798 research outputs found
Flag Hilbert schemes, colored projectors and Khovanov-Rozansky homology
We construct a categorification of the maximal commutative subalgebra of the type A Hecke algebra. Specifically, we propose a monoidal functor from the (symmetric) monoidal category of coherent sheaves on the flag Hilbert scheme to the (non-symmetric) monoidal category of Soergel bimodules. The adjoint of this functor allows one to match the Hochschild homology of any braid with the Euler characteristic of a sheaf on the flag Hilbert scheme. The categorified Jones-Wenzl projectors studied by Abel, Elias and Hogancamp are idempotents in the category of Soergel bimodules, and they correspond to the renormalized Koszul complexes of the torus fixed points on the flag Hilbert scheme. As a consequence, we conjecture that the endomorphism algebras of the categorified projectors correspond to the dg algebras of functions on affine charts of the flag Hilbert schemes. We define a family of differentials dN on these dg algebras and conjecture that their homology matches that of the glN projectors, generalizing earlier conjectures of the first and third authors with Oblomkov and Shende
Dualities in integrable systems and N=2 theories
We discuss dualities of the integrable dynamics behind the exact solution to
the N=2 SUSY YM theory. It is shown that T duality in the string theory is
related to the separation of variables procedure in dynamical system. We argue
that there are analogues of S duality as well as 3d mirror symmetry in the
many-body systems of Hitchin type governing low-energy effective actions.Comment: 16 pages, Latex, Talk given at QFTHEP-99, Moscow, May 27-June
The Classical -Matrix for the Relativistic Ruijsenaars-Schneider System
We compute the classical -matrix for the relativistic generalization of
the Calogero-Moser model, or Ruijsenaars-Schneider model, at all values of the
speed-of-light parameter . We connect it with the non-relativistic
Calogero-Moser -matrix and the
sine-Gordon soliton limit.Comment: LaTeX file, no figures, 8 page
Coadjoint orbits of the Virasoro algebra and the global Liouville equation
The classification of the coadjoint orbits of the Virasoro algebra is
reviewed and is then applied to analyze the so-called global Liouville
equation. The review is self-contained, elementary and is tailor-made for the
application. It is well-known that the Liouville equation for a smooth, real
field under periodic boundary condition is a reduction of the SL(2,R)
WZNW model on the cylinder, where the WZNW field g in SL(2,R) is restricted to
be Gauss decomposable. If one drops this restriction, the Hamiltonian reduction
yields, for the field where is a constant,
what we call the global Liouville equation. Corresponding to the winding number
of the SL(2,R) WZNW model there is a topological invariant in the reduced
theory, given by the number of zeros of Q over a period. By the substitution
, the Liouville theory for a smooth is recovered in
the trivial topological sector. The nontrivial topological sectors can be
viewed as singular sectors of the Liouville theory that contain blowing-up
solutions in terms of . Since the global Liouville equation is
conformally invariant, its solutions can be described by explicitly listing
those solutions for which the stress-energy tensor belongs to a set of
representatives of the Virasoro coadjoint orbits chosen by convention. This
direct method permits to study the `coadjoint orbit content' of the topological
sectors as well as the behaviour of the energy in the sectors. The analysis
confirms that the trivial topological sector contains special orbits with
hyperbolic monodromy and shows that the energy is bounded from below in this
sector only.Comment: Plain TEX, 48 pages, final version to appear in IJMP
A structural view of maximal green sequences
We initiate a new approach to maximal green sequences by considering them up
to an equivalence relation. This reveals extra structure, since the set of
equivalence classes of maximal green sequences of an algebra carries
interesting partial orders. We show that the equivalence relation may be
defined in several equivalent ways. We likewise define three conjecturally
equivalent partial orders on the set of equivalence classes, and prove some of
the implications between them. In the case of Nakayama algebras we prove that
these three partial orders indeed coincide.Comment: 72 pages, 8 figure
The Effects of Smolt Stocking Strategies on Migratory Path Selection of Adult Atlantic Salmon in the Penobscot River, Maine
Understanding the homing behavior of Atlantic salmon Salmo salar is vital to the restoration program employed on the Penobscot River, Maine. To produce significant adult returns, managers currently stock hatchery-raised smolts in specific river sections, providing smolts the opportunity to imprint on chemical signals and enabling their return to productive spawning and rearing habitat as adults. In this study, we used observational evidence from passive integrated transponder telemetry to determine whether adults returning from smolt stockings behaved in a way that suggested strong homing to smolt stocking locations. Adults returning from smolt stocking locations located in or at the mouth of the Piscataquis River were more likely to be detected as entering the Piscataquis River than were adults returning from the upper Penobscot River smolt stocking locations. In general, returning adult Atlantic salmon that had been stocked near or in tributaries as smolts chose a path more quickly than those that had been stocked in more downstream or main-stem locations. These results suggest that Atlantic salmon smolts should be stocked at specific sites with superior habitat for spawning kind juvenile survival to capitalize on the strong homing tendency in adults. This technique call also be utilized to allow for natural selection and the development of localized stocks
The structure functions of longitudinal virtual photon at low virtualities
The structure functions of longitudinal virtual photon at low virtualities
are calculated in the framework of chiral pertubation theory(ChPT) in the zero
and first order of ChPT. It is assumed that the virtuality of target
longitudinal photon is much less than the virtuality of the hard projectile
photon and both are less than the characteristic ChPT scale.Comment: 16 pages, 8 figure
On spin chains and field theories
We point out that the existence of global symmetries in a field theory is not
an essential ingredient in its relation with an integrable model. We describe
an obvious construction which, given an integrable spin chain, yields a field
theory whose 1-loop scale transformations are generated by the spin chain
Hamiltonian. We also identify a necessary condition for a given field theory to
be related to an integrable spin chain.
As an example, we describe an anisotropic and parity-breaking generalization
of the XXZ Heisenberg spin chain and its associated field theory. The system
has no nonabelian global symmetries and generally does not admit a
supersymmetric extension without the introduction of more propagating bosonic
fields. For the case of a 2-state chain we find the spectrum and the
eigenstates. For certain values of its coupling constants the field theory
associated to this general type of chain is the bosonic sector of the
Leigh-Strassler deformation of N=4 SYM theory.Comment: 22 pages, Latex; v2. typos correcte
Non-relativistic electron-electron interaction in a Maxwell-Chern-Simons-Proca model endowed with a timelike Lorentz-violating background
A planar Maxwell-Chern-Simons-Proca model endowed with a Lorentz-violating
background is taken as framework to investigate the electron-electron
interaction. The Dirac sector is introduced exhibiting a Yukawa and a minimal
coupling with the scalar and the gauge fields, respectively. The the
electron-electron interaction is then exactly evaluated as the Fourier
transform of the Moller scattering amplitude (carried out in the
non-relativistic limit) for the case of a purely time-like background. The
interaction potential exhibits a totally screened behavior far from the origin
as consequence of massive character of the physical mediators. The total
interaction (scalar plus gauge potential) can always be attractive, revealing
that this model may lead to the formation of electron-electron bound states.Comment: 14 pages, 4 figures, style revtex. To appear in International Journal
Modern Physics
Integrability in QCD and beyond
Yang--Mills theories in four space-time dimensions possess a hidden symmetry
which does not exhibit itself as a symmetry of classical Lagrangians but is
only revealed on the quantum level. It turns out that the effective Yang--Mills
dynamics in several important limits is described by completely integrable
systems that prove to be related to the celebrated Heisenberg spin chain and
its generalizations. In this review we explain the general phenomenon of
complete integrability and its realization in several different situations. As
a prime example, we consider in some detail the scale dependence of composite
(Wilson) operators in QCD and super-Yang--Mills (SYM) theories. High-energy
(Regge) behavior of scattering amplitudes in QCD is also discussed and provides
one with another realization of the same phenomenon that differs, however, from
the first example in essential details. As the third example, we address the
low-energy effective action in a N=2 SYM theory which, contrary to the previous
two cases, corresponds to a classical integrable model. Finally, we include a
short overview of recent attempts to use gauge/string duality in order to
relate integrability of Yang--Mills dynamics with the hidden symmetry of a
string theory on a curved background.Comment: 87 pages, 4 figures; minor stylistic changes, references added. To be
published in the memorial volume 'From Fields to Strings: Circumnavigating
Theoretical Phyiscs', World Scientific, 2004. Dedicated to the memory of Ian
Koga
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