Let $V$ be a vertex algebra. The traditional non-commutative Wick formula is a tool to calculate term like $[a_\lambda:bc:]$. However, I need to calculate terms of the form $[:ab:_\lambda c]$. I found a formula in Arakawa's note $$[:ab:_\lambda c]=:(e^{T\partial_\lambda}a)[b_\lambda c]:+:(e^{T\partial_\lambda}b)[a_\lambda c]:+\int_0^\lambda[b_\mu[a_{\lambda-\mu}c]]d\mu.$$ However, I don't understand the meaning of $e^{T\partial_\lambda}a$. How does $\partial_\lambda$ acts on $a$? Moreover, I want to know the correct sign in the super version of this identity. Are there any references that provide all sign stuff?
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1$\begingroup$ Take a look at (!.38) in "Finite vs. Affine W algebras" by De Sole and Kac. Not sure if this question belongs here. arxiv.org/abs/math-ph/0511055 $\endgroup$– Reimundo HeluaniMay 7, 2023 at 22:04
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$\begingroup$ @ReimundoHeluani Thanks, that's exactly what I want. $\endgroup$– EstwaldMay 8, 2023 at 1:52
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