String Theory and Integrable Systems
Org: Lisa Jeffrey (Toronto), Boris Khesin (Toronto) and Rob Myers (Perimeter Inst.)
- YULY BILLIG, Carleton University
Representations of extended affine algebras and hierarchies
of non-linear PDEs
Extended affine Lie algebras are higher rank generalizations of affine
Kac-Moody algebras. In this talk I will discuss recent developments
in the representation theory of these algebras, and their connections
with hierarchies of soliton PDEs.
- FREDDY CACHAZO, Institute for Advanced Study
One-Loop Amplitudes of Gluons in N=4 super Yang-Mills
Very recently several new techniques in perturbative gauge theory have
been introduced. At one-loop, any amplitude of gluons in N=4 super
Yang-Mills can be written as a linear combination of known scalar box
integrals with coefficients that are rational functions. Using a
generalization of unitarity cuts, in particular quadruple cuts, any
coefficient can be easily written as the product of four tree-level
amplitudes. Therefore, this new technique solves the problem of
computing one-loop amplitudes in N=4 super-Yang-Mills.
- EMANUEL DIACONESCU, Rutgers University
A Vertex Formalism for Local Ruled Surfaces
- JACQUES DISTLER, Physics Department, University of Texas at Austin, Austin,
Chiral Rings for (0,2) Models
The Topological A-Model is a "twisted" version of the N=(2,2)
supersymmetric s-model, with target space, X. The ring of
observables of the A-Model is isomorphic to a certain deformation of
the cohomology ring of X. I would like to present a generalization
of this structure to the case of N=(0,2) supersymmetry. The data
will consist of X and a rank-r holomorphic vector bundle V® X,
satisfying Ùr V = KX.
I will explain, first, from the point of view of the twisted
supersymmetric s-model, why a finite-dimensional
graded-commutative ring exists. And I will explain, in a few examples,
how quantum effect deform the ring structure.
This is joint work with Allan Adams and Morten Ernebjerg.
- DMITRI KOROTKIN, Concordia University, 7141 Sherbrooke West, Montreal, Quebec
Tau-functions on spaces of abelian differentials and
determinants of Laplacians in flat metrics with conical
singularities over Riemann surfaces
We define a natural analog of the Jimbo-Miwa tau-function on
different strata of the space of holomorphic differentials over
Riemann surfaces. We compute the tau-functions in terms of higher
genus generalization of Dedekind eta-function. The developed
formalism is applied to rigorously compute the determinants of Laplace
operators over Riemann surfaces in flat metrics with conical
singularities. The holomorphic factorization formula for such
determinants gives the higher genus generalization of genus one
expression by Ray-Singer.
This is a joint work with Alexey Kokotov.
- SHAHN MAJID, Queen Mary, University of London
Semiclassicalisaton of quantum differentials and Poisson
We semiclassicalise the standard notion of differential calculus in
noncommutative geometry on algebras and quantum groups. We show in
the symplectic case that the infinitesimal data for a differential
calculus is a symplectic connection, and interpret its curvature as
lowest order nonassociativity of the exterior algebra. In the
Poisson-Lie group case we study left-covariant infinitesimal data in
terms of partial connections. We show that the moduli space of
bicovariant infinitesimal data for quasitriangular Poisson-Lie groups
has a canonical reference point which is flat in the triangular case.
Using a theorem of Kostant, we completely determine the moduli space
when the Lie algebra is simple: the canonical partial connection is
the unique point for other than sln, n > 2, when the moduli
space is 1-dimensional. This proves that the deformation-theoretic
exterior algebra on standard quantum groups must be nonassociative and
we provide it as a super-quasiHopf algebra. More generally, we show
that many standard quantisations in physics including of coadjoint
orbits (such as fuzzy spheres) have naturally nonassociative
differential structures. Our methods also quantise quasi-Poisson
manifolds of interest in string theory.
Mostly joint work with E. J. Beggs.
- GERARD MISIOLEK, Mathematics, University of Notre Dame, IN 46556, USA
Recent well-posedness results for the CH equation
I will describe some recent results on analyticity and ill-posedness.
- ANDREW NEITZKE, Harvard University, Cambridge, MA
BPS Microstates and the Open Topological String Wave
I will describe recent joint work with Mina Aganagic and Cumrun Vafa,
which reinterprets the square of the open topological string wave
function (also known as the generating function for open
Gromov-Witten invariants) in terms of counting supersymmetric
microstates localized on a stringy defect in a gravitational theory in
4 dimensions. I will also sketch the sense in which the wave function
property of the topological string, which plays a crucial role in this
work, is related to integrability.
- RONEN PLESSER, Duke University
Linear Sigma Models and Coulomb Branches
- MATSUO SATO, Department of Physics and Astronomy, University of Rochester,
Rochester, NY 14627-0171, USA)
Integrability of the AdS5 ×S5 Superstring
We study integrability aspects of superstrings on AdS5 ×S5. We show that a one parameter family of flat currents, which is
gauge equivalent to that obtained by Bena, Polchinski and Roiban, is
manifestly invariant under a generalized Z4 transformation. This
symmetry is expected to simplify analysis of the currents because the
Z4 transformation is an automorphism of PSU(2,2|4), the isometry
in the theory.
We perform the canonical analysis of the theory. Especially we
calculate the Poisson bracket of the currents. This bracket results
in an algebra which includes a Schwinger term. Because of the
Schwinger term, more work is needed in understanding the quantum
integrability properties of the system.
- MICHAEL SHAPIRO, Michigan State University, East Lansing, MI 48823
Cluster algebras and Poisson Geometry
We describe a Poisson structure compatible with a cluster algebra
structure. In particular case of cluster algebra formed by Penner
coordinates on the decorated Teichmuller space that leads to a known
Weil-Petersson symplectic form a Teichmuller space.
- JACEK SZMIGIELSKI, Department of Mathematics and Statistics, University of
Saskatchewan, 106 Wiggins Rd., Saskatoon, S7N 5E6
Degasperis-Procesi peakons and the discrete cubic string
We use an inverse scattering approach to study multi-peakon solutions
of the Degasperis-Procesi (DP) equation, an integrable PDE similar to
the Camassa-Holm shallow water equation. The spectral problem
associated to the DP equation is equivalent under a change of
variables to what we call the cubic string problem, which is a third
order non-selfadjoint generalization of the well-known equation
describing the vibrational modes of an inhomogeneous string attached
at its ends.
For the discrete cubic string (analogous to a string consisting of n
point masses) we solve explicitly the inverse spectral problem of
reconstructing the mass distribution from suitable spectral data, and
this leads to explicit formulas for the general n-peakon solution of
the DP equation. Central to our study of the inverse problem is a
peculiar type of simultaneous rational approximation of the two Weyl
functions of the cubic string, similar to classical Padé-Hermite
approximation but with lower order of approximation and an additional
symmetry condition instead. The results obtained are intriguing and
nontrivial generalizations of classical facts from the theory of
Stieltjes continued fractions and orthogonal polynomials.
This talk is based on joint work with Hans Lundmark (Linköping
University, Sweden) which, under the same title, appeared recently
(International Mathematics Research Papers, vol. 2005, 2, 53-116).
- KIRILL VANINSKY, Michigan State University
Poisson structures on meromorphic functions defined on
Riemann surfaces and classical integrable models
In 1988 Atiyah and Hitchin introduced a Poisson bracket (PB) on
meromorphic functions defined on the Riemann sphere.
Can one replace the Riemann sphere by a Riemann surface of genus
g > 0? Are there other natural Poisson structures?
We survey recent progress in these problems. It is based on the
theory of classical completely integrable systems.