Expectation value of $\mathrm{T}\overline{\mathrm{T}}$ operator in curved spacetimes

We study the expectation value of the $ \mathrm{T}\overline{\mathrm{T}} $ operator in maximally symmetric spacetimes. We define an diffeomorphism invariant biscalar whose coinciding limit gives the expectation value of the $ \mathrm{T}\overline{\mathrm{T}} $ operator. We show that this biscalar is a constant in flat spacetime, which reproduces Zamolodchikov’s result in 2004. For spacetimes with non-zero curvature, we show that this is no longer true and the expectation value of the $ \mathrm{T}\overline{\mathrm{T}} $ operator depends on both the one- and two-point functions of the stress-energy tensor.