Invariance under worldsheet diffeomorphisms

Proved

PolyakovAction.diffeomorphism_invariance

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

mathematical-physicsstring-theory

Let $\varphi$ be a change of worldsheet coordinates $\sigma=\varphi(\tilde\sigma)$ that is injective and differentiable on an open set $s$ , with Jacobian matrix $J=\partial\sigma/\partial\tilde\sigma$ . Transform the worldsheet metric as a tensor, $\tilde h_{ab}(\tilde\sigma)=\frac{\partial\sigma^c}{\partial\tilde\sigma^a}\frac{\partial\sigma^d}{\partial\tilde\sigma^b}h_{cd}(\varphi(\tilde\sigma))$ , and the embedding as a scalar, $\tilde X=X\circ\varphi$ . If $X$ is differentiable on $\varphi(s)$ , then

S[\tilde h,\tilde X;\,s]=S[h,X;\,\varphi(s)].

This is the source's local symmetry under worldsheet diffeomorphisms: $h^{ab}$ transforms with the Jacobian, $\tilde h=J^2h$ , and $\sqrt{-\tilde h}\,d^2\tilde\sigma$ compensates the change of measure.

Formalization Note. The transformed metric is written as $J^{\mathsf T}h(\varphi(\tilde\sigma))J$ with $J$ the matrix of the derivative $\varphi'(\tilde\sigma)$ ; this is equivalent to the source's transformation of the inverse metric $h^{ab}$ . No invertibility of $J$ is assumed.

Preamble

import Mathlib
import Definitions.Def_PolyakovAction_Defs

open MeasureTheory Filter Asymptotics
open scoped Matrix Topology

Formal statement

namespace PolyakovAction

theorem diffeomorphism_invariance {D : ℕ} (T : ℝ)
    (g : Spacetime D → Matrix (Fin D) (Fin D) ℝ)
    (h : Worldsheet → Matrix (Fin 2) (Fin 2) ℝ) (X : Worldsheet → Spacetime D)
    (φ : Worldsheet → Worldsheet) (φ' : Worldsheet → (Worldsheet →L[ℝ] Worldsheet))
    (s : Set Worldsheet) (hs : IsOpen s) (hφ : ∀ x ∈ s, HasFDerivAt φ (φ' x) x)
    (hinj : Set.InjOn φ s) (hX : ∀ x ∈ s, DifferentiableAt ℝ X (φ x)) :
    polyakovAction T g
        (fun σ' => (LinearMap.toMatrix' (φ' σ').toLinearMap)ᵀ * h (φ σ')
          * LinearMap.toMatrix' (φ' σ').toLinearMap)
        (X ∘ φ) s
      = polyakovAction T g h X (φ '' s) := 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 every $D$ , real $T$ , target metric $g:\mathbb R^D\to\mathbb R^{D\times D}$ , worldsheet metric $h$ , map $X:\mathbb R^2\to\mathbb R^D$ , map $\varphi:\mathbb R^2\to\mathbb R^2$ , family of linear maps $\varphi'(x):\mathbb R^2\to\mathbb R^2$ and set $s\subseteq\mathbb R^2$ , assume

$s$ is open;
for every $x\in s$ , $\varphi$ is Fréchet differentiable at $x$ with derivative $\varphi'(x)$ ;
$\varphi$ is injective on $s$ ;
for every $x\in s$ , $X$ is differentiable at $\varphi(x)$ .

Let $J(\tilde\sigma)$ be the $2\times2$ matrix of $\varphi'(\tilde\sigma)$ in the standard basis ( $J_{ca}=$ $c$ -th component of $\varphi'(\tilde\sigma)e_a$ ) and $\tilde h(\tilde\sigma)=J(\tilde\sigma)^{\mathsf T}\,h(\varphi(\tilde\sigma))\,J(\tilde\sigma)$ . Then

S_P\big(T,g,\tilde h,\,X\circ\varphi,\,s\big)=S_P\big(T,g,h,X,\varphi(s)\big),

where $\varphi(s)$ is the image of $s$ . No integrability, invertibility of $J$ , or signature of $h$ is assumed.

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).

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