Open string wavefunctions in flux compactifications

TL;DR

The paper addresses how background fluxes and D-brane magnetization modify open-string spectra in type I flux vacua with SU(3) structure, by solving flux-modified Dirac and Laplace equations on twisted tori and elliptic fibrations. The authors develop a representation-theoretic framework, organizing KK excitations into irreducible unitary representations of the Kaloper–Myers algebra and its extensions, and compute 2- and 3-point couplings to compare with 4d effective supergravity. They provide explicit spectral results for gauge bosons, scalars, and fermions in vanishing and non-vanishing μ-term backgrounds, including Landau-like degeneracies and flux-induced mass shifts, with wavefunctions expressed in terms of theta and Riemann theta functions. The work also clarifies the link between 10d wavefunctions and 4d SUSY multiplets, and maps the Type I setups to T-dual Type IIB vacua with corresponding fluxes, offering a concrete open-closed string dictionary for these flux compactifications.

Abstract

We consider compactifications of type I supergravity on manifolds with SU(3) structure, in the presence of RR fluxes and magnetized D9-branes, and analyze the generalized Dirac and Laplace-Beltrami operators associated to the D9-brane worldvolume fields. These compactifications are T-dual to standard type IIB toroidal orientifolds with NSNS and RR 3-form fluxes and D3/D7 branes. By using techniques of representation theory and harmonic analysis, the spectrum of open string wavefunctions can be computed for Lie groups and their quotients, as we illustrate with explicit twisted tori examples. We find a correspondence between irreducible unitary representations of the Kaloper-Myers algebra and families of Kaluza-Klein excitations. We perform the computation of 2- and 3-point couplings for matter fields in the above flux compactifications, and compare our results with those of 4d effective supergravity.

Figures (4)

• Figure 1: Spectra of massive gauge bosons in a fluxless toroidal compactification (left) and in the fluxed example at hand (right), in the regime $R_1R_2\gg M R_6$. The mass scale introduced by the fluxes is given by $\varepsilon = M R_6/\pi R_1R_2$.
• Figure 2: $|B^{(k,\delta_1,\delta_4)}_{n,k_3,k_6}|^2$ for $k=0,1$, $k_6M=0,1,2$, $n=k$ and arbitrary $\delta_1, \delta_4,$ and $k_3$, in the plane $x^i=0, \ \ i=3\ldots 6$. The normalization has been left unfixed.
• Figure 3: Mass spectra for the complex scalar modes $\xi_3$ and $\xi_{\pm}$ excited along the fiber with same momentum $|k_6|$ in the example with vanishing $\mu$-terms. Continuous red lines relate states with same $n$ and $k_6<0$, whereas dashed blue lines relate states with same $n$ and $k_6>0$. We have labeled the energy levels by $n^{s_{k_6M}}$. The spectrum of gauge boson excitations coincide with the one of $\xi_3$. The flux mass scale is given by $\varepsilon=\frac{MR_6}{\pi R_1R_2}$, whereas $\Delta_{k_3,k_6}^2$ is defined in (\ref{['delta']}). We have also indicated the number of real scalars at each energy level, for fixed $s_{k_6}$, $s_{k_3}$.
• Figure 4: Open string wavefunction for a D7-brane winding mode in the T-dual type IIB flux picture. Even if both ends of the open string sit on the same point in the internal space, they feel a different B-field due to the presence of the NSNS flux $H_3$ and the extended nature of the winding mode. As a result, D7-brane winding modes behave as open strings that end on D7-branes with different magnetizations, and so do their wavefunctions.