406 research outputs found
Solving matrix models using holomorphy
We investigate the relationship between supersymmetric gauge theories with
moduli spaces and matrix models. Particular attention is given to situations
where the moduli space gets quantum corrected. These corrections are controlled
by holomorphy. It is argued that these quantum deformations give rise to
non-trivial relations for generalized resolvents that must hold in the
associated matrix model. These relations allow to solve a sector of the
associated matrix model in a similar way to a one-matrix model, by studying a
curve that encodes the generalized resolvents. At the level of loop equations
for the matrix model, the situations with a moduli space can sometimes be
considered as a degeneration of an infinite set of linear equations, and the
quantum moduli space encodes the consistency conditions for these equations to
have a solution.Comment: 38 pages, JHEP style, 1 figur
Boundary relations and generalized resolvents of symmetric operators
The Kre\u{\i}n-Naimark formula provides a parametrization of all selfadjoint
exit space extensions of a, not necessarily densely defined, symmetric
operator, in terms of maximal dissipative (in \dC_+) holomorphic linear
relations on the parameter space (the so-called Nevanlinna families). The new
notion of a boundary relation makes it possible to interpret these parameter
families as Weyl families of boundary relations and to establish a simple
coupling method to construct the generalized resolvents from the given
parameter family. The general version of the coupling method is introduced and
the role of boundary relations and their Weyl families for the
Kre\u{\i}n-Naimark formula is investigated and explained.Comment: 47 page
Compressions of Resolvents and Maximal Radius of Regularity
Suppose that is left-invertible in for all , where is an open subset of the complex plane. Then an
operator-valued function is a left resolvent of in if
and only if has an extension , the resolvent of which is a
dilation of of a particular form. Generalized resolvents exist on
every open set , with included in the regular domain of . This
implies a formula for the maximal radius of regularity of in terms of the
spectral radius of its generalized inverses. A solution to an open problem
raised by J. Zem\'anek is obtained.Comment: 15 pages, to appear in Trans. Amer. Math. So
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