This paper gives methods for understanding invariants of symplectic quotients. The symplectic quotients that we consider are compact symplectic manifolds (or more generally orbifolds), which arise as the symplectic quotients of a symplectic manifold by a compact torus. A companion paper [23] examines symplectic quotients by a nonabelian group, showing how to reduce to the maximal torus. Through...

In this paper, we establish two results concerning algebraic (C,+)-actions on C. First of all, let φ be an algebraic (C,+)-action on C3. By a result of Miyanishi, its ring of invariants is isomorphic to C[t1, t2]. If f1, f2 generate this ring, the quotient map of φ is the map F : C3 → C2, x 7→ (f1(x), f2(x)). By using some topological arguments, we prove that F is always surjective. Secondly we...

In this paper we reopen the discussion of gauging the two-dimensional off-shell (2, 2) supersymmetric sigma models written in terms of semichiral superfields. The associated target space geometry of this particular sigma model is generalized Kähler (or bi-hermitean with two noncommuting complex structures). The gauging of the isometries of the sigma model is now done by coupling the semichiral ...

If X is a separable Banach space, we consider the existence of non-trivial twisted sums 0 → C(K) → Y → X → 0, where K = [0, 1] or ωω . For the case K = [0, 1] we show that there exists a twisted sum whose quotient map is strictly singular if and only if X contains no copy of `1. If K = ωω we prove an analogue of a theorem of Johnson and Zippin (for K = [0, 1]) by showing that all such twisted s...

The quantum modular invariant jqt(θ) of θ ∈R is defined as a discontinuous PGL2(Z)-invariant multi-valued map using the distance-to-the-nearest-integer function ‖ · ‖. For θ ∈Q it is shown that jqt(θ) =∞ and for quadratic irrationalities PARI/GP experiments suggest that jqt(θ) is a finite set. In the case of the golden mean φ, we produce explicit formulas involving weighted versions of the Roge...

Let g be a complex semisimple Lie algebra, and f : g → G\\g the adjoint quotient map. Springer theory of Weyl group representations can be seen as the study of the singularities of f . In this paper, we give a generalization of Springer theory to visible, polar representations. It is a class of rational representations of reductive groups over C, for which the invariant theory works by analogy ...

Let M be a connected compact quantizable Kähler manifold equipped with a Hamiltonian action of a connected compact Lie group G. Let M//G = φ(0)/G = M0 be the symplectic quotient at value 0 of the moment map φ. The space M0 may in general not be smooth. It is known that, as vector spaces, there is a natural isomorphism between the quantum Hilbert space over M0 and the G-invariant subspace of the...

where Gn, denoted by Bn by Goncharov, is placed at degree 1. To save space we here only point out that Gn(F ) are quotient groups of Z[P 1 F ] and refer the interested readers to [G2, p.49] for the detailed definition of these groups. On the other hand, currently there are two versions of higher Chow groups available: simplicial and cubical, which are isomorphic (cf. [Le]). We will recall the c...

In the paper 10] (see also 2]), the authors construct a holomorphic map from the modular curve X(k) = H 2 =?(k), where ?(k) is the congruence subgroup of ? = PSL(2; Z), into the projective space C P (k?3 2) , via theta functions. This holomorphic map is equivariant with respect to the action of PSL(2; Z=k) on both factors, and extends to the conformal compactiication X(k). Thus, X(k) can be exh...

We generalize the results of hep-th/0701214 about the covariant c-map to include the perturbative quantum corrections. We also perform explicitly the superconformal quotient from the hyperkähler cone obtained by the quantum c-map to the quaternion-Kähler space, which is the moduli space of hypermultiplets. As a result, the perturbatively corrected metric on the moduli space is found in a simpli...