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Nil Wrapped D4 Pullback Metric And Determinant

Physical scope

Exact induced metric and DBI determinant for the stated smooth D4 embedding in the adopted Nil metric block; assumes a four-dimensional mostly-plus string-frame metric in zero-shift cosmic-time form ds_{4,s}^2=-dt^2+h_ij(t,x)dx^i dx^j with smooth positive-definite h and retains only homogeneous u1(t).

Statement

For the embedding $ι_{u}$ of Wrapped D4 Embedding In Nil Coordinates, with smooth $u_{1} = u_{1} (t) > 0$ , $u_{2} = σ \in R / Z$ , $x^{'} = x_{0}^{'}$ , fixed ${\tilde{q}}_{0}$ , and $d s_{4, s}^{2} = - d t^{2} + h_{i j} (t, x) d x^{i} d x^{j}$ where $h_{i j}$ is smooth and positive definite, the induced metric is block diagonal between $X_{4}$ and $σ$ , with $(ι_{u}^{*} G^{(s)})_{μ ν} = g_{4, s, μ ν} + α^{'} L_{u_{1}}^{2} \partial_{μ} u_{1} \partial_{ν} u_{1}, (ι_{u}^{*} G^{(s)})_{σ σ} = α^{'} (L_{u_{2}}^{2} + M^{2} L_{x}^{2} u_{1}^{2}) .$ Consequently, for $α^{'} L_{u_{1}}^{2} {\dot{u}}_{1}^{2} < 1$ , $\sqrt{- \det (ι_{u}^{*} G^{(s)})} = \sqrt{- g_{4, s}} \sqrt{α^{'} (L_{u_{2}}^{2} + M^{2} L_{x}^{2} u_{1}^{2})} \sqrt{1 - α^{'} L_{u_{1}}^{2} {\dot{u}}_{1}^{2}} .$

Statement References

Silverstein and Westphal, §2.3

Facts & Assumptions

Given: The full product metric and lattice and the embedding in [F1]–[F2], with $X_{4} = I \times Σ_{3}$ , smooth $u_{1} (t)$ , and the zero-shift mostly-plus string-frame metric $d s_{4, s}^{2} = - d t^{2} + h_{i j} (t, x) d x^{i} d x^{j}$ with smooth positive-definite spatial metric $h$ .

[F1]

Silverstein Westphal Nil Background Data For Wrapped D4 Reduction: $d s_{N^{3}}^{2} = α^{'} (L_{u_{1}}^{2} d u_{1}^{2} + L_{u_{2}}^{2} d u_{2}^{2} + L_{x}^{2} (d x^{'} + M u_{1} d u_{2})^{2})$ , with the isomorphic second Nil factor held at the fixed point in the embedding.

[F2]

Wrapped D4 Embedding In Nil Coordinates: $u_{2} = σ$ , $x^{'} = x_{0}^{'}$ , the second Nil coordinate is fixed, and $u_{1} = u_{1} (t)$ is smooth on $I$ .

Proof

1.1givenF1F2algebra

Pulling back the Nil one-forms by [F2] gives $ι_{u}^{*} d u_{1} = \partial_{μ} u_{1} d x^{μ}$ , $ι_{u}^{*} d u_{2} = d σ$ , and $ι_{u}^{*} (d x^{'} + M u_{1} d u_{2}) = M u_{1} d σ$ . The metric [F1] therefore yields the two displayed blocks, with no mixed $d x^{μ} d σ$ term.

2.1givenstep 1.1algebra∎

Write $v_{μ} = \partial_{μ} u_{1}$ and $a = α^{'} L_{u_{1}}^{2}$ . Since $g_{4, s}$ is invertible, factor $\det (g_{4, s} + a v v^{T}) = \det (g_{4, s}) \det (I + a g_{4, s}^{- 1} v v^{T})$ . If $v = 0$ the second determinant is $1$ . If $v \neq 0$ , let $w = g_{4, s}^{- 1} v \neq 0$ and use a basis whose first vector is $w$ ; the rank-one map $w v^{T}$ has only its first row possibly nonzero, so $I + a w v^{T}$ has determinant $1 + a v^{T} w$ . Thus the determinant factor is $1 + α^{'} L_{u_{1}}^{2} g_{4, s}^{μ ν} \partial_{μ} u_{1} \partial_{ν} u_{1} = 1 - α^{'} L_{u_{1}}^{2} {\dot{u}}_{1}^{2}$ in the assumed zero-shift cosmic-time metric. Multiplying by the positive $σ$ -block determinant proves the square-root formula on the stated Lorentzian branch.

Nil Wrapped D4 Pullback Metric And Determinant

Statement

Statement References

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources