I am studying Batalin-Vilkovisky formalism. I am a little bit confused on what an odd (or even) symplectic structure is (i know what the degree of the underlying 2-form is). I can not find a clear definition anywhere. I read that an even bilinear form $B$ is even if $B(x,x)=0$ for all $x$ and odd if it exists $x$ such that $B(x,x)=1$ but this does not seem to me to be the right definition for a symplectic form. Could someone enlighten me on this point?
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1$\begingroup$ What source do you use? Textbook or review article. $\endgroup$– DanielCCommented May 22, 2021 at 10:23
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$\begingroup$ @DanielC i use both $\endgroup$– FarCommented May 22, 2021 at 11:03
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$\begingroup$ Hi Far. Welcome to Phys.SE. Which references? Which pages? $\endgroup$– Qmechanic ♦Commented May 23, 2021 at 10:38
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1 Answer
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An even (odd) symplectic 2-form $\omega~=~\frac{1}{2}\mathrm{d}z^I ~\omega_{IJ}(z)~ \mathrm{d}z^J $ on a supermanifold means a Grassmann-even (Grassmann-odd) closed non-degenerate 2-form, respectively.
There is a straightforward generalization to even and odd Poisson structures on supermanifolds.
The Hamiltonian Batalin-Fradkin-Vilkovisky (BFV) formalism uses an even Poisson bracket, while the Lagrangian Batalin-Vilkovisky (BV) formalism uses an odd Poisson bracket (also known as an antibracket).
answered May 22, 2021 at 10:43