Manuscript XX-B@1 · XX-B.tex · section · 56d708a5

Bundles, pairings, and gauge generator

The coupled Euler operator and the normalized action Hessian › Bundles, pairings, and gauge generator · lines 450–488

XX-B@1#eq:pair0, XX-B@1#eq:pair1, XX-B@1#eq:K, XX-B@1#eq:Kadj

Let

V = S^{2} T^{*} M \oplus R, W = T^{*} M .

Use $g ˉ$ to identify vector and covector gauge descriptors. For $z = (h, φ)$ , $w = (j, ω)$ , and $a, b \in W$ , the integrated pairings are

(z, w)_{0} (a, b)_{1} = \int_{M} (h_{μν} j^{μν} + φ ω) d V_{g ˉ}, = \int_{M} a_{μ} b^{μ} d V_{g ˉ}, d V_{g ˉ} = a (t)^{3} d t d^{3} x .

These pairings are nondegenerate, not positive definite. Complexification uses their Hermitian versions.

Set $v_{μ} = \partial_{μ} ϕ ˉ$ . The gauge generator is

K ξ = (2 \nabla_{(μ} ξ_{ν)}, v_{μ} ξ^{μ}),

where $\nabla$ is the background covariant derivative. Its adjoint is

K^{†} (h, φ)_{ν} = - 2 \nabla^{μ} h_{μν} + v_{ν} φ .

Displayed formulas in this passage

XX-B@1#eq:pair0 · line 461
$(z, w)_{0} (a, b)_{1} = \int_{M} (h_{μν} j^{μν} + φ ω) d V_{g ˉ}, = \int_{M} a_{μ} b^{μ} d V_{g ˉ}, d V_{g ˉ} = a (t)^{3} d t d^{3} x .$
XX-B@1#eq:pair1 · line 461
$(z, w)_{0} (a, b)_{1} = \int_{M} (h_{μν} j^{μν} + φ ω) d V_{g ˉ}, = \int_{M} a_{μ} b^{μ} d V_{g ˉ}, d V_{g ˉ} = a (t)^{3} d t d^{3} x .$
XX-B@1#eq:K · line 477
$K ξ = (2 \nabla_{(μ} ξ_{ν)}, v_{μ} ξ^{μ}),$
XX-B@1#eq:Kadj · line 483
$K^{†} (h, φ)_{ν} = - 2 \nabla^{μ} h_{μν} + v_{ν} φ .$
unlabeled · line 453
$V = S^{2} T^{*} M \oplus R, W = T^{*} M .$