2,624 research outputs found
Cohomological Hall algebra of a symmetric quiver
In the paper \cite{KS}, Kontsevich and Soibelman in particular associate to
each finite quiver with a set of vertices the so-called Cohomological
Hall algebra \cH, which is -graded. Its graded component
\cH_{\gamma} is defined as cohomology of Artin moduli stack of
representations with dimension vector The product comes from natural
correspondences which parameterize extensions of representations. In the case
of symmetric quiver, one can refine the grading to and
modify the product by a sign to get a super-commutative algebra (\cH,\star)
(with parity induced by -grading). It is conjectured in \cite{KS} that in
this case the algebra (\cH\otimes\Q,\star) is free super-commutative
generated by a -graded vector space of the form
V=V^{prim}\otimes\Q[x], where is a variable of bidegree and all the spaces
are
finite-dimensional. In this paper we prove this conjecture (Theorem 1.1).
We also prove some explicit bounds on pairs for which
(Theorem 1.2). Passing to generating functions, we
obtain the positivity result for quantum Donaldson-Thomas invariants, which was
used by S. Mozgovoy to prove Kac's conjecture for quivers with sufficiently
many loops \cite{M}. Finally, we mention a connection with the paper of Reineke
\cite{R}.Comment: 16 pages, no figures; a reference adde
Condensates of Strongly-interacting Atoms and Dynamically Generated Dimers
In a system of atoms with large positive scattering length, weakly-bound
diatomic molecules (dimers) are generated dynamically by the strong
interactions between the atoms. If the atoms are modeled by a quantum field
theory with an atom field only, condensates of dimers cannot be described by
the mean-field approximation because there is no field associated with the
dimers. We develop a method for describing dimer condensates in such a model
based on the one-particle-irreducible (1PI) effective action. We construct an
equivalent 1PI effective action that depends not only on the classical atom
field but also on a classical dimer field. The method is illustrated by
applying it to the many-body behavior of bosonic atoms with large scattering
length at zero temperature using an approximation in which the 2-atom amplitude
is treated exactly but irreducible -atom amplitudes for are
neglected. The two 1PI effective actions give identical results for the atom
superfluid phase, but the one with a classical dimer field is much more
convenient for describing the dimer superfluid phase. The results are also
compared with previous work on the Bose gas near a Feshbach resonance.Comment: 10 figure
- …