Manuscript X-I-ST@1 · X-I-ST.tex · section · 9745fd40

Multilocal functionals and antifields

The relative local differential and its signed BV lift › Multilocal functionals and antifields · lines 727–775

X-I-ST@1#eq:bracket, X-I-ST@1#eq:H

Let $F_{ml} (U)$ be the supercommutative algebra generated by compactly supported local functionals under ordinary multiplication. A product of two single integrals is generally multilocal, not another single local integral.

Local actions are horizontal four-forms modulo horizontal divergences. Their Euler derivatives act on $F_{ml} (U)$ . This is the local variational interpretation of the classical BV bracket FRBV.

Adjoin antifields with

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

They are dual tensor densities. Fix the canonical odd bracket by

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

with the odd biderivation and integration-by-parts rules.

For the even Hamiltonian acting on the left, $QF = (S, F)$ , define

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

Explicitly,

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-I-ST@1#eq:bracket · line 746
$(Φ^{A} (x), Φ_{B}^{*} (y)) = δ^{A}_{B} δ (x - y), (F, G) = - (- 1)^{(∣ F ∣ + 1) (∣ G ∣ + 1)} (G, F),$
X-I-ST@1#eq:H · line 756
$H_{s} = A \sum (- 1)^{∣ Φ^{A} ∣} \int Φ_{A}^{*} s Φ^{A} d^{4} x .$
unlabeled · line 740
$∣ Φ_{A}^{*} ∣ = ∣ Φ^{A} ∣ + 1 (mod 2), gh (Φ_{A}^{*}) = - 1 - gh (Φ^{A}) .$
unlabeled · line 762
$H_{s} = \int [r^{* μν} ((K c)_{μν} + κ (L_{c} (h + r))_{μν} + κ^{2} (L_{c} k)_{μν}) + κ ϕ^{*} c^{μ} \partial_{μ} ϕ - κ c_{μ}^{*} c^{ν} \partial_{ν} c^{μ} - c ˉ^{* μ} b_{μ}] d^{4} x .$

Multilocal functionals and antifields

Displayed formulas in this passage

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