Abstract
Studies of Eigenstate Thermalization Hypothesis (ETH) in two-dimensional CFTs call for calculation of the expectation values of local operators in highly excited energy eigenstates. This can be done efficiently by representing zero modes of these operators in terms of the Virasoro algebra generators. In this paper we present a pedagogical introduction explaining how this calculation can be performed analytically or using computer algebra. We illustrate the computation of zero modes by a number of examples and list explicit expressions for all local operators from the vacuum family with the dimension of less or equal than eight. Finally, we derive an explicit expression for the quantum KdV generator Q7 in terms of the Virasoro algebra generators. The obtained results can be used for quantitative studies of ETH at finite value of central charge.
Highlights
Zero modes of the operator productIf we rewrite the commutator in terms of the mode expansion e−inw[An, B(w)] = m[An, Bm]e−(n+m)w and keep in mind that both operators A and B are “built” out of stress-energy tensor, both An and Bn will be some normal ordered polynomials in Li such that the total sum of indexes is equal to n
Where fA is a smooth function of its arguments, Qk are the conserved charges, and Qk(Ei) are the charge values associated with an individual energy eigenstate |Ei
Studies of Eigenstate Thermalization Hypothesis (ETH) in two-dimensional CFTs call for calculation of the expectation values of local operators in highly excited energy eigenstates
Summary
If we rewrite the commutator in terms of the mode expansion e−inw[An, B(w)] = m[An, Bm]e−(n+m)w and keep in mind that both operators A and B are “built” out of stress-energy tensor, both An and Bn will be some normal ordered polynomials in Li such that the total sum of indexes is equal to n. We only note that the finite number of terms contributing in the sum over k in (2.10) provide a crucial simplification
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