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Modeling bundle-valued forms on the path space with a curved iterated integral

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Abstract

The usual iterated integral map given by Chen produces an equivalence between the two-sided bar complex on differential forms and the de Rham complex on the path space. This map fails to make sense when considering the curved differential graded algebra of bundle-valued forms with a covariant derivative induced by a connection. In this paper, we define a curved version of Chen’s iterated integral that incorporates parallel transport and maps an analog of the two-sided bar construction on bundle-valued forms to bundle-valued forms on the path space. This iterated integral is proven to be a homotopy equivalence of curved differential graded algebras, and for real-valued forms it factors through the usual Chen iterated integral.

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Notes

  1. These columns represent the “time-slots” \(0=t_0\le t_1 \le \cdots \le t_n \le t_{n+1}=1\) over which we will eventually integrate when considering the iterated integral for differential forms.

  2. The relation \(\sim _b\) is required to have \(c_z\) be well defined with respect to the relation \(\sim _a\).

  3. Technically, F is only piecewise smooth and PM was defined as the space of smooth paths, but one can modify F with a bump function so that it is smooth.

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Acknowledgements

C.G. would like to thank the Max-Planck-Institut für Mathematik in Bonn, for their support during his visit, as well as St. Joseph’s College for their support with a Faculty Summer Research Grant. The authors would also like to thank Thomas Tradler for inspiring the use of zigzags, as well as Jorge Florez, Joey Hirsch, and Scott Wilson for their helpful comments regarding curved cohomology. Finally, the authors thank the anonymous referees for their numerous suggestions and corrections that improved the paper.

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Authors and Affiliations

  1. SUNY New Paltz, 1 Hawk Dr, New Paltz, NY, 12561, USA

    Cheyne Glass

  2. Long Island University, 720 Northern Blvd, Brookville, NY, 11548, USA

    Corbett Redden

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  1. Cheyne Glass
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  2. Corbett Redden
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Correspondence to Corbett Redden.

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Communicated by Scott Wilson.

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Glass, C., Redden, C. Modeling bundle-valued forms on the path space with a curved iterated integral. J. Homotopy Relat. Struct. 17, 309–353 (2022). https://doi.org/10.1007/s40062-022-00306-x

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  • DOI: https://doi.org/10.1007/s40062-022-00306-x

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