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Journal articles on the topic 'G-bundles'

Author: Grafiati

Published: 14 March 2023

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1

MURRAY, MICHAEL K., and RAYMOND F. VOZZO. "CIRCLE ACTIONS, CENTRAL EXTENSIONS AND STRING STRUCTURES." International Journal of Geometric Methods in Modern Physics 07, no. 06 (September 2010): 1065–92. http://dx.doi.org/10.1142/s0219887810004725.

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The caloron correspondence can be understood as an equivalence of categories between G-bundles over circle bundles and LG ⋊ρ S1-bundles where LG is the group of smooth loops in G. We use it, and lifting bundle gerbes, to derive an explicit differential form based formula for the (real) string class of an LG ⋊ρ S1-bundle.

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2

Choe, Insong, and George H. Hitching. "Non-defectivity of Grassmannian bundles over a curve." International Journal of Mathematics 27, no. 07 (June 2016): 1640002. http://dx.doi.org/10.1142/s0129167x16400024.

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Let [Formula: see text] be the Grassmann bundle of two-planes associated to a general bundle [Formula: see text] over a curve [Formula: see text]. We prove that an embedding of [Formula: see text] by a certain twist of the relative Plücker map is not secant defective. This yields a new and more geometric proof of the Hirschowitz-type bound on the isotropic Segre invariant for maximal isotropic sub-bundles of orthogonal bundles over [Formula: see text], analogous to those given for vector bundles and symplectic bundles in [I. Choe and G. H. Hitching, Secant varieties and Hirschowitz bound on vector bundles over a curve, Manuscripta Math. 133 (2010) 465–477, I. Choe and G. H. Hitching, Lagrangian sub-bundles of symplectic vector bundles over a curve, Math. Proc. Cambridge Phil. Soc. 153 (2012) 193–214]. From the non-defectivity, we also deduce an interesting feature of a general orthogonal bundle of even rank over [Formula: see text], contrasting with the classical and symplectic cases: a general maximal isotropic sub-bundle of maximal degree intersects at least one other such sub-bundle in positive rank.

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3

Deaconu, Valentin, Alex Kumjian, and Birant Ramazan. "Fell bundles associated to groupoid morphisms." MATHEMATICA SCANDINAVICA 102, no. 2 (June 1, 2008): 305. http://dx.doi.org/10.7146/math.scand.a-15064.

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Given a continuous open surjective morphism $\pi :G\rightarrow H$ of étale groupoids with amenable kernel, we construct a Fell bundle $E$ over $H$ and prove that its $C^*$-algebra $C^*_r(E)$ is isomorphic to $C^*_r(G)$. This is related to results of Fell concerning $C^*$-algebraic bundles over groups. The case $H=X$, a locally compact space, was treated earlier by Ramazan. We conclude that $C^*_r(G)$ is strongly Morita equivalent to a crossed product, the $C^*$-algebra of a Fell bundle arising from an action of the groupoid $H$ on a $C^*$-bundle over $H^0$. We apply the theory to groupoid morphisms obtained from extensions of dynamical systems and from morphisms of directed graphs with the path lifting property. We also prove a structure theorem for abelian Fell bundles.

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4

BISWAS, INDRANIL, and GÜNTHER TRAUTMANN. "A CRITERION FOR HOMOGENEOUS PRINCIPAL BUNDLES." International Journal of Mathematics 21, no. 12 (December 2010): 1633–38. http://dx.doi.org/10.1142/s0129167x10006689.

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We consider principal bundles over G/P, where P is a parabolic subgroup of a semi-simple and simply connected linear algebraic group G defined over ℂ. We prove that a holomorphic principal H-bundle EH → G/P, where H is a complex reductive group, and is homogeneous if the adjoint vector bundle ad (EH) is homogeneous. Fix a faithful H-module V. We also show that EH is homogeneous if the vector bundle EH ×H V associated to it for the H-module V is homogeneous.

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5

Perego, Arvid. "Kobayashi—Hitchin correspondence for twisted vector bundles." Complex Manifolds 8, no. 1 (January 1, 2021): 1–95. http://dx.doi.org/10.1515/coma-2020-0107.

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Abstract We prove the Kobayashi—Hitchin correspondence and the approximate Kobayashi—Hitchin correspondence for twisted holomorphic vector bundles on compact Kähler manifolds. More precisely, if X is a compact manifold and g is a Gauduchon metric on X, a twisted holomorphic vector bundle on X is g−polystable if and only if it is g−Hermite-Einstein, and if X is a compact Kähler manifold and g is a Kähler metric on X, then a twisted holomorphic vector bundle on X is g−semistable if and only if it is approximate g−Hermite-Einstein.

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6

Lashof, R. "Homogeneous Hamiltonian G-Bundles." Proceedings of the American Mathematical Society 121, no. 2 (June 1994): 599. http://dx.doi.org/10.2307/2160442.

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7

Lashof, R. "Homogeneous Hamiltonian $G$-bundles." Proceedings of the American Mathematical Society 121, no. 2 (February 1, 1994): 599. http://dx.doi.org/10.1090/s0002-9939-1994-1218116-6.

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8

Mozgovoy, Sergey, and Olivier Schiffmann. "Counting Higgs bundles and type quiver bundles." Compositio Mathematica 156, no. 4 (February 27, 2020): 744–69. http://dx.doi.org/10.1112/s0010437x20007010.

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We prove a closed formula counting semistable twisted (or meromorphic) Higgs bundles of fixed rank and degree over a smooth projective curve of genus $g$ defined over a finite field, when the twisting line bundle degree is at least $2g-2$ (this includes the case of usual Higgs bundles). This yields a closed expression for the Donaldson–Thomas invariants of the moduli spaces of twisted Higgs bundles. We similarly deal with twisted quiver sheaves of type $A$ (finite or affine), obtaining in particular a Harder–Narasimhan-type formula counting semistable $U(p,q)$-Higgs bundles over a smooth projective curve defined over a finite field.

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9

Krepski, Derek. "Central Extensions of Loop Groups and Obstruction to Pre-Quantization." Canadian Mathematical Bulletin 56, no. 1 (March 1, 2013): 116–26. http://dx.doi.org/10.4153/cmb-2011-131-6.

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AbstractAn explicit construction of a pre-quantumline bundle for themoduli space of flat G-bundles over a Riemann surface is given, where G is any non-simply connected compact simple Lie group. This work helps to explain a curious coincidence previously observed between Toledano Laredo's work classifying central extensions of loop groups LG and the author's previous work on the obstruction to pre-quantization of the moduli space of flat G-bundles.

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10

BISWAS, INDRANIL, and JOHN LOFTIN. "HERMITIAN–EINSTEIN CONNECTIONS ON PRINCIPAL BUNDLES OVER FLAT AFFINE MANIFOLDS." International Journal of Mathematics 23, no. 04 (April 2012): 1250039. http://dx.doi.org/10.1142/s0129167x12500395.

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Let M be a compact connected special flat affine manifold without boundary equipped with a Gauduchon metric g and a covariant constant volume form. Let G be either a connected reductive complex linear algebraic group or the real locus of a split real form of a complex reductive group. We prove that a flat principal G-bundle EG over M admits a Hermitian–Einstein structure if and only if EG is polystable. A polystable flat principal G-bundle over M admits a unique Hermitian–Einstein connection. We also prove the existence and uniqueness of a Harder–Narasimhan filtration for flat vector bundles over M. We prove a Bogomolov type inequality for semistable vector bundles under the assumption that the Gauduchon metric g is astheno-Kähler.

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11

BISWAS, INDRANIL, and TOMÁS L. GÓMEZ. "HECKE CORRESPONDENCE FOR SYMPLECTIC BUNDLES WITH APPLICATION TO THE PICARD BUNDLES." International Journal of Mathematics 17, no. 01 (January 2006): 45–63. http://dx.doi.org/10.1142/s0129167x06003357.

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We construct a Hecke correspondence for a moduli space of symplectic vector bundles over a curve. As an application we prove the following. Let X be a complex smooth projective curve of genus g(X) > 2 and L a line bundle over X. Let [Formula: see text] be the moduli space parametrizing stable pairs of the form (E,φ), where E is a vector bundle of rank 2n over X and φ : E ⊗ E → L a skew-symmetric nondegenerate bilinear form on the fibers of E. If deg (E) ≥ 4n(g(X)-1), then there is a projectivized Picard bundle on [Formula: see text], which is a projective bundle whose fiber over any point [Formula: see text] is ℙ(H0(X,E)). We prove that this projective bundle is stable.

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12

Biswas, Indranil, Oscar García-Prada, Jacques Hurtubise, and Steven Rayan. "Principal co-Higgs bundles on ℙ1." Proceedings of the Edinburgh Mathematical Society 63, no. 2 (March 5, 2020): 512–30. http://dx.doi.org/10.1017/s0013091520000024.

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AbstractFor complex connected, reductive, affine, algebraic groups G, we give a Lie-theoretic characterization of the semistability of principal G-co-Higgs bundles on the complex projective line ℙ1 in terms of the simple roots of a Borel subgroup of G. We describe a stratification of the moduli space in terms of the Harder–Narasimhan type of the underlying bundle.

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13

Ionescu, Marius, and Dana P. Dilliams. "A classic Morita equivalence result for Fell bundle $C^*$-algebras." MATHEMATICA SCANDINAVICA 108, no. 2 (June 1, 2011): 251. http://dx.doi.org/10.7146/math.scand.a-15170.

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We show how to extend a classic Morita Equivalence Result of Green's to the $C^*$-algebras of Fell bundles over transitive groupoids. Specifically, we show that if $p:{\mathcal B}\to G$ is a saturated Fell bundle over a transitive groupoid $G$ with stability group $H=G(u)$ at $u\in G^{(0)}$, then $C^* (G,{\mathcal B})$ is Morita equivalent to $C^*(H,{\mathcal C})$, where ${\mathcal C}={\mathcal B}_{| H}$. As an application, we show that if $p:{\mathcal B}\to G$ is a Fell bundle over a group $G$ and if there is a continuous $G$-equivariant map $\sigma:$ Prim $A\to G/H$, where $A=B(e)$ is the $C^*$-algebra of $\mathcal B$ and $H$ is a closed subgroup, then $C^*(G,{\mathcal B})$ is Morita equivalent to $C^* (H,{\mathcal C}^{I})$ where ${\mathcal C}^{I}$ is a Fell bundle over $H$ whose fibres are $A/I$-$A/I$-imprimitivity bimodules and $I=\bigcap\{ P:\sigma(P)=eH\}$. Green's result is a special case of our application to bundles over groups.

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14

Singh, Mahender. "Equivariant Maps from Stiefel Bundles to Vector Bundles." Proceedings of the Edinburgh Mathematical Society 60, no. 1 (June 1, 2016): 231–50. http://dx.doi.org/10.1017/s0013091515000541.

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AbstractLet E → B be a fibre bundle and let Eʹ → B be a vector bundle. Let G be a compact Lie group acting fibre preservingly and freely on both E and Eʹ – 0, where 0 is the zero section of Eʹ → B. Let f : E → Eʹ be a fibre-preserving G-equivariant map and let Zf = {x ∈ E | f(x) = 0} be the zero set of f. In this paper we give a lower bound for the cohomological dimension of the zero set Zf when a fibre of E → B is a real Stiefel manifold with a free ℤ/2-action or a complex Stiefel manifold with a free 𝕊1-action. This generalizes a well-known result of Dold for sphere bundles equipped with free involutions.

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15

BISWAS, INDRANIL. "PRINCIPAL BUNDLES ON RATIONALLY CONNECTED FIBRATIONS OVER ABELIAN VARIETIES." International Journal of Mathematics 20, no. 02 (February 2009): 167–88. http://dx.doi.org/10.1142/s0129167x0900525x.

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Let f : M → A be a smooth surjective algebraic morphism, where M is a connected complex projective manifold and A a complex abelian variety, such that all the fibers of f are rationally connected. We show that an algebraic principal G-bundle EG over M admits a flat holomorphic connection if EG admits a holomorphic connection; here G is any connected reductive linear algebraic group defined over ℂ. We also show that EG admits a holomorphic connection if and only if any of the following three statements holds. (1) The principal G-bundle EG is semistable, c2( ad (EG)) = 0, and all the line bundles associated to EG for the characters of G have vanishing rational first Chern class. (2) There is an algebraic principal G-bundle E'G on A such that f*E'G = EG, and all the translations of E'G by elements of A are isomorphic to E'G itself. (3) There is a finite étale Galois cover [Formula: see text] and a reduction of structure group [Formula: see text] to a Borel subgroup B ⊂ G such that all the line bundles associated to ÊB for the characters of B have vanishing rational first Chern class. In particular, the above three statements are equivalent.

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16

DEICKE, KLAUS. "EXTERIOR EQUIVALENCE FOR POINTWISE UNITARY COACTIONS." International Journal of Mathematics 12, no. 01 (February 2001): 63–79. http://dx.doi.org/10.1142/s0129167x01000630.

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Let G be a second countable locally compact group and A a separable continuous trace C*-algebra. To each pointwise unitary coaction δ of G on A one can associate a proper G-bundle [Formula: see text], π × μ → π. We show that two pointwise unitary coactions δ and ∊ of G on A are exterior equivalent if and only if the proper G-bundles [Formula: see text] and [Formula: see text] are isomorphic. Thus, if A is stable, there exists a bijection between the isomorphism classes of proper G-bundles over [Formula: see text] and the exterior equivalence classes of pointwise unitary coactions of G on A. Moreover, when G is abelian we recover a theorem of Olesen and Raeburn.

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17

Martens, Johan, and Michael Thaddeus. "Compactifications of reductive groups as moduli stacks of bundles." Compositio Mathematica 152, no. 1 (August 18, 2015): 62–98. http://dx.doi.org/10.1112/s0010437x15007484.

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Let $G$ be a split reductive group. We introduce the moduli problem of bundle chains parametrizing framed principal $G$-bundles on chains of lines. Any fan supported in a Weyl chamber determines a stability condition on bundle chains. Its moduli stack provides an equivariant toroidal compactification of $G$. All toric orbifolds may be thus obtained. Moreover, we get a canonical compactification of any semisimple $G$, which agrees with the wonderful compactification in the adjoint case, but which in other cases is an orbifold. Finally, we describe the connections with Losev–Manin’s spaces of weighted pointed curves and with Kausz’s compactification of $GL_{n}$.

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18

Bhosle, Usha N. "Moduli of parabolic $G$-bundles." Bulletin of the American Mathematical Society 20, no. 1 (January 1, 1989): 45–49. http://dx.doi.org/10.1090/s0273-0979-1989-15693-0.

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19

Bleecker, David D. "Symmetry breakdown in G-bundles." Physica D: Nonlinear Phenomena 17, no. 3 (December 1985): 257–78. http://dx.doi.org/10.1016/0167-2789(85)90210-6.

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20

Heinloth, Jochen. "Uniformization of $${\mathcal {G}}$$ -bundles." Mathematische Annalen 347, no. 3 (November 18, 2009): 499–528. http://dx.doi.org/10.1007/s00208-009-0443-4.

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21

Florentino, Carlos, and Thomas Ludsteck. "Unipotent Schottky bundles on Riemann surfaces and complex tori." International Journal of Mathematics 25, no. 06 (June 2014): 1450056. http://dx.doi.org/10.1142/s0129167x14500566.

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We study a natural map from representations of a free (respectively, free abelian) group of rank g in GL r(ℂ), to holomorphic vector bundles of degree zero over a compact Riemann surface X of genus g (respectively, complex torus X of dimension g). This map defines what is called a Schottky functor. Our main result is that this functor induces an equivalence between the category of unipotent representations of Schottky groups and the category of unipotent vector bundles on X. We also show that, over a complex torus, any vector or principal bundle with a flat holomorphic connection is Schottky.

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22

SAN MARTIN, LUIZ A. B., and LUCAS SECO. "Morse and Lyapunov spectra and dynamics on flag bundles." Ergodic Theory and Dynamical Systems 30, no. 3 (June 23, 2009): 893–922. http://dx.doi.org/10.1017/s0143385709000285.

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AbstractIn this paper we study characteristic exponents of flows in relation with the dynamics of flows on flag bundles. The starting point is a flow on a principal bundle with semi-simple group G. Projection against the Iwasawa decomposition G=KAN defines an additive cocycle over the flow with values in 𝔞=log A. Its Lyapunov exponents (limits along trajectories) and Morse exponents (limits along chains) are studied. A symmetric property of these spectral sets is proved, namely invariance under the Weyl group. We also prove that these sets are located in certain Weyl chambers, defined from the dynamics on the associated flag bundles. As a special case linear flows on vector bundles are considered.

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23

Choe, Insong, and George H. Hitching. "Maximal isotropic subbundles of orthogonal bundles of odd rank over a curve." International Journal of Mathematics 26, no. 13 (December 2015): 1550106. http://dx.doi.org/10.1142/s0129167x15501062.

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An orthogonal bundle over a curve has an isotropic Segre invariant determined by the maximal degree of a maximal isotropic subbundle. This invariant and the induced stratifications on moduli spaces of orthogonal bundles were studied for bundles of even rank in [I. Choe and G. H. Hitching, A stratification on the moduli space of symplectic and orthogonal bundles over a curve, Internat. J. Math. 25(5) (2014), Article ID: 1450047, 27pp.]. In this paper, we obtain analogous results for bundles of odd rank. We compute the sharp upper bound on the isotropic Segre invariant. Also we show the irreducibility of the induced strata on the moduli spaces of orthogonal bundles of odd rank, and compute their dimensions.

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Plaszczyk, Mariusz. "The natural transformations between r-th order prolongation of tangent and cotangent bundles over Riemannian manifolds." Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica 69, no. 1 (November 30, 2015): 91. http://dx.doi.org/10.17951/a.2015.69.1.91.

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If (M, g) is a Riemannian manifold then there is the well-known base preserving vector bundle isomorphism TM → T*M given by v → g(v, –) between the tangent TM and the cotangent T*M bundles of M. In the present note first we generalize this isomorphism to the one J<sup>r</sup>TM → J<sup>r</sup>T*M between the r-th order prolongation J<sup>r</sup>TM of tangent TM and the r-th order prolongation J<sup>r</sup>T*M of cotangent T*M bundles of M. Further we describe all base preserving vector bundle maps D<sub>M</sub>(g) : J<sup>r</sup>TM → J<sup>r</sup>T*M depending on a Riemannian metric g in terms of natural (in g) tensor fields on M.

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25

Azhar, Sayed Waqar, Fujun Xu, Yinnan Zhang, and Yiping Qiu. "Fabrication and mechanical properties of flaxseed fiber bundle-reinforced polybutylene succinate composites." Journal of Industrial Textiles 50, no. 1 (January 4, 2019): 98–113. http://dx.doi.org/10.1177/1528083718821876.

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Flaxseed plants are widely grown globally due to the beneficial seed oil derivatives for human and animal consumption and other industrial uses. However, plentiful flaxseed straws are annually burnt after the harvesting of seeds, lacking utilization of the abundant flaxseed fibers, resulting in wastage of a valuable fiber resource and drastic increase in environmental pollution. In this study, initially the chemical composition and mechanical property of flaxseed fiber bundle were investigated, which resulted as 40.11% cellulose, 28.27% hemi-cellulose, 15.08% lignin, 6.3% pectin, 3.1% wax, and the tensile strength of 1.14 cN/dTex. The surface modification treatment was carried out with concentrations of 10 g/L and 20 g/L sodium hydroxide (NaOH). Later, flaxseed fiber bundles reinforced Polybutylene Succinate (PBS) resin composites were fabricated by thermal compression method. The tensile strength of untreated flaxseed fiber bundle/PBS composites was 78.2 MPa, while the flexural strength of 20 g/L NaOH treated flaxseed fiber bundle/PBS composites showed 84% increment from 26.70 MPa to 49.16 MPa. The scanning electron microscopy (SEM) images revealed significantly rougher surface morphology and stronger interfacial bonding of the alkali treated fiber bundles with matrix. The good mechanical properties observed demonstrated the absolute potential of resultant composites reinforced by flaxseed fiber bundles for utilization in the civil and industrial applications.

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Rajput, Kishore Shankarsinh, and Vidya Shivram Patil. "Structure and development of cortical bundles in Couroupita guianensis Aubl. (Lecythidaceae)." Anales de Biología, no. 38 (June 27, 2016): 95–102. http://dx.doi.org/10.6018/analesbio.38.10.

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El desarrollo de haces corticales, en ramas y pedúnculos de Couroupita guianensis (Lecythidaceae), comienza cerca del meristemo apical concomitante con los haces vasculares normales. Cada haz cortical llega a estar rodeado por una vaina de fibras que, a menudo, mostraba la presencia de una capa gelatinosa (fibras G). A medida que avanza el crecimiento, cada haz se puede dividir en dos o tres haces. Algunos de los haces son mayores y muestran elementos vasculares bien diferenciados debido a su asociación con frutos en desarrollo, mientras que los más pequeños, con pocos vasos, pueden ser trazas foliares o de yemas de flores que caen antes de la fecundación. El xilema secundario del haz cortical está compuesto de vasos, fibras y células del parénquima axial, mientras que el floema consiste en tubos cribosos, células de acompañamiento y células del parénquima axial.The development of cortical bundles, in the branches and peduncles of Couroupita guianensis (Lecythidaceae), initiates close to the apical meristem concomitant with the normal vascular bundles. Each cortical bundle becomes surrounded by a sheath of fibres, which most often showed presence of gelatinous layer (G-fibres). As growth progresses, theses bundle may divide into two-three bundles. Some of the bundles are larger and show well differentiated vascular elements due to their association with developing fruits while narrower bundles, with few vessels, may be leaf traces or flower bud traces that fell down before fertilization. The secondary xylem of cortical bundle is composed of vessels, fibres and axial parenchyma cells while phloem consists of sieve tubes, companion cells and axial parenchyma cells.

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Abbassi, Mohamed Tahar Kadaoui, and Ibrahim Lakrini. "On the completeness of total spaces of horizontally conformal submersions." Communications in Mathematics 29, no. 3 (December 1, 2021): 493–504. http://dx.doi.org/10.2478/cm-2021-0031.

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Abstract In this paper, we address the completeness problem of certain classes of Riemannian metrics on vector bundles. We first establish a general result on the completeness of the total space of a vector bundle when the projection is a horizontally conformal submersion with a bound condition on the dilation function, and in particular when it is a Riemannian submersion. This allows us to give completeness results for spherically symmetric metrics on vector bundle manifolds and eventually for the class of Cheeger-Gromoll and generalized Cheeger-Gromoll metrics on vector bundle manifolds. Moreover, we study the completeness of a subclass of g-natural metrics on tangent bundles and we extend the results to the case of unit tangent sphere bundles. Our proofs are mainly based on techniques of metric topology and on the Hopf-Rinow theorem.

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28

Wu, Jing, Heng Wang, Xuan Guo, and Jiong Chen. "Cofilin-mediated actin dynamics promotes actin bundle formation during Drosophila bristle development." Molecular Biology of the Cell 27, no. 16 (August 15, 2016): 2554–64. http://dx.doi.org/10.1091/mbc.e16-02-0084.

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The actin bundle is an array of linear actin filaments cross-linked by actin-bundling proteins, but its assembly and dynamics are not as well understood as those of the branched actin network. Here we used the Drosophila bristle as a model system to study actin bundle formation. We found that cofilin, a major actin disassembly factor of the branched actin network, promotes the formation and positioning of actin bundles in the developing bristles. Loss of function of cofilin or AIP1, a cofactor of cofilin, each resulted in increased F-actin levels and severe defects in actin bundle organization, with the defects from cofilin deficiency being more severe. Further analyses revealed that cofilin likely regulates actin bundle formation and positioning by the following means. First, cofilin promotes a large G-actin pool both locally and globally, likely ensuring rapid actin polymerization for bundle initiation and growth. Second, cofilin limits the size of a nonbundled actin-myosin network to regulate the positioning of actin bundles. Third, cofilin prevents incorrect assembly of branched and myosin-associated actin filament into bundles. Together these results demonstrate that the interaction between the dynamic dendritic actin network and the assembling actin bundles is critical for actin bundle formation and needs to be closely regulated.

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29

LI, LINGGUANG. "ON A CONJECTURE OF LAN–SHENG–ZUO ON SEMISTABLE HIGGS BUNDLES: RANK 3 CASE." International Journal of Mathematics 25, no. 02 (February 2014): 1450013. http://dx.doi.org/10.1142/s0129167x1450013x.

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Let X be a smooth projective curve of genus g over an algebraically closed field k of characteristic p > 2. We prove that any rank 3 nilpotent semistable Higgs bundle (E, θ) on X is a strongly semistable Higgs bundle. This gives a partially affirmative answer to a conjecture of Lan–Sheng–Zuo [Semistable Higgs bundles and representations of algebraic fundamental groups: positive characteristic case, preprint (2012), arXiv:1210.8280][(Very recently, A. Langer [Semistable modules over Lie algebroids in positive characteristic, preprint (2013), arXiv:1311.2794] and independently Lan–Sheng–Yang–Zuo [Semistable Higgs bundles of small ranks are strongly Higgs semistable, preprint (2013), arXiv:1311.2405] have proven the conjecture for ranks less than or equal to p case.)] In addition, we prove a tensor product theorem for strongly semistable Higgs bundles with p satisfying some bounds (Theorem 4.3). From this we reprove a tensor theorem for semistable Higgs bundles on the condition that the Lan–Sheng–Zuo conjecture holds (Corollary 4.4).

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Gopal, Suhasini R., Yvonne T. Lee, Ruben Stepanyan, Brian M. McDermott, and Kumar N. Alagramam. "Unconventional secretory pathway activation restores hair cell mechanotransduction in an USH3A model." Proceedings of the National Academy of Sciences 116, no. 22 (May 16, 2019): 11000–11009. http://dx.doi.org/10.1073/pnas.1817500116.

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The pathogenic variant c.144T>G (p.N48K) in the clarin1 gene (CLRN1) results in progressive loss of vision and hearing in Usher syndrome IIIA (USH3A) patients. CLRN1 is predicted to be an essential protein in hair bundles, the mechanosensory structure of hair cells critical for hearing and balance. When expressed in animal models, CLRN1 localizes to the hair bundle, whereas glycosylation-deficient CLRN1N48K aggregates in the endoplasmic reticulum, with only a fraction reaching the bundle. We hypothesized that the small amount of CLRN1N48K that reaches the hair bundle does so via an unconventional secretory pathway and that activation of this pathway could be therapeutic. Using genetic and pharmacological approaches, we find that clarin1 knockout (clrn1KO/KO) zebrafish that express the CLRN1c.144T>G pathogenic variant display progressive hair cell dysfunction, and that CLRN1N48K is trafficked to the hair bundle via the GRASP55 cargo-dependent unconventional secretory pathway (GCUSP). On expression of GRASP55 mRNA, or on exposure to the drug artemisinin (which activates GCUSP), the localization of CLRN1N48K to the hair bundles was enhanced. Artemisinin treatment also effectively restored hair cell mechanotransduction and attenuated progressive hair cell dysfunction in clrn1KO/KO larvae that express CLRN1c.144T>G, highlighting the potential of artemisinin to prevent sensory loss in CLRN1c.144T>G patients.

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31

SZAMOTULSKI, MARCIN, and DOROTA MARCINIAK. "TOTAL SPACE OF ABELIAN GERBES." International Journal of Modern Physics A 24, no. 15 (June 20, 2009): 2877–88. http://dx.doi.org/10.1142/s0217751x09046229.

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We present a generalization of the construction of a principal G-bundle from a one Čech cocycle to the case of higher abelian gerbes. We prove that the sheaf of local sections of the associated bundle to a higher abelian gerbe is isomorphic to the sheaf of sections of the gerbe itself. Our main result states that equivalence classes of higher abelian gerbes are in bijection with isomorphism classes of the corresponding bundles. We also present topological characterization of those bundles. In the last section, we show that the usual notion of Ehresmann connection leads to the gerbe connection for higher ℂ*-gerbes.

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32

Kempf, George R. "Rank g Picard Bundles are Stable." American Journal of Mathematics 112, no. 3 (June 1990): 397. http://dx.doi.org/10.2307/2374748.

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33

Laszlo, Yves. "About $G$-bundles over elliptic curves." Annales de l’institut Fourier 48, no. 2 (1998): 413–24. http://dx.doi.org/10.5802/aif.1623.

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34

Corlette, Kevin. "Flat $G$-bundles with canonical metrics." Journal of Differential Geometry 28, no. 3 (1988): 361–82. http://dx.doi.org/10.4310/jdg/1214442469.

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35

Friedman, Robert, John W. Morgan, and Edward Witten. "Principal $G$-bundles over elliptic curves." Mathematical Research Letters 5, no. 1 (1998): 97–118. http://dx.doi.org/10.4310/mrl.1998.v5.n1.a8.

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36

Abbassi, Mohamed Tahar Kadaoui, Noura Amri, and Giovanni Calvaruso. "g ‐natural symmetries on tangent bundles." Mathematische Nachrichten 293, no. 10 (July 8, 2020): 1873–87. http://dx.doi.org/10.1002/mana.201900158.

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37

BALACHANDRAN, A. P., G. MARMO, A. SIMONI, and G. SPARANO. "QUANTUM BUNDLES AND THEIR SYMMETRIES." International Journal of Modern Physics A 07, no. 08 (March 30, 1992): 1641–67. http://dx.doi.org/10.1142/s0217751x92000715.

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Wave functions in the domain of observables such as the Hamiltonian are not always smooth functions on the classical configuration space Q. Rather, they are often best regarded as functions on a G bundle EG over Q or as sections of an associated bundle. If H is a classical group which acts on Q, its quantum version HG, which acts on EG, is not always H, but an extension of H by G. A powerful and physically transparent construction of EG and HG, where G= U(1) and H1(Q, Z)=0, has been developed using the path space [Formula: see text]. ([Formula: see text] consists of paths on Q from a fixed point.) In this paper we show how to construct EG and HG when G is U(1) or U(1)×π1(Q) and there is no restriction on H1(Q, Z). The method is illustrated with concrete examples, such as a system of charges and monopoles. We argue also that [Formula: see text] is a sort of superbundle from which a large variety of bundles can be obtained by imposing suitable equivalence relations.

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38

Pal, Sarbeswar. "Moduli of Rank 2 Stable Bundles and Hecke Curves." Canadian Mathematical Bulletin 59, no. 4 (December 1, 2016): 865–77. http://dx.doi.org/10.4153/cmb-2016-058-9.

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AbstractLet X be a smooth projective curve of arbitrary genus g > 3 over the complex numbers. In this short note we will show that the moduli space of rank 2 stable vector bundles with determinant isomorphic to Lx , where Lx denotes the line bundle corresponding to a point x ∊ X, is isomorphic to a certain variety of lines in the moduli space of S-equivalence classes of semistable bundles of rank 2 with trivial determinant.

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Ionescu, Marius, Alex Kumjian, Aidan Sims, and Dana P. Williams. "A stabilization theorem for Fell bundles over groupoids." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148, no. 1 (October 17, 2017): 79–100. http://dx.doi.org/10.1017/s0308210517000129.

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We study the C*-algebras associated with upper semi-continuous Fell bundles over second-countable Hausdorff groupoids. Based on ideas going back to the Packer–Raeburn ‘stabilization trick’, we construct from each such bundle a groupoid dynamical system whose associated Fell bundle is equivalent to the original bundle. The upshot is that the full and reduced C*-algebras of any saturated upper semi-continuous Fell bundle are stably isomorphic to the full and reduced crossed products of an associated dynamical system. We apply our results to describe the lattice of ideals of the C*-algebra of a continuous Fell bundle by applying Renault's results about the ideals of the C*-algebras of groupoid crossed products. In particular, we discuss simplicity of the Fell-bundle C*-algebra of a bundle over G in terms of an action, described by Ionescu and Williams, of G on the primitive-ideal space of the C*-algebra of the part of the bundle sitting over the unit space. We finish with some applications to twisted k-graph algebras, where the components of our results become more concrete.

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40

Ali, Sahadat. "Prolongation of Tensor Fields and G-Structures in Tangent Bundles of Second Order." Journal of the Tensor Society 9, no. 01 (June 30, 2009): 77–81. http://dx.doi.org/10.56424/jts.v9i01.10563.

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Tangent and cotangent bundles have been defined and studied by Yano, Ishihara, Patterson and others. Duggal gives the notion of GF-structure, which plays an important role in the differentiable manifold [1]. R. Nivas and Ali have studied the existence of GF-structure and generalized contact structure on the tangent bundle and some interesting results have been obtained for such structures [2]. Prolongation of tensor fields, almost complex and almost product structures have been defined and studied by Yano, Ishihara [3] and others whereas Das [4] and Morimoto [5] have studied the prolongation of F-structure and G-structures respectively to the tangent bundles. In the present paper problems of prolongation in tangent bundle of second order and few results on GF, fa(3, −1) and generalized contact structures have been discussed.

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Kouotchop Wamba, P. M., G. F. Wankap Nono, and A. Ntyam. "Prolongations of G-structures related to Weil bundles and some applications." Extracta Mathematicae 37, no. 1 (June 1, 2022): 111–38. http://dx.doi.org/10.17398/2605-5686.37.1.111.

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Let M be a smooth manifold of dimension m ≥ 1 and P be a G-structure on M , where G is a Lie subgroup of linear group GL(m). In [8], it has been defined the prolongations of G-structures related to tangent functor of higher order and some properties have been established. The aim of this paper is to generalize these prolongations to a Weil bundles. More precisely, we study the prolongations of G-structures on a manifold M , to its Weil bundle TAM (A is a Weil algebra) and we establish some properties. In particular, we characterize the canonical tensor fields induced by the A-prolongation of some classical G-structures.

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BISWAS, I., T. GÓMEZ, and V. MUÑOZ. "Torelli theorem for the moduli space of framed bundles." Mathematical Proceedings of the Cambridge Philosophical Society 148, no. 3 (November 26, 2009): 409–23. http://dx.doi.org/10.1017/s0305004109990417.

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AbstractLet X be an irreducible smooth complex projective curve of genus g ≥ 2, and let x ∈ X be a fixed point. Fix r > 1, and assume that g > 2 if r = 2. A framed bundle is a pair (E, φ), where E is coherent sheaf on X of rank r and fixed determinant ξ, and φ: Ex → r is a non–zero homomorphism. There is a notion of (semi)stability for framed bundles depending on a parameter τ > 0, which gives rise to the moduli space of τ–semistable framed bundles τ. We prove a Torelli theorem for τ, for τ > 0 small enough, meaning, the isomorphism class of the one–pointed curve (X, x), and also the integer r, are uniquely determined by the isomorphism class of the variety τ.

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43

Aschieri, Paolo, Rita Fioresi, and Emanuele Latini. "Quantum Principal Bundles on Projective Bases." Communications in Mathematical Physics 382, no. 3 (March 2021): 1691–724. http://dx.doi.org/10.1007/s00220-021-03985-4.

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AbstractThe purpose of this paper is to propose a sheaf theoretic approach to the theory of quantum principal bundles over non affine bases. We study noncommutative principal bundles corresponding to $$G \rightarrow G/P$$ G → G / P , where G is a semisimple group and P a parabolic subgroup.

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44

Marian, Alina, Dragos Oprea, Rahul Pandharipande, Aaron Pixton, and Dimitri Zvonkine. "The Chern character of the Verlinde bundle over ℳ¯ g,n." Journal für die reine und angewandte Mathematik (Crelles Journal) 2017, no. 732 (November 1, 2017): 147–63. http://dx.doi.org/10.1515/crelle-2015-0003.

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Abstract We prove an explicit formula for the total Chern character of the Verlinde bundle of conformal blocks over \overline{\mathcal{M}}_{g,n} in terms of tautological classes. The Chern characters of the Verlinde bundles define a semisimple CohFT (the ranks, given by the Verlinde formula, determine a semisimple fusion algebra). According to Teleman’s classification of semisimple CohFTs, there exists an element of Givental’s group transforming the fusion algebra into the CohFT. We determine the element using the first Chern class of the Verlinde bundle on the interior {\mathcal{M}}_{g,n} and the projective flatness of the Hitchin connection.

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45

Hong, Sungpyo, Jin Ho Kwak, and Jaeun Lee. "Bipartite graph bundles with connected fibres." Bulletin of the Australian Mathematical Society 59, no. 1 (February 1999): 153–61. http://dx.doi.org/10.1017/s0004972700032718.

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Let G be a finite connected simple graph. The isomorphism classes of graph bundles and graph coverings over G have been enumerated by Kwak and Lee. Recently, Archdeacon and others characterised bipartite coverings of G and enumerated the isomorphism classes of regular 2p-fold bipartite coverings of G, when G is nonbipartite. In this paper, we characterise bipartite graph bundles over G and derive some enumeration formulas of the isomorphism classes of them when the fibre is a connected bipartite graph. As an application, we compute the exact numbers of the isomorphism classes of bipartite graph bundles over G when the fibre is the path Pn or the cycle Cn.

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46

Heinloth, Jochen. "Semistable reduction for $G$-bundles on curves." Journal of Algebraic Geometry 17, no. 1 (January 1, 2008): 167–83. http://dx.doi.org/10.1090/s1056-3911-07-00476-6.

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Zelaci, Hacen. "On very stablity of principal G-bundles." Geometriae Dedicata 204, no. 1 (April 8, 2019): 165–73. http://dx.doi.org/10.1007/s10711-019-00447-z.

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48

Biswas, Indranil. "Invariant and homogeneous bundles on G/Γ." Advances in Mathematics 232, no. 1 (January 2013): 327–34. http://dx.doi.org/10.1016/j.aim.2012.08.018.

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Muñoz, Roberto, Gianluca Occhetta, and Luis E. Solá Conde. "Rank two Fano bundles on G(1,4)." Journal of Pure and Applied Algebra 216, no. 10 (October 2012): 2269–73. http://dx.doi.org/10.1016/j.jpaa.2012.03.006.

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50

Langer, Adrian. "Semistable principal $G$ -bundles in positive characteristic." Duke Mathematical Journal 128, no. 3 (June 2005): 511–40. http://dx.doi.org/10.1215/s0012-7094-04-12833-7.

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