Arbitrary Riemann Surfaces Research Articles

Framed parabolic sheaves on a trinion

We consider for structure groups [Formula: see text] a densely defined toric structure on the moduli space of framed parabolic sheaves on a three-punctured sphere, which degenerates to an actual toric structure. In combination with previous degeneration results, these extend to similar moduli for arbitrary Riemann surfaces.

Effective gravitational action for 2D massive Majorana fermions on arbitrary genus Riemann surfaces

We explore the effective gravitational action for two-dimensional massive Euclidean Majorana fermions in a small mass expansion, continuing and completing the study initiated in a previous paper [1]. We perform a detailed analysis of local zeta functions, heat kernels, and Green’s functions of the Dirac operator on arbitrary Riemann surfaces. We obtain the full expansion of the effective gravitational action to all orders in m2. For genus one and larger, this requires the understanding of the role of the zero-modes of the (massless) Dirac operator which is worked out.Besides the Liouville action, at order m0, which only involves the background metric and the conformal factor σ, the various contributions to the effective gravitational action at higher orders in m2 can be expressed in terms of integrals of the renormalized Green’s function at coinciding points of the squared (massless) Dirac operator, as well as of higher Green’s functions. In particular, at order m2, these contributions can be re-written as a term ∫ e2σσ characteristic of the Mabuchi action, much as for 2D massive scalars, as well as several other terms that are multi-local in the conformal factor σ and involve the Green’s functions of the massless Dirac operator and the renormalized Green’s function, but for the background metric only, and certain area-like parameters related to the zero-modes.

Diagrams and harmonic maps, revisited

We extend many known results for harmonic maps from the 2-sphere into a Grassmannian to harmonic maps of finite uniton number from an arbitrary Riemann surface. Our method relies on a new theory of nilpotent cycles arising from the diagrams of F.E. Burstall and the second author associated to such harmonic maps; these properties arise from a criterion for finiteness of the uniton number found recently by the authors with A. Aleman. Applications include a new classification result on minimal surfaces of constant curvature and a constancy result for finite type harmonic maps.

3-Dimensional mixed BF theory and Hitchin’s integrable system

The affine Gaudin model, associated with an untwisted affine Kac–Moody algebra, is known to arise from a certain gauge fixing of 4-dimensional mixed topological–holomorphic Chern–Simons theory in the Hamiltonian framework. We show that the finite Gaudin model, associated with a finite-dimensional semisimple Lie algebra, or more generally the tamely ramified Hitchin system on an arbitrary Riemann surface, can likewise be obtained from a similar gauge fixing of 3-dimensional mixed BF theory in the Hamiltonian framework.

Abelianization of Matrix Orthogonal Polynomials

Abstract The main goal of the paper is to connect matrix polynomial biorthogonality on a contour in the plane with a suitable notion of scalar, multi-point Padé approximation on an arbitrary Riemann surface endowed with a rational map to the Riemann sphere. To this end we introduce an appropriate notion of (scalar) multi-point Padé approximation on a Riemann surface and corresponding notion of biorthogonality of sections of the semi-canonical bundle (half-differentials). Several examples are offered in illustration of the new notions.

Minimal $(D,D)$ conformal matter and generalizations of the van Diejen model

We consider supersymmetric surface defects in compactifications of the 6d6d minimal (D_{N+3},D_{N+3})(DN+3,DN+3) conformal matter theories on a punctured Riemann surface. For the case of N=1N=1 such defects are introduced into the supersymmetric index computations by an action of the BC_1\,(\sim A_1\sim C_1)BC1(∼A1∼C1) van Diejen model. We (re)derive this fact using three different field theoretic descriptions of the four dimensional models. The three field theoretic descriptions are naturally associated with algebras A_{N=1}AN=1, C_{N=1}CN=1, and (A_1)^{N=1}(A1)N=1. The indices of these 4d4d theories give rise to three different Kernel functions for the BC_1BC1 van Diejen model. We then consider the generalizations with N>1N>1. The operators introducing defects into the index computations are certain A_{N}AN, C_NCN, and (A_1)^{N}(A1)N generalizations of the van Diejen model. The three different generalizations are directly related to three different effective gauge theory descriptions one can obtain by compactifying the minimal (D_{N+3},D_{N+3})(DN+3,DN+3) conformal matter theories on a circle to five dimensions. We explicitly compute the operators for the A_NAN case, and derive various properties these operators have to satisfy as a consequence of 4d4d dualities following from the geometric setup. In some cases we are able to verify these properties which in turn serve as checks of said dualities. As a by-product of our constructions we also discuss a simple Lagrangian description of a theory corresponding to compactification on a sphere with three maximal punctures of the minimal (D_5,D_5)(D5,D5) conformal matter and as consequence give explicit Lagrangian constructions of compactifications of this 6d SCFT on arbitrary Riemann surfaces.

The Heights Theorem for infinite Riemann surfaces

Marden and Strebel established the Heights Theorem for integrable holomorphic quadratic differentials on parabolic Riemann surfaces. We extends the validity of the Heights Theorem to all surfaces whose fundamental group is of the first kind. In fact, we establish a more general result: the horizontal map which assigns to each integrable holomorphic quadratic differential a measured lamination obtained by straightening the horizontal trajectories of the quadratic differential is injective for an arbitrary Riemann surface with a conformal hyperbolic metric. This was established by Strebel in the case of the unit disk. When a hyperbolic surface has a bounded geodesic pants decomposition, the horizontal map assigns a bounded measured lamination to each integrable holomorphic quadratic differential. When surface has a sequence of closed geodesics whose lengths go to zero, then there exists an integrable holomorphic quadratic differential whose horizontal measured lamination is not bounded.

Toverline{T} -deformation of q-Yang-Mills theory

We derive the Toverline{T} -perturbed version of two-dimensional q-deformed Yang-Mills theory on an arbitrary Riemann surface by coupling the unperturbed theory in the first order formalism to Jackiw-Teitelboim gravity. We show that the Toverline{T} -deformation results in a breakdown of the connection with a Chern-Simons theory on a Seifert manifold, and of the large N factorization into chiral and anti-chiral sectors. For the U(N) gauge theory on the sphere, we show that the large N phase transition persists, and that it is of third order and induced by instantons. The effect of the Toverline{T} -deformation is to decrease the critical value of the ’t Hooft coupling, and also to extend the class of line bundles for which the phase transition occurs. The same results are shown to hold for (q, t)-deformed Yang-Mills theory. We also explicitly evaluate the entanglement entropy in the large N limit of Yang-Mills theory, showing that the Toverline{T} -deformation decreases the contribution of the Boltzmann entropy.

On mappings with finite length distortion on Riemann surfaces

The present paper is a natural continuation of our previous paper (2017) on the boundary behavior of mappings in the Sobolev classes on Riemann surfaces, where the reader will be able to find the corresponding historic comments and a discussion of many definitions and relevant results. The given paper was devoted to the theory of the boundary behavior of mappings with finite distortion by Iwaniec on Riemannian surfaces first introduced for the plane in the paper of Iwaniec T. and Sverak V. (1993) On mappings with integrable dilatation and then extended to the spatial case in the monograph of Iwaniec T. and Martin G. (2001) devoted to Geometric function theory and non-linear analysis. At the present paper, it is developed the theory of the boundary behavior of the so--called mappings with finite length distortion first introduced in the paper of Martio O., Ryazanov V., Srebro U. and Yakubov~E. (2004) in the spatial case, see also Chapter 8 in their monograph (2009) on Moduli in modern mapping theory. As it was shown in the paper of Kovtonyuk D., Petkov I. and Ryazanov V. (2017) On the boundary behavior of mappings with finite distortion in the plane, such mappings, generally speaking, are not mappings with finite distortion by Iwaniec because their first partial derivatives can be not locally integrable. At the same time, this class is a generalization of the known class of mappings with bounded distortion by Martio--Vaisala from their paper (1988). Moreover, this class contains as a subclass the so-called finitely bi-Lipschitz mappings introduced for the spatial case in the paper of Kovtonyuk D. and Ryazanov V. (2011) On the boundary behavior of generalized quasi-isometries, that in turn are a natural generalization of the well-known classes of bi-Lipschitz mappings as well as isometries and quasi-isometries. In the research of the local and boundary behavior of mappings with finite length distortion in the spatial case, the key fact was that they satisfy some modulus inequalities which was a motivation for the consideration more wide classes of mappings, in particular, the Q-homeomorphisms (2005) and the mappings with finite area distortion (2008). Hence it is natural that under the research of mappings with finite length distortion on Riemann surfaces we start from establishing the corresponding modulus inequalities that are the main tool for us. On this basis, we prove here a series of criteria in terms of dilatations for the continuous and homeomorphic extension to the boundary of the mappings with finite length distortion between domains on arbitrary Riemann surfaces.

Plemelj\u2013Sokhotski isomorphism for quasicircles in Riemann surfaces and the Schiffer operators

Let R be a compact Riemann surface and Gamma be a Jordan curve separating R into connected components Sigma _1 and Sigma _2. We consider Calderón–Zygmund type operators T(Sigma _1,Sigma _k) taking the space of L^2 anti-holomorphic one-forms on Sigma _1 to the space of L^2 holomorphic one-forms on Sigma _k for k=1,2, which we call the Schiffer operators. We extend results of Max Schiffer and others, which were confined to analytic Jordan curves Gamma , to general quasicircles, and prove new identities for adjoints of the Schiffer operators. Furthermore, let V be the space of anti-holomorphic one-forms which are orthogonal to L^2 anti-holomorphic one-forms on R with respect to the inner product on Sigma _1. We show that the restriction of the Schiffer operator T(Sigma _1,Sigma _2) to V is an isomorphism onto the set of exact holomorphic one-forms on Sigma _2. Using the relation between this Schiffer operator and a Cauchy-type integral involving Green’s function, we also derive a jump decomposition (on arbitrary Riemann surfaces) for quasicircles and initial data which are boundary values of Dirichlet-bounded harmonic functions and satisfy the classical algebraic constraints. In particular we show that the jump operator is an isomorphism on the subspace determined by these constraints.

Gauged sigma models and magnetic Skyrmions

We define a gauged non-linear sigma model for a 2-sphere valued field and a SU(2)SU(2) connection on an arbitrary Riemann surface whose energy functional reduces to that for critically coupled magnetic skyrmions in the plane, with arbitrary Dzyaloshinskii-Moriya interaction, for a suitably chosen gauge field. We use the interplay of unitary and holomorphic structures to derive a general solution of the first order Bogomol’nyi equation of the model for any given connection. We illustrate this formula with examples, and also point out applications to the study of impurities.

Relationship between the Discrete and Resonance Spectrum for the Laplace Operator on a Noncompact Hyperbolic Riemann Surface

We consider arbitrary noncompact hyperbolic Riemann surfaces of finite area. For such surfaces, we obtain identities relating the discrete spectrum of the Laplace operator to the resonance spectrum (formed by the poles of the scattering matrix). These identities depend on the choice of a test function. We indicate a class of admissible test functions and consider two examples corresponding to specific choices of the test function.

Affine Gaudin models and hypergeometric functions on affine opers

We conjecture that quantum Gaudin models in affine types admit families of higher Hamiltonians, labelled by the (countably infinite set of) exponents, whose eigenvalues are given by functions on a space of meromorphic opers associated with the Langlands dual Lie algebra. This is in direct analogy with the situation in finite types. However, in stark contrast to finite types, we prove that in affine types such functions take the form of hypergeometric integrals, over cycles of a twisted homology defined by the levels of the modules at the marked points. That result prompts the further conjecture that the Hamiltonians themselves are naturally expressed as such integrals.We go on to describe the space of meromorphic affine opers on an arbitrary Riemann surface. We prove that it fibres over the space of meromorphic connections on the canonical line bundle Ω. Each fibre is isomorphic to the direct product of the space of sections of the square of Ω with the direct product, over the exponents j not equal to 1, of the twisted cohomology of the jth tensor power of Ω.

E‐String Theory on Riemann Surfaces

AbstractWe study compactifications of the 6d E‐string theory, the theory of a small E8 instanton, to four dimensions. In particular we identify field theories in four dimensions corresponding to compactifications on arbitrary Riemann surfaces with punctures and with arbitrary non‐abelian flat connections as well as fluxes for the abelian sub‐groups of the E8 flavor symmetry. This sheds light on emergent symmetries in a number of 4d SCFTs (including the ‘E7 surprise’ theory) as well as leads to new predictions for a large number of 4‐dimensional exceptional dualities and symmetries.

ON THE DISTRIBUTION OF ZEROS OF THE DERIVATIVE OF SELBERG’S ZETA FUNCTION ASSOCIATED TO FINITE VOLUME RIEMANN SURFACES

We study the distribution of zeros of the derivative of the Selberg zeta function associated to a noncompact, finite volume hyperbolic Riemann surface $M$. Actually, we study the zeros of $(Z_{M}H_{M})^{\prime }$, where $Z_{M}$ is the Selberg zeta function and $H_{M}$ is the Dirichlet series component of the scattering matrix, both associated to an arbitrary finite volume hyperbolic Riemann surface $M$. Our main results address finiteness of number of zeros of $(Z_{M}H_{M})^{\prime }$ in the half-plane $\operatorname{Re}(s)<1/2$, an asymptotic count for the vertical distribution of zeros, and an asymptotic count for the horizontal distance of zeros. One realization of the spectral analysis of the Laplacian is the location of the zeros of $Z_{M}$, or, equivalently, the zeros of $Z_{M}H_{M}$. Our analysis yields an invariant $A_{M}$ which appears in the vertical and weighted vertical distribution of zeros of $(Z_{M}H_{M})^{\prime }$, and we show that $A_{M}$ has different values for surfaces associated to two topologically equivalent yet different arithmetically defined Fuchsian groups. We view this aspect of our main theorem as indicating the existence of further spectral phenomena which provides an additional refinement within the set of arithmetically defined Fuchsian groups.

Explicit Hodge-Type Decomposition on Projective Complete Intersections

We construct an explicit homotopy formula for the \(\bar{\partial }\)-complex on a reduced complete intersection subvariety \(V\subset {\mathbb {C}}{\mathbb {P}}^n\). This formula can be interpreted as an explicit Hodge-type decomposition for residual currents on V. As a first application of this formula we obtained the explicit Hodge decomposition on arbitrary Riemann surfaces.

Entanglement Entropy in 2D non-abelian pure gauge theory

We compute the Entanglement Entropy (EE) of a bipartition in 2D pure non-abelian U(N) gauge theory. We obtain a general expression for EE on an arbitrary Riemann surface. We find that due to area-preserving diffeomorphism symmetry EE does not depend on the size of the subsystem, but only on the number of disjoint intervals defining the bipartition.In the strong coupling limit on a torus we find that the scaling of the EE at small temperature is given by S(T)−S(0)=O(mgapTe−mgapT), which is similar to the scaling for the matter fields recently derived in literature. In the large N limit we compute all of the Renyi entropies and identify the Douglas–Kazakov phase transition.

Computation of Quasi-Conformal Surface Maps Using Discrete Beltrami Flow

The manipulation of surface homeomorphisms is an important aspect in three-dimensional modeling and surface processing. Every homeomorphic surface map can be considered as a quasi-conformal map, with its local nonconformal distortion given by its Beltrami differential. As a generalization of conformal maps, quasi-conformal maps are of great interest in mathematical study and real applications. Efficient and accurate computational construction of desirable quasi-conformal maps between general surfaces is crucial. However, in the literature we have reviewed, all existing computational works on construction of quasi-conformal maps to or from a compact domain require global parametrization onto the plane and are difficult to directly apply to maps between arbitrary surfaces. This work fills the gap by proposing to compute quasi-conformal homeomorphisms between arbitrary Riemann surfaces using discrete Beltrami flow, which is a vector field corresponding to the adjustment to the intrinsic Beltrami differential...

Decomposition of integrable holomorphic quadratic differential on Riemann surface of infinite type

In this paper, decomposition of the integrable holomorphic quadratic differential on Riemann surface of infinite type is studied. Hubbard, Schleicher and Shishikura gave a thick-thin decomposition on Riemann surface of finite type with an integrable holomorphic quadratic differential and they thought their result might be generalized to arbitrary hyperbolic Riemann surface of infinite type. We confirm what they thought is right and give a proposition (Proposition 2.2) of its own interest.

Exponential Thurston maps and limits of quadratic differentials

We give a topological characterization of postsingularly finite topological exponential maps, i.e., universal covers g : C → C ∖ { 0 } g\colon \mathbb {C}\to \mathbb {C}\setminus \{0\} such that 0 0 has a finite orbit. Such a map either is Thurston equivalent to a unique holomorphic exponential map λ e z \lambda e^z or it has a topological obstruction called a degenerate Levy cycle. This is the first analog of Thurston’s topological characterization theorem of rational maps, as published by Douady and Hubbard, for the case of infinite degree. One main tool is a theorem about the distribution of mass of an integrable quadratic differential with a given number of poles, providing an almost compact space of models for the entire mass of quadratic differentials. This theorem is given for arbitrary Riemann surfaces of finite type in a uniform way.