Continuous Hochschild Cohomology and Formality
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866917168896016384 |
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| author | Antweiler, Patrick |
| author_facet | Antweiler, Patrick |
| contents | We define the appropriate homological setting to study deformation theory of complete locally convex (curved) dg-algebras based on Positselski's contraderived categories. We define the corresponding Hochschild complex controlling deformations and prove formality theorems for the Fréchet algebras of smooth functions on a manifold, the de Rham algebra and for the Dolbeault algebra of a complex manifold. In the latter case, the Hochschild cohomology is equivalent to Kontsevich's extended deformation complex, the Hochschild cohomology of the derived category in case $X$ is a smooth projective variety and to Gualtieri's deformation complex of $X$ viewed as generalized complex manifold. We also compute the continuous Hochschild cohomology for various categories of matrix factorisations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_21090 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Continuous Hochschild Cohomology and Formality Antweiler, Patrick Quantum Algebra Mathematical Physics Category Theory We define the appropriate homological setting to study deformation theory of complete locally convex (curved) dg-algebras based on Positselski's contraderived categories. We define the corresponding Hochschild complex controlling deformations and prove formality theorems for the Fréchet algebras of smooth functions on a manifold, the de Rham algebra and for the Dolbeault algebra of a complex manifold. In the latter case, the Hochschild cohomology is equivalent to Kontsevich's extended deformation complex, the Hochschild cohomology of the derived category in case $X$ is a smooth projective variety and to Gualtieri's deformation complex of $X$ viewed as generalized complex manifold. We also compute the continuous Hochschild cohomology for various categories of matrix factorisations. |
| title | Continuous Hochschild Cohomology and Formality |
| topic | Quantum Algebra Mathematical Physics Category Theory |
| url | https://arxiv.org/abs/2512.21090 |