Manuscript X-J-ST@1 · X-J-ST.tex · section · 8f0d1a36

Fields and field differential

The retained classical interface and its recovery › Fields and field differential · lines 395–427

X-J-ST@1#eq:smetric, X-J-ST@1#eq:sscalar, X-J-ST@1#eq:saux

Use the independent fields

Φ = (r_{μν}, ϕ, c^{μ}, c ˉ_{μ}, b_{μ}),

of respective parities $0, 0, 1, 1, 0$ and ghost numbers $0, 0, 1, - 1, 0$ . The auxiliary field $b$ remains independent.

With fixed smooth backgrounds $h, k$ , write

q ˉ = η + κh + κ^{2} k, q = q ˉ + κ r .

The field differential is

sr s ϕ s c ˉ = K c + κ L_{c} (h + r) + κ^{2} L_{c} k, = κ c^{μ} \partial_{μ} ϕ, = b, s c^{μ} s b = κ c^{ν} \partial_{ν} c^{μ}, = 0, s h = s k = 0,

where

(K c)_{μν} = \partial_{μ} c_{ν} + \partial_{ν} c_{μ} .

The rules are left graded derivations commuting with total derivatives.

Displayed formulas in this passage

X-J-ST@1#eq:smetric · line 410
$sr s ϕ s c ˉ = K c + κ L_{c} (h + r) + κ^{2} L_{c} k, = κ c^{μ} \partial_{μ} ϕ, = b, s c^{μ} s b = κ c^{ν} \partial_{ν} c^{μ}, = 0, s h = s k = 0,$
X-J-ST@1#eq:sscalar · line 410
$sr s ϕ s c ˉ = K c + κ L_{c} (h + r) + κ^{2} L_{c} k, = κ c^{μ} \partial_{μ} ϕ, = b, s c^{μ} s b = κ c^{ν} \partial_{ν} c^{μ}, = 0, s h = s k = 0,$
X-J-ST@1#eq:saux · line 410
$sr s ϕ s c ˉ = K c + κ L_{c} (h + r) + κ^{2} L_{c} k, = κ c^{μ} \partial_{μ} ϕ, = b, s c^{μ} s b = κ c^{ν} \partial_{ν} c^{μ}, = 0, s h = s k = 0,$
unlabeled · line 398
$Φ = (r_{μν}, ϕ, c^{μ}, c ˉ_{μ}, b_{μ}),$
unlabeled · line 405
$q ˉ = η + κh + κ^{2} k, q = q ˉ + κ r .$
unlabeled · line 422
$(K c)_{μν} = \partial_{μ} c_{ν} + \partial_{ν} c_{μ} .$