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I am interested in the calculation of total Chern class of Hodge bundle. I am aware that there is a way by the Grothendieck-Riemann-Roch formula, however, I read that this is also a cohomological field theory. I did not know how to produce the R-matrix for this case. Is there any hint or reference for this?

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A reference would be Pandharipande's ICM address Cohomological field theory calculations, section 1.3. The $R$-matrix is $$ R(z)=\exp\bigg(-\sum_{k=1}^\infty \frac{B_{2k}}{2k(2k-1)}z^{2k-1}\bigg). $$

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