Misaligned supersymmetry and open strings

Abstract

The study of non-supersymmetric string theories is shedding light on an important corner of the string landscape and might ultimately explain why, so far, we did not observe supersymmetry in our universe. We review how misaligned supersymmetry in closed-string theories leads to a cancellation between bosons and fermions even in non-supersymmetric string theories. We then show that the same cancellation takes place for open strings by studying an anti-Dp-brane placed on top of an Op-plane in type II string theory. Misaligned supersymmetry consists in cancellations between bosons and fermions at different energy levels, in such a way that the averaged number of states grows at a rate dominated by a factor $$ {\mathrm{e}}^{C_{\mathrm{e}\mathrm{ff}}\sqrt{n}} $$ e C eff n , with Ceff< Ctot, where Ctot is the inverse Hagedorn temperature. We prove the previously conjectured complete cancellation, i.e. we prove that Ceff = 0, for a vast class of models.

Modular invariance, misalignment and finiteness in non-supersymmetric strings

“…In ref. [5], we extended the re-sults to open-string setups and proved the conjecture that all exponential growth is cancelled in an appropriately defined sector-average. However, there was no clear, direct relationship between the sector-average and finite physical observables.…”

“…For example, for an anti-Dp-brane on an Op-plane, after a few manipulations detailed in ref. [5], one arrives at the exponentially growing oscillations pictured in fig. 1.…”

“…In ref. [5], for a class of theories, the cancellation of the envelope functions in the sectoraverage has been shown at all orders in the Hardy-Ramanujan-Rademacher sum. This applies to both closed and open strings.…”

“…Instead, these theories have an exponentially growing oscillation between the net number of bosons and fermions at each mass level. A simple example realising this property is made up by an anti-Dp-brane on top of an Op-plane [5], whose spectrum is sketched in fig. 1.…”

“…As discussed by refs. [5,25], these coefficients are particularly simple to determine if the partition function can be expressed as a quotient of Dedekind η-functions, since they can be obtained via a simple Hardy-Ramanujan-Rademacher expansion. After a review of such expansion in subsection 2.1, in subsection 2.2 we review the idea of sector-average, which is the tool commonly used in the literature to discuss the presence of misaligned supersymmetry.…”