SU(N) gauge fields on a cylindrical spacetime are canonically quantized via two routes revealing almost equivalent but different quantizations. After removing all continuous gauge degrees of freedom, the canonical coordinate Aμ (in the Cartan subalgebra h) is quantized. The compact route, as in lattice gauge theory, quantizes the Wilson loop W , projecting out gauge invariant wavefunctions on t...

Let L be the Lie algebra of a simple algebraic group defined over F and let H be a split Cartan subalgebra of L. Let R = (X,Φ, Y,Φ∨) be the root datum of L, so that H = Y ⊗ F, and let 〈·, ·〉 : Φ × Φ∨ 7→ Z be the corresponding bilinear form. This bilinear form induces a linear form on the roots of L by defining α : h 7→ P i〈α, yi〉ti, where h = P i yi⊗ ti. Given a root α, we define the multiplici...

We fix a base field k of characteristic not dividing l. We suppose that k contains a primitive l-th root of unity ζ , and fix it. Starting from these data, one defines certain category C. Its objects are finite dimensional X-graded k-vector spaces equipped with an action of Lusztig’s ”small” quantum group u (cf. [L2]) such that the action of its Cartan subalgebra is compatible with the X-gradin...

The structure constants of quantum Lie algebras depend on a quantum deformation parameter q and they reduce to the classical structure constants of a Lie algebra at q = 1. We explain the relationship between the structure constants of quantum Lie algebras and quantum Clebsch-Gordan coefficients for adjoint ⊗ adjoint → adjoint. We present a practical method for the determination of these quantum...

We discuss the bosonization of non-relativistic fermions in one space dimension in terms of bilocal operators which are naturally related to the generators of W infinity algebra. The resulting system is analogous to the problem of a spin in a magnetic field for the group W -infinity. The new dynamical variables turn out to be W -infinity group elements valued in the coset W -infinity/H where H ...

Let the notation be as in the previous lecture. Let G be any complex semisimple Lie group unless otherwise specified. Let us recall that we are looking for the differential operators D1, · · · , Dr on the maximal torus which commute with the operator Q(q ∂ ∂q )−ΣQkkqk. Our approach will be to construct integrable system, called Toda lattice, with the Hamiltonian Q(q ∂ ∂q ) − ΣQkkqk. Then the op...

Let 9 be a reductive complex Lie algebra, with adjoint group G, Cartan subalgebra ~ and Weyl group W. Then G acts naturally on the algebra of polynomial functions &'(g) and hence on the ring of differential operators with polynomial coefficients, .97(g). Similarly, W acts on ~ and hence on .97(~). In [BC2], Harish-Chandra defined an algebra homomorphism J : .97(g)G -t .97(~)w. Recently, Wallach...

In the framework of McKay correspondence we determine, for every finite subgroup Γ of SL4C, how the finite dimensional irreducible representations of SL4C decompose under the action of Γ. Let h be a Cartan subalgebra of sl4C and let ̟1, ̟2, ̟3 be the corresponding fundamental weights. For (p, q, r) ∈ N, the restriction πp,q,r|Γ of the irreducible representation πp,q,r of highest weight p̟1 + q̟2 + r̟...

QED 2 with mass and N flavors of fermions is constructed using Euclidean path integrals. The fermion masses are treated perturbatively and the convergence of the mass perturbation series is proven for a finite space-time cutoff. The expectation functional is decomposed into clustering θ-vacua and their properties are compared to the θ-vacua of QCD for zero fermion mass. The sector that is creat...

We consider Yang-Mills theories with general gauge groups G and twists on the four torus. We find consistent boundary conditions for gauge fields in all instanton sectors. An extended Abelian projection with respect to the Polyakov loop operator is presented, where A 0 is independent of time and in the Cartan subalgebra. Fundamental domains for the gauge fixed A 0 are constructed for arbitrary ...