BV rigidity in the string topology of compact Lie groups

Abstract

Let G be a connected compact Lie group and R a commutative ring over which H_*(G;R) is an exterior algebra on odd primitive generators. We prove a rigidity theorem: the Batalin-Vilkovisky (BV) algebra ℍ_*(LG;R) of Chas and Sullivan on the homology of the free loop space is that of the product of odd spheres with the classical exponents of G, tensored with the group ring of the torsion of π₁(G). Conversely, over an integral domain the graded algebra ℍ_*(LG;R) already determines the exponents and the group ring, hence the BV algebra. The isomorphism with the product of spheres is algebraic: for Sp(n), n≥2, and SU(n), n≥3, no map of spaces induces it over ℤ. If moreover G is simply connected, then ℍ_*(LG;R), as a BV algebra, is the Hochschild cohomology of H^*(G;R) with the BV operator induced by Poincaré duality.

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BibTeX RIS

BibTeX

@misc{mukilraj2026rigidity,
  title = {{BV rigidity in the string topology of compact Lie groups}},
  author = {Mukilraj K},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.04164},
  eprint = {2610.04164},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.04164},
}

RIS

TY  - UNPB
TI  - BV rigidity in the string topology of compact Lie groups
AU  - Mukilraj K
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.04164
UR  - https://arxiv.org/abs/2610.04164
AB  - Let G be a connected compact Lie group and R a commutative ring over which H_*(G;R) is an exterior algebra on odd primitive generators. We prove a rigidity theorem: the Batalin-Vilkovisky (BV) algebra ℍ_*(LG;R) of Chas and Sullivan on the homology of the free loop space is that of the product of odd spheres with the classical exponents of G, tensored with the group ring of the torsion of π_1(G). Conversely, over an integral domain the graded algebra ℍ_*(LG;R) already determines the exponents and the group ring, hence the BV algebra. The isomorphism with the product of spheres is algebraic: for Sp(n), n≥2, and SU(n), n≥3, no map of spaces induces it over ℤ. If moreover G is simply connected, then ℍ_*(LG;R), as a BV algebra, is the Hochschild cohomology of H^*(G;R) with the BV operator induced by Poincaré duality.
ER  -

Abstract

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