D4-brane monodromy inflation as a model proposal

Let $I\subseteq \mathbb{R}$ be a nonempty open interval, let ${\mathrm{\Sigma }}_3$ be a smooth oriented boundaryless three-manifold, and set $X_4=I\times {\mathrm{\Sigma }}_3$. Choose a spatial chart $x:U_{\mathrm{\Sigma }}\to U\subseteq {\mathbb{R}}^3$, with $U_{\mathrm{\Sigma }}\subseteq {\mathrm{\Sigma }}_3$ and $U$ open. On this chart the four-dimensional metric factor in the adopted ten-dimensional string-frame background is $d{s}_{4,s}^2=-d{t}^2+h_{ij}(t,x)d{x}^{i}d{x}^j,$ where $h_{ij}\in C^{\mathrm{\infty }}(I\times U)$ is positive definite at every point; the dot denotes $d\mathrm{/}dt$. The chart is zero-shift and $t$ is proper time along its coordinate observers. The subscript $s$ denotes string frame.

In the Nil quotient, the base coordinate has period one under the lattice generator ${\tau }_1(u_1,u_2,x^{\mathrm{\prime }})=(u_1+1,u_2,x^{\mathrm{\prime }}-M{u}_2)$. Thus $u_1$ itself is a real coordinate on the covering space, not a period-one real-valued function; its quotient base coordinate is $[u_1]\in \mathbb{R}\mathrm{/}\mathbb{Z}$, with the stated simultaneous $x^{\mathrm{\prime }}$ shift. In this example choose the positive real lift $u\in C^{\mathrm{\infty }}(I,(0,\mathrm{\infty }))$ of that base coordinate. Let $M\in {\mathbb{Z}}_{>0}$ be the integer Nil twist, $L_x,L_u>0$ dimensionless size moduli, $\beta >0$ the dimensionless anisotropy, $g_s:=e^{{\mathrm{\Phi }}_{\mathrm{\infty }}}>0$ the constant dimensionless string coupling, and ${\alpha }^{\mathrm{\prime }}>0$ the inverse string-tension parameter with dimensions of length squared. Set $L:=(L_u^2{L}_x{)}^{1\mathrm{/}3}$, $L_{u_1}=L_u\mathrm{/}\sqrt{\beta }$, and $L_{u_2}=L_u\sqrt{\beta }$.

Let $S^1=\mathbb{R}\mathrm{/}\mathbb{Z}$ with coordinate $\sigma$ of period one. The local worldvolume is $W_5=X_4\times S^1$, with the D4 embedding fixed by $u_1=u(t)$, $u_2=\sigma$, $x^{\mathrm{\prime }}=x_0^{\mathrm{\prime }}\in \mathbb{R}\mathrm{/}\mathbb{Z}$, and a fixed point ${\tilde{q}}_0$ in the second Nil factor, as in Wrapped D4 Embedding In Nil Coordinates. The deck transformation above records the brane-position monodromy; this local DBI calculation does not construct a global orientifold. Use $\mathrm{\hslash }=c=1$, so $u,M,L_x,L_u,L,\beta ,g_s$ and the positive frame factor $\mathrm{\Omega }$ are dimensionless, ${\varphi }_s$ and ${\varphi }_E$ have mass dimension one, and four-dimensional potentials have mass dimension four.

Use the single-Abelian-probe, fixed-modulus, constant-dilaton, zero-pulled-back-$B_2$, zero-$F_A$, negligible-backreaction branch of the cited geometric DBI term. Define $u_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}:=\frac{\sqrt{\beta }{L}_u}{M{L}_x},ϵ(t):=\frac{{\alpha }^{\mathrm{\prime }}{L}_u^2\dot{u}(t{)}^2}{\beta }\mathrm{.}$ For this branch the local D4 supplier gives the positive source-frame kinetic coefficient and potential $K(u)=\frac{L_u^3}{(2\pi {)}^4{g}_s{\alpha }^{\mathrm{\prime }}\sqrt{\beta }}{(1+\frac{M^2{L}_x^2{u}^2}{\beta L_u^2})}^{1\mathrm{/}2},V_s(u)=\frac{L_u\sqrt{\beta }}{(2\pi {)}^4{g}_s{\alpha }^{\mathrm{\prime }2}}{(1+\frac{M^2{L}_x^2{u}^2}{\beta L_u^2})}^{1\mathrm{/}2}\mathrm{.}$ Assume $ϵ\ll 1$ and use the large-position asymptotic regime $u\mathrm{/}{u}_{\mathrm{c}\mathrm{r}\mathrm{i}\mathrm{t}}\to \mathrm{\infty }$. By the local D4 theorem, the source-frame canonical field is fixed, not freely shifted, by ${\varphi }_s(u):={\int }_0^u\sqrt{K(v)}dv,{\varphi }_s(0)=0.$ In particular, ${\varphi }_s\sim C_{\varphi }{u}^{3\mathrm{/}2}$ and $V_s\sim V_{0}u$, so $V_s({\varphi }_s)\sim {\mu }_s{\varphi }_s^{2\mathrm{/}3}$ in this asymptotic regime. For the Einstein-frame scalar EFT, supply a positive constant $\mathrm{\Omega }$ as frame data, with $g_E={\mathrm{\Omega }}^2{g}_{4,s}$; fixed moduli and dilaton make this factor field-independent in this example. The source sections do not fix the numerical value of this frame map. If the source metric is separately identified with the Einstein-frame metric, that choice is $\mathrm{\Omega }=1$.

The conditional Einstein-frame conclusion is $V_E({\varphi }_E)\sim {\mathrm{\Omega }}^{-10\mathrm{/}3}{\mu }_s{\varphi }_E^{2\mathrm{/}3},$ where ${\varphi }_E={\mathrm{\Omega }}^{-1}{\varphi }_s$ with the corresponding canonical origin ${\varphi }_E(0)=0$, and ${\mu }_s={\left(\frac{3}{2}\right)}^{2\mathrm{/}3}(2\pi {)}^{-8\mathrm{/}3}{M}^{2\mathrm{/}3}{\beta }^{1\mathrm{/}3}\frac{L_x\mathrm{/}L}{{\alpha }^{\mathrm{\prime }5\mathrm{/}3}{g}_s^{2\mathrm{/}3}}\mathrm{.}$ Here ${\mu }_s$ has mass dimension $10\mathrm{/}3$, so the potential has dimension four. This is the local geometric DBI contribution only; it makes no claim of a globally controlled compactification, moduli stabilization, slow roll, an observational fit, or an empirical result.

Four-dimensional Einstein-frame multi-scalar cosmology EFT data

Adopted two-derivative multi-scalar cosmology EFT

Silverstein Westphal Nil Background Data For Wrapped D4 Reduction

Wrapped Nil D4 DBI Reduction And Large Field Potential

Silverstein and Westphal, arXiv:0803.3085v2

Cicoli et al., arXiv:2303.04819v2