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Finite-Dimensional Q-Exact Deformation Invariance

Statement

Let $M$ be an oriented smooth manifold of finite dimension $n$ , with integration cycle $M$ , smooth positive volume form $μ$ , graded algebra $A = Ω^{∙} (M; C)$ , and integral $I_{Γ} (F) = \int_{M} F_{[n]}$ of the degree- $n$ part. Let $Q$ be a smooth odd derivation of $A$ , let $S, O$ be smooth even forms, let $V$ be a smooth odd form, and let $J \subseteq R$ be a nonempty open interval. Suppose $Q S = Q O = 0$ , $(Q^{2} V) e^{- S - t Q V} = 0$ for every $t \in J$ , and $I_{Γ} (Q (O V e^{- S - t Q V})) = 0$ for every $t \in J$ , including the zero-boundary-flux condition. Write $(O e^{- S - t Q V})_{[n]} = f_{t} μ$ . Suppose that $f_{t}$ is integrable for each $t \in J$ , is differentiable in $t$ almost everywhere with measurable derivative, and that for every compact $K \subset J$ one integrable majorant $g_{K}$ bounds both $∣ f_{t} ∣$ and $∣ \partial_{t} f_{t} ∣$ for all $t \in K$ , outside one null set. Then

$Z (t) : = I_{Γ} (O e^{- S - t Q V}) = \int_{M} f_{t} d μ$

is constant on $J$ , and for every $t \in J$ ,

$Z^{'} (t) = - I_{Γ} (O (Q V) e^{- S - t Q V}) = - I_{Γ} (Q (O V e^{- S - t Q V})) = 0.$

In particular, the conclusion is finite-dimensional and applies only at finite parameter values in $J$ ; it gives no $t \to \infty$ saddle-point limit and no identity for an infinite-dimensional continuum functional integral.

Statement references

Schwarz and Zaboronsky, §§2–3

Pestun and Zabzine, §1

Folland, Theorem 2.27(b)

Lebl, Proposition 4.2.6

Facts & Assumptions

Given: $M, μ, A, I_{Γ}, Q, S, O, V, J$ satisfying the Statement, with $Q$ -invariant integration on each $O V e^{- S - t Q V}$ , the stated $Q^{2} V$ condition, and the compact-parameter integrable majorants.

[F1]

Finite-Dimensional Q-Exact Localization Data: $Γ = M$ , $A = Ω^{∙} (M; C)$ , $I_{Γ} (F) = \int_{M} F_{[n]}$ , and $Q$ is an odd derivation with the graded Leibniz rule. For a complex coefficient depending on real $t$ , $\partial_{t} f_{t} (x)$ is the limit in $C$ of $(f_{t + h} (x) - f_{t} (x)) / h$ as real $h \to 0$ . $Q$ -invariance is the integrated identity $I_{Γ} (Q F) = 0$ on the specified admissible class; zero divergence together with zero boundary flux is a sufficient condition for obtaining that identity by graded integration by parts, and the two conditions are distinct.

[F2]

Differentiation under the integral sign: for a measurable family $f (x, t)$ whose sections are integrable, whose parameter derivative exists almost everywhere and is measurable, and whose derivative has one $L^{1}$ majorant uniform on an open parameter interval, the integral $F (t) = \int f (x, t) d μ (x)$ is differentiable and $F^{'} (t) = \int \partial_{t} f (x, t) d μ (x)$ .

[F3]

The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set: for a real-valued function $u$ at an interior point $t$ , $u^{'} (t)$ is the finite limit of $(u (t + h) - u (t)) / h$ as $h \to 0$ , $h \neq 0$ .

[F4]

A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant: a real-valued function continuous on an interval and differentiable at every interior point with derivative zero throughout is constant on that interval.

Proof

1.1givenF1algebra

Put $E_{t} = e^{- S - t Q V}$ . Since $Q V$ is even, coefficientwise differentiation of its exponential gives $\partial_{t} E_{t} = - (Q V) E_{t}$ . The top-degree projection is linear and $O$ is independent of $t$ , so $(\partial_{t} f_{t}) μ = - [O (Q V) E_{t}]_{[n]}$ .

1.2givenF1algebra

The assumptions $Q S = 0$ and $(Q^{2} V) E_{t} = 0$ give $Q E_{t} = - (Q S + t Q^{2} V) E_{t} = 0$ . The graded Leibniz rule, $Q O = 0$ , and the odd parity of $V$ then give $Q (O V E_{t}) = (Q O) V E_{t} + O (Q V) E_{t} - O V (Q E_{t}) = O (Q V) E_{t}$ .

2.1givenF2step 1.1

Fix $t_{0} \in J$ and choose $a < t_{0} < b$ with $[a, b] \subset J$ . The majorant $g_{[a, b]}$ makes every $f_{t}$ integrable on $(a, b)$ and bounds its measurable parameter derivative there by one $L^{1} (M, μ)$ function. Applying [F2] on $(a, b)$ , and using step 1.1, gives $Z^{'} (t_{0}) = - I_{Γ} (O (Q V) E_{t_{0}})$ .

3.1givenF1step 2.1step 1.2

Substituting step 1.2 into the derivative formula of step 2.1 yields $Z^{'} (t_{0}) = - I_{Γ} (Q (O V E_{t_{0}})) = 0$ by the exact integrated $Q$ -invariance identity assumed in the Statement for $F = O V E_{t_{0}}$ . Since $t_{0}$ was arbitrary, $Z^{'}$ vanishes throughout $J$ .

4.1F3F4step 3.1∎

Write $Z = u + i v$ with real functions $u, v$ . Step 3.1 implies that both real difference quotients have limit zero at every $t \in J$ . By [F3], for each $t$ and all sufficiently small nonzero $h$ , $∣ u (t + h) - u (t) ∣ < ∣ h ∣$ and $∣ v (t + h) - v (t) ∣ < ∣ h ∣$ ; hence $u$ and $v$ are continuous at every $t$ . Applying [F4] to each component shows that both are constant on $J$ . Therefore $Z$ is constant on $J$ .

Finite-Dimensional Q-Exact Deformation Invariance

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources