs equation of Pakuliak-Perelomov
type. It is shown that the instanton induced prepotential reduces to that of
the SU(2) gauge theory coupled with a massive adjoint hypermultiplet.Comment: revtex, 15 pages, to be published in Journal of Mathematical Physic
Dual WDVV Equations in N=2 Supersymmetric Yang-Mills Theory
This paper studies the dual form of Witten-Dijkgraaf-Verlinde-Verlinde (WDVV)
equations in N=2 supersymmetric Yang-Mills theory by applying a duality
transformation to WDVV equations. The dual WDVV equations called in this paper
are non-linear differential equations satisfied by dual prepotential and are
found to have the same form with the original WDVV equations. However, in
contrast with the case of weak coupling calculus, the perturbative part of dual
prepotential itself does not satisfy the dual WDVV equations. Nevertheless, it
is possible to show that the non-perturbative part of dual prepotential can be
determined from dual WDVV equations, provided the perturbative part is given.
As an example, the SU(4) case is presented. The non-perturbative dual
prepotential derived in this way is consistent to the dual prepotential
obtained by D'Hoker and Phong.Comment: misprints are corrected, revtex, 10 page
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.Comment: revised version, appendix E is enlarged, references are added, latex,
32 pages, 1 figure, documentstyle[12pt,eclepsf,a4]{article