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A T-Duality of Non-Supersymmetric Heterotic Strings and an implication for Topological Modular Forms
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abstract
Motivated by recent developments connecting non-supersymmetric heterotic string theory to the theory of Topological Modular Forms (TMF), we show that the worldsheet theory with central charge $(17,\frac{3}{2})$ obtained by fibering the $(E_8)_1 \times (E_8)_1$ current algebra over the $\mathit{N}$=(0,1) sigma model on $S^{1}$ with antiperiodic spin structure (such that the two $E_8$ factors are exchanged as we go around the circle), is continuously connected to the $(E_8)_2$ theory in the Gaiotto$-$Johnson-Freyd$-$Witten sense of going "up and down the RG trajectories". Combined with the work of Tachikawa and Yamashita, this furnishes a physical derivation of the fact that the $(E_8)_2$ theory corresponds to the unique nontrivial torsion element $[(E_8)_2]$ of $\mathsf{TMF}^{31}$ with zero mod-2 elliptic genus.
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Non-supersymmetric branes and discrete topological terms
A direct computation from the tentative NS5-brane spectrum gives the opposite Z3 topological term (1/3 vs 2/3) from Tachikawa-Zhang, so the spectrum or the inflow interpretation must change.
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