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For most aspherical Seifert-fibered 3-manifolds $M$, the space of Seifert fiberings $\mathrm{\text{SF}}(M)$ is known to have contractible components [McCullough and Soma, The Smale conjecture for Seifert fibered spaces with hyperbolic base orbifold, J. Differential Geom.93(2) (2013) 327–353]. It is also known that the space of Hopf fiberings of the three-sphere is noncontractible [DeTurck et al.., Homotopy type of the moduli space of fibrations of the three-sphere by simple closed curves, in progress]. We provide the second example of a non-aspherical 3-manifold $M$ such that $\mathrm{\text{SF}}(M)$ has noncontractible components. In particular, we show that $\mathrm{\text{SF}}(S^1\times S^2)$ has two connected components which are both homotopy equivalent to the identity-based loop space $\mathrm{\Omega }{S}^3$. We also utilize the reduced product suspension of James [Reduced product spaces, Ann. of Math. (2)62 (1955) 170–197] to exhibit second homology generators for both connected components of $\mathrm{\text{SF}}(S^1\times S^2)$.

Quantization needs evaluation of all of states of a quantized object rather than its stationary states with respect to its energy. In this paper, we have investigated moduli of a quantized elastica, a quantized loop with an energy functional associated with the Schwarz derivative, on a Riemann sphere ℙ. Then it is proved that its moduli space is decomposed to a set of equivalent classes determined by flows obeying the Korteweg-de Vries (KdV) hierarchy which conserve the energy. Since the flow obeying the KdV hierarchy has a natural topology, it induces topology in the moduli space . Using the topology, is classified.

Studies on a loop space in the category of topological spaces Top are well-established and its cohomological properties are well-known. As the moduli space of a quantized elastica can be regarded as a loop space in the category of differential geometry DGeom, we also proved an existence of a functor between a triangle category related to a loop space in Top and that in DGeom using the induced topology.

As Euler investigated the elliptic integrals and its moduli by observing a shape of classical elastica on , this paper devotes relations between hyperelliptic curves and a quantized elastica on ℙ as an extension of Euler's perspective of elastica.

Motivated by the computation of loop space quantum mechanics as indicated in [14], here we seek a better understanding of the tubular geometry of loop space ${ℒ}{ℳ}$ corresponding to a Riemannian manifold ${ℳ}$ around the submanifold of vanishing loops. Our approach is to first compute the tubular metric of ${({{ℳ}}^{2N+1})}_C$ around the diagonal submanifold, where ${({{ℳ}}^N)}_C$ is the Cartesian product of $N$ copies of ${ℳ}$ with a cyclic ordering. This gives an infinite sequence of tubular metrics such that the one relevant to ${ℒ}{ℳ}$ can be obtained by taking the limit $N\to \infty$. Such metrics are computed by adopting an indirect method where the general tubular expansion theorem of [21] is crucially used. We discuss how the complete reparametrization isometry of loop space arises in the large-$N$ limit and verify that the corresponding Killing equation is satisfied to all orders in tubular expansion. These tubular metrics can alternatively be interpreted as some natural Riemannian metrics on certain bundles of tangent spaces of ${ℳ}$ which, for ${ℳ}\times {ℳ}$, is the tangent bundle $T{ℳ}$.

This paper is a step towards realizing T-duality and Hori formulae for loop spaces. Here, we prove T-duality and Hori formulae for winding $q$-loop spaces, which are infinite dimensional subspaces of loop spaces.

The loop space Lℙ₁ of the Riemann sphere consisting of all C^k or Sobolev W^{k, p} maps S¹ → ℙ₁ is an infinite dimensional complex manifold. We compute the Picard group pic(Lℙ₁) of holomorphic line bundles on Lℙ₁ as an infinite dimensional complex Lie group with Lie algebra the Dolbeault group H^{0, 1}(Lℙ₁). The group G of Möbius transformations and its loop group LG act on Lℙ₁. We prove that an element of pic(Lℙ₁) is LG-fixed if it is G-fixed, thus completely answering the question of Millson and Zombro about the G-equivariant projective embedding of Lℙ₁.

I review an extension of the ADHMN construction of monopoles to M-brane models. This extended construction gives a map from solutions to the Basu–Harvey equation to solutions to the self-dual string equation transgressed to loop space. Loop spaces appear in fact quite naturally in M-brane models. This is demonstrated by translating a recently proposed M5-brane model to loop space. Finally, I comment on some recent developments related to the loop space approach to M-brane models.

We study and clarify in a reduced dynamical model for $\mathrm{\text{QCD}}(SU(\infty ))$ called Bollini–Giambiagi model and defined by constant gauge fields Yang–Mills path integral several concepts on the validity of string representations on $\mathrm{\text{QCD}}(SU(\infty ))$ and the confinement problem.

Virtual knots arise in the study of Gauss diagrams and Vassiliev invariants of usual knots. The group of virtual braids on n strings VB_n and its Burau representation to GL_nℤ[t,t^-1] also can be considered. The homological properties of the series of groups VB_n and its Burau representation are studied. The following splitting of infinite loop spaces is proved for the plus-construction of the classifying space of the virtual braid group on the infinite number of strings:

We prove that the rack and quandle spaces of links in 3-manifolds, considered only as topological spaces (disregarding their cubical structure), are closely related to certain subspaces of the loop spaces on the 3-manifold, which we call the vertical and the straight loop space of the link. Using these models we prove that the homotopy type of the non-augmented rack and quandle spaces of a link L in a 3-manifold M depends essentially only on the homotopy type of the pair (M,M -L).

Let M be a complex manifold and let PM ≔ C^∞([0, 1], M) be space of smooth paths over M. We prove that the induced almost complex structure on PM is weak integrable by extending the result of Indranil Biswas and Saikat Chatterjee of [Geometric structures on path spaces, Int. J. Geom. Meth. Mod. Phys.8(7) (2011) 1553–1569]. Further we prove that if M is smooth manifold with corner and N is any complex manifold then induced almost complex structure 𝔍 on Fréchet manifold C^∞(M, N) is weak integrable.

Suppose M be the projective limit of weak symplectic Banach manifolds {(M_i, ϕ_ij)}_{i, j∈ℕ}, where M_i are modeled over reflexive Banach space and σ is compatible with the projective system (defined in the article). We associate to each point x ∈ M, a Fréchet space H_x. We prove that if H_x are locally identical, then with certain smoothness and boundedness condition, there exists a Darboux chart for the weak symplectic structure.

Given an ascending sequence of weak symplectic Banach manifolds on which the Darboux Theorem is true, we can ask about conditions under which the Darboux Theorem is also true on the direct limit. We will show that, in general, without very strong conditions, the answer is negative. In particular, we give an example of an ascending symplectic Banach manifolds on which the Darboux Theorem is true but not on the direct limit. In the second part, we illustrate this discussion in the context of an ascending sequence of Sobolev manifolds of loops in symplectic finite-dimensional manifolds. This context gives rise to an example of direct limit of weak symplectic Banach manifolds on which the Darboux Theorem is true around any point.