Manuscript X-H-ST@1 · X-H-ST.tex · section · 943aa27e

Antifields and a generator-level sign check

Stage B: an explicit relative classical master action › Antifields and a generator-level sign check · lines 771–812

X-H-ST@1#eq:Hs, X-H-ST@1#eq:Hsexplicit

Adjoin antifields $Φ_{A}^{*}$ with

∣ Φ_{A}^{*} ∣ = ∣ Φ^{A} ∣ + 1 (mod 2), gh (Φ_{A}^{*}) = - 1 - gh (Φ^{A}) .

They are dual tensor densities of the appropriate bundle type.

Use the canonical odd bracket with

(Φ^{A} (x), Φ_{B}^{*} (y)) = δ^{A}_{B} δ (x - y), (F, G) = - (- 1)^{(∣ F ∣ + 1) (∣ G ∣ + 1)} (G, F),

extended as an odd biderivation and by integration by parts on local functionals. For an even Hamiltonian $S$ , its left action is $Q F = (S, F)$ .

Let $R^{A} = s Φ^{A}$ . Define

H_{s} = A \sum (- 1)^{∣ Φ^{A} ∣} \int Φ_{A}^{*} R^{A} d^{4} x .

In components this is

H_{s} = \int {r^{* μν} ((K c)_{μν} + κ (L_{c} (h + r))_{μν} + κ^{2} (L_{c} k)_{μν}) + κ ϕ^{*} c^{μ} \partial_{μ} ϕ - κ c_{μ}^{*} c^{ν} \partial_{ν} c^{μ} - c ˉ^{* μ} b_{μ}} d^{4} x .

Displayed formulas in this passage

X-H-ST@1#eq:Hs · line 794
$H_{s} = A \sum (- 1)^{∣ Φ^{A} ∣} \int Φ_{A}^{*} R^{A} d^{4} x .$
X-H-ST@1#eq:Hsexplicit · line 800
$H_{s} = \int {r^{* μν} ((K c)_{μν} + κ (L_{c} (h + r))_{μν} + κ^{2} (L_{c} k)_{μν}) + κ ϕ^{*} c^{μ} \partial_{μ} ϕ - κ c_{μ}^{*} c^{ν} \partial_{ν} c^{μ} - c ˉ^{* μ} b_{μ}} d^{4} x .$
unlabeled · line 774
$∣ Φ_{A}^{*} ∣ = ∣ Φ^{A} ∣ + 1 (mod 2), gh (Φ_{A}^{*}) = - 1 - gh (Φ^{A}) .$
unlabeled · line 782
$(Φ^{A} (x), Φ_{B}^{*} (y)) = δ^{A}_{B} δ (x - y), (F, G) = - (- 1)^{(∣ F ∣ + 1) (∣ G ∣ + 1)} (G, F),$