Double Fibration Research Articles

Harmonic maps and biharmonic maps for double fibrations of compact Lie groups

Harmonic maps and biharmonic maps for double fibrations of compact Lie groups

Extra-special quotients of surface braid groups and double Kodaira fibrations with small signature

We study some special systems of generators on finite groups, introduced in previous work by the first author and called diagonal double Kodaira structures, in order to investigate finite non-abelian quotients of the pure braid group on two strands {mathsf {P}}_2(Sigma _b), where Sigma _b is a closed Riemann surface of genus b. In particular, we prove that, if a finite group G admits a diagonal double Kodaira structure, then |G|ge 32, and equality holds if and only if G is extra-special. In the last section, as a geometrical application of our algebraic results, we construct two 3-dimensional families of double Kodaira fibrations having signature 16. Such surfaces are different from the ones recently constructed by Lee, Lönne and Rollenske and, as far as we know, they provide the first examples of positive-dimensional families of double Kodaira fibrations with small signature.

Surface braid groups, Finite Heisenberg covers and double Kodaira fibrations

We exhibit new examples of double Kodaira fibrations by using finite Galois covers of a product $\Sigma_b \times \Sigma_b$, where $\Sigma_b$ is a smooth projective curve of genus $b \geq 2$. Each cover is obtained by providing an explicit group epimorphism from the pure braid group $\mathsf{P}_2(\Sigma_b)$ to some finite Heisenberg group. In this way, we are able to show that every curve of genus $b$ is the base of a double Kodaira fibration; moreover, the number of pairwise non-isomorphic Kodaira fibred surfaces fibering over a fixed curve $\Sigma_b$ is at least $\boldsymbol{\omega}(b+1)$, where $\boldsymbol{\omega} \colon \mathbb{N} \to \mathbb{N}$ stands for the arithmetic function counting the number of distinct prime factors of a positive integer. As a particular case of our general construction, we obtain a real $4$-manifold of signature $144$ that can be realized as a real surface bundle over a surface of genus $2$, with fibre genus $325$, in two different ways. This provides (to our knowledge) the first "double" solution to a problem from Kirby's list in low-dimensional topology.

The Radon transform for double fibrations of semisimple symmetric spaces

The Radon transform for double fibrations of semisimple symmetric spaces

The Penrose Transform and the Exactness of the Tangential $$\pmb {k}$$-Cauchy–Fueter Complex on the Heisenberg Group

The k-Cauchy–Fueter complex in quaternionic analysis was generalized to the Heisenberg group in a previous paper (Ren et al., in Adv Appl Clifford Algebras 30(2): Paper No. 20, 2020). But it is not known whether this differential complex is exact or not. In this paper, we apply the Penrose transform (the twistor method) to a double fibration of homogeneous spaces of $$\mathrm{SO}(2N,{\mathbb {C}})$$ to prove its exactness on the Heisenberg group.

Double Kodaira fibrations with small signature

Kodaira fibrations are surfaces of general type with a non-isotrivial fibration, which are differentiable fiber bundles. They are known to have positive signature divisible by [Formula: see text]. Examples are known only with signature 16 and more. We review approaches to construct examples of low signature which admit two independent fibrations. Special attention is paid to ramified covers of product of curves which we analyze by studying the monodromy action for bundles of punctured curves. As a by-product, we obtain a classification of all fix-point-free automorphisms on curves of genus at most [Formula: see text].

The range of the Radon transform on the real hyperbolic Grassmann manifold

Let Γ n k be the space of all the k-dimensional totally geodesic submanifolds of the n-dimensional real hyperbolic space where 1 ≤ k ≤ n − 1. We prove that the Radon transform R for double fibrations of the real hyperbolic Grassmann manifolds Γ n p and Γ n q with respect to the inclusion incidence relations maps C ∞0 (Γn p ) bijectively onto the space of all the functions in C ∞0 (Γn q ) which satisfy a certain system of linear partial differential equations explicitly constructed from the left infinitesimal action of the transformation group when 0 ≤ p < q ≤ n − 1 and dim Γ n p < dim Γ n q . Our approach is based on the generalized method of gnomonic projections. We also treat the dual Radon transform R.

Aerobatics of Flying Saucers

Starting from the observation that a flying saucer is a nonholonomic mechanical system whose 5-dimensional configuration space is a contact manifold, we show how to enrich this space with a number of geometric structures by imposing further nonlinear restrictions on the saucer’s velocity. These restrictions define certain ‘manœuvres’ of the saucer, which we call ‘attacking,’ ‘landing,’ or ‘$$G_2$$ mode’ manœuvres, and which equip its configuration space with three kinds of flat parabolic geometry in five dimensions. The attacking manœuvre corresponds to the flat Legendrean contact structure, the landing manœuvre corresponds to the flat hypersurface type CR structure with Levi form of signature (1, 1), and the most complicated $$G_2$$ manœuvre corresponds to the contact Engel structure (Engel in C R Acad Sci 116:786–788, 1893; Mano et al. in The geometry of marked contact twisted cubic structures, 2018, arXiv:1809.06455) with split real form of the exceptional Lie group $$G_2$$ as its symmetries. A celebrated double fibration relating the two nonequivalent flat 5-dimensional parabolic $$G_2$$ geometries is used to construct a ‘$$G_2$$ joystick,’ consisting of two balls of radii in ratio $$1\!:\!3$$ that transforms the difficult $$G_2$$ manœuvre into the pilot’s action of rolling one of joystick’s balls on the other without slipping nor twisting.

Some remarks on the Fuglede p–modulus

We study the properties of extremal function for the Fuglede's p–modulus of a measure family Σ. Based on the recent result concerning p–modulus showing the existence of certain Borel measure nΣ on Σ, we give an interpretation of nΣ in some simple cases. In particular, we consider the case of disjoint supports and a natural family of measures associated with a double fibration.

Dihedral symmetries of gauge theories from dual Calabi-Yau threefolds

Recent studies (arXiv:1610.07916, arXiv:1711.07921, arXiv:1807.00186) of six-dimensional supersymmetric gauge theories that are engineered by a class of toric Calabi-Yau threefolds $X_{N,M}$, have uncovered a vast web of dualities. In this paper we analyse consequences of these dualities from the perspective of the partition functions $\mathcal{Z}_{N,M}$ (or the free energy $\mathcal{F}_{N,M}$) of these theories. Focusing on the case $M=1$, we find that the latter is invariant under the group $\mathbb{G}(N)\times S_N$: here $S_N$ corresponds to the Weyl group of the largest gauge group that can be engineered from $X_{N,1}$ and $\mathbb{G}(N)$ is a dihedral group, which acts in an intrinsically non-perturbative fashion and which is of infinite order for $N\geq 4$. We give an explicit representation of $\mathbb{G}(N)$ as a matrix group that is freely generated by two elements which act naturally on a specific basis of the K\"ahler moduli space of $X_{N,1}$. While we show the invariance of $\mathcal{Z}_{N,1}$ under $\mathbb{G}(N)\times S_N$ in full generality, we provide explicit checks by series expansions of $\mathcal{F}_{N,1}$ for a large number of examples. We also comment on the relation of $\mathbb{G}(N)$ to the modular group that arises due to the geometry of $X_{N,1}$ as a double elliptic fibration, as well as T-duality of Little String Theories that are constructed from $X_{N,1}$.

Quadratic Capelli operators and Okounkov polynomials

Let $Z$ be the symmetric cone of $r \times r$ positive definite Hermitian matrices over a real division algebra $\mathbb F$. Then $Z$ admits a natural family of invariant differential operators -- the Capelli operators $C_\lambda$ -- indexed by partitions $\lambda$ of length at most $r$, whose eigenvalues are given by specialization of Knop--Sahi interpolation polynomials. In this paper we consider a double fibration $Y \longleftarrow X \longrightarrow Z$ where $Y$ is the Grassmanian of $r$-dimensional subspaces of $\mathbb F^n $ with $n \geq 2r$. Using this we construct a family of invariant differential operators $D_{\lambda,s}$ on $Y$ that we refer to as quadratic Capelli operators. Our main result shows that the eigenvalues of the $D_{\lambda,s}$ are given by specializations of Okounkov interpolation polynomials.

Dual little strings from F-theory and flop transitions

A particular two-parameter class of little string theories can be described by M parallel M5-branes probing a transverse affine AN − 1 singularity. We previously discussed the duality between the theories labelled by (N, M) and (M, N). In this work, we propose that these two are in fact only part of a larger web of dual theories. We provide evidence that the theories labelled by (N, M) and left(frac{NM}{k},kright) are dual to each other, where k = gcd(N,M). To argue for this duality, we use a geometric realization of these little string theories in terms of F-theory compactifications on toric, non-compact Calabi-Yau threefolds XN,M which have a double elliptic fibration structure. We show explicitly for a number of examples that XNM/k,k is part of the extended moduli space of XN,M, i.e. the two are related through symmetry transformations and flop transitions. By working out the full duality map, we provide a simple check at the level of the free energy of little string theories.

Heterotic supergravity with internal almost-Kähler spaces; instantons for SO(32), or E8 × E8, gauge groups; and deformed black holes with soliton, quasiperiodic and/or pattern-forming structures

Heterotic supergravity with (1 + 3)-dimensional domain wall configurations and (warped) internal, six dimensional, almost-Kähler manifolds are studied. Considering ten dimensional spacetimes with nonholonomic distributions and conventional double fibrations, 2 + 2 + ... = 2 + 2 + 3 + 3, and associated SU(3) structures on internal space, we generalize for real, internal, almost symplectic gravitational structures the constructions with gravitational and gauge instantons of tanh-kink type [, ]. They include the first corrections to the heterotic supergravity action, parameterized in a form to imply nonholonomic deformations of the Yang–Mills sector and corresponding Bianchi identities. We show how it is possible to construct a variety of solutions depending on the type of nonholonomic distributions and deformations of ‘prime’ instanton configurations characterized by two real supercharges. This corresponds to supersymmetric, nonholonomic manifolds from the four dimensional point of view. Our method provides a unified description of embedding nonholonomically deformed tanh-kink-type instantons into half-BPS solutions of heterotic supergravity. This allows us to elaborate new geometric methods of constructing exact solutions of motion equations, with first order corrections to the heterotic supergravity. Such a formalism is applied for general and/or warped almost-Kähler configurations, which allows us to generate nontrivial (1 + 3)-d domain walls and black hole deformations determined by quasiperiodic internal space structures. This formalism is utilized in our associated publication [] in order to construct and study generic off-diagonal nonholonomic deformations of the Kerr metric, encoding contributions from heterotic supergravity.

Higher-Dimensional Representations of $$\mathsf {SL}_2$$ SL 2 and its Real Forms Via Plücker Embedding

The paper provides a geometric perspective on the inclusion map \(\mathfrak {sl}_2\cong \!\mathfrak {so}_3\!\subset \! \mathfrak {so}_n\) realized via the Plucker embedding. It relates higher-dimensional representations of the complex Lie group \(\mathsf {SO}_3\) and its real forms in terms of \(\mathsf {SO}(n)\) and \(\mathsf {SO}(p,q)\) transformations to different subspaces described by means of the well-known double fibrations used in twistor theory. Moreover, explicit matrix realizations and various factorization techniques, such as Euler and Wigner decompositions, are constructed in this generalized setting. Examples are provided for \(n=3,4\) and 5 in the context of special relativity, classical and quantum mechanics.

&amp;lt;i&amp;gt;CMB&amp;lt;/i&amp;gt;—A Geometric, Lorentz Invariant Model in Non-Expanding Lobachevskian Universe with a Black Body Spectral Distribution Function

In the present paper, based on Lobachevskian (hyperbolic) static geometry, we present (as an alternative to the existing Big Bang model of CMB) a geometric model of CMB in a Lobachevskian static universe as a homogeneous space of horospheres. It is shown that from the point of view of physics, a horosphere is an electromagnetic wavefront in Lobachevskian space. The presented model of CMB is an Lorentz invariant object, possesses observable properties of isotropy and homogeneity for all observers scattered across the Lobachevskian universe, and has a black body spectrum. The Lorentz invariance of CMB implies a mathematical equation for cosmological redshift for all z. The global picture of CMB, described solely in terms of the Lorentz group—SL(2C), is an infinite union of double sided quotient spaces (double fibration of the Lorentz group) taken over all parabolic stabilizers P⊂SL(2C). The local picture of CMB (as seen by us from Earth) is a Grassmannian space of an infinite union all horospheres containing origin o∈L3, equivalent to a projective plane RP2. The space of electromagnetic wavefronts has a natural identification with the boundary at infinity (an absolute) of Lobachevskian universe. In this way, it is possible to regard the CMB as a reference at infinity (an absolute reference) and consequently to define an absolute motion and absolute rest with respect to CMB, viewed as an infinitely remote reference.

Self-duality and self-similarity of little string orbifolds

We study a class of ${\cal N}=(1,0)$ little string theories obtained from orbifolds of M-brane configurations. These are realised in two different ways that are dual to each other: either as $M$ parallel M5-branes probing a transverse $A_{N-1}$ singularity or $N$ M5-branes probing an $A_{M-1}$ singularity. These backgrounds can further be dualised into toric, non-compact Calabi-Yau threefolds $X_{N,M}$ which have double elliptic fibrations and thus give a natural geometric description of T-duality of the little string theories. The little string partition functions are captured by the topological string partition function of $X_{N,M}$. We analyse in detail the free energies $\Sigma_{N,M}$ associated with the latter in a special region in the K\"ahler moduli space of $X_{N,M}$ and discover a remarkable property: in the Nekrasov-Shatashvili-limit, $\Sigma_{N,M}$ is identical to $NM$ times $\Sigma_{1,1}$. This entails that the BPS degeneracies for any $(N,M)$ can uniquely be reconstructed from the $(N,M)=(1,1)$ configuration, a property we refer to as self-similarity. Moreover, as $\Sigma_{1,1}$ is known to display a number of recursive structures, BPS degeneracies of little string configurations for arbitrary $(N,M)$ as well acquire additional symmetries. These symmetries suggest that in this special region the two little string theories described above are self-dual under T-duality.

Partial extensions of jets and the polar distribution on Grassmannians of non-maximal integral elements

We study an intrinsic distribution, called polar, on the space of l-dimensional integral elements of the higher order contact structure on jet spaces. The main result establishes that this exterior differential system is the prolongation of a natural system of PDEs, named pasting conditions, on sections of the bundle of partial jet extensions. Informally, a partial jet extension is a kth order jet with additional (k+1)st order information along l of the n possible directions. A choice of partial extensions of a jet into all possible l-directions satisfies the pasting conditions if the extensions coincide along pairwise intersecting l-directions.We further show that prolonging the polar distribution once more yields the space of (l,n)-dimensional integral flags with its double fibration distribution. When l>1 the exterior differential system is holonomic, stabilizing after one further prolongation.The proof starts form the space of integral flags, constructing the tower of prolongations by reduction.

The twisted Cartesian model for the double path fibration

AbstractIn this paper, the notion of a truncating twisting function from a cubical set to a permutahedral set and the corresponding notion of a twisted Cartesian product of these sets are introduced. The latter becomes a permutocubical set that models, in particular, the path fibration on a loop space. The chain complex of this twisted Cartesian product is in fact a comultiplicative twisted tensor product of cubical chains of base and permutahedral chains of fibre. This construction is formalized as a theory of twisted tensor products for Hirsch algebras.

Holomorphic Double Fibration and the Mapping Problems of Classical Domains

Holomorphic Double Fibration and the Mapping Problems of Classical Domains

On twistor transformations and invariant differential operator of simple Lie group G2(2)

The twistor transformations associated to the simple Lie group G2 are described explicitly. We consider the double fibration \documentclass[12pt]{minimal}\begin{document}${\rm G}_2/P_2 \xleftarrow {\eta } {{\rm G}_2/B} \xrightarrow {\tau }{\rm G}_2/P_1$\end{document}G2/P2←ηG2/B→τG2/P1, where P1 and P2 are two parabolic subgroups of G2 and B is a Borel subgroup, and its local version: \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}^*_2 \xleftarrow {\eta } \mathcal {F} \xrightarrow {\tau } \mathcal {H}_1$\end{document}H2*←ηF→τH1, where \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}_1$\end{document}H1 is the Heisenberg group of dimension 5 embedded in the coset space G2/P1, \documentclass[12pt]{minimal}\begin{document}$\mathcal {F} = \mathbb {CP}^1 \times \mathcal {H}_1$\end{document}F=CP1×H1 and \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}^*_2$\end{document}H2* contains the nilpotent Lie group \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}_2$\end{document}H2 of step three. The Baker-Campbell-Hausdorff formula is used to parametrize the coset spaces, coordinates charts, their transition functions and the fibers of the projection η as complex curves. We write down the relative De-Rham sequence on \documentclass[12pt]{minimal}\begin{document}$\mathcal {F}$\end{document}F along the fibers and push it down to \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}_1$\end{document}H1 to get a family of matrix-valued differential operators \documentclass[12pt]{minimal}\begin{document}${\mathscr D}_k$\end{document}Dk. Then we establish a kind of Penrose correspondence for G2: the kernel of \documentclass[12pt]{minimal}\begin{document}$\mathscr{D}_k$\end{document}Dk is isomorphic to the first cohomology of the sheaf \documentclass[12pt]{minimal}\begin{document}$\mathscr{O} (-k )$\end{document}O(−k) over \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}^*_2$\end{document}H2*. We also give the Penrose-type integral transformation u = Pf for \documentclass[12pt]{minimal}\begin{document}$f\in \mathscr{O} (-k )$\end{document}f∈O(−k), which gives solutions to equations \documentclass[12pt]{minimal}\begin{document}${\mathscr D}_ku=0$\end{document}Dku=0. When restricted to the real Heisenberg group, the differential operators are invariant under the action of G2(2). Exchanging P1 and P2, we derive corresponding results on \documentclass[12pt]{minimal}\begin{document}$\mathcal {H}_2$\end{document}H2.