Abstract
We consider so-called Yang-Baxter deformations of bosonic string sigma- models, based on an R-matrix solving the (modified) classical Yang-Baxter equation. It is known that a unimodularity condition on R is sufficient for Weyl invariance at least to two loops (first order in α′). Here we ask what the necessary condition is. We find that in cases where the matrix (G + B)mn, constructed from the metric and B-field of the undeformed background, is degenerate the unimodularity condition arising at one loop can be replaced by weaker conditions. We further show that for non-unimodular deformations satisfying the one-loop conditions the Weyl invariance extends at least to two loops (first order in α′). The calculations are simplified by working in an O(D, D)-covariant doubled formulation.
Highlights
JHEP10(2020)065 rise to a goodgravity background [15, 16]
We find that in cases where the matrix (G + B)mn, constructed from the metric and B-field of the undeformed background, is degenerate the unimodularity condition arising at one loop can be replaced by weaker conditions
We will show that the α -correction to the equations of motion can be cast in a form that is manifestly invariant under non-unimodular YB deformations satisfying the one-loop Weyl invariance conditions of the previous section
Summary
At the first order in α the double Lorentz transformations get corrected to [28]. where a = b = −α for the bosonic string and a = −α , b = 0 for the heterotic string (a = b = 0 for type II). At the first order in α the double Lorentz transformations get corrected to [28]. The variation (2.23) is of the form δ = δ0 +aδ and a short calculation gives for the projections of the generalized fluxes appearing in the lowest order action (2.11), (2.12). At lowest order in α the action is Lorentz invariant. We must find terms of order α whose lowest order Lorentz transformation cancels the terms on the r.h.s. The first term on the second line must be canceled by the variation of a term of the form F ABC tr ∂AFB(−)FC(−) and we find. Using (2.28) we see that the last two terms on the second line come from the variation of tr R(−)ABR(A−B) and the. The expression for R(+) is obtained by reversing the projections in an obvious way. These expressions agree with the ones written in [29] but are much more compact
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