5,238 research outputs found
Singularities, Lax degeneracies and Maslov indices of the periodic Toda chain
The n-particle periodic Toda chain is a well known example of an integrable
but nonseparable Hamiltonian system in R^{2n}. We show that Sigma_k, the k-fold
singularities of the Toda chain, ie points where there exist k independent
linear relations amongst the gradients of the integrals of motion, coincide
with points where there are k (doubly) degenerate eigenvalues of
representatives L and Lbar of the two inequivalent classes of Lax matrices
(corresponding to degenerate periodic or antiperiodic solutions of the
associated second-order difference equation). The singularities are shown to be
nondegenerate, so that Sigma_k is a codimension-2k symplectic submanifold.
Sigma_k is shown to be of elliptic type, and the frequencies of transverse
oscillations under Hamiltonians which fix Sigma_k are computed in terms of
spectral data of the Lax matrices. If mu(C) is the (even) Maslov index of a
closed curve C in the regular component of R^{2n}, then (-1)^{\mu(C)/2} is
given by the product of the holonomies (equal to +/- 1) of the even- (or odd-)
indexed eigenvector bundles of L and Lmat.Comment: 25 pages; published versio
Phases of N=1 Supersymmetric SO/Sp Gauge Theories via Matrix Model
We extend the results of Cachazo, Seiberg and Witten to N=1 supersymmetric
gauge theories with gauge groups SO(2N), SO(2N+1) and Sp(2N). By taking the
superpotential which is an arbitrary polynomial of adjoint matter \Phi as a
small perturbation of N=2 gauge theories, we examine the singular points
preserving N=1 supersymmetry in the moduli space where mutually local monopoles
become massless. We derive the matrix model complex curve for the whole range
of the degree of perturbed superpotential. Then we determine a generalized
Konishi anomaly equation implying the orientifold contribution. We turn to the
multiplication map and the confinement index K and describe both Coulomb branch
and confining branch. In particular, we construct a multiplication map from
SO(2N+1) to SO(2KN-K+2) where K is an even integer as well as a multiplication
map from SO(2N) to SO(2KN-2K+2) (K is a positive integer), a map from SO(2N+1)
to SO(2KN-K+2) (K is an odd integer) and a map from Sp(2N) to Sp(2KN+2K-2).
Finally we analyze some examples which show some duality: the same moduli space
has two different semiclassical limits corresponding to distinct gauge groups.Comment: 55pp; two paragraphs in page 19 added to clarify the relation between
confinement index and multiplication map index, refs added and to appear in
JHEP; Konishi anomaly equations corrected and some comments on the
degenerated cases for SO(7) and SO(8) adde
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