The Powers of Monodromy

TL;DR

The paper develops a UV-complete framework for large-field inflation via axion monodromy in string theory, showing how flux couplings to axions yield super-Planckian field excursions while preserving a sub-Planckian periodicity. It demonstrates that backreaction of the inflationary energy on stabilized moduli can flatten the inflaton potential, reducing the effective exponent from a fiducial $p_0$ to $p<p_0$, with explicit constructions giving $p=3,2,4/3,2/3$ and corresponding tensor-to-scalar ratios $r$ in the range $0.04$–$0.20$. The authors provide concrete realizations in type IIB string theory on products of tori or Riemann surfaces, where the inflaton arises from NS-NS $B$-field axions via couplings like $|F_1\wedge B\wedge B|^2$, and also discuss dual monodromies in complex structure moduli space via mirror symmetry. The work emphasizes that moduli stabilization need not be rigidly decoupled from inflation; instead, the inflationary energy can drive controlled shifts in moduli to yield a family of flattening mechanisms, yielding a predictive landscape for $r$ and related observables and suggesting directions for future study of complex-structure monodromies and oscillatory features in the potential.

Abstract

Flux couplings to string theory axions yield super-Planckian field ranges along which the axion potential energy grows. At the same time, other aspects of the physics remain essentially unchanged along these large displacements, respecting a discrete shift symmetry with a sub-Planckian period. After a general overview of this monodromy effect and its application to large-field inflation, we present new classes of specific models of monodromy inflation, with monomial potentials $μ^{4-p}φ^p$. A key simplification in these models is that the inflaton potential energy plays a leading role in moduli stabilization during inflation. The resulting inflaton-dependent shifts in the moduli fields lead to an effective flattening of the inflaton potential, i.e. a reduction of the exponent from a fiducial value $p_0$ to $p<p_0$. We focus on examples arising in compactifications of type IIB string theory on products of tori or Riemann surfaces, where the inflaton descends from the NS-NS two-form potential $B_2$, with monodromy induced by a coupling to the R-R field strength $F_1$. In this setting we exhibit models with $p=2/3,4/3,2,$ and $3$, corresponding to predictions for the tensor-to-scalar ratio of $r\approx 0.04, 0.09, 0.13,$ and $0.2$, respectively. Using mirror symmetry, we also motivate a second class of examples with the role of the axions played by the real parts of complex structure moduli, with fluxes inducing monodromy.

Figures (3)

• Figure 1: On the left, a sketch of a large field range with new effects --- such as altered couplings or new light states --- appearing after each displacement of order $\sim M_P$, parameterizing our ignorance of quantum gravity. Such features could arise both at the classical and the quantum level. On the right, the structure of the potential along axion directions (and their various duals) in string theory. The whole structure has a sub-Planckian period $f$, but on each branch the field can reach a large field range. The potential energy grows with each cycle around the underlying period $f$, while other conditions --- such as the spectrum of branes wrapping the cycles threaded by the higher dimensional potential fields yielding axions --- remain the same each time around. The result is a radiatively-stable potential as in chaotic inflation with a monomial potential.
• Figure 2: An example of a very symmetric Riemann surface configuration, with the loci along which various sectors of 7-branes sit marked in blue. As drawn, the 7-branes lie on contractible cycles, thereby automatically satisfying Gauss's law constraints. To create a Riemann surface with additional symmetry, we can impose periodic boundary conditions, cutting out holes where marked by single or double slashes and identifying them as indicated. In that case each 7-brane at one location needs to be balanced by an antibrane elsewhere, a configuration also consistent with the setup in Saltman. The $F_1$ flux and legs of the $B$ field described in the text lie along the nontrivial $a$-cycles and $b$-cycles of the manifold. When microscopic consistency conditions from the orientifold projection require components of $B$ to vanish at the positions of the 7-branes, this can be achieved via suitable linear combinations as in (\ref{['Bcycles']}).
• Figure 3: The orientations of some of the ingredients. The 7-branes lie on the blue cycles in figure \ref{['RSfig']}, while the $B$ field legs and $F_1$ lie on appropriate combinations of $a$- or $b$-cycles around the handles. For example, the field $B^{(1)}$ has both legs parallel to the $7"$ sector of 7-branes, but $B^{(1)}$ vanishes at the position of the $7"$ branes if we take a linear combination with opposite orientations around the cycles on either side of each $7"$ brane on the Riemann surface $\Sigma_1$ depicted in figure \ref{['RSfig']}.