differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, lin ...

and

mathematical physics Mathematical physics is the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the de ...

(especially

string theory In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. String theory describes how these strings propagate through space and intera ...

), the Polyakov formula expresses the conformal variation of the zeta functional determinant of a

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

. Proposed by

Alexander Markovich Polyakov Alexander Markovich Polyakov (; born 27 September 1945) is a Russian theoretical physicist, formerly at the Landau Institute in Moscow and, since 1989, at Princeton University, where he is the Joseph Henry Professor of Physics Emeritus. Importa ...

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 Riemannian geometry and pseudo-Riemannian geometry, curvature invariants are scalar quantities constructed from tensors that represent curvature. These tensors are usually the Riemann tensor, the Weyl tensor, the Ricci tensor and tensors formed ...

. In particular, whereas the

functional determinant In functional analysis, a branch of mathematics, it is sometimes possible to generalize the notion of the determinant of a square matrix of finite order (representing a linear transformation from a finite-dimensional vector space to itself) to the ...

itself is prohibitively difficult to work with in general, its conformal variation can be written down explicitly.

References

* * Conformal geometry Spectral theory String theory {{geometry-stub