Article · Wikipedia archive · Last revised Jul 15, 2026

Polyakov formula

In differential geometry and mathematical physics, 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.

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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.1 Proposed by Alexander Markovich Polyakov, this formula arose in the study of the quantum theory of strings.2 The corresponding density is local, and therefore is a Riemannian curvature invariant.3 In particular, whereas the functional determinant itself is prohibitively difficult to work with in general, its conformal variation can be written down explicitly.1

References

References

  1. Osgood, B.; Phillips, R.; Sarnak, P. (1988), "Extremals of determinants of Laplacians", Journal of Functional Analysis, 80 (1): 148–211, doi:10.1016/0022-1236(88)90070-5
  2. 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
  3. 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