# Almost complex structure on path space

10 Jan 2013-International Journal of Geometric Methods in Modern Physics (World Scientific Publishing Company)-Vol. 10, Iss: 03, pp 1220034

TL;DR: In this article, it was shown 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. Geom. Meth. Phys.

Abstract: 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 Frechet manifold C∞(M, N) is weak integrable.

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TL;DR: In this article, the limits of Banach tensor structures with Frechet structures and adapted connections to $G$-structures in both frameworks are studied. But the authors focus on the case where the connection between the two structures is not projective.

Abstract: We endow projective (resp. direct) limits of Banach tensor structures with Frechet (resp. convenient) structures and study adapted connections to $G$-structures in both frameworks. This situation is illustrated by a lot of examples.

3 citations

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TL;DR: For a finite dimensional symplectic manifold with a weak symplectic form, this paper showed that the loop space over the manifold admits the Darboux chart for the weak structure of the manifold.

Abstract: For a finite dimensional symplectic manifold $(M,\omega)$ with a symplectic form $\omega$, corresponding loop space ($LM=C^\infty(S^1,M)$) admits a weak symplectic form $\Omega^\omega$. We prove that the loop space over $\mbr^n$ admits Darboux chart for the weak symplectic structure $\Omega^\omega$. Further, we show that inclusion map from the symplectic cohomology (as defined by Kriegl and Michor \cite{KM}) of the loop space over $\mathbb R^n$ to the De Rham cohomology of the loop space is an isomorphism.

3 citations

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TL;DR: In this paper, it was shown that geodesics on path space yield a double category, and that if the path space is complete, then the double category can be used to identify paths under backtrack equivalence.

Abstract: Let $M$ be a Riemannian manifold and ${\mathcal P}M$ be the space of all smooth paths on $M$. We describe geodesics on path space ${\mathcal P}M$. Normal neighbourhood structure on ${\mathcal P}M$ has been discussed. We identify paths on $M$ under "back-track" equivalence. Under this identification we show that if $M$ is complete, then geodesics on path space yield a double category.We gave a physical interpretation of this double category.

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TL;DR: In this article, the authors identify paths on a Riemannian manifold under back-track equivalence and show that if M is complete, then geodesics on the path space yield a double category.

Abstract: Let M be a Riemannian manifold and 𝒫M be the space of all smooth paths on M. We describe geodesics on path space 𝒫M. Normal neighborhoods on 𝒫M have been discussed. We identify paths on M under “back-track” equivalence. Under this identification, we show that if M is complete, then geodesics on the path space yield a double category. This double category has a natural interpretation in terms of the worldsheets generated by freely moving (without any external force) strings.

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01 Jan 1993

TL;DR: In this article, a 3-dimensional analogue of the Kostant-Weil theory of line bundles is presented, where the curvature of a fiber bundle becomes a three-dimensional form.

Abstract: This book deals with the differential geometry of manifolds, loop spaces, line bundles and groupoids, and the relations of this geometry to mathematical physics. Recent developments in mathematical physics (e.g., in knot theory, gauge theory and topological quantum field theory) have led mathematicians and physicists to look for new geometric structures on manifolds and to seek a synthesis of ideas from geometry, topology and category theory. In this spirit this book develops the differential geometry associated to the topology and obstruction theory of certain fibre bundles (more precisely, associated to gerbes). The new theory is a 3-dimensional analogue of the familiar Kostant-Weil theory of line bundles. In particular the curvature now becomes a 3-form. Applications presented in the book involve anomaly line bundles on loop spaces and anomaly functionals, central extensions of loop groups, Kaehler geometry of the space of knots, Cheeger-Chern-Simons secondary characteristic classes, and group cohomology. Finally, the last chapter deals with the Dirac monopole and Dirac's quantization of the electrical charge. The book will be of interest to topologists, geometers, Lie theorists and mathematical physicists, as well as to operator algebraists. It is written for graduate students and researchers, and will be an excellent textbook. It has a self-contained introduction to the theory of sheaves and their cohomology, line bundles and geometric prequantization a la Kostant-Souriau.

936 citations

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TL;DR: In particular, this paper showed that a generalized (almost) complex structure on a C∞ manifold can be constructed from the geometric structures on the manifold, and that such a generalized complex structure can be used to construct a generalized almost complex structure.

Abstract: Let M be a C∞ manifold, and let ${\mathcal P}M$ be the space of all smooth maps from [0, 1] to M. We investigate geometric structures on ${\mathcal P}M$ constructed from the geometric structures on M. In particular, we show that a generalized (almost) complex structure on M produce a generalized (almost) complex structure on ${\mathcal P}M$.

7 citations