A_infty algebra

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A_infty algebra

attachments/Pasted%20image%2020220207231003.png Here

First relation \(m_1^2=0\) makes \(A\) a chain complex
Second relation makes multiplication a chain morphism
Third relation makes \(m_2\) associative up to chain homotopy
Higher relations impose homotopy coherence

See Gerstenhaber bracket, coalgebra, Maurer-Cartan equations. attachments/Pasted%20image%2020220323182816.png

pre-CY structures

Somehow give rise to TQFTs. attachments/Pasted%20image%2020220323183021.png

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Gerstenhaber bracket in -A_infty algebra
L infty algebra

For A_infty algebras, the role of equations of motion is played by the MC equation \begin{align*} \sum_{n \geq 1} \mu_{n}(a, \ldots, a)=0 \end{align*}

Considering gauge theories for all \({\operatorname{U}}_n\) at the same time: it is more convenient to work with \(\mathrm{A}_{\infty}\) instead of \(\mathrm{L}_{\infty}\) algebras. See A_infty algebra.

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DGA

See A_infty algebra:

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