A construction of the Deligne-Mumford orbifold

Abstract

The Deligne–Mumford moduli space is the space \overline{\mathcal{M}}_{g,n} of isomorphism classes of stable nodal Riemann surfaces of arithmetic genus g with n marked points. A marked nodal Riemann surface is stable if and only if its isomorphism group is finite. We introduce the notion of a universal unfolding of a marked nodal Riemann surface and show that it exists if and only if the surface is stable. A natural construction based on the existence of universal unfoldings endows the Deligne–Mumford moduli space with an orbifold structure. We include a proof of compactness. Our proofs use the methods of differential geometry rather than algebraic geometry.

Quaternionic Dirac oscillator

“…The universality of this approach has been apprised in many different fields [25,26], see also section 3 in [27], where a 'topological normal form' for molecules is introduced, and the number of the redistributed energy levels is related to the Chern number. It should be placed next to two other groups of results in the literature: (i) the 'folk' theorems 'Fredholm index = spectral flow' for rather general families of self-adjoint operators on Hilbert spaces [28,29], and (ii) the index theorems 'Fredholm index = Bott index', see [30][31][32], and chapter 11 in [33], or 'Fredholm index = Chern number c k (Λ)' if the slow phase space P is compact (see [21,28,34,35] or chapter 12 in [33]) and we can describe the spectral flow using topological invariants of individual 'bands' or Λ-bundles of semi-quantum eigenstates over P at non-critical values of formal parameter α = 0, cf section 1.2 of [16].…”