Non-compact Surfaces Research Articles

Flavor symmetry of 5d SCFTs. Part I. General setup

A large class of 5d superconformal field theories (SCFTs) can be constructed by integrating out BPS particles from 6d SCFTs compactified on a circle. We describe a general method for extracting the flavor symmetry of any 5d SCFT lying in this class. For this purpose, we utilize the geometric engineering of 5d mathcal{N} = 1 theories in M-theory, where the flavor symmetry is encoded in a collection of non-compact surfaces.

Some Aspects of Positive Kernel Method of Quantization

We discuss various aspects of the positive kernel method of quantization of the one-parameter groups tau _t in text{ Aut }(P,vartheta ) of automorphisms of a G-principal bundle P(G,pi ,M) with a fixed connection form vartheta on its total space P. We show that the generator {hat{F}} of the unitary flow U_t = e^{it {hat{F}}} being the quantization of tau _t is realized by a generalized Kirillov–Kostant–Souriau operator whose domain consists of sections of some vector bundle over M, which are defined by a suitable positive kernel. This method of quantization applied to the case when G=hbox {GL}(N,{mathbb {C}}) and M is a non-compact Riemann surface leads to quantization of the arbitrary holomorphic flow tau _t^{mathrm{hol}} in text{ Aut }(P,vartheta ). For the above case, we present the integral decompositions of the positive kernels on Ptimes P invariant with respect to the flows tau _t^{mathrm{hol}} in terms of the spectral measure of {hat{F}}. These decompositions generalize the ones given by Bochner’s Theorem for the positive kernels on {mathbb {C}} times {mathbb {C}} invariant with respect to the one-parameter groups of translations of complex plane.

A nonexistence result for CMC surfaces in hyperbolic 3-manifolds

We prove that a complete hyperbolic 3-manifold of finite volume does not admit a properly embedded noncompact surface of finite topology with constant mean curvature greater than or equal to 1.

Dessins D’enfants and Some Holomorphic Structures on the Loch Ness Monster

Abstract The theory of dessins d’enfants on compact Riemann surfaces, which are bipartite maps on compact orientable surfaces, are combinatorial objects used to study branched covers between compact Riemann surfaces and the absolute Galois group of the field of rational numbers. In this paper, we show how this theory is naturally extended to non-compact orientable surfaces and, in particular, we observe that the Loch Ness monster (LNM; the surface of infinite genus with exactly one end) admits infinitely many regular dessins d’enfants (either chiral or reflexive). In addition, we study different holomorphic structures on the LNM, which come from homology covers of compact Riemann surfaces, and infinite hyperelliptic and infinite superelliptic curves.

Quadratic Hecke Sums and Mass Equidistribution

AbstractWe consider the analogue of the quantum unique ergodicity conjecture for holomorphic Hecke eigenforms on compact arithmetic hyperbolic surfaces. We show that this conjecture follows from nontrivial bounds for Hecke eigenvalues summed over quadratic progressions. Our reduction provides an analogue for the compact case of a criterion established by Luo–Sarnak for the case of the non-compact modular surface. The novelty is that known proofs of such criteria have depended crucially upon Fourier expansions, which are not available in the compact case. Unconditionally, we establish a twisted variant of the Holowinsky–Soundararajan theorem involving restrictions of normalized Hilbert modular forms arising via base change.

Geometric Schottky groups and non-compact hyperbolic surfaces with infinite genus

The topological type of a non-compact Riemann surface is determined by its ends space and the ends having infinite genus. In this paper for a non-compact Riemann Surface Sm,s with s ends and exactly m of them with infinite genus, such that m,s∈N and 1<m≤s, we give a precise description of the infinite set of generators of a Fuchsian (geometric Schottky) group Γm,s such that the quotient space H/Γm,s is homeomorphic to Sm,s and has infinite area. For this construction, we exhibit a hyperbolic polygon with an infinite number of sides and give a collection of Mobius transformations identifying the sides in pairs.

A Thurston boundary for infinite-dimensional Teichmüller spaces

For a compact surface $X_0$, Thurston introduced a compactification of its Teichm\"uller space $\mathcal T(X_0)$ by completing it with a boundary $\mathcal{PML}(X_0)$ consisting of projective measured geodesic laminations. We introduce a similar bordification for the Teichm\"uller space $\mathcal T(X_0)$ of a noncompact Riemann surface $X_0$, using the technical tool of geodesic currents. The lack of compactness requires the introduction of certain uniformity conditions which were unnecessary for compact surfaces. A technical step, providing a convergence result for earthquake paths in $\mathcal T(X_0)$, may be of independent interest.

Super-zeta functions and regularized determinants associated with cofinite Fuchsian groups with finite-dimensional unitary representations

Let M be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $$\chi $$ denote a finite dimensional unitary representation of the fundamental group of M. Let $$\Delta $$ denote the hyperbolic Laplacian which acts on smooth sections of the flat bundle over M associated with $$\chi $$ . From the spectral theory of $$\Delta $$ , there are three distinct sequences of numbers: the first coming from the eigenvalues of $$L^{2}$$ eigenfunctions, the second coming from resonances associated with the continuous spectrum, and the third being the set of negative integers. Using these sequences of spectral data, we employ the super-zeta approach to regularization and introduce two super-zeta functions, $$\mathcal {Z}_-(s,z)$$ and $$\mathcal {Z}_+(s,z)$$ that encode the spectrum of $$\Delta $$ in such a way that they can be used to define the regularized determinant of $$\Delta -z(1-z)I$$ . The resulting formula for the regularized determinant of $$\Delta -z(1-z)I$$ in terms of the Selberg zeta function, see Theorem 5.3, encodes the symmetry $$z\leftrightarrow 1-z$$ .

Irreducible self-adjoint representations of quantum Teichmüller space and the phase constants

Quantization of the Teichmüller space of a non-compact Riemann surface has emerged in 1980s as an approach to three dimensional quantum gravity. For any choice of an ideal triangulation of the surface, Thurston’s shear coordinate functions on the edges form a coordinate system for the Teichmüller space, and they should be replaced by suitable self-adjoint operators on a Hilbert space. Upon a change of triangulations, one must construct a unitary operator between the Hilbert spaces intertwining the quantum coordinate operators and satisfying the composition identities up to multiplicative phase constants. In the well-known construction by Chekhov, Fock and Goncharov, the quantum coordinate operators form a family of reducible representations, and the phase constants are all trivial. In the present paper, we employ the harmonic–analytic theory of the Shale–Weil intertwiners for the Schrödinger representations, as well as Faddeev–Kashaev’s quantum dilogarithm function, to construct a family of irreducible representations of the quantum shear coordinate functions and the corresponding intertwiners for the changes of triangulations. The phase constants are explicitly computed and described by the Maslov indices of the Lagrangian subspaces of a symplectic vector space, and by the pentagon relation of the flips of triangulations. The present work may generalize to the cluster X-varieties.

The length of a shortest closed geodesic on a surface of finite area

In this paper we prove new upper bounds for the length of a shortest closed geodesic, denoted $l(M)$, on a complete, non-compact Riemannian surface $M$ of finite area $A$. We will show that $l(M) \leq 4\sqrt{2A}$ on a manifold with one end, thus improving the prior estimate of C. B. Croke, who first established that $l(M) \leq 31 \sqrt{A}$. Additionally, for a surface with at least two ends we show that $l(M) \leq 2\sqrt{2A}$, improving the prior estimate of Croke that $l(M) \leq (12+3\sqrt{2})\sqrt{A}$.

Earthquakes and Graftings of Hyperbolic Surface laminations

Abstract We study compact hyperbolic surface laminations. These are a generalisation of closed hyperbolic surfaces, which appear to be more suited to the study of Teichmüller theory than arbitrary non-compact surfaces. We show that the Teichmüller space of any non-trivial hyperbolic surface lamination is infinite dimensional. In order to prove this result, we study the theory of deformations of hyperbolic surfaces, and we derive a new formula for the derivative of the length of a simple closed geodesic with respect to the action of grafting. This formula complements those derived by McMullen [ 24], in terms of the Weil–Petersson metric, and by Wolpert [ 34], for the case of earthquakes.

The determinant of the Lax–Phillips scattering operator

Let $M$ denote a finite volume, non-compact Riemann surface without elliptic points, and let $B$ denote the Lax-Phillips scattering operator. Using the superzeta function approach due to Voros, we define a Hurwitz-type zeta function $\zeta^{\pm}_{B}(s,z)$ constructed from the resonances associated to $zI -[ (1/2)I \pm B]$. We prove the meromorphic continuation in $s$ of $\zeta^{\pm}_{B}(s,z)$ and, using the special value at $s=0$, define a determinant of the operators $zI -[ (1/2)I \pm B]$. We obtain expressions for Selberg's zeta function and the determinant of the scattering matrix in terms of the operator determinants.

A Prime Geodesic Theorem of Gallagher Type for Riemann Surfaces

We consider a cofinite Fuchsian group of the first kind with finitely many inequivalent parabolic elements and a unitary multiplier system of an arbitrary weight on it. Through the Gallagher–Koyama approach to the prime geodesic theorem on the corresponding noncompact hyperbolic surface, we reduce the exponent in the error term from $$\frac{3}{4}$$ to $$\frac{7}{10}$$ outside a set of finite logarithmic measure. Recent advances in results of the latter type and the methods applied are briefly discussed.

Foliations with non-compact leaves on surfaces

The paper studies non-compact surfaces obtained by gluing strips R × (−1, 1) with at most countably many boundary intervals along some of these intervals. Every such strip possesses a foliation by parallel lines, which gives a foliation on the resulting surface. It is proved that the identity path component of the group of homeomorphisms of that foliation is contractible.

On the capacity and depth of compact surfaces

K. Borsuk in 1979, at the Topological Conference in Moscow, introduced the concept of capacity and depth of a compactum. In this paper we compute the capacity and depth of compact surfaces. We show that the capacity and depth of every compact orientable surface of genus $$g\ge 0$$ is equal to $$g+2$$. Also, we prove that the capacity and depth of a compact non-orientable surface of genus $$g>0$$ is $$[\frac{g}{2}]+2$$.

Jordan surface theorem for simple closed SST-surfaces

The present paper proposes a new type of Jordan surface theorem for simple closed SST-surfaces. To do this work, we use space set topological (referred to as SST-, for short) structures on compact or non-compact surfaces in Rn. Indeed, the SST-structure is a special kind of Alexandroff topological structure and further, it is indeed a locally finite (LF-, for brevity) topological one. Owing to the LF-topological structure, it can efficiently be used in studying objects from the viewpoints of pure and applied topology such as discrete geometry, combinatorial topology, computational topology, digital topology, digital geometry and so on. The present paper first establishes an SST-structure of a subdivided abstract cell (SAC-, for brevity) complex on X(⊂Rn) using a generalized face to face tessellation of X⊂Rn. Second, the paper introduces a strong SST-homeomorphism to classify SST-surfaces. Finally, unlike the typical Jordan surface theorem under Hausdorff topology or polyhedral geometry, the current theorem is formulated by using Alexandroff topological or SST-structure. Thus this theorem has strong advantages being compared with those associated with Euclidean topology, digital topology and polygonal geometry. Besides, based on the SST-structure on an SAC-complex, the paper suggests some remarks on digital manifolds, Khalimsky manifolds, (n−1)-dimensional polyhedral pseudomanifolds and so on.

On Infinitely generated Fuchsian groups of the Loch Ness monster, the Cantor tree and the Blooming Cantor tree

AbstractIn this paper, for a non-compact Riemman surface S homeomorphic to either: the Infinite Loch Ness monster, the Cantor tree and the Blooming Cantor tree, we give a precise description of an infinite set of generators of a Fuchsian group Γ &lt; PSL(2, ℝ), such that the quotient space ℍ/Γ is a hyperbolic Riemann surface homeomorphic to S. For each one of these constructions, we exhibit a hyperbolic polygon with an infinite number of sides and give a collection of Mobius transformations identifying the sides in pairs.

Projective properties of Lorentzian surfaces

We investigate projective properties of Lorentzian surfaces. In particular, we prove that if T is a non-flat torus, then the index of its isometry group in its projective group is at most two. We also prove that any topologically finite non-compact surface can be endowed with a metric having a non-isometric projective transformation of infinite order.

Renormalization of integrals of products of Eisenstein series and analytic continuation of representations

We combine Zagier's theory of renormalizable automorphic functions on the hyperbolic plane with the analytic continuation of representations of SL(2,R) due to Bernstein and Reznikov to study triple products of Eisenstein series of arbitrary (in particular, non-arithmetic) non-compact finite-volume hyperbolic surfaces.

Optimal Morse–Smale Flows with Singularities on the Boundary of a Surface

We consider the optimal flows on noncompact surfaces with boundary, which have a minimum number of fixed points and all these points lie on the boundary of the surface. It is proved that the flow is optimal if it has a single sink and a single source. We describe the structures of the optimal flows on a simply connected region, on a Mobius strip, on a torus with hole, and on a Klein bottle with hole.