In differential geometry and mathematical physics (especially string theory), the Polyakov formula expresses the conformal variation of the zeta functional determinant of a Riemannian manifold. Proposed by Alexander Markovich Polyakov this formula arose in the study of the quantum theory of strings. The corresponding density is local, and therefore is a Riemannian curvature invariant. In particular, whereas the functional determinant itself is prohibitively difficult to work with in general, its conformal variation can be written down explicitly.

References

• Polyakov, Alexander (1981), "Quantum geometry of bosonic strings", Physics Letters B, 103 (3): 207–210, Bibcode:1981PhLB..103..207P, doi:10.1016/0370-2693(81)90743-7

• Branson, Thomas (2007), "Q-curvature, spectral invariants, and representation theory" (PDF), Symmetry, Integrability and Geometry: Methods and Applications, 3: 090, arXiv:0709.2471, Bibcode:2007SIGMA...3..090B, doi:10.3842/SIGMA.2007.090, S2CID 14629173