Monthly Archives: octobre 2021

Doubled diffeomorphisms and the generalised Ricci curvature

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TracingCurves

I was asked a question the other week about the idea of a doubled diffeomorphism, such as that found in double field theory. A nice way to approach the concept is to start with dualised linearised gravity [1]. That is to say, we start with a theory considering only the field $latex h_{ij}(x^{mu}, x^a, tilde{x}_a) &fg=000000 &s=2$. This field transforms under normal linearised diffeomorphism as

$latex delta h_{ij} = partial_i epsilon_j + partial_j epsilon_i (1)&fg=000000 &s=2$

and, under the dual diffeomorphism as

$latex tilde{delta} h_{ij} = tilde{partial}_i tilde{epsilon}_j + tilde{partial}_j tilde{epsilon}_i. (2)&fg=000000 &s=2$

Now, take the basic Einstein-Hilbert action

$latex S_{EH} = frac{1}{2k^2} int sqrt{-g} R, (3)&fg=000000 &s=2$

and expand to quadratic order in the fluctuation field $latex h_{ij}(x) = g_{ij} – eta_{ij} &fg=000000 &s=2$. Just think of standard linearised gravity with the following familiar quadratic action

$latex S^2_{EH} = frac{1}{2k^2} int dx [frac{1}{4} h^{ij}partial^2 h_{ij} – frac{1}{4} h partial^2…

Voir l’article original 1 669 mots de plus