### 6 lectures, one hour each

**Prof. Mikhail SHIFMAN*** (University of Minnesota (USA)
et Laboratoire de Physique Théorique (Orsay)

**Polyakov’s Confinement in Three and Four Dimensions**

3 Mercredis [**8, 15 et 22 oct**.] et 3 Vendredis [**10, 17 et 24 oct**]
de 15h à 16h, à Orsay - LPT (Bat.210), AMPHI-1

This lecture course begins with a pedagogical presentation of
the Polyakov Abelian confinement in three dimensions.
The crucial theoretical ingredients are dualization and
instantons-monopoles whose contribution is responsible for
dual photon mass generation. Dual photon in this theory is a
(pseudo)scalar phase field which lives on .
Confining strings are equivalent to domain lines, with
vortices of the field at the endpoints. In the original
formulation (before dualization) the vortices are electric charges.
Confinement of electric charges is explicit and occurs at weak coupling
which makes it fully calculable. Introduction of massless quarks
suppresses Polyakov’s confinement.

In the second part of the lecture course I discuss recent developments
which may have serious implications for four-dimensional QCD-like
theories.
If we formulate QCD-like theories on with a small
radius of the compactified dimension *and*
stabilize the center symmetry, then we deal with a weakly coupled gauge
theory,
with SU broken down to U.
The low-energy theory confines \`*a* la Polyakov, even
in spite of massless quarks that can be present in the low-energy spectrum.
Moreover, with one quark flavor, a discrete chiral symmetry of the
theory is spontaneously
broken. One can argue that the transition to large (i.e. to
four-dimensional
one flavor QCD ) is smooth. If so, one can treat qualitatively, and,
perhaps, even semiquantitavily,
basic regularities of four-dimensional QCD.

With two or more flavors we get a remarkable example of of a dynamical
gauge theory with confinement
but without chiral symmetry breaking.

—

* Prof. M. Shifman est un des 5 lauréats des prestigieuses Chaires
internationales de recherche Blaise Pascal 2007.