International Journal of Modern Physics A

We present a self-contained formalism for calculating the background solution, the linearized solutions and a class of generalized Frobenius-like solutions to a system of scale-invariant differential equations.

We first cast the scale-invariant model into its equidimensional and autonomous forms, find its fixed points, and then obtain power-law background solutions. After linearizing about these fixed points, we find a second linearized solution, which provides a distinct collection of power laws characterizing the deviations from the fixed point. We prove that generically there will be a region surrounding the fixed point in which the complete general solution can be represented as a generalized Frobenius-like power series with exponents that are integer multiples of the exponents arising in the linearized problem. While discussions of the linearized system are common, and one can often find a discussion of power-series with integer exponents, power series with irrational (indeed complex) exponents are much rarer in the extant literature. The Frobenius-like series we encounter can be viewed as a variant of the rarely-discussed Liapunov expansion theorem (not to be confused with the more commonly encountered Liapunov functions and Liapunov exponents).

As specific examples we apply these ideas to Newtonian and relativistic isothermal stars and construct two separate power series with the overlapping radius of convergence. The second of these power series solutions represents an expansion around "spatial infinity," and in realistic models it is this second power series that gives information about the stellar core, and the damped oscillations in core mass and core radius as the central pressure goes to infinity. The power-series solutions we obtain extend classical results; as exemplified for instance by the work of Lane, Emden, and Chandrasekhar in the Newtonian case, and that of Harrison, Thorne, Wakano, and Wheeler in the relativistic case.

We also indicate how to extend these ideas to situations where fixed points may not exist — either due to "monotone" flow or due to the presence of limit cycles. Monotone flow generically leads to logarithmic deviations from scaling, while limit cycles generally lead to discrete self-similar solutions.

In this paper we show the equivalence of various (nonthreshold) bound state solutions of branes, or equivalently branes in background potentials, in ten- and eleven-dimensional supergravity. We compare solutions obtained in two very different ways. One method uses a zero mode analysis to make an ansatz which makes it possible to solve the full nonlinear supergravity equations. The other method utilizes T duality techniques to turn on the fields on the brane. To be specific, in eleven dimensions we show the equivalence for the (M2,M5) bound state, or equivalently an M5-brane in a C₍₃₎ field, where we also consider the (MW,M2,M2′,M5) solution, which can be obtained from the (M2,M5) bound state by a boost. In ten dimensions we show the equivalence for the ((F,D1),D3) bound states as well as the bound states of (p,q) five-branes with lower dimensional branes in type IIB, corresponding to D3-branes in B₍₂₎ and C₍₂₎ fields and (p,q) five-branes in B₍₂₎, C₍₂₎ and C₍₄₎ fields. We also comment on the recently proposed V duality related to infinitesimally boosted solutions.

Following recent advances in the local theory of current-algebraic orbifolds, we study various geometric properties of the general WZW orbifold, the general coset orbifold and a large class of (nonlinear) sigma model orbifolds. Phase-space geometry is emphasized for the WZW orbifolds — while for the sigma model orbifolds we construct the corresponding sigma model orbifold action, which includes the previously-known general WZW orbifold action and general coset orbifold action as special cases. We focus throughout on the twisted Einstein tensors with diagonal monodromy, including the twisted Einstein metric, the twisted B field and the twisted torsion field of each orbifold sector. Finally, we present strong evidence for a conjectured set of twisted Einstein equations which should describe those sigma model orbifolds in this class which are also one-loop conformal.

Discretized non-Abelian gauge theories living on finite group spaces G are defined by means of a geometric action ∫ Tr F ∧ _*F. This technique is extended to obtain discrete versions of the Born–Infeld action. The discretizations are in 1–1 correspondence with differential calculi on finite groups.

A consistency condition for duality invariance of the discretized field equations is derived for discretized U(1) actions S[F] living on a four-dimensional Abelian G. Discretized electromagnetism satisfies this condition and therefore admits duality rotations.

Yang–Mills and Born–Infeld theories are also considered on product spaces M^D×G, and we find the corresponding field theories on M^D after Kaluza–Klein reduction on the G discrete internal spaces. We examine in detail the case G=Z_N, and discuss the limit N→∞.

A self-contained review on the noncommutative differential geometry of finite groups is included.