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

Domain and background split

The relative local differential and its signed BV lift › Domain and background split · lines 644–678

X-I-ST@1#eq:split, X-I-ST@1#eq:sr, X-I-ST@1#eq:sphi, X-I-ST@1#eq:snonminimal

Retain fixed smooth classical backgrounds $h, k$ , and put

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

Neither $q$ nor $q ˉ$ is the host metric $G$ .

Let $P (U) [[κ]]$ be the algebra of local polynomial jet expressions with supercommutative coefficients. Formal inverse-metric and determinant coefficients are allowed; each fixed coefficient has finite field degree and finite jet order.

Define

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

Use the left graded Leibniz rule. These are the relative field rules of X-H-ST, restated rather than inferred from its title XHST. In particular,

s = s_{0} + κ s_{1} + κ^{2} s_{2}, s q = κ L_{c} q .

Displayed formulas in this passage

X-I-ST@1#eq:split · line 647
$q ˉ = η + κh + κ^{2} k, q = q ˉ + κ r .$
X-I-ST@1#eq:sr · line 661
$sr s ϕ s c ˉ = K c + κ L_{c} (h + r) + κ^{2} L_{c} k, = κ c^{μ} \partial_{μ} ϕ, s c^{μ} = κ c^{ν} \partial_{ν} c^{μ}, = b, s b = 0, s h = s k = 0.$
X-I-ST@1#eq:sphi · line 661
$sr s ϕ s c ˉ = K c + κ L_{c} (h + r) + κ^{2} L_{c} k, = κ c^{μ} \partial_{μ} ϕ, s c^{μ} = κ c^{ν} \partial_{ν} c^{μ}, = b, s b = 0, s h = s k = 0.$
X-I-ST@1#eq:snonminimal · line 661
$sr s ϕ s c ˉ = K c + κ L_{c} (h + r) + κ^{2} L_{c} k, = κ c^{μ} \partial_{μ} ϕ, s c^{μ} = κ c^{ν} \partial_{ν} c^{μ}, = b, s b = 0, s h = s k = 0.$
unlabeled · line 674
$s = s_{0} + κ s_{1} + κ^{2} s_{2}, s q = κ L_{c} q .$

Domain and background split

Displayed formulas in this passage

Nearby in this section