Orbifold Riemann Surfaces Research Articles

Correction to: Local index theorem for orbifold Riemann surfaces

Correction to: Local index theorem for orbifold Riemann surfaces

Resolvent Trace Formula and Determinants of n Laplacians on Orbifold Riemann Surfaces

For $n$ a nonnegative integer, we consider the $n$-Laplacian $\Delta_n$ acting on the space of $n$-differentials on a confinite Riemann surface $X$ which has ramification points. The trace formula for the resolvent kernel is developed along the line \`a la Selberg. Using the trace formula, we compute the regularized determinant of $\Delta_n+s(s+2n-1)$, from which we deduce the regularized determinant of $\Delta_n$, denoted by $\det\!'\Delta_n$. Taking into account the contribution from the absolutely continuous spectrum, $\det\!'\Delta_n$ is equal to a constant $\mathcal{C}_n$ times $Z(n)$ when $n\geq 2$. Here $Z(s)$ is the Selberg zeta function of $X$. When $n=0$ or $n=1$, $Z(n)$ is replaced by the leading coefficient of the Taylor expansion of $Z(s)$ around $s=0$ and $s=1$ respectively. The constants $\mathcal{C}_n$ are calculated explicitly. They depend on the genus, the number of cusps, as well as the ramification indices, but is independent of the moduli parameters.

Local index theorem for orbifold Riemann surfaces

We derive a local index theorem in Quillen's form for families of Cauchy-Riemann operators on orbifold Riemann surfaces (or Riemann orbisurfaces) that are quotients of the hyperbolic plane by the action of cofinite finitely generated Fuchsian groups. Each conical point (or a conjugacy class of primitive elliptic elements in the Fuchsian group) gives rise to an extra term in the local index theorem that is proportional to the symplectic form of a new K\"{a}hler metric on the moduli space of Riemann orbisurfaces. We find a simple formula for a local K\"{a}hler potential of the elliptic metric and show that when the order of elliptic element becomes large, the elliptic metric converges to the cuspidal one corresponding to a puncture on the orbisurface (or a conjugacy class of primitive parabolic elements). We also give a simple example of a relation between the elliptic metric and special values of Selberg's zeta function.

Seifert fibering operators in 3d mathcal{N}=2 theories

We study 3d mathcal{N}=2 supersymmetric gauge theories on closed oriented Seifert manifolds — circle bundles over an orbifold Riemann surface —, with a gauge group G given by a product of simply-connected and/or unitary Lie groups. Our main result is an exact formula for the supersymmetric partition function on any Seifert manifold, generalizing previous results on lens spaces. We explain how the result for an arbitrary Seifert geometry can be obtained by combining simple building blocks, the “fibering operators.” These operators are half-BPS line defects, whose insertion along the S1 fiber has the effect of changing the topology of the Seifert fibration. We also point out that most supersymmetric partition functions on Seifert manifolds admit a discrete refinement, corresponding to the freedom in choosing a three-dimensional spin structure. As a strong consistency check on our result, we show that the Seifert partition functions match exactly across infrared dualities. The duality relations are given by intricate (and seemingly new) mathematical identities, which we tested numerically. Finally, we discuss in detail the supersymmetric partition function on the lens space L(p, q)b with rational squashing parameter b2 ∈ ℚ, comparing our formalism to previous results, and explaining the relationship between the fibering operators and the three-dimensional holomorphic blocks.

On Kawai theorem for orbifold Riemann surfaces

We prove a generalization of Kawai theorem for the case of orbifold Riemann surface. The computation is based on a formula for the differential of a holomorphic map from the cotangent bundle of the Teichmuller space to the $${\mathrm {PSL}}(2,\mathbb {C})$$ -character variety, which allows to evaluate explicitly the pullback of Goldman symplectic form in the spirit of Riemann bilinear relations. As a corollary, we obtain a generalization of Goldman’s theorem that the pullback of Goldman symplectic form on the $${\mathrm {PSL}}(2,\mathbb {R})$$ -character variety is a symplectic form of the Weil–Petersson metric on the Teichmuller space.

On symplectic vortex equations over a compact orbifold Riemann surface

Making use of theory of differentiable stacks, we study symplectic vortex equations over a compact orbifold Riemann surface. We discuss the category of representable morphisms from a compact orbifold Riemann surface to a quotient stack. After that we define symplectic vortex equations over a compact orbifold Riemann surface. We also discuss the moduli space of solutions to the equations for linear actions of the circle group on the complex plane.

The Witten–Reshetikhin–Turaev invariants of finite order mapping tori I

We formulate the Asymptotic Expansion Conjecture for the Witten–Reshetikhin–Turaev quantum invariants of closed oriented 3-manifolds. For of finite order mapping tori, we study these quantum invariants via the geometric gauge theory approach to the corresponding quantum representations and prove, using a version of the Lefschetz–Riemann–Roch theorem due to Baum, Fulton, MacPherson & Quart, that the quantum invariants can be expressed as a sum over the components of the moduli space of flat connections on the mapping torus. Moreover, we show that the term corresponding to a component is a polynomial in the level k, weighted by a complex phase, which is k times the Chern–Simons invariant corresponding to the component. We express the coefficients of these polynomials in terms of cohomological pairings on the fixed point set of the moduli space of flat connections on the surface. We explicitly describe the fixed point set in terms of moduli spaces of the quotient orbifold Riemann surface and for the smooth components we express the aforementioned coefficients in terms of the known generators of the cohomology ring. We provide an explicit formula in terms of the Seifert invariants of the mapping torus for the contributions from each of the smooth components. We further establish that the Asymptotic Expansion Conjecture for manifolds and we verify the Growth Rate Conjecture for these finite order mapping tori.

One-relator Kähler groups

We prove that a one-relator group $G$ is K\"ahler if and only if either $G$ is finite cyclic or $G$ is isomorphic to the fundamental group of a compact orbifold Riemann surface of genus $g > 0$ with at most one cone point of order $n$: $$< a_1\, b_1\, \,...\, a_g\, b_g\, \mid\, (\prod_{i=1}^g [a_i\, b_i])^n>\, .$$

Orbifold Riemann surfaces and geodesic algebras

We study the Teichmüller theory of Riemann surfaces with orbifold points of order 2 using the fat graph technique. The previously developed technique of quantization, classical and quantum mapping-class group transformations, and Poisson and quantum algebras of geodesic functions is applicable to the surfaces with orbifold points. We describe classical and quantum braid group relations for particular sets of geodesic functions corresponding to An and Dn algebras and describe their central elements for the Poisson and quantum algebras.

Dn quivers from branes

D-branes can end on orbifold planes if the action of the orbifold group includes (-1)^{F_L}. We consider configurations of D-branes ending on such orbifolds and study the low-energy theory on their worldvolume. We apply our results to gauge theories with eight supercharges in three and four dimensions. We explain how mirror symmetry for N=4 d=3 gauge theories with gauge group Sp(k) and matter in the antisymmetric tensor and fundamental representations follows from S-duality of IIB string theory. We argue that some of these theories have hidden Fayet-Iliopoulos deformations, not visible classically. We also study a class of finite N=2 d=4 theories (so-called D_n quiver theories) and find their exact solution. The integrable model corresponding to the exact solution is a Hitchin system on an orbifold Riemann surface. We also give a simple derivation of the S-duality group of these theories based on their relationship to SO(2n) instantons on R^2\times T^2.