Abstracts
Tetsuji Miwa, "Fermionic structure in the XXZ model"
For the XXZ model, we introduce annihilation and creation operators
acting on the space of quasi-local operators acting on the quantum spin 1/2 chain.
They satisfy canonical anti-commutation relations. The exponential
formula for the vacuum expectation values of the quasi-local operators
are given in terms of these operators.
Evgeni Sklyanin, "Combinatorics of generalized Bethe Ansatz"
Since Faddeev and Takhtajan's (1981) work on the completeness of
Bethe Ansatz for SU(2)-invariant spin chain the Bethe Ansatz keeps being a
source of interesting and nontrivial combinatorial problems. We present the
results on counting the solutions of generalised Bethe equations of XXZ type
where the 2-particle phase factor is a ratio of two trigonometric
polynomials. We show that the solutions are enumerated by Fuss-Catalan
numbers that grow at exponential rate, unlike the polynomial growth rate
associated with the standard XXZ chain.
Alain Connes, "Gravity and the Standard Model: the spectral model"
Noncommutative geometry gives a spectral reformulation
of Riemannian geometry where the analogue of the CKM
(Cabbibo-Kobayashi-Maskawa) matrix provides a complementary
invariant to the spectrum of the Dirac operator to obtain a
complete geometric invariant. This framework is close to
the physical information about the universe which is mostly
of spectral nature, and more versatile since it allows one to
encode a fine structure of space-time. It leads to a remarkably
simple form of the Lagrangian of gravity coupled with the
Standard Model of elementary particles including the Higgs
and massive neutrinos with see-saw mechanism. A test of a
possible exactness of the model at unification scale is
provided by several predictions including the masses of
the top quark and the Higgs as well as the gravitational sector.
Edward Witten, "Quantization And Topological Field Theory"
I will describe the use of two-dimensional
topological field theory to gain a new perspective on the
quantization of classical phase spaces.
Samson Shatashvili, "Supersymmetric Vacua and Quantum Integrability"
Supersymmetric vacua of two dimensional N=4 gauge theories with
matter, softly broken by the twisted masses down to N=2, are shown to be
in one-to-one correspondence with the eigenstates of integrable spin chain
Hamiltonians. Examples include: the Heisenberg SU(2) XXX spin chain is
mapped to the two dimensional U(N) theory with fundamental
hypermultiplets, the XXZ spin chain is mapped to the analogous three
dimensional super-Yang-Mills theory compactified on a circle, the XYZ
spin chain and the eight-vertex model are related to the four dimensional
theory compactified ona torus. A consequence of our duality is the
isomorphism of the quantum cohomology ring of various quiver varieties
and the ring of quantum integrals of motion of various spin chains.
In all these cases the duality extends to any spin group, representations,
boundary conditions, and inhomogeneity, including Sinh-Gordon and
non-linear Schrödinger models as well as dynamical spin chains like
Hubbard model. A lift to a four dimensional N=2*
theory on S2 leads to (instanton-) corrected Bethe equations.
Talk is based on joint paper with N. Nekrasov.
Leon Takhtajan, "Quantum Field Theories on Algebraic Curves"
We formulate quantum field theories of additive, charged, and multiplicative bosons
on an algebraic curve X over an algebraically closed constant field k of characteristic zero.
This formulation, based on Witten's 1987-88 original proposal, uses Serre’s adelic interpretation
of the cohomology. We will show how to develop “differential and integral calculus” on an algebraic
curve X and to introduce algebraic analogs of additive and multiplicative multi- valued functions on
X. Corresponding quantum field theories are defined uniquely if one extends the global symmetries —
Witten’s additive and multiplicative Ward identities — from the k-vector space of rational functions
on X to the k-vector space of additive multi-valued functions, and from the multiplicative group of
rational functions on X to the group of multiplicative multi-valued functions on X. The quantum field
theory of additive bosons is naturally associated with the algebraic de Rham theorem and the
generalized residue theorem, and the quantum field theory of multiplicative bosons — with the
generalized A. Weil reciprocity law.
Roman Jackiw, "Fractional Charge in Planar Systems"
Quantum theory has shown us that dynamical
entities, which take continuous values in classical theory, may
posses only discrete "quantized" values - for example angular
momentum, energy, etc. Here I describe a further quantum phenomenon
affecting quantities that in a
classical physics description take on discrete values - for example
particle number. In the quantum theory this integrality can be lost,
becoming replaced by fractional or even continuous values. Physical
realization of this "quantum weirdness" is described.
Jürg Fröhlich, "Are there mysteries in quantum mechanics?"
I review some of the puzzling features of quantum mechanics and attempt
to show how we presently think they are resolved.
I start with some comments on the point of view brought forward by the
late Moshe Flato and Ludwig Faddeev that most fundamental theories
discovered in the 20th Century arise from precursor theories by
"deformation" and sketch how one can extend this point of view to
explaining the atomistic constitution of matter. I then explain why
quantum mechanics is intrinsically probabilistic and develop an
interpretation of quantum mechanics based on a calculus of "frequencies
of histories of events". I draw attention to the important role played
by decoherence.