Wrapped D4 Embedding In Nil Coordinates

Let $I\subseteq \mathbb{R}$ be an open time interval, let ${\mathrm{\Sigma }}_3$ be a smooth oriented boundaryless three-manifold, and set $X_4=I\times {\mathrm{\Sigma }}_3$ with product orientation and coordinates $(t,x^i)$. Thus $X_4$ is a smooth boundaryless spacetime factor with a specified time coordinate. Let $S^1=\mathbb{R}\mathrm{/}\mathbb{Z}$ have coordinate $\sigma$ of period one. Choose a smooth function $u_1:I\to (0,\mathrm{\infty })$, a fixed $x_0^{\mathrm{\prime }}\in \mathbb{R}\mathrm{/}\mathbb{Z}$, and a fixed point ${\tilde{q}}_0\in {\tilde{N}}^3$ of the second Nil factor defined in Silverstein Westphal Nil Background Data For Wrapped D4 Reduction. Define $W_5=X_4\times S^1$ and ${\iota }_u:W_5⟶X^{10}:=X_4\times N^3\times {\tilde{N}}^3,(x,\sigma )⟼(x,u_1(t),u_2=\sigma ,x^{\mathrm{\prime }}=x_0^{\mathrm{\prime }},{\tilde{q}}_0)\mathrm{.}$ The $\sigma$-circle closes by ${\tau }_2$, so this is a well-defined smooth embedding in the full ten-dimensional product. Give $W_5$ the product orientation from $X_4$ and increasing $\sigma$. The derivative of $u_1(t)$ is the only collective-coordinate velocity retained. The term “pullback” means ${\iota }_u^{\ast }$ on tensors and differential forms.

The embedding is proper: for compact $K\subset X^{10}$, its projection $K_4$ to $X_4$ is compact, and ${\iota }_u^{-1}(K)$ is a closed subset of $K_4\times S^1$, hence compact.

Silverstein Westphal Nil Background Data For Wrapped D4 Reduction

Embedded smooth submanifolds with boundary

This is a local representative in the covering-space coordinate description, with the second Nil factor held at ${\tilde{q}}_0$; it does not specify a global orientifold image or total RR charge.