Loop Space Research Articles

Homology transfer products on free loop spaces: orientation reversal on spheres

Homology transfer products on free loop spaces: orientation reversal on spheres

Shear Bond Strength of Different Types of Cement Used for Bonding Band and Loop Space Maintainer

Shear Bond Strength of Different Types of Cement Used for Bonding Band and Loop Space Maintainer

The fermionic integral on loop space and the Pfaffian line bundle

As the loop space of a Riemannian manifold is infinite-dimensional, it is a non-trivial problem to make sense of the “top degree component” of a differential form on it. In this paper, we show that a formula from finite dimensions generalizes to assign a sensible “top degree component” to certain composite forms, obtained by wedging with the exponential (in the exterior algebra) of the canonical presymplectic 2-form on the loop space. This construction is a crucial ingredient for the definition of the supersymmetric path integral on the loop space.

The principle of maximum entropy and the principle of minimum influence from the position of models in different sections

With involvement of different sections of mathematics: the tor of operations in topology and a theory of combinations, an approximation of function of two variables by the sum of products of function of one variable Soboleva in spaces, infinite-to-one spaces of loops when reviewing a transfer, the theorem of Grotendika-Rimana-Rokha for nonspecial varieties, etc., consider some plants correspond them with a position of models, it are show that these models by comparison with a condition of system with the maximum entropy correspond to a principle of least action, was rather convincingly correlat among themselves. Therefore the purpose of the paper - to show a similarity of interaction of correspond plants in thermodynamics, a mechanics and mathematics. Thus, this purpose, could lead to some Applications of more detailed study of various dynamic systems where from correspond models it are possible to pass to a writing of algorithms and further to a writing of programs. For example when reviewing interaction of geodynamic components of the Earth (atmospheres, was more its than the upper stratum - ionospheres). In the Inference it are formulate that when reviewing dynamic systems, with a position are more their than more detailed study, "interosculation" of various areas of mathematical knowledge where the maximum principle of an entropy corresponded to a principle of least action are necessary and these principles should appear with a position of some "governor".

Eilenberg–Moore spectral sequences converging to the homology of the four fold loop spaces of compact simple Lie groups

Eilenberg–Moore spectral sequences converging to the homology of the four fold loop spaces of compact simple Lie groups

Symplectic cohomology and a conjecture of Viterbo

We identify a new class of closed smooth manifolds for which there exists a uniform bound on the Lagrangian spectral norm of Hamiltonian deformations of the zero section in a unit cotangent disk bundle. This settles a well-known conjecture of Viterbo from 2007 as the special case of \(T^n,\) which has been completely open for \(n>1\). Our methods are different and more intrinsic than those of the previous work of the author first settling the case \(n=1\). The new class of manifolds is defined in topological terms involving the Chas–Sullivan algebra and the BV-operator on the homology of the free loop space. It contains spheres and is closed under products. We discuss generalizations and various applications, to \(C^0\) symplectic topology in particular.

Children’s satisfaction and survival rate of fiber-reinforced space maintainer versus band and loop space maintainer in children with premature loss of maxillary primary first molar

Background: Band and loop space maintainer (BLSM) are well tolerated and long-lasting appliances, but it comes with several drawbacks, including longer lab times, more frequent cement loss dislodgements, longer visits, and the inability to avoid the rotation or tilting of neighboring teeth. These limitations motivate the search for alternative appliance such as fiber-reinforced space maintainers. Aim of the study: Evaluation of children’s satisfaction and survival rate of fiber-reinforced space maintainer versus band and loop space maintainer in children with premature loss of maxillary primary 1st molar. Materials and Methods: Thirty patients, randomly divided into two groups (n=15). BLSM group: which included (15) patients who receive band and loop space maintainer. FRSM group: which included (15) patients who were received fiber-reinforced space maintainer. The patients were evaluated for satisfaction based on Likert -type scale and survival rate. Results: The FRSM group showed a higher satisfaction score than BLSM group. For survival rate the fiber-reinforced space maintainers showed a greater success rate than band and loop space maintainers. Conclusions: The FRSM can be used as an alternative to BLSM regarding clinical performance and patient’s satisfaction in children with premature loss of maxillary 1st primary molars.

Equivariant higher Dixmier-Douady theory for circle actions on UHF-algebras

We develop an equivariant Dixmier-Douady theory for locally trivial bundles of C⁎-algebras with fibre D⊗K equipped with a fibrewise T-action, where T denotes the circle group and D=End(V)⊗∞ for a T-representation V. In particular, we show that the group of T-equivariant ⁎-automorphisms AutT(D⊗K) is an infinite loop space giving rise to a cohomology theory ED,T⁎(X). Isomorphism classes of equivariant bundles then form a group with respect to the fibrewise tensor product that is isomorphic to ED,T1(X)≅[X,BAutT(D⊗K)]. We compute this group for tori and compare the case D=C to the equivariant Brauer group for trivial actions on the base space.

Feynman–Kac formula for perturbations of order le 1, and noncommutative geometry

Let Q be a differential operator of order le 1 on a complex metric vector bundle mathscr {E}rightarrow mathscr {M} with metric connection nabla over a possibly noncompact Riemannian manifold mathscr {M}. Under very mild regularity assumptions on Q that guarantee that nabla ^{dagger }nabla /2+Q canonically induces a holomorphic semigroup mathrm {e}^{-zH^{nabla }_{Q}} in Gamma _{L^2}(mathscr {M},mathscr {E}) (where z runs through a complex sector which contains [0,infty )), we prove an explicit Feynman–Kac type formula for mathrm {e}^{-tH^{nabla }_{Q}}, t>0, generalizing the standard self-adjoint theory where Q is a self-adjoint zeroth order operator. For compact mathscr {M}’s we combine this formula with Berezin integration to derive a Feynman–Kac type formula for an operator trace of the form TrV~∫0te-sHV∇Pe-(t-s)HV∇ds,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mathrm {Tr}\\left( \\widetilde{V}\\int ^t_0\\mathrm {e}^{-sH^{\ abla }_{V}}P\\mathrm {e}^{-(t-s)H^{\ abla }_{V}}\\mathrm {d}s\\right) , \\end{aligned}$$\\end{document}where V,widetilde{V} are of zeroth order and P is of order le 1. These formulae are then used to obtain a probabilistic representations of the lower order terms of the equivariant Chern character (a differential graded extension of the JLO-cocycle) of a compact even-dimensional Riemannian spin manifold, which in combination with cyclic homology play a crucial role in the context of the Duistermaat–Heckmann localization formula on the loop space of such a manifold.

Dualized gravity beyond linear approximation

We will construct a loop space formalism for general relativity, and construct the Polyakov variables as connections for such a loop space. We will use these Polyakov variables to construct a dual theory of gravity beyond linear approximation. It will be demonstrated that this loop space duality reduces to the Hodge duality for linearized gravity. Furthermore, a loop space curvature will be constructed from this Polyakov variable. It will be shown that this loop space curvature vanishes in the absence of topological defects, and so it can be used to investigate gravitational monopoles. We will also construct the suitable monopole charge for such gravitational monopoles.

A detailed look on actions on Hochschild complexes especially the degree 1 coproduct and actions on loop spaces

We explain our previous results about Hochschild actions (2007/2008) pertaining, in particular, to the coproduct, which appeared in a different form in a work by Goresky and Hingston (2009), and provide a fresh look at the results. We recall the general action, specialize to the aforementioned coproduct and prove that the assumption of commutativity, made for convenience in a previous article (2008), is not needed. We give detailed background material on loop spaces, Hochschild complexes and dualizations, and discuss details and extensions of these techniques which work for all operations given in two previous articles (2007/2008). With respect to loop spaces, we show that the coproduct is well defined modulo constant loops and going one step further that in the case of a graded Gorenstein Frobenius algebra, the coproduct is well defined on the reduced normalized Hochschild complex. We discuss several other aspects such as “time reversal” duality and several homotopies of operations induced by it. This provides a cohomology operation which is a homotopy of the anti-symmetrization of the coproduct. The obstruction again vanishes on the reduced normalized Hochschild complex if the Frobenius algebra is graded Gorenstein. Further structures such as “animation”, the BV structure, a coloring for operations on chains and cochains, and a Gerstenhaber double bracket are briefly treated.

The positive scalar curvature cobordism category

We prove that many spaces of positive scalar curvature (psc) metrics have the homotopy type of infinite loop spaces. Our result in particular applies to the path component of the round metric inside R+(Sd) if d≥6. To achieve that goal, we study the cobordism category of manifolds with positive scalar curvature. Under suitable connectivity conditions, we can identify the homotopy fiber of the forgetful map from the psc cobordism category to the ordinary cobordism category with a delooping of spaces of psc metrics. This uses a version of Quillen’s Theorem B and instances of the Gromov–Lawson surgery theorem. We extend some of the surgery arguments by Galatius and the second-named author to the psc setting to pass between different connectivity conditions. Segal’s theory of Γ-spaces is then used to construct the claimed infinite loop space structures. The cobordism category viewpoint also illuminates the action of diffeomorphism groups on spaces of psc metrics. We show that under mild hypotheses on the manifold, the action map from the diffeomorphism group to the homotopy automorphisms of the spaces of psc metrics factors through the Madsen–Tillmann spectrum. This implies a strong rigidity theorem for the action map when the manifold has trivial rational Pontryagin classes. A delooped version of the Atiyah–Singer index theorem proved by the first-named author is, moreover, used to show that the secondary index invariant to real K-theory is an infinite loop map. These ideas also give a new proof of the main result of our previous work with Botvinnik.

Loop space decompositions of (2n-2)-connected (4n-1)-dimensional Poincaré Duality complexes

Beben and Wu showed that if M is a (2n-2)-connected (4n-1)-dimensional Poincaré Duality complex such that nge 3 and H^{2n}(M;{{mathbb {Z}}}) consists only of odd torsion, then Omega M can be decomposed up to homotopy as a product of simpler, well-studied spaces. We use a result from Beben and Theriault (Doc Math 27:183-211, 2022) to greatly simplify and enhance Beben and Wu’s work and to extend it in various directions.

Complex manifolds of Sobolev mappings and a Hartogs-type theorem in loop spaces

ABSTRACT We recall the complex structure on the generalized loop spaces , where S is a compact real manifold with boundary and X is a complex manifold, and prove a Hartogs-type extension theorem for holomorphic maps from certain domains in generalized loop spaces.

Global algebraic K‐theory

We introduce a global equivariant refinement of algebraic K-theory; here `global equivariant' refers to simultaneous and compatible actions of all finite groups. Our construction turns a specific kind of categorical input data into a global $\Omega$-spectrum that keeps track of genuine $G$-equivariant infinite loop spaces, for all finite groups $G$. The resulting global algebraic K-theory spectrum is a rigid way of packaging the representation K-theory, or `Swan K-theory' into one highly structured object.

Tate’s thesis in the de Rham setting

We calculate the category of D D -modules on the loop space of the affine line in coherent terms. Specifically, we find that this category is derived equivalent to the category of ind-coherent sheaves on the moduli space of rank one de Rham local systems with a flat section. Our result establishes a conjecture coming out of the 3 d 3d mirror symmetry program, which obtains new compatibilities for the geometric Langlands program from rich dualities of QFTs that are themselves obtained from string theory conjectures.

A Zero-Curvature Representation of Electromagnetism and the Conservation of Electric Charge

We show that the laws of electromagnetism in $(D+1)$-dimensional Minkowski space-time $\mathcal{M}$, explicitly for $D=1$, $2$ and $3$, can be obtained from an integral representation of the zero-curvature equation in the corresponding loop space $\mathcal{L}^{(D-1)}(\mathcal{M})$. The conservation of the electric charge can be seen as the result of a hidden symmetry in this representation of the dynamical equations.

H-space and loop space structures for intermediate curvatures

For dimensions [Formula: see text] and [Formula: see text], we show that the space of metrics of [Formula: see text]-positive Ricci curvature on the sphere [Formula: see text] has the structure of an [Formula: see text]-space with a homotopy commutative, homotopy associative product operation. We further show, using the theory of operads and results of Boardman, Vogt and May, that the path component of this space containing the round metric is weakly homotopy equivalent to an [Formula: see text]-fold loop space.

Comparison of fit accuracy between conventional and CAD/CAM-fabricated band-loop space maintainers.

The aim of this in vitro study was to evaluate the band fit of metallic band-loop space maintainers fabricated by conventional and 3D printing technologies for clinical applications. Sixteen digital intraoral impressions were selected, and resin models were printed with a 3D printer. Forty band-loop space maintainers were fabricated (20 each for both the conventional and digital groups). To evaluate the band fit of conventional and CAD/CAM-fabricated space maintainers, cement space was visualized with low-viscosity condensation silicone material on resin models. Digital evaluation was carried out using dual scan method by superimposing two scans of resin models covered with or without a silicone layer on the abutment tooth. The mean quadratic deviation (root mean square, RMS) was calculated. The RMS values between the groups were evaluated with Student's t test. Statistical significance was set at p<.05. No statistical significance was found between the conventional and digital groups (p=.56). Within the limitations of this study, the band fit of conventional and CAD/CAM-fabricated space maintainers had no difference. Future research into CAD/CAM-fabricated appliances from different perspectives such as fracture strength and patient comfort is warranted.

Invariance of the Goresky–Hingston algebra on reduced Hochschild homology

We prove that two quasi-isomorphic simply connected differential graded associative Frobenius algebras have isomorphic Goresky–Hingston algebras on their reduced Hochschild homology. Our proof is based on relating the Goresky–Hingston algebra on reduced Hochschild homology to the singular Hochschild cohomology algebra. For any simply connected oriented closed manifold M $M$ of dimension k $k$ , the Goresky–Hingston algebra on reduced Hochschild homology induces an algebra structure of degree k − 1 $k-1$ on H ¯ ∗ ( L M ; Q ) $\overline{\mathrm{H}}^*(LM;\mathbb {Q})$ , the reduced rational cohomology of the free loop space of M $M$ . As a consequence of our algebraic result, we deduce that the isomorphism class of the induced algebra structure on H ¯ ∗ ( L M ; Q ) $\overline{\mathrm{H}}^*(LM;\mathbb {Q})$ is an invariant of the homotopy type of M $M$ .