Affine Orbifolds Research Articles

Quiver asymptotics: free chiral ring

The large N generating functions for the counting of chiral operators in , four-dimensional quiver gauge theories have previously been obtained in terms of the weighted adjacency matrix of the quiver diagram. We introduce the methods of multi-variate asymptotic analysis to study this counting in the limit of large charges. We describe a Hagedorn phase transition associated with the asymptotics, which refines and generalizes known results on the 2-matrix harmonic oscillator. Explicit results are obtained for two infinite classes of quiver theories, namely the generalized clover quivers and affine orbifold quivers.

An analog of Chern’s conjecture for the Euler–Satake characteristic of affine orbifolds

S.S. Chern conjectured that the Euler characteristic of every closed affine manifold has to vanish. We present an analog of this conjecture stating that the Euler–Satake characteristic of any compact affine orbifold is equal to zero. We prove that Chern’s conjecture is equivalent to its analog for the Euler–Satake characteristic of compact affine orbifolds, not necessarily effective. This fact allowed us to extend to orbifolds sufficient conditions for Chern’s conjecture proved by Klingler and Kostant–Sullivan. Thus, we prove that, if an n-dimensional compact affine orbifold N is complete or if its holonomy group belongs to the special linear group SL(n,R), then the Euler–Satake characteristic of N has to vanish. An application to pseudo-Riemannian orbifolds is considered. We give examples of orbifolds from the class under investigation. In particular, we construct an example of a compact incomplete affine orbifold with vanishing Euler–Satake characteristic, the holonomy group of which is not contained in SL(n,R).

On Affine Orbifold Nets Associated with Outer Automorphisms

We construct solitons in affine orbifold nets associated with outer automorphisms, and we show that our construction gives all the twisted representations of the fixed point subnet. This allows us to settle a number of questions concerning such orbifold constructions.

A note on hyperbolic leaves and wild laminations of rational functions

We study the affine orbifold laminations that were constructed by Lyubich and Minsky (J. Differential Geom. 47(1) (1997), pp. 17–94). An important question left open in the original construction is whether these laminations are always locally compact. We show that this is not the case. The counterexample we construct has the property that the regular leaf space contains (many) hyperbolic leaves that intersect the Julia set; whether this can happen was itself a question raised by Lyubich and Minsky.

Proper groupoids and momentum maps: Linearization, affinity, and convexity

We show that proper Lie groupoids are locally linearizable. As a consequence, the orbit space of a proper Lie groupoid is a smooth orbispace (a Hausdorff space which locally looks like the quotient of a vector space by a linear compact Lie group action). In the case of proper (quasi-)symplectic groupoids, the orbit space admits a natural integral affine structure, which makes it into an affine orbifold with locally convex polyhedral boundary, and the local structure near each boundary point is isomorphic to that of a Weyl chamber of a compact Lie group. We then apply these results to the study of momentum maps of Hamiltonian actions of proper (quasi-)symplectic groupoids, and show that these momentum maps preserve natural transverse affine structures with local convexity properties. Many convexity theorems in the literature can be recovered from this last statement and some elementary results about affine maps.

Affine Orbifolds and Rational Conformal Field Theory Extensions of W 1+∞

Chiral orbifold models are defined as gauge field theories with a finite gauge group $\Gamma$. We start with a conformal current algebra A associated with a connected compact Lie group G and a negative definite integral invariant bilinear form on its Lie algebra. Any finite group $\Gamma$ of inner automorphisms or A (in particular, any finite subgroup of G) gives rise to a gauge theory with a chiral subalgebra $A^{\Gamma}\subset A$ of local observables invariant under $\Gamma$. A set of positive energy $A^{\Gamma}$ modules is constructed whose characters span, under some assumptions on $\Gamma$, a finite dimensional unitary representation of $SL(2,Z)$. We compute their asymptotic dimensions (thus singling out the nontrivial orbifold modules) and find explicit formulae for the modular transformations and hence, for the fusion rules. As an application we construct a family of rational conformal field theory (RCFT) extensions of $W_{1+\infty}$ that appear to provide a bridge between two approaches to the quantum Hall effect.

Rational conformal field theory extensions of W1+∞ in terms of bilocal fields

The rational conformal field theory (RCFT) extensions of W_{1+infinity} at c=1 are in one-to-one correspondence with 1-dimensional integral lattices L(m). Each extension is associated with a pair of oppositely charged ``vertex operators" of charge square m in N. Their product defines a bilocal field V_m(z_1,z_2) whose expansion in powers of z_{12}=z_1-z_2 gives rise to a series of (neutral) local quasiprimary fields V^l(z,m) (of dimension l+1). The associated bilocal exponential of a normalized current generates the W_{1+infinity} algebra spanned by the V^l(z,1) (and the unit operator). The extension of this construction to higher (integer) values of the central charge c is also considered. Applications to a quantum Hall system require computing characters (i.e., chiral partition functions) depending not just on the modular parameter tau, but also on a chemical potential zeta. We compute such a zeta dependence of orbifold characters, thus extending the range of applications of a recent study of affine orbifolds.