Odd Signature Research Articles

Closed geodesics and Frøyshov invariants of hyperbolic three-manifolds

Frøyshov invariants are numerical invariants of rational homology three-spheres derived from gradings in monopole Floer homology. In the past few years, they have been employed to solve a wide range of problems in 3- and 4-dimensional topology. In this paper, we look at connections with hyperbolic geometry for the class of minimal L -spaces. In particular, we study relations between Frøyshov invariants and closed geodesics using ideas from analytic number theory. We discuss two main applications of our approach. First, we derive explicit upper bounds for the Frøyshov invariants of minimal hyperbolic L -spaces purely in terms of volume and injectivity radius. Second, we describe an algorithm to compute Frøyshov invariants of minimal L -spaces in terms of data arising from hyperbolic geometry. As a concrete example of our method, we compute the Frøyshov invariants for all spin ^{c} structures on the Seifert–Weber dodecahedral space. Along the way, we also prove several results about the eta invariants of the odd signature and Dirac operators on hyperbolic three-manifolds which might be of independent interest.

On liftings of modular forms and Weil representations

Abstract We give an explicit construction of lifting maps from integral and half-integral modular forms to vector-valued modular forms for Weil representations associated with arbitrary isotropic subgroups and finite quadratic modules of even and odd signature. This construction provides an explicit and general extension of previous work by Scheithauer and Zhang.

The Musical Geometry of Genes: Generating Rhythms from DNA

DNA encodes all sorts of information that makes us human, but, aside from encoding genes, could DNA also encode for a mapping of musical rhythms in a very abstract way? This project sought to generate rhythms out of DNA and compose a musical piece out of a gene's rhythmic sequence. Computational rules inspired by geometric analyses of rhythms guided the mapping of DNA's molecular structure into rhythmic timelines and melodic scales; these basic structures were then used to compose a song according to the sickle cell gene DNA sequence. The rhythms generated by this ‘genetic analysis’ alternate pleasantly between even and odd time signatures.

Automorphism groups of Walecki tournaments with zero and odd signatures

Walecki tournaments were defined by Alspach in 1966. They form a class of regular tournaments that posses a natural Hamilton directed cycle decomposition. It has been conjectured by Kelly in 1964 that every regular tournament possesses such a decomposition. Therefore Walecki tournaments speak in favor of the conjecture. A second interest in Walecki tournaments arises from the mapping between cycles of the complementing circular shift register and isomorphism classes of Walecki tournaments. The problem of enumerating non-isomorphic Walecki tournaments has not been solved to date. We characterize the arc structure of Walecki tournaments whose corresponding binary sequences have zero and odd signature. Automorphism groups are determined for zero signature Walecki tournaments and for odd signature Walecki tournaments with the zero signature Walecki subtournaments. Walecki tournaments possess a broad range of subtournaments isomorphic to some Walecki tournament. Subtournaments of odd signature Walecki tournaments induced by the outsets of the central vertex are proven to be either regular or almost regular.

On the minimal field of definition of rational maps: rational maps of odd signature

On the minimal field of definition of rational maps: rational maps of odd signature

The refined analytic torsion and a well-posed boundary condition for the odd signature operator

Abstract In this paper we discuss the refined analytic torsion on an odd dimensional compact oriented Riemannian manifold with boundary under some assumption. For this purpose we introduce two boundary conditions which are complementary to each other and well-posed for the odd signature operator B in the sense of Seeley. We then show that the zeta-determinants of B 2 and eta-invariants of B subject to these boundary conditions are well defined by using the method of the asymptotic expansions of the traces of the heat kernels. We use these facts to define the refined analytic torsion on a compact manifold with boundary and show that it is invariant on the change of metrics in the interior of the manifold. We finally describe the refined analytic torsion under these boundary conditions as an element of the determinant line.

Lefschetz fixed point formula on a compact Riemannian manifold with boundary for some boundary conditions

In Huang and Lee (in The Refined Analytic Torsion and a Well-posed Boundary Condition for the Odd Signature Operator, 2010) introduced a pair of new de Rham complexes on a compact oriented Riemannian manifold with boundary by using a pair of global boundary conditions to discuss the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the Lefschetz fixed point formula on these complexes with respect to a smooth map which is a local isometry on the boundary and has only simple fixed points. For this purpose we are going to use the heat kernel method for the Lefschetz fixed point formula.

Parity-odd correlators of diffuse gamma-rays and intergalactic magnetic fields

We develop the connection between intergalactic helical magnetic fields and parity odd signatures in the diffuse gamma ray sky. We find that the location and the amplitude of a peak in a parity odd correlator, $Q(R)$, can be used to infer the normal and helical power spectra of the intergalactic magnetic field. When applied to Fermi-LAT data, the amplitude of the observed peak in $Q(R)$ gives $\sim 10^{-14}~{\rm G}$ intergalactic magnetic field strength, which is consistent with an earlier independent estimate that only used the peak location (Tashiro et al. 2014). We discuss features in the observed $Q(R)$ that further support the intergalactic magnetic field hypothesis and make predictions for future tests.

Index type invariants for twisted signature complexes and homotopy invariance

AbstractFor a closed, oriented, odd dimensional manifold X, we define the rho invariant ρ(X,${\cal E}$,H) for the twisted odd signature operator valued in a flat hermitian vector bundle ${\cal E}$, where H = ∑ ij+1H2j+1 is an odd-degree closed differential form on X and H2j+1 is a real-valued differential form of degree 2j+1. We show that ρ(X,${\cal E}$,H) is independent of the choice of metrics on X and ${\cal E}$ and of the representative H in the cohomology class [H]. We establish some basic functorial properties of the twisted rho invariant. We express the twisted eta invariant in terms of spectral flow and the usual eta invariant. In particular, we get a simple expression for it on closed oriented 3-dimensional manifolds with a degree three flux form. A core technique used is our analogue of the Atiyah–Patodi–Singer theorem, which we establish for the twisted signature operator on a compact, oriented manifold with boundary. The homotopy invariance of the rho invariant ρ(X,${\cal E}$,H) is more delicate to establish, and is settled under further hypotheses on the fundamental group of X.

The gluing formula of the zeta-determinants of Dirac Laplacians for certain boundary conditions

The odd signature operator is a Dirac operator which acts on the space of differential forms of all degrees and whose square is the usual Laplacian. We extend the result see ( J. Geom. Phys. 57 (2007) 1951–1976) to prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the boundary conditions $\mathcal{P}_{-,\mathcal{L}_{0}}$, $\mathcal{P}_{+,\mathcal{L}_{1}}$. We next consider a double of de Rham complexes consisting of differential forms of all degrees with the absolute and relative boundary conditions. Using a similar method, we prove the gluing formula of the zeta-determinants of Laplacians acting on differential forms of all degrees with respect to the absolute and relative boundary conditions.

The comparison of two constructions of the refined analytic torsion on compact manifolds with boundary

The refined analytic torsion on compact Riemannian manifolds with boundary has been discussed by B. Vertman (Vertman, 2009, 2008) and the authors (Huang and Lee, 2010, 2012) but these two constructions are completely different. Vertman used a double of de Rham complexes consisting of the minimal and maximal closed extensions of a flat connection and the authors used well-posed boundary conditions P−,L0, P+,L1 for the odd signature operator. In this paper we compare these two constructions by using the BFK-gluing formula for zeta-determinants, the adiabatic method for stretching cylinder part near boundary and the result for comparison of eta invariants in Huang and Lee (2012) when the odd signature operator comes from a Hermitian flat connection.

Fields of moduli and fields of definition of odd signature curves

Let X be a smooth projective curve of genus \({g \geq 2}\) defined over a field K. We show that X can be defined over its field of moduli KX if the signature of the covering \({X \rightarrow X/ Aut(X)}\) is of type \({(0;c_1,\dots,c_k)}\) , where some ci appears an odd number of times. This result is applied to cyclic q-gonal curves and to plane quartics.

The gluing formula of the refined analytic torsion for an acyclic Hermitian connection

In the previous study by Huang and Lee (arXiv:1004.1753) we introduced the well-posed boundary conditions \({{\mathcal P}_{-, {\mathcal L}_{0}}}\) and \({{\mathcal P}_{+, {\mathcal L}_{1}}}\) for the odd signature operator to define the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the gluing formula of the refined analytic torsion for an acyclic Hermitian connection with respect to the boundary conditions \({{\mathcal P}_{-, {\mathcal L}_{0}}}\) and \({{\mathcal P}_{+, {\mathcal L}_{1}}}\). In this case the refined analytic torsion consists of the Ray-Singer analytic torsion, the eta invariant and the values of the zeta functions at zero. We first compare the Ray-Singer analytic torsion and eta invariant subject to the boundary condition \({{\mathcal P}_{-, {\mathcal L}_{0}}}\) or \({{\mathcal P}_{+, {\mathcal L}_{1}}}\) with the Ray-Singer analytic torsion subject to the relative (or absolute) boundary condition and eta invariant subject to the APS boundary condition on a compact manifold with boundary. Using these results together with the well known gluing formula of the Ray-Singer analytic torsion subject to the relative and absolute boundary conditions and eta invariant subject to the APS boundary condition, we obtain the main result.

Eta invariant and Selberg zeta function of odd type over convex co-compact hyperbolic manifolds

We show meromorphic extension and give a complete description of the divisors of a Selberg zeta function of odd type Z Γ , Σ o ( λ ) associated to the spinor bundle Σ on an odd dimensional convex co-compact hyperbolic manifold Γ \\ H 2 n + 1 . As a byproduct we do a full analysis of the spectral and scattering theory of the Dirac operator on asymptotically hyperbolic manifolds. We show that there is a natural eta invariant η ( D ) associated to the Dirac operator D over a convex co-compact hyperbolic manifold Γ \\ H 2 n + 1 and that exp ( π i η ( D ) ) = Z Γ , Σ o ( 0 ) , thus extending Millson's formula to this setting. Under some assumption on the exponent of convergence of Poincaré series for the group Γ, we also define an eta invariant for the odd signature operator, and we show that for Schottky 3-dimensional hyperbolic manifolds it gives the argument of a holomorphic function which appears in the Zograf factorization formula relating two natural Kähler potentials for Weil–Petersson metric on Schottky space.

Exotic smooth structures on small 4-manifolds with odd signatures

For every integer $k\geq 2$, we construct infinite families of mutually nondiffeomorphic irreducible smooth structures on the topological $4$-manifolds $(2k-1)(S^2\times S^2)$ and $(2k-1)(\CP#\CPb)$, the connected sums of $2k-1$ copies of $S^2\times S^2$ and $\CP#\CPb$.

Refined analytic torsion

Given an acyclic representation $\alpha$ of the fundamental group of a compact oriented odd-dimensional manifold, which is close enough to an acyclic unitary representation, we define a refinement $T_\alpha$ of the Ray-Singer torsion associated to $\alpha$, which can be viewed as the analytic counterpart of the refined combinatorial torsion introduced by Turaev. $T_\alpha$ is equal to the graded determinant of the odd signature operator up to a correction term, the metric anomaly, needed to make it independent of the choice of the Riemannian metric. $T_\alpha$ is a holomorphic function on the space of such representations of the fundamental group. When $\alpha$ is a unitary representation, the absolute value of $T_\alpha$ is equal to the Ray-Singer torsion and the phase of $T_\alpha$ is proportional to the $\eta$-invariant of the odd signature operator. The fact that the Ray-Singer torsion and the $\eta$-invariant can be combined into one holomorphic function allows one to use methods of complex analysis to study both invariants. In particular, using these methods we compute the quotient of the refined analytic torsion and Turaev’s refinement of the combinatorial torsion generalizing in this way the classical Cheeger-M¨uller theorem. As an application, we extend and improve a result of Farber about the relationship between the Farber-Turaev absolute torsion and the $\eta$-invariant. As part of our construction of $T_\alpha$ we prove several new results about determinants and $\eta$-invariants of non self-adjoint elliptic operators.

J--Glueballs and a Low Odderon Intercept

We report an odderon Regge trajectory emerging from a field theoretical Coulomb gauge QCD model for the odd signature J(PC) (P = C = -1) glueball states. The trajectory intercept is clearly smaller than the Pomeron and even the omega trajectory's intercept which provides an explanation for the nonobservation of the odderon in high energy scattering data. To further support this result we compare to glueball lattice data and also perform calculations with an alternative model based upon an exact Hamiltonian diagonalization for three constituent gluons.

A splitting formula for the spectral flow of the odd signature operator on 3–manifolds coupled to a path ofSU(2) connections

We establish a splitting formula for the spectral flow of the odd signature operator on a closed 3-manifold M coupled to a path of SU(2) connections, provided M = S cup X, where S is the solid torus. It describes the spectral flow on M in terms of the spectral flow on S, the spectral flow on X (with certain Atiyah-Patodi-Singer boundary conditions), and two correction terms which depend only on the endpoints. Our result improves on other splitting theorems by removing assumptions on the non-resonance level of the odd signature operator or the dimension of the kernel of the tangential operator, and allows progress towards a conjecture by Lisa Jeffrey in her work on Witten's 3-manifold invariants in the context of the asymptotic expansion conjecture.

Calderón Projector for the Hessian of the Perturbed Chern–Simons Function on a 3‐Manifold with Boundary

The existence and continuity for the Calderón projector of the perturbed odd signature operator on a 3‐manifold is established. As an application we give a new proof of a result of Taubes relating the modulo 2 spectral flow of a family of operators on a homology 3‐sphere with the difference in local intersection numbers of the character varieties coming from a Heegard decomposition. 2000 Mathematics Subject Classification 57M27 (primary), 58J32 (secondary).

The [eta]-invariant, Maslov index, and spectral flow for Dirac-type operators on manifolds with boundary

Several proofs have been published of the Mod Z gluing formula for the eta-invariant of a Dirac operator. However, so far the integer contribution to the gluing formula for the eta-invariant is left obscure in the literature. In this article we present a gluing formula for the eta-invariant which expresses the integer contribution as a triple index involving the boundary conditions and the Calderon projectors of the two parts of the decomposition. The main ingredients of our presentation are the Scott-Wojciechowski theorem for the determinant of a Dirac operator on a manifold with boundary and the approach of Bruning-Lesch to the mod Z gluing formula. Our presentation includes careful constructions of the Maslov index and triple index in a symplectic Hilbert space. As a byproduct we give intuitively appealing proofs of two theorems of Nicolaescu on the spectral flow of Dirac operators. As an application of our methods, we carry out a detailed analysis of the eta-invariant of the odd signature operator coupled to a flat connection using adiabatic methods. This is used to extend the definition of the Atiyah-Patodi-Singer rho-invariant to manifolds with boundary. We derive a ``non-additivity'' formula for the Atiyah-Patodi-Singer rho-invariant and relate it to Wall's non-additivity formula for the signature of even-dimensional manifolds.