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Centre Bernoulli
 Ecole Polytechnique Fédérale de Lausanne

Mathematical Physics: from XX to XXI century

in honor of the 75th birthday of Ludwig Faddeev

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.