In cohomological field theories we have nilpotent $Q$ operator and one can define Lagrangian as $L=[Q,V]$ and energy momentum tensor $T_{\alpha \beta} = [Q,G_{\alpha \beta}]$. So these quantities are defined as commutators (or anti commutators) of some operators with Q. One also has this relation $\delta_{\epsilon} \mathcal{O}= i\epsilon[Q,\mathcal{O}]$.

My question is: If someone gave me a Lagrangian or energy-momentum tensor for the cohomological field theory how would I proceed to find the operator $Q$ and/or the other two operators $V$ and $G_{\alpha \beta}$ ? Is this possible without guessing?


You have to guess (or rather, choose). The Lagrangian does not uniquely determine $Q$. In fact, there are Lagrangians (with appropriately corresponding supersymmetric QFTs) which admits continuous families of Qs. N=4 SYM is one example.

$V$ and $G_{\alpha\beta}$ are only determined up to $Q$-commutators. Another choice here.

More generally, I don't think there's an algorithm to find all of the symmetries of some class of a given Lagrangian.

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  • $\begingroup$ Can you point me to a reference where this family of operators is derived in N=4 SYM ? $\endgroup$ – Caims Jan 12 '17 at 23:54
  • $\begingroup$ It's discussed in Kapustin & Witten, Electric-Magnetic Duality And The Geometric Langlands Program, arxiv.org/abs/hep-th/0604151 . $\endgroup$ – user1504 Jan 12 '17 at 23:58

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