Non-Abelian Born-Infeld versus String Theory

TL;DR

This paper investigates whether the non-abelian Born-Infeld action, defined with Tseytlin's symmetrized trace, reproduces the string-theory mass spectra of D-brane configurations with constant magnetic flux. By computing quadratic fluctuations around constant background fields on tori and comparing to spectra of intersecting D-branes in several cases (including D2 on $T^2$, D4 on $T^4$, and BPS setups on $T^6$), the authors demonstrate that agreement with string theory holds only in two special configurations; in general, the match fails starting at order $F^6$ and higher. They develop a framework to implement the symmetrized trace in non-abelian backgrounds, and find that the discrepancy persists even when carefully accounting for zero-point energies and flux-induced momentum shifts, except in particular limits or for single wrapped branes where the prescription reduces to an ordinary trace. The results underscore the necessity of higher-order non-abelian corrections or alternative trace prescriptions to reconcile the BI action with full string dynamics, guiding future work in constructing a consistent non-abelian BI framework compatible with T-duality and string amplitudes.

Abstract

Motivated by the results of Hashimoto and Taylor, we perform a detailed study of the mass spectrum of the non-abelian Born-Infeld theory, defined by the symmetrized trace prescription, on tori with constant magnetic fields turned on. Subsequently, we compare this for several cases to the mass spectrum of intersecting D-branes. Exact agreement is found in only two cases: BPS configurations on the four-torus and coinciding tilted branes. Finally we investigate the fluctuation dynamics of an arbitrarily wrapped Dp-brane with flux.

Figures (1)

• Figure 1: A D2-brane wrapped 6 times around the torus $\mathbb{R}^2 / \Lambda$ can be represented as a quotient $\mathbb{R}^2/\Sigma$ with $\Sigma$ a rank 6 sublattice of $\Lambda$. For the lattice shown here, the classification parameters $m_{ij}$ introduced in the text are $m_{11}=3$, $m_{22}=2$, $m_{21}=1$, $m_{12}=0$. With the indicated sheet labeling $1,\ldots,6$, the closed path $e_1$ acts as the permutation $(123)(456)$ on the sheets (so $e_1[1]=2$, $e_1[2]=3$, $e_1[3]=1$ etc.), while $e_2$ acts as $(143625)$. From this diagram, one can also read off the winding numbers $\ell_i$ that appear further on in the momentum quantization condition: for example the off-diagonal mode generated by $E_{16}$ has $\ell_1=2, \ell_2=1$.