arXiv subjects
Publications and source records attributed to Robert Carroll.
We sketch some of the different roles played by Whitham times in connection with averaging, adiabatic invariants, soliton theory, Hamiltonian structures, Seiberg-Witten theory, isomonodromy problems, Hitchin systems, WDVV and Picard-Fuchs equations, renormalization, soft susy breaking, etc.
Some observations are made about energy-time uncertainty and spin in the context of trajectories as in Faraggi-Matone or Floyd.
A number of formulas are displayed concerning Whitham theory for a simple example of pure N=2 susy YM with gauge group SU(2). In particular this serves to illuminate the role of Lambda and T derivatives and the interaction with prepotentials F based on Seiberg-Witten and Whitham theory.
We organize and review some material from various sources about prepotentials, Riemann surfaces and kernels, WDVV, and the renormalization group, provide some further connections and information, and indicate some directions and problems.
We collect a number of facts and conjectures concerning Whitham theory and the renormalization group. Some explicit relations and problems are indicated in the context of N=2 susy Yang-Mills.
For certain situations relations are indicated between the space-wave function duality of Faraggi-Matone, enhanced dispersionless KdV, and Whitham dynamics for appropriate hyperelliptic Riemann surfaces related to Seiberg-Witten theory. This paper gives refinements of hep-th/9702138 and some new ideas.
We show how the WDVV equations and the DZM system can be characterized via a background family of functions.
In a finite zone KdV context we show relations between the duality variables of Faraggi-Matone and those involved in Seiberg-Witten type duality.
Further formulas are presented involving quantum mechanics, thermodynamics, and integrable systems. Modifications of dispersionless theory are developed.
Some formulas and speculations are presented relative to integrable systems and quantum mechanics.
We survey some topics involving the Whitham equations, concentrating on the role of the product of the wave function and its adjoint in averaging and in producing Cauchy kernels and differentials on Riemann surfaces. There are also some new results.
Basic quantities related to 2-D gravity, such as Polyakov extrinsic action, Nambu-Goto action, geometrical action, and Euler characteristic are studied using generalized Weierstrass-Enneper (GWE) inducing of surfaces. Connection of the GWE inducing with conformal immersion is made and varius aspects of the theory are shown to be invariant under the modified Veselov-Novikov hierarchy of flows. The geometry of certain surfaces is shown to be connected with the dynamics of infinite and finite dimensional integrable systems. Connections to Liouville-Beltrami gravity are indicated.
For certain situations we give a geometrical background for quasiclassical KP calculations based on an explicit connection to quantum mechanics and the collapse of coherent states to coadjoint orbits for classical operators.
Dispersionless Hirota type equations are extracted from the dispersionless limit of the Fay differential identity due to Takasaki- Takebe. A few other results are sketched between inverse scattering, dKdV, and gravity.