29,756 research outputs found
Dimensional Mutation and Spacelike Singularities
I argue that string theory compactified on a Riemann surface crosses over at
small volume to a higher dimensional background of supercritical string theory.
Several concrete measures of the count of degrees of freedom of the theory
yield the consistent result that at finite volume, the effective dimensionality
is increased by an amount of order for a surface of genus and volume
in string units. This arises in part from an exponentially growing density
of states of winding modes supported by the fundamental group, and passes an
interesting test of modular invariance. Further evidence for a plethora of
examples with the spacelike singularity replaced by a higher dimensional phase
arises from the fact that the sigma model on a Riemann surface can be naturally
completed by many gauged linear sigma models, whose RG flows approximate time
evolution in the full string backgrounds arising from this in the limit of
large dimensionality. In recent examples of spacelike singularity resolution by
tachyon condensation, the singularity is ultimately replaced by a phase with
all modes becoming heavy and decoupling. In the present case, the opposite
behavior ensues: more light degrees of freedom arise in the small radius
regime. I comment on the emerging zoology of cosmological singularities that
results.Comment: 15 pages, harvmac big. v2: 18 pages, harvmac big; added computation
of density of states and modular invariance check, enhanced discussion of
multiplicity of solutions all sharing the feature of increased density of
states, added reference
Rationally Isomorphic Hermitian Forms and Torsors of Some Non-Reductive Groups
Let be a semilocal Dedekind domain. Under certain assumptions, we show
that two (not necessarily unimodular) hermitian forms over an -algebra with
involution, which are rationally ismorphic and have isomorphic semisimple
coradicals, are in fact isomorphic. The same result is also obtained for
quadratic forms equipped with an action of a finite group. The results have
cohomological restatements that resemble the Grothendieck--Serre conjecture,
except the group schemes involved are not reductive. We show that these group
schemes are closely related to group schemes arising in Bruhat--Tits theory.Comment: 27 pages. Changes from previous version: Section 5 was split into two
sections, several proofs have been simplified, other mild modification
A new proof of a Nordgren, Rosenthal and Wintrobe Theorem on universal operators
A striking result by Nordgren, Rosenthal and Wintrobe states that the Invariant Subspace Problem is equivalent to the fact that any minimal invariant subspace for a composition operator Cφ induced by a hyperbolic automorphism φ of the unit disc D acting on the classical Hardy space H² is one dimensional. We provide a completely different proof of Nordgren, Rosenthal and Wintrobe’s Theorem based on analytic Toeplitz operators
A hyperbolic universal operator commuting with a compact operator
A Hilbert space operator is called universal (in the sense of Rota) if every operator on the Hilbert space is similar to a multiple of the restriction of the universal operator to one of its invariant subspaces. We exhibit an analytic Toeplitz operator whose adjoint is universal in the sense of Rota and commutes with a non-trivial, quasinilpotent, injective, compact operator with dense range, but unlike other examples, it acts on the Bergman space instead of the Hardy space and this operator is associated with a `hyperbolic' composition operator
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