Antifields and the antibracket

For each field ${\Phi }^A$ introduce an antifield ${\Phi }_A^{\ast }$ of opposite parity and ghost number $\mathrm{gh}({\Phi }_A^{\ast })=-1-\mathrm{gh}({\Phi }^A)$:

Antifields are smooth sections compactly supported in ${M_T}^{\circ }$; on the relative space below, $c^{\ast }$ is a smooth section over $W$. For functionals $F,G$ of fields and antifields, the antibracket is

with the left/right functional derivatives and the graded Jacobi identity of HT (Ch. 15); on local functionals the derivatives are Euler derivatives and the bracket is again local GPS,FRBV. Antifield degree is polynomial degree in the ${\Phi }_A^{\ast }$.