3,903 research outputs found
Integral Grothendieck-Riemann-Roch theorem
We show that, in characteristic zero, the obvious integral version of the
Grothendieck-Riemann-Roch formula obtained by clearing the denominators of the
Todd and Chern characters is true (without having to divide the Chow groups by
their torsion subgroups). The proof introduces an alternative to Grothendieck's
strategy: we use resolution of singularities and the weak factorization theorem
for birational maps.Comment: 24 page
Schur Q-functions and degeneracy locus formulas for morphisms with symmetries
We give closed-form formulas for the fundamental classes of degeneracy loci
associated with vector bundle maps given locally by (not necessary square)
matrices which are symmetric (resp. skew-symmetric) w.r.t. the main diagonal.
Our description uses essentially Schur Q-polynomials of a bundle, and is based
on a certain push-forward formula for these polynomials in a Grassmann bundle.Comment: 22 pages, AMSTEX, misprints corrected, exposition improved. to appear
in the Proceedings of Intersection Theory Conference in Bologna, "Progress in
Mathematics", Birkhause
In the Flower Crowned Green Mountains of Vermont
https://digitalcommons.library.umaine.edu/mmb-vp/6811/thumbnail.jp
Classical and quantum communication without a shared reference frame
We show that communication without a shared reference frame is possible using
entangled states. Both classical and quantum information can be communicated
with perfect fidelity without a shared reference frame at a rate that
asymptotically approaches one classical bit or one encoded qubit per
transmitted qubit. We present an optical scheme to communicate classical bits
without a shared reference frame using entangled photon pairs and linear
optical Bell state measurements.Comment: 4 pages, published versio
Stringy K-theory and the Chern character
For a finite group G acting on a smooth projective variety X, we construct
two new G-equivariant rings: first the stringy K-theory of X, and second the
stringy cohomology of X. For a smooth Deligne-Mumford stack Y we also construct
a new ring called the full orbifold K-theory of Y. For a global quotient
Y=[X/G], the ring of G-invariants of the stringy K-theory of X is a subalgebra
of the full orbifold K-theory of the the stack Y and is linearly isomorphic to
the ``orbifold K-theory'' of Adem-Ruan (and hence Atiyah-Segal), but carries a
different, ``quantum,'' product, which respects the natural group grading. We
prove there is a ring isomorphism, the stringy Chern character, from stringy
K-theory to stringy cohomology, and a ring homomorphism from full orbifold
K-theory to Chen-Ruan orbifold cohomology. These Chern characters satisfy
Grothendieck-Riemann-Roch for etale maps.
We prove that stringy cohomology is isomorphic to Fantechi and Goettsche's
construction. Since our constructions do not use complex curves, stable maps,
admissible covers, or moduli spaces, our results simplify the definitions of
Fantechi-Goettsche's ring, of Chen-Ruan's orbifold cohomology, and of
Abramovich-Graber-Vistoli's orbifold Chow.
We conclude by showing that a K-theoretic version of Ruan's Hyper-Kaehler
Resolution Conjecture holds for symmetric products.
Our results hold both in the algebro-geometric category and in the
topological category for equivariant almost complex manifolds.Comment: Exposition improved and additional details provided. To appear in
Inventiones Mathematica
On the number of representations providing noiseless subsystems
This paper studies the combinatoric structure of the set of all
representations, up to equivalence, of a finite-dimensional semisimple Lie
algebra. This has intrinsic interest as a previously unsolved problem in
representation theory, and also has applications to the understanding of
quantum decoherence. We prove that for Hilbert spaces of sufficiently high
dimension, decoherence-free subspaces exist for almost all representations of
the error algebra. For decoherence-free subsystems, we plot the function
which is the fraction of all -dimensional quantum systems which
preserve bits of information through DF subsystems, and note that this
function fits an inverse beta distribution. The mathematical tools which arise
include techniques from classical number theory.Comment: 17 pp, 4 figs, accepted for Physical Review
On the integral cohomology of smooth toric varieties
Let be a smooth, not necessarily compact toric variety. We show
that a certain complex, defined in terms of the fan , computes the
integral cohomology of , including the module structure over the
homology of the torus. In some cases we can also give the product. As a
corollary we obtain that the cycle map from Chow groups to integral Borel-Moore
homology is split injective for smooth toric varieties. Another result is that
the differential algebra of singular cochains on the Borel construction of
is formal.Comment: 10 page
Standard model plethystics
We study the vacuum geometry prescribed by the gauge invariant operators of the minimal supersymmetric standard model via the plethystic program. This is achieved by using several tricks to perform the highly computationally challenging Molien-Weyl integral, from which we extract the Hilbert series, encoding the invariants of the geometry at all degrees. The fully refined Hilbert series is presented as the explicit sum of 1422 rational functions. We found a good choice of weights to unrefine the Hilbert series into a rational function of a single variable, from which we can read off the dimension and the degree of the vacuum moduli space of the minimal supersymmetric standard model gauge invariants. All data in Mathematica format are also presented
Grothendieck groups and a categorification of additive invariants
A topologically-invariant and additive homology class is mostly not a natural
transformation as it is. In this paper we discuss turning such a homology class
into a natural transformation; i.e., a "categorification" of it. In a general
categorical set-up we introduce a generalized relative Grothendieck group from
a cospan of functors of categories and also consider a categorification of
additive invariants on objects. As an example, we obtain a general theory of
characteristic homology classes of singular varieties.Comment: 27 pages, to appear in International J. Mathematic
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