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Question is as mentioned in the title:

Are there any introductory notes on deformation theory that are easier to read for differential geometers?

I am learning about differential graded Lie algebras (and $L_\infty$-algebras). I am aware of Marco Manetti's notes Deformation theory via differential graded Lie algebras.

Are there any other notes that are easier to read for people who are trained in differential geometry?

I know some commutative algebra, for example definition (and few properties) of Noetherian, Artinian rings. I can recall (or read) some more commutative algebra if required.

I read from the paper From Lie Theory to Deformation Theory and Quantization that

"Deformation Theory is a natural generalization of Lie Theory, from Lie groups and their linearization, Lie algebras, to differential graded Lie algebras and their higher order deformations, quantum groups."

So, this gave some hope that there would be some notes from the point of view of Lie theory/differential geometry.

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  • $\begingroup$ I am attending trimester program on higher structures indico.math.cnrs.fr/event/7893.. In first week they planned for course on "derived deformation theory".. The above question is an attempt to gather some references and ideas so that I can appreciate that crash course.. $\endgroup$
    – Praphulla Koushik
    Commented Mar 19, 2023 at 16:03
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    $\begingroup$ @LSpice edited. I noted that quote but did not write down the details of the paper.. so, wrote as "somewhere". NoW I edIted.. $\endgroup$
    – Praphulla Koushik
    Commented Mar 19, 2023 at 17:37

3 Answers 3

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A classical subject in deformation theory with a differential-geometric flavour is that of deformation quantization. There are several good introductory references I can recommend, all of which introduce the Maurer–Cartan formalism for DG Lie algebras in deformation theory:

  • Cattaneo, Alberto S., Formality and star products (lecture notes taken by D. Indelicato), Gutt, Simone (ed.) et al., Poisson geometry, deformation quantisation and group representations. Cambridge: Cambridge University Press (ISBN 0-521-61505-4/pbk). London Mathematical Society Lecture Note Series 323, 79-144 (2005). ZBL1077.53074.

  • [some later chapters in French] Cattaneo, Alberto; Keller, Bernhard; Torossian, Charles; Bruguières, Alain, Déformation, quantification, théorie de Lie, Panoramas et Synthèses 20. Paris: Société Mathématique de France (ISBN 2-85629-183-X/pbk). vii, 186 p. (2005). ZBL1093.53095.

  • [in German] Waldmann, Stefan, Poisson-Geometrie und Deformationsquantisierung, Berlin: Springer (ISBN 978-3-540-72517-6/pbk). xii, 612 p. (2007). ZBL1139.53001.

  • Esposito, Chiara, Formality theory. From Poisson structures to deformation quantization, SpringerBriefs in Mathematical Physics 2. Cham: Springer (ISBN 978-3-319-09289-8/pbk; 978-3-319-09290-4/ebook). xii, 90 p. (2015). ZBL1301.81003.

Another reference for deformation quantization, taking a Lie-theoretic point of view (but whose prerequisites are only "basic linear algebra and differential geometry"), is

Besides Manetti's new book

which centers around deformations of compact complex manifolds, there is also a freely available shorter "book" (a 183-page article) by Manetti

where you can also find differential forms in the main examples.

For the passage from the DG Lie algebra / L$_\infty$ algebra viewpoint to derived deformation theory see this question.

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  • $\begingroup$ Many thanks for such details… I can not read French or German. So, I am left with only two options. I did not heard about any of the two references you mentioned about Alberto Catteno’s work or Chiara Esposito’s work. I will see them.. many thanks again.. $\endgroup$
    – Praphulla Koushik
    Commented Mar 23, 2023 at 1:49
  • $\begingroup$ @PraphullaKoushik The first four sections in the second reference, which are already sufficient to get a good overview of deformation quantization (and which cover the introduction to deformation theory) are actually in English. $\endgroup$
    – Severin Barmeier
    Commented Mar 23, 2023 at 5:12
  • $\begingroup$ In particular, the second to fourth sections by Bernhard Keller are available on his webpage webusers.imj-prg.fr/~bernhard.keller/publ/emalca.pdf $\endgroup$
    – Severin Barmeier
    Commented Mar 23, 2023 at 5:49
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    $\begingroup$ I should really thank you for suggesting Chiara Esposito's work. I ready first 14 pages.. That is very nicely written.. May be other works are also very well written but, I thought I should say :D $\endgroup$
    – Praphulla Koushik
    Commented Mar 23, 2023 at 13:56
  • $\begingroup$ I will See Bernhard Keller's notes as you suggested. I will check that.. $\endgroup$
    – Praphulla Koushik
    Commented Mar 23, 2023 at 14:02
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Marco Manetti has recently expanded the 22-page lecture notes from 1999 mentioned in the OP into a 574-page text book: Lie Methods in Deformation Theory (2022).
It is advertised as "being the first book to apply the differential graded Lie approach to deformation theory of complex manifolds".
I have the impression the pdf can be downloaded freely from Springer (but perhaps that capability is tied to my IP range).

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  • $\begingroup$ Many thanks. First google search gave me that result but, I could not download it from my institute page. So, was looking for other resources. I heard from some people that Marco Manetti writing is very good. This may turn out to be the only option for me, but, I will wait for some days and see if others have any other suggestions.. $\endgroup$
    – Praphulla Koushik
    Commented Mar 20, 2023 at 2:09
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    $\begingroup$ It's not free, but that your university or institute subscribed Springer (with a high annual fee). $\endgroup$
    – Z. M
    Commented Mar 20, 2023 at 15:11
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Maybe some physics-oriented introductions will be also helpful: https://link.springer.com/book/10.1007/978-3-031-05122-7 (Kontsevich’s Deformation Quantization and Quantum Field Theory, by Nima Moshayedi), https://staff.fnwi.uva.nl/j.deboer/education/projects/projects/beentjes.pdf ( An introduction to deformation quantization after Kontsevich, bachelor thesis by Sjoerd Beentjes).

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  • $\begingroup$ Many thanks. Both of them seems to be useful for me.. $\endgroup$
    – Praphulla Koushik
    Commented Mar 23, 2023 at 14:08

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