Variation of $\sqrt{-h}$ with respect to $h^{ab}$

Proved

PolyakovAction.sqrt_neg_det_variation

by Lucas · Oct 1, 2026 · Mathlib 0df444a (Lean v4.33.1)

mathematical-physicsstring-theory

Let $h^{ab}$ be a symmetric $2\times2$ matrix with negative determinant and $h_{ab}$ its inverse, so $h=\det h_{ab}<0$ . For every variation $\delta h^{ab}$ ,

\delta\sqrt{-h}=-\frac12\sqrt{-h}\,h_{ab}\,\delta h^{ab},

in the sense that $t\mapsto\sqrt{-\det\big((h^{ab}+t\,\delta h^{ab})^{-1}\big)}$ has derivative $-\frac12\sqrt{-h}\,h_{ab}\delta h^{ab}$ at $t=0$ .

This is the identity used in the source ("Knowing also that") to compute the variational derivative of the action with respect to the inverse metric.

Formalization Note. The variation is formalized as a directional derivative in the space of inverse metrics.

Preamble

import Mathlib
import Definitions.Def_PolyakovAction_Defs

open MeasureTheory Filter Asymptotics
open scoped Matrix Topology

Formal statement

namespace PolyakovAction

theorem sqrt_neg_det_variation (k δk : Matrix (Fin 2) (Fin 2) ℝ) (hk : k.det < 0)
    (hksymm : kᵀ = k) :
    HasDerivAt (fun t : ℝ => Real.sqrt (-((k + t • δk)⁻¹).det))
      (-(1 / 2) * Real.sqrt (-(k⁻¹).det) * ∑ a, ∑ b, k⁻¹ a b * δk a b) 0 := by sorry

end PolyakovAction

Source

Wikipedia, "Polyakov action", revision oldid=1292212311 (https://en.wikipedia.org/w/index.php?title=Polyakov_action&oldid=1292212311)

Read-back

What the Lean code literally says, in plain math · Aristotle (Harmonic)

Non-blind read-back — not independent testimony. This read-back was written by the same agent (Aristotle, by Harmonic) that drafted the Lean statements of this proposal, at the account owner's explicit instruction. Its author knew the source material and the intended meaning of the code while writing it, so it is not a blind audit and must not be mistaken for independent testimony. Reviewers should compare it against the Lean code themselves.

Throughout, $\sigma=(\sigma^0,\sigma^1)\in\mathbb R^2$ is a worldsheet point (index $0$ is $\tau$ , index $1$ is $\sigma$ ), $X:\mathbb R^2\to\mathbb R^D$ is the embedding, $\partial_a X^\mu(\sigma)$ is the $\mu$ -th component of the Fréchet derivative of $X$ at $\sigma$ applied to the $a$ -th standard basis vector (it is $0$ wherever $X$ is not differentiable), $G_{ab}(\sigma)=\sum_{\mu,\nu} g_{\mu\nu}(X(\sigma))\,\partial_aX^\mu\,\partial_bX^\nu$ is the induced metric, $h^{ab}$ denotes the entries of the matrix inverse $(h_{ab})^{-1}$ (Mathlib's inverse, which is the zero matrix when $\det h=0$ ), and $\sqrt{\cdot}$ is the real square root, which returns $0$ on negative inputs. The Polyakov density is $\mathcal L_P=\sqrt{-\det h}\,\sum_{a,b}h^{ab}G_{ab}$ , the Polyakov action on a set $U$ is $S_P[U]=\frac T2\int_U\mathcal L_P\,d^2\sigma$ (Lebesgue measure; Bochner integral, equal to $0$ if the integrand is not integrable on $U$ ), and the Nambu–Goto action is $S_{NG}[U]=T\int_U\sqrt{-\det G}\,d^2\sigma$ . The stress–energy tensor is $T_{ab}=T\big(G_{ab}-\tfrac12 h_{ab}\sum_{c,d}h^{cd}G_{cd}\big)$ , and $\eta^{(n)}=\mathrm{diag}(1,-1,\dots,-1)$ is the $n\times n$ Minkowski matrix.

Statement. For all real $2\times2$ matrices $k$ and $\delta k$ such that $\det k<0$ and $k^{\mathsf T}=k$ , the real function

t\mapsto\sqrt{-\det\big((k+t\,\delta k)^{-1}\big)}

has derivative at $t=0$ equal to

-\frac12\,\sqrt{-\det(k^{-1})}\;\sum_{a,b}(k^{-1})_{ab}\,\delta k_{ab}.

Here $(\cdot)^{-1}$ is the matrix inverse (zero for singular matrices) and $\sqrt{\cdot}$ is $0$ on negative inputs. No symmetry is assumed on $\delta k$ .

Human review

Endorsed by Shuze Chen · Oct 8, 2026
Confirmed by the moderator at approval.
Endorsed by Lucas · Oct 8, 2026
Confirmed by the mission captain (proposal self-audit).

View graph

Variation of −h\sqrt{-h}−h​ with respect to habh^{ab}hab

Variation of $\sqrt{-h}$ with respect to $h^{ab}$