03.09.26 · modern-geometry / spin-geometry

Mathai-Quillen formalism and universal Thom forms

shipped3 tiersLean: none

Anchor (Master): Mathai-Quillen 1986 *Superconnections, Thom classes, and equivariant differential forms* (Topology 25, originator paper); Bismut 1986 *The Atiyah-Singer index theorem for families of Dirac operators* (Invent. Math. 83, parallel superconnection story); Atiyah-Jeffrey 1990 *Topological Lagrangians and cohomology* (J. Geom. Phys. 7, infinite-dimensional MQ); Witten 1988 *Topological sigma models* (Comm. Math. Phys. 118, TQFT application); Berline-Getzler-Vergne 1992 *Heat Kernels and Dirac Operators* Ch. 1 + Ch. 7 §7.7; Bott-Tu *Differential Forms in Algebraic Topology* §6 + §11 (classical Thom theory)

Intuition [Beginner]

A vector bundle has a Thom class: a cohomology class on the total space, supported near the zero section, whose integral along each fibre is one. The Thom class is the universal carrier of integration-along-the-fibre. Most constructions of the Thom class give it as an abstract cohomology class, or as a current concentrated on the zero section. The Mathai-Quillen formula is a recipe that produces an actual smooth differential form representing it.

The recipe is built around a Gaussian. The form has a factor like $e^{- ∣ v ∣^{2} /2}$ that decays rapidly away from the zero section, so the form is concentrated near the zero section without being a distribution. Multiplied against this Gaussian are pieces built from the connection and curvature of the bundle, packaged in a way that the resulting form is closed and integrates to one along each fibre.

The payoff is that you can now write the Euler class of a bundle as a single explicit formula in the connection and curvature, and the same formula extends to infinite-dimensional settings, where it powers the path-integral construction of topological field theory.

Visual [Beginner]

A vector bundle over a base manifold, drawn as a fattening of the base: above each base point sits a copy of a vector space (the fibre). The Mathai-Quillen form is drawn as a Gaussian bump sitting on each fibre, peaked at the zero of the fibre, decaying outward, and integrating to one over each fibre. The shape of the bump deforms slightly from fibre to fibre to track the connection and curvature of the bundle.

Restricting the Mathai-Quillen form to the zero section recovers the Euler class of the bundle as a Chern-Weil polynomial in the curvature. Integrating the form over each fibre gives the constant function one on the base.

Worked example [Beginner]

Take the simplest case: a rank-two real vector bundle $V \to M$ where the base $M$ is a single point. Then $V$ is just a copy of two-dimensional space $R^{2}$ , with coordinates $(v_{1}, v_{2})$ , and the connection and curvature are absent (a single point has no curvature). The Mathai-Quillen form on $V = R^{2}$ reduces to

U = \frac{1}{2 π} e^{- (v_{1}^{2} + v_{2}^{2}) /2} d v_{1} \land d v_{2} .

Step 1. The form is supported away from the zero of the fibre only through its Gaussian factor; the wedge of $d v_{1}$ and $d v_{2}$ picks out the area form on $R^{2}$ .

Step 2. Compute the area-weighted Gaussian on $R^{2}$ using polar coordinates $(r, θ)$ . The angular variable contributes a factor $2 π$ . The radial variable contributes the area-Gaussian, which evaluates to $1$ in standard tables (the area-Gaussian $e^{- r^{2} /2} r d r$ from $0$ to infinity has value $1$ ). The normalisation $1/ (2 π)$ cancels the $2 π$ from the angle, leaving $1$ as the fibre area-integral of $U$ .

Step 3. The form is closed because it has top degree two on the two-dimensional fibre and there is no base direction to differentiate into.

What this shows is that the Mathai-Quillen recipe, in the rank-two single-point case, is the standard Gaussian normalised area form. The bundle version dresses this Gaussian up with extra pieces that record the connection and the curvature of the bundle, while keeping the same essential structure: rapid decay away from the zero section, fibre integral equal to one.

Check your understanding [Beginner]

Formal definition [Intermediate+]

Berezin integral. Let $W$ be an oriented Euclidean vector space of dimension $2 n$ with orthonormal basis $e_{1}, \dots, e_{2 n}$ . The Berezin integral $\int^{Ber} : Λ^{*} W \to R$ is the linear functional that picks out the top-degree component: for $ω = \sum_{I} ω_{I} e_{I} \in Λ^{*} W$ with multi-index $I = (i_{1} < \dots < i_{k})$ , $$ \int^{\mathrm{Ber}} \omega := \omega_{(1, 2, \ldots, 2n)}, $$ the coefficient of the volume element $e_{1} \land \dots \land e_{2 n}$ . Equivalently, $\int^{Ber} α = α / (e_{1} \land \dots \land e_{2 n})$ for $α \in Λ^{2 n} W$ , and zero on lower-degree components. The Berezin integral is the fermionic analogue of the Gaussian integral on a vector space: for $A$ a skew-symmetric $2 n \times 2 n$ matrix, $$ \int^{\mathrm{Ber}} e^{-\tfrac{1}{2} \langle e, A e\rangle} = \mathrm{Pf}(A), $$ the Pfaffian of $A$ , where $⟨ e, A e ⟩ = \sum_{i, j} A_{ij} e_{i} \land e_{j} \in Λ^{2} W$ ^{[Berline-Getzler-Vergne]}.

The Mathai-Quillen Thom form. Let $V \to M$ be a smooth oriented real vector bundle of even rank $2 n$ with a Euclidean metric $⟨ \cdot, \cdot ⟩_{V}$ and a metric-compatible connection $\nabla = \nabla^{V}$ . Let $π : V \to M$ denote the bundle projection. The connection has curvature $R = R^{V} \in Ω^{2} (M, so (V))$ , taking values in the bundle of skew-symmetric endomorphisms of $V$ .

On the total space $V$ there is a tautological section of $π^{*} V \to V$ : at a point $v \in V$ lying over $x \in M$ , the fibre of $π^{*} V$ is $V_{x}$ and the tautological section sends $v$ to $v \in V_{x}$ . Write this section as $v \in Γ (V, π^{*} V)$ . Its covariant derivative $\nabla v \in Ω^{1} (V, π^{*} V)$ is the natural one-form-valued section: along vertical directions it is the identity, and along horizontal directions it captures the connection.

Definition (Mathai-Quillen Thom form). The Mathai-Quillen form is the differential form $U (V) \in Ω^{2 n} (V)$ defined by $$ U(V) := \frac{1}{(2\pi)^n} \int^{\mathrm{Ber}} \exp!\left(-\tfrac{1}{2}|\mathbf{v}|^2 - \tfrac{1}{2} \nabla \mathbf{v} \wedge \nabla \mathbf{v} - R^V\right), $$ where the Berezin integral is taken fibrewise on $Λ^{*} (π^{*} V)$ , and $∣ v ∣^{2} \in C^{\infty} (V)$ is the squared norm of the tautological section, while $\nabla v \land \nabla v$ and $R^{V}$ are elements of $Ω^{*} (V, Λ^{2} π^{*} V)$ paired with the Berezin variable ^{[Mathai-Quillen 1986]}.

The exponent inside is a polynomial of total form-degree two combined with a function: explicitly, $- \frac{1}{2} ∣ v ∣^{2}$ is form-degree zero, $- \frac{1}{2} \nabla v \land \nabla v$ is form-degree two valued in $Λ^{2} π^{*} V$ (the wedge is taken in both the de Rham and exterior-algebra factors), and $- R^{V}$ is form-degree two on the base, valued in $Λ^{2} V ≅ so (V)$ via the skew-symmetric identification.

Cohomology setting. The form $U (V)$ has compact support in the vertical (fibre) direction because of the Gaussian factor $e^{- ∣ v ∣^{2} /2}$ . It therefore lives in the compactly-vertical-support de Rham complex $Ω_{c v}^{*} (V)$ , whose cohomology $H_{c v}^{*} (V; R)$ admits a Thom isomorphism with $H^{*- 2 n} (M; R)$ ^{[Bott-Tu §6]}.

Sign convention. We follow Mathai-Quillen 1986 in writing $exp (- ∣ v ∣^{2} /2)$ rather than $exp (- ∣ v ∣^{2})$ and absorbing the factor of $1/2$ into the Gaussian. Different conventions (Berline-Getzler-Vergne, Cordes-Moore-Ramgoolam) differ by a redefinition of the Berezin variable and an overall normalisation in $(2 π)^{n}$ ; the cohomology class is unaffected. The Euclidean metric on $V$ supplies the identification $Λ^{2} V ≅ so (V)$ via $e_{i} \land e_{j} \leftrightarrow E_{ij}$ (the skew matrix unit), making $R^{V}$ a $Λ^{2} π^{*} V$ -valued two-form.

Counterexamples to common slips

The Mathai-Quillen form is not the same as the Pfaffian form $Pf (R^{V} /2 π)$ on the base. The Pfaffian is the Euler form on $M$ ; $U (V)$ is a form on the total space $V$ that restricts (along the zero section $s_{0} : M \to V$ ) to the Euler form: $s_{0}^{*} U (V) = Pf (R^{V} /2 π) = e (V)$ .
The form lives in compactly-vertically-supported cohomology $H_{c v}^{*} (V)$ , not in ordinary cohomology $H^{*} (V)$ . In ordinary $H^{*} (V) ≅ H^{*} (M)$ (because $V$ deformation retracts to $M$ ), the form is just exact; the Thom-class structure lives in the compact-vertical-support variant.
The Berezin variable $e$ (basis of $Λ^{*} π^{*} V$ ) is not the same as the fibre coordinate $v$ . The Berezin variable is fermionic — it lives in the exterior algebra and obeys $e_{i} \land e_{j} = - e_{j} \land e_{i}$ — while $v$ is the bosonic fibre coordinate (the tautological section). They are paired against each other in the MQ formula, but they are conceptually distinct.

Key theorem with proof [Intermediate+]

Theorem (Mathai-Quillen 1986). Let $V \to M$ be an oriented real vector bundle of even rank $2 n$ with Euclidean metric and compatible connection. The Mathai-Quillen form $U (V) \in Ω^{2 n} (V)$ satisfies:

(i) $U (V)$ is closed: $d U (V) = 0$ in $\Omega^(V)$.*

(ii) $U (V)$ is compactly supported in the vertical direction: for any base-compact set $K \subset M$ , the restriction $U (V) ∣_{π^{- 1} (K)}$ has compact support.

(iii) The fibre integral is normalised to one: $$ \pi_* U(V) = 1 \in \Omega^0(M), $$ where $\pi_: \Omega^{+2n}_{cv}(V) \to \Omega^(M)$ is integration along the fibre.*

(iv) The pullback along the zero section $s_{0} : M \to V$ recovers the Chern-Weil Euler form, $$ s_0^* U(V) = \mathrm{Pf}!\left(\frac{R^V}{2\pi}\right) = e(V) \in \Omega^{2n}(M). $$

Consequently, $U (V)$ represents the Thom class of $V$ in $H_{c v}^{2 n} (V; R)$ , and pushing it forward gives the Euler class.

Proof. Write $U (V)$ as a Berezin integral $U (V) = (2 π)^{- n} \int^{Ber} e^{- A}$ with the exponent $$ \mathcal{A} = \tfrac{1}{2}|\mathbf{v}|^2 + \tfrac{1}{2} \nabla \mathbf{v} \wedge \nabla \mathbf{v} + R^V \in \Omega^(V, \Lambda^ \pi^* V), $$ where each term is a $Λ^{*} π^{*} V$ -valued form on $V$ .

Step 1: closedness. Compute $d U (V)$ by moving the exterior differential past the Berezin integral (which is $C^{\infty}$ -linear) and through the exponential: $$ d e^{-\mathcal{A}} = -d\mathcal{A} \cdot e^{-\mathcal{A}}. $$ The three pieces of $d A$ are $$ d!\left(\tfrac{1}{2}|\mathbf{v}|^2\right) = \langle \mathbf{v}, \nabla \mathbf{v}\rangle, \quad d!\left(\tfrac{1}{2} \nabla \mathbf{v} \wedge \nabla \mathbf{v}\right) = \langle \nabla \mathbf{v}, R^V \mathbf{v}\rangle, \quad d(R^V) = 0 $$ using the metric compatibility $d ⟨ \cdot, \cdot ⟩ = ⟨ \nabla \cdot, \cdot ⟩ + ⟨ \cdot, \nabla \cdot ⟩$ , the second Bianchi identity $\nabla R^{V} = 0$ , and the identity $d (\nabla v) = R^{V} v$ (the curvature of the tautological section). Combining, $$ d\mathcal{A} = \langle \mathbf{v}, \nabla \mathbf{v}\rangle + \langle \nabla \mathbf{v}, R^V \mathbf{v}\rangle. $$

After Berezin integration, the contribution $⟨ v, \nabla v ⟩$ pairs against the Berezin variable in a degree that, when combined with the rest of the expansion, produces a perfect $\nabla$ -derivative inside the Berezin integral and vanishes by the analogue of integration by parts for the Berezin variable. Concretely, the Berezin integral satisfies the identity $$ \int^{\mathrm{Ber}} \partial_{e_i} (\cdots) = 0 $$ for any $Λ^{*}$ -polynomial argument, where $\partial_{e_{i}}$ is the contraction by $e_{i}^{*}$ . The two non-vanishing terms in $d A$ are jointly a contraction-derivative of a single polynomial in $v$ and the Berezin variable, hence the Berezin integral of $d A \cdot e^{- A}$ vanishes ^{[Mathai-Quillen 1986; ref: TODO_REF Berline-Getzler-Vergne Ch. 1]}.

Step 2: vertical compact support. The function $∣ v ∣^{2}$ is the squared norm in the fibre and grows quadratically as $∣ v ∣ \to \infty$ . The factor $e^{- ∣ v ∣^{2} /2}$ then decays faster than any polynomial along the fibre, and the polynomial $e^{- \nabla v \land \nabla v /2 - R^{V}}$ (truncated by Berezin integration to top exterior degree) is at most polynomial in $v$ . The product is therefore Schwartz along each fibre; in particular it has compact support up to rapid decay, which is sufficient for the Thom-class normalisation.

Step 3: fibre integral. Fix a base point $x \in M$ and a trivialisation of $V$ near $x$ in which $R^{V} = 0$ at $x$ . In this trivialisation $\nabla v = d v$ on the fibre, and the Mathai-Quillen form restricted to the fibre $V_{x} = R^{2 n}$ becomes $$ U(V)|{V_x} = \frac{1}{(2\pi)^n} \int^{\mathrm{Ber}} e^{-|\mathbf{v}|^2/2 - d\mathbf{v} \wedge d\mathbf{v} /2}. $$ Expand the Berezin variable as a basis and compute the Berezin integral fibrewise: only the top-exterior-degree component survives, which is $$ U(V)|{V_x} = \frac{1}{(2\pi)^n} e^{-|\mathbf{v}|^2/2}, dv_1 \wedge dv_2 \wedge \cdots \wedge dv_{2n}. $$ The integral over $V_{x} = R^{2 n}$ is the standard Gaussian: $$ \int_{\mathbb{R}^{2n}} \frac{1}{(2\pi)^n} e^{-|\mathbf{v}|^2/2}, d^{2n}\mathbf{v} = 1. $$ By the Bianchi-identity argument of Step 1, the curvature contribution to the fibre integral is exact in the fibre direction (a perfect fibre-divergence), and vanishes upon integration. The general-base fibre integral is therefore one, independent of $x$ ^{[Mathai-Quillen 1986]}.

Step 4: zero-section pullback. Pull $U (V)$ back along $s_{0} : M \to V$ . The tautological section $v$ vanishes along the image of $s_{0}$ : $s_{0}^{*} v = 0$ . Therefore $s_{0}^{*} ∣ v ∣^{2} = 0$ and $s_{0}^{*} \nabla v = 0$ , and only the curvature piece $- R^{V}$ in the exponent survives: $$ s_0^* U(V) = \frac{1}{(2\pi)^n} \int^{\mathrm{Ber}} e^{-R^V}. $$ The Berezin integral of $e^{- R^{V}}$ , with $R^{V}$ regarded as a $Λ^{2} V$ -valued two-form on $M$ identified with a skew-symmetric two-form on $V$ via the metric, is the Pfaffian of $- R^{V}$ : $$ \int^{\mathrm{Ber}} e^{-R^V} = \mathrm{Pf}(-R^V) = (-1)^n \mathrm{Pf}(R^V), $$ and absorbing the sign into the orientation convention (the Pfaffian depends on orientation, and the Mathai-Quillen normalisation fixes the sign so that the Euler class is positive on a positively-oriented sphere bundle), we get $$ s_0^* U(V) = \mathrm{Pf}!\left(\frac{R^V}{2\pi}\right) \in \Omega^{2n}(M). $$ This is the Chern-Weil representative of the Euler class $e (V) \in H^{2 n} (M; R)$ ^{[Mathai-Quillen 1986; ref: TODO_REF Bott-Tu §11]}.

Combining (i)-(iv), $U (V)$ is a closed compactly-vertical-support form representing the Thom class of $V$ . The standard identity $π_{*} (π^{*} α \land Th (V)) = α$ for $α \in H^{*} (M)$ specialises at $α = 1$ to $π_{*} Th (V) = 1$ , matching (iii). And the standard identity $s_{0}^{*} Th (V) = e (V)$ matches (iv). $□$

Bridge. The Mathai-Quillen form builds toward the entire equivariant-cohomology and infinite-dimensional-Thom-class machinery, and identifies the explicit Gaussian-shape representative of $Th (V)$ with the Chern-Weil Euler form after restriction to the zero section. The foundational reason it works is exactly that the Berezin integral has the same algebraic structure as the Gaussian integral: a vanishing-derivative identity ( $\int^{Ber} \partial = 0$ ) parallel to integration by parts, and a quadratic-form evaluation ( $\int^{Ber} e^{- A} = Pf (A)$ ) parallel to $\int e^{- x^{2} /2} d x = 2 π$ . This is exactly why the MQ construction is the odd analog of the Bismut superconnection of 03.09.23 — both arise from the same supersymmetric Gaussian template, with the bosonic fibre coordinate $v$ and the fermionic Berezin variable $e$ pairing to produce a closed differential form whose Berezin integral yields a topological invariant.

The bridge is the recognition that a fermionic Gaussian integral on $Λ^{*} V$ produces the Pfaffian, identifies the Pfaffian with the Euler class, and generalises the classical Bott-Tu global-angular-form Thom representative 03.04.09 to all even-rank oriented bundles in one explicit formula. Putting these together, the central insight is that the Berezin integral is the correct fermionic substitute for the fibre integral, and the same recipe appears again in 03.09.21 (family / equivariant index theory) as the equivariant Mathai-Quillen form, and is dual to the K-theoretic Thom class 03.08.03 whose Chern character it computes.

Exercises [Intermediate+]

Exercise 3 (medium, symbolic).

For the rank-two case $V \to M$ with connection one-form $ω \in Ω^{1} (M, so (2)) = Ω^{1} (M, i R)$ and curvature $R^{V} = d ω \in Ω^{2} (M, i R)$ , write the Mathai-Quillen form $U (V)$ in coordinates $(v_{1}, v_{2})$ on the fibre. Show that $$ U(V) = \frac{1}{2\pi}, e^{-|\mathbf{v}|^2/2}!\left(dv_1 \wedge dv_2 + \nabla v_1 \wedge dv_2 - dv_1 \wedge \nabla v_2 + (\nabla v_1 \wedge \nabla v_2 - R^V_{12})\right), $$ where $\nabla v_{i}$ are the covariant differentials of the components.

Hint

For rank two, the Berezin variable has only two generators $e_{1}, e_{2}$ , and the Berezin integral picks out the coefficient of $e_{1} \land e_{2}$ . Expand $e^{- ∣ v ∣^{2} /2 - \nabla v \land \nabla v /2 - R^{V}}$ as a polynomial in $e_{1} \land e_{2}$ and read off the coefficient.

Answer

For rank two with $V$ -basis $e_{1}, e_{2}$ , write the tautological section $v = v_{1} e_{1} + v_{2} e_{2}$ , the covariant derivative $\nabla v = \nabla v_{1} \cdot e_{1} + \nabla v_{2} \cdot e_{2}$ , and the curvature $R^{V} = R_{12}^{V} e_{1} \land e_{2}$ . The exponent in MQ is $$ \mathcal{A} = \tfrac{1}{2}(v_1^2 + v_2^2) + \nabla v_1 \wedge \nabla v_2 \cdot e_1 \wedge e_2 + R^V_{12}, e_1 \wedge e_2. $$ (Note: $\nabla v \land \nabla v = 2 \nabla v_{1} \land \nabla v_{2} \cdot e_{1} \land e_{2}$ before the $1/2$ in $A$ .) Expanding $e^{- A} = e^{- (v_{1}^{2} + v_{2}^{2}) /2} (1 - (\nabla v_{1} \land \nabla v_{2} + R_{12}^{V}) e_{1} \land e_{2} + \dots)$ and projecting onto the Berezin coefficient of $e_{1} \land e_{2}$ gives the coefficient $- (\nabla v_{1} \land \nabla v_{2} + R_{12}^{V})$ . The MQ normalisation $(2 π)^{- 1}$ and additional cross terms from the wedge product with $\nabla v \cdot d v$ produce the displayed formula. Rubric: full credit for the expansion plus Berezin projection plus identification of cross terms.

Exercise 4 (medium, short-answer).

Why does the Mathai-Quillen form $U (V)$ have vertical compact support rather than ordinary compact support on the total space $V$ ?

Hint

The Gaussian decay $e^{- ∣ v ∣^{2} /2}$ controls only the fibre direction. The base direction is uncontrolled.

Answer

The Gaussian factor $e^{- ∣ v ∣^{2} /2}$ in $U (V)$ depends only on the fibre norm $∣ v ∣^{2}$ — a function on $V$ that grows quadratically along each fibre. This decay forces compact support along each fibre but says nothing about the base $M$ : if $M$ is non-compact, $U (V)$ is supported on all of $π^{- 1} (M) = V$ in the base direction. Hence $U (V)$ lies in the compactly-vertical-support de Rham complex $Ω_{c v}^{*} (V)$ — forms with compact support after restriction to any $π^{- 1} (K)$ for $K \subset M$ compact — not in $Ω_{c}^{*} (V)$ . This is precisely the right complex for the Thom isomorphism: $H_{c v}^{*} (V) ≅ H^{*- 2 n} (M)$ . The compactly-vertical class is what pulls back to a class in $H^{*- 2 n} (M)$ under fibre integration. Rubric: full credit for the fibre-vs-base distinction plus the correct cohomology complex.

Exercise 5 (medium, short-answer).

State the Mathai-Quillen form for the universal bundle $E O (2 n) \to B O (2 n)$ over the classifying space, and explain why the construction is genuinely universal — that is, naturally compatible with bundle pullbacks.

Hint

The universal bundle has a universal connection (e.g., the connection induced from the Stiefel manifold), and the MQ formula is functorial under pullback of (bundle, connection).

Answer

Take the universal rank- $2 n$ oriented bundle $E S O (2 n) \to B S O (2 n)$ (the standard model: $E S O (2 n) = V_{2 n} (R^{\infty})$ a Stiefel-like manifold, $B S O (2 n)$ the classifying space of oriented rank- $2 n$ bundles). Equip it with a universal connection $\nabla^{univ}$ (existence: any choice of $S O (2 n)$ -invariant connection on the universal Stiefel bundle gives one). The Mathai-Quillen form $U_{univ}$ on the total space is the explicit form $(2 π)^{- n} \int^{Ber} exp (- ∣ v ∣^{2} /2 - \nabla^{univ} v \land \nabla^{univ} v /2 - R^{univ})$ .

For any classifying map $f : M \to B S O (2 n)$ pulling back to a bundle $V = f^{*} E S O (2 n) \to M$ , the Mathai-Quillen form $U (V)$ on $V$ is the pullback of $U_{univ}$ along the bundle map $\tilde{f} : V \to E S O (2 n)$ that covers $f$ : $U (V) = \tilde{f}^{*} U_{univ}$ . This is genuine universality at the level of differential forms, not just cohomology classes — the form itself transforms covariantly. The construction is universal because the MQ recipe uses only the connection and curvature, both of which pull back functorially; no choice on the base is required beyond a choice of universal connection. Rubric: full credit for the universal form plus the pullback identity plus the functoriality remark.

Exercise 6 (hard, short-answer).

Show that the cohomology class of $U (V)$ in $H_{c v}^{2 n} (V; R)$ is independent of the choice of connection $\nabla$ on $V$ .

Hint

Let $\nabla_{0}$ and $\nabla_{1}$ be two metric-compatible connections and form the linear path $\nabla_{t} = (1 - t) \nabla_{0} + t \nabla_{1}$ . Compute $\frac{d}{d t} U (V; \nabla_{t})$ and exhibit a transgression form.

Answer

Let $\nabla_{t} = \nabla_{0} + t α$ with $α = \nabla_{1} - \nabla_{0} \in Ω^{1} (M, so (V))$ for $t \in [0, 1]$ . Write $U_{t} = U (V; \nabla_{t})$ and compute $$ \frac{d U_t}{dt} = -\frac{1}{(2\pi)^n} \int^{\mathrm{Ber}} \frac{d \mathcal{A}_t}{dt}, e^{-\mathcal{A}_t}. $$ Compute $\frac{d A _{t}}{d t}$ : only the connection-dependent pieces $\nabla_{t} v \land \nabla_{t} v /2$ and $R_{t}^{V}$ contribute, giving $\frac{d}{d t} (\nabla_{t} v) \land \nabla_{t} v + \frac{d R _{t}}{d t}$ . Using $\frac{d}{d t} (\nabla_{t} v) = α (v)$ and $\frac{d R _{t}}{d t} = d α + [\nabla_{t}, α]$ , one finds that $\frac{d A _{t}}{d t}$ is a graded commutator in $\nabla_{t}$ and a one-parameter family of operators. The Berezin-integral analogue of the supertrace cyclicity then yields $$ \frac{d U_t}{dt} = -d, T_t $$ for an explicit transgression form $T_{t} \in Ω^{2 n - 1} (V)$ , also of vertical compact support. Integrating from $t = 0$ to $t = 1$ gives $U_{1} - U_{0} = - d \int_{0}^{1} T_{t} d t$ , exact in $Ω_{c v}^{*} (V)$ . The cohomology class $[U (V)] \in H_{c v}^{2 n} (V; R)$ is therefore independent of the connection. Rubric: full credit for the linear-path argument plus the transgression identity plus the conclusion of cohomological independence.

Exercise 7 (hard, short-answer).

Apply the Mathai-Quillen form to the tautological line bundle $O (- 1) \to CP^{1}$ (viewed as a rank-two real bundle). Compute $\int_{CP^{1}} s_{0}^{*} U (O (- 1))$ and verify it matches the Euler number.

Hint

The first Chern class of $O (- 1)$ over $CP^{1}$ is the negative of the hyperplane class, so the underlying rank-two real bundle has Euler class $- 1$ .

Answer

The complex line bundle $O (- 1) \to CP^{1}$ has $c_{1} = - [pt]$ where $[pt] \in H^{2} (CP^{1}; Z)$ is the point class. As a real bundle, its Euler class equals its first Chern class via the canonical complex orientation: $e (O (- 1)_{R}) = c_{1} (O (- 1)) = - [pt]$ . By the Mathai-Quillen theorem, $s_{0}^{*} U (O (- 1)) = e (O (- 1)_{R})$ in $H^{2} (CP^{1}; R)$ . Integrating over $CP^{1}$ : $$ \int_{\mathbb{CP}^1} s_0^* U(\mathcal{O}(-1)) = \int_{\mathbb{CP}^1} e(\mathcal{O}(-1)_{\mathbb{R}}) = -1, $$ matching the Euler number $- 1$ of the tautological line bundle. (Equivalently, $O (1)$ has Euler number $+ 1$ , and the question would invert sign for the hyperplane bundle.) Rubric: full credit for the Chern-class identification plus the integration plus the sign-tracking.

Exercise 8 (hard, short-answer).

The equivariant Mathai-Quillen form refines $U (V)$ to a closed form in the equivariant cohomology of $V$ for a compact Lie group $G$ acting on $V$ by bundle automorphisms. State the equivariant Mathai-Quillen form in the Cartan model and explain its role in equivariant localisation.

Hint

In the Cartan model, the equivariant differential is $d_{X} = d - ι_{X^{*}}$ where $X^{*}$ is the fundamental vector field for $X \in g$ . The equivariant MQ form replaces the curvature $R^{V}$ with the equivariant curvature $R^{V} - μ_{X}^{V}$ involving the moment map.

Answer

Let $G$ act on $V \to M$ preserving the metric and connection. The equivariant curvature in the Cartan model is $$ R^V_G(X) = R^V - \mu^V(X) \in (\mathrm{Sym}, \mathfrak{g}^* \otimes \Omega^(M, \mathfrak{so}(V)))^G, $$ where $μ^{V} : g \to Γ (so (V))$ is the moment-map endomorphism (the infinitesimal action on $V$ ). The equivariant Mathai-Quillen form is $$ U_G(V) = \frac{1}{(2\pi)^n} \int^{\mathrm{Ber}} \exp!\left(-\tfrac{1}{2}|\mathbf{v}|^2 - \tfrac{1}{2} \nabla \mathbf{v} \wedge \nabla \mathbf{v} - R^V_G(X)\right) $$ in the Cartan model $(\mathrm{Sym}, \mathfrak{g}^ \otimes \Omega^(V))^G $, c l ose d u n d er t h e C a r t an d i f f er e n t ia l$ d_X = d - \iota_{X^} $. A f t er z er o - sec t i o n p u l l ba c k,$ s_0^* U_G(V) = \mathrm{Pf}(R^V_G(X)/2\pi)$, the equivariant Euler class.

Role in localisation: the equivariant Atiyah-Bott-Berline-Vergne localisation theorem expresses the equivariant integral of a closed form $α$ as a sum of contributions from the fixed-point set of the action. The equivariant MQ form is the explicit closed-form representative of the equivariant Thom class that makes localisation a closed-form computation rather than a cohomological identity. This is the source of the modern path-integral approach to localisation in supersymmetric gauge theories (Pestun, Nekrasov, Hori-Vafa). Rubric: full credit for the Cartan-model formula plus the equivariant Euler-class identification plus the localisation application.

Lean formalization [Intermediate+]

lean_status: none — Mathlib lacks the Berezin-integral infrastructure and the compactly-vertical-support de Rham complex needed to state the Mathai-Quillen form. A formal route would build:

The Berezin integral $\int^{Ber} : Λ^{*} W \to R$ on the exterior algebra of a Euclidean vector space, packaged as a $C^{\infty} (M)$ -linear extension to $Λ^{*} V \to C^{\infty} (M)$ for a vector bundle $V \to M$ .
The Pfaffian identity $\int^{Ber} e^{- A} = Pf (A)$ for skew-symmetric $A$ .
The compactly-vertical-support de Rham complex $Ω_{c v}^{*} (V)$ and the Thom isomorphism $H_{c v}^{*} (V) ≅ H^{*- 2 n} (M)$ .
The Mathai-Quillen form $U (V) \in Ω_{c v}^{2 n} (V)$ and its closedness, normalisation, and zero-section pullback identities.
The equivariant refinement via the Cartan model.

Each is a substantial Mathlib contribution. The conceptual content most amenable to near-term formalisation is the algebraic part (items 1-2), packaging Berezin integration as BerezinIntegral in LinearAlgebra.ExteriorAlgebra. The differential-geometric part (items 3-5) needs the compactly-vertical-support cohomology theory first, which Mathlib does not currently have in any form.

Advanced results [Master]

Mathai-Quillen as universal Thom form. The Mathai-Quillen formula on the universal bundle $E S O (2 n) \to B S O (2 n)$ provides an explicit differential-form representative of the universal Thom class. Choose a universal connection on the universal bundle (existence is classical: any $S O (2 n)$ -invariant connection on the universal Stiefel bundle descends). The Mathai-Quillen form $U_{univ}$ on the total space $E S O (2 n)$ then pulls back, via any classifying map $f : M \to B S O (2 n)$ , to the Mathai-Quillen form $U (V) = \tilde{f}^{*} U_{univ}$ of the bundle $V = f^{*} E S O (2 n)$ . This is a genuine universal-property statement at the level of differential forms, not merely a cohomological identification ^{[Mathai-Quillen 1986]}. The universal form realises the Chern-Weil approach to characteristic classes: the universal Euler class $e_{univ} = s_{0}^{*} U_{univ} \in H^{2 n} (B S O (2 n); R)$ is the Pfaffian polynomial in the universal curvature.

The Mathai-Quillen formalism as the odd analog of the Bismut superconnection. Mathai and Quillen's 1986 paper appeared in the same year as Bismut's 1986 family-index theorem, both built on Quillen's 1985 superconnection framework. The Bismut construction of 03.09.23 applies a superconnection $A_{t} = \nabla^{E} + t D + \dots$ to the infinite-dimensional bundle $π_{*} E$ of fibre solutions, producing a Chern character form on the base of a fibration. The Mathai-Quillen construction applies a superconnection $A = π^{*} \nabla^{V} + i c (v)$ to the finite-dimensional pullback bundle $π^{*} V \to V$ , producing the Thom form on the total space. Both constructions are evaluations of the same template — supertrace of an exponential of a superconnection's square — with $A^{2}$ playing the role of a Gaussian in the path-integral interpretation. The fermionic content (Berezin variable, Clifford action) carries the form-degree information, the bosonic content (fibre coordinate, Brownian motion path) carries the support information. The MQ form is the finite-dimensional, odd-Clifford-action analog of the Bismut family Chern character.

Atiyah-Jeffrey: TQFT as Euler class of infinite-dimensional bundles. Atiyah and Jeffrey 1990 Topological Lagrangians and cohomology (J. Geom. Phys. 7, 119-136) recognised that the Mathai-Quillen formula, applied formally to an infinite-dimensional vector bundle, reproduces the path-integral integrand of Donaldson-Witten topological gauge theory. The Donaldson invariants of a four-manifold $X$ are computed by a path integral over the space $A / G$ of gauge equivalence classes of connections, with a Mathai-Quillen-shape integrand that formally computes the Euler class of an infinite-dimensional vector bundle whose zero locus is the moduli space of anti-self-dual connections. The path integral collapses to an integral over the finite-dimensional zero locus by the Mathai-Quillen mechanism: the Gaussian factor concentrates the path measure on the zero locus, and the Berezin integral over fermions produces the appropriate Euler density. Witten 1988 Topological sigma models (Comm. Math. Phys. 118, 411-449) had previously identified the topological-sigma-model partition function as such an Euler-class integral; Atiyah-Jeffrey gave the rigorous Mathai-Quillen interpretation.

Equivariant Mathai-Quillen form. When a compact Lie group $G$ acts on $V \to M$ preserving the metric and connection, the Mathai-Quillen construction lifts to the Cartan model of equivariant cohomology. The equivariant curvature $R_{G}^{V} (X) = R^{V} - μ^{V} (X)$ (with $μ^{V} : g \to Γ (so (V))$ the bundle moment map) replaces $R^{V}$ in the exponent, and the equivariant Mathai-Quillen form $U_{G} (V)$ is closed in $(Sym g^{*} \otimes Ω^{*} (V))^{G}$ under the Cartan differential. The zero-section pullback is the equivariant Euler class. The equivariant MQ form is the central object of equivariant localisation in modern supersymmetric gauge theory: Duistermaat-Heckman, Pestun 2012 on $N = 2$ gauge theory on $S^{4}$ , Nekrasov's instanton partition function, and the Hori-Vafa mirror-symmetry-via-localisation framework all depend on the explicit equivariant MQ representative.

Kalkman model and BRST. Kalkman 1993 BRST model for equivariant cohomology and representatives for the equivariant Thom class (Comm. Math. Phys. 153, 447-463) gave an alternative presentation of the equivariant Mathai-Quillen form via the BRST cohomology model for equivariant cohomology. The Kalkman formula introduces ghost fields for the gauge symmetry, producing a manifestly BRST-closed integrand that physicists found more natural. The Kalkman form is cohomologically equivalent to the Cartan-model MQ form but more useful in path-integral constructions where the gauge fixing has substantive content beyond the identity.

Bismut-Goette-Kotschick connections. The Mathai-Quillen form on a flat bundle (in the sense of 03.09.23's Bismut-Lott extension) refines the Bismut-Lott analytic torsion to a form on a parametrising space. Bismut, Goette, and Kotschick used Mathai-Quillen-style transgressions to construct secondary characteristic classes for flat bundles in differential K-theory.

Synthesis. The Mathai-Quillen formalism identifies the algebraic Berezin integral with the geometric Thom class, and the central insight is that a fermionic Gaussian on $Λ^{*} V$ realises the Pfaffian as the Euler form. The foundational reason it works is exactly that the Berezin integral has the same algebraic structure as a bosonic Gaussian integral: a vanishing-derivative property and a quadratic-form Pfaffian evaluation. This is exactly why the MQ construction is the odd analog of the Bismut superconnection 03.09.23, why equivariant localisation in TQFT computes finite-dimensional integrals from path integrals, and why the Atiyah-Jeffrey realisation of Donaldson-Witten theory as an infinite-dimensional Euler class is a structural identification, not a heuristic analogy. Putting these together, the MQ form generalises the Bott-Tu Thom representative 03.04.09 to an explicit Gaussian-shape form on every even-rank oriented bundle, identifies the Chern-Weil Euler form 03.06.06 with the zero-section pullback, and is dual to the K-theoretic Thom class 03.08.03 in the Chern-character sense — the bridge between the differential-form and K-theoretic pictures is precisely the MQ form. The same construction generalises to the equivariant setting via the Cartan model, to infinite-dimensional settings via Atiyah-Jeffrey's path-integral interpretation, and to differential-K-theoretic settings via Bismut-Goette-Kotschick, each time producing the canonical closed-form representative of the appropriate Thom/Euler class. The pattern recurs throughout modern geometric analysis: a finite-dimensional Gaussian integral, paired with a fermionic Berezin integral, identifies an analytic object with a topological invariant in one explicit formula.

Full proof set [Master]

Proposition (Berezin integral and Pfaffian). Let $W$ be a Euclidean vector space of dimension $2 n$ and let $A \in so (W)$ be a skew-symmetric endomorphism. Identify $A$ with the two-form $\sum_{i < j} A_{ij} e_{i} \land e_{j} \in Λ^{2} W$ . Then $$ \int^{\mathrm{Ber}} e^{-A} = \mathrm{Pf}(A). $$

Proof. Block-diagonalise $A$ in an orthonormal basis: there exist real numbers $λ_{1}, \dots, λ_{n}$ such that, after orthonormal change of basis, $A = diag (λ_{1} J, \dots, λ_{n} J)$ where $J = (0 - 1 10)$ . Then $A \in Λ^{2} W$ has the form $$ A = \lambda_1, e_1 \wedge e_2 + \lambda_2, e_3 \wedge e_4 + \cdots + \lambda_n, e_{2n-1} \wedge e_{2n}. $$ The terms $λ_{k} e_{2 k - 1} \land e_{2 k}$ commute pairwise in $Λ^{*}$ (each is even-degree), so $$ e^{-A} = \prod_{k=1}^n e^{-\lambda_k e_{2k-1} \wedge e_{2k}} = \prod_{k=1}^n (1 - \lambda_k e_{2k-1} \wedge e_{2k}), $$ using $(e_{2 k - 1} \land e_{2 k})^{2} = 0$ in $Λ^{*} W$ . The Berezin integral picks out the coefficient of $e_{1} \land e_{2} \land \dots \land e_{2 n}$ : $$ \int^{\mathrm{Ber}} e^{-A} = (-1)^n \lambda_1 \lambda_2 \cdots \lambda_n = \mathrm{Pf}(A), $$ with the sign convention $Pf (A) = (- 1)^{n} \prod λ_{k}$ for the block-diagonal form (the absolute value $∣ Pf (A) ∣ = ∣ det A ∣$ holds in any case; the sign is fixed by orientation). $□$

Proposition (closedness of the Mathai-Quillen form). Let $V \to M$ be a smooth oriented Euclidean vector bundle of even rank $2 n$ with metric-compatible connection $\nabla$ . The Mathai-Quillen form $U (V)$ satisfies $d U (V) = 0 \in Ω^{2 n + 1} (V)$ .

Proof. Write $U (V) = (2 π)^{- n} \int^{Ber} e^{- A}$ with $A = ∣ v ∣^{2} /2 + \nabla v \land \nabla v /2 + R^{V}$ . Compute $$ d(e^{-\mathcal{A}}) = -d\mathcal{A} \cdot e^{-\mathcal{A}}, $$ using that $A$ takes values in a graded-commutative algebra (functions and even-degree exterior forms commute with each other). Compute the three pieces of $d A$ separately.

Piece 1: $d (∣ v ∣^{2} /2) = ⟨ \nabla v, v ⟩$ , using the metric compatibility $d ⟨ s, t ⟩ = ⟨ \nabla s, t ⟩ + ⟨ s, \nabla t ⟩$ and that $\nabla$ on the function $∣ v ∣^{2}$ acts via $v$ paired with $\nabla v$ .

Piece 2: $d (\nabla v \land \nabla v /2)$ . The connection on $π^{*} V$ has curvature equal to $π^{*} R^{V}$ , so $\nabla (\nabla v) = π^{*} R^{V} \cdot v$ . Use the Leibniz identity $d (\nabla v \land \nabla v /2) = \nabla (\nabla v) \land \nabla v = R^{V} v \land \nabla v$ , where the right-hand side pairs the curvature-acting-on- $v$ with the covariant differential of $v$ via the Berezin variable.

Piece 3: $d (R^{V}) = 0$ by the second Bianchi identity.

Combining, $$ d\mathcal{A} = \langle \nabla \mathbf{v}, \mathbf{v}\rangle + \langle R^V \mathbf{v}, \nabla \mathbf{v}\rangle = \langle \mathbf{v} + R^V \mathbf{v}, \nabla\mathbf{v}\rangle. $$

Now the Berezin integral $\int^{Ber}$ satisfies the integration-by-parts identity $$ \int^{\mathrm{Ber}} \iota_{e_i^}(\alpha) = 0 \quad \text{for all } \alpha \in \Lambda^ W, $$ where $ι_{e_{i}^{*}}$ is the contraction by the dual vector $e_{i}^{*}$ — this is because contraction reduces exterior degree, and the Berezin integral only sees top degree. Inside the Berezin integral, $d A \cdot e^{- A}$ is a contraction of a polynomial of mixed exterior degree; the linear pairing $⟨ v + R^{V} v, \nabla v ⟩ = \sum_{i} (v_{i} + (R^{V} v)_{i}) \nabla v_{i}$ becomes (after expressing $\nabla v$ in the Berezin variable as $\sum_{i} \nabla v_{i} \otimes e_{i}$ ) a contraction operator $ι_{e_{i}^{*}}$ applied to a polynomial expression. The Berezin integral of this contraction-derivative vanishes: $$ \int^{\mathrm{Ber}} d\mathcal{A} \cdot e^{-\mathcal{A}} = 0. $$

Therefore $d U (V) = - (2 π)^{- n} \int^{Ber} d A \cdot e^{- A} = 0$ . The detailed computation in coordinates appears in Mathai-Quillen 1986 §3 and Berline-Getzler-Vergne Ch. 1 §1.6 ^{[Mathai-Quillen 1986; ref: TODO_REF Berline-Getzler-Vergne]}. $□$

Proposition (fibre integral normalisation). Under the hypotheses of the previous proposition, $\pi_ U(V) = 1 \in \Omega^0(M) $, w h er e$ \pi_*: \Omega^_{cv}(V) \to \Omega^{-2n}(M)$ is integration along the fibre.*

Proof. Pointwise on the base: fix $x \in M$ and work in normal coordinates on $V$ centred at $x$ , in which $\nabla = d$ at $x$ and $R^{V} ∣_{x} = 0$ to first order. Then on the fibre $V_{x}$ the MQ form reduces to $$ U(V)|{V_x} = \frac{1}{(2\pi)^n} \int^{\mathrm{Ber}} e^{-|\mathbf{v}|^2/2 - d\mathbf{v}\wedge d\mathbf{v}/2} = \frac{1}{(2\pi)^n} e^{-|\mathbf{v}|^2/2}, dv_1\wedge\cdots\wedge dv{2n}. $$ The Berezin integration projects onto the top exterior degree, picking out the volume form on the fibre. Integration over the fibre $V_{x} = R^{2 n}$ is the standard Gaussian: $$ \int_{V_x} U(V)|{V_x} = \frac{1}{(2\pi)^n} \int{\mathbb{R}^{2n}} e^{-|\mathbf{v}|^2/2}, d^{2n}\mathbf{v} = \frac{1}{(2\pi)^n} \cdot (2\pi)^n = 1. $$ For a general connection and curvature, the higher-curvature pieces in $U (V)$ contribute to the fibre integral as exact terms (a fibre-divergence) by the Berezin-IBP argument used in the closedness proof, and integrate to zero. Hence $π_{*} U (V) = 1$ at every point of $M$ ^{[Mathai-Quillen 1986]}. $□$

Proposition (zero-section pullback gives Euler class). Under the same hypotheses, $s_0^ U(V) = \mathrm{Pf}(R^V/2\pi) = e(V) \in \Omega^{2n}(M)$.*

Proof. The zero section $s_{0} : M \to V$ pulls back the tautological section to zero: $s_{0}^{*} v = 0$ and $s_{0}^{*} (\nabla v) = 0$ . Substituting into the MQ exponent, $$ s_0^* \mathcal{A} = 0 + 0 + R^V = R^V \in \Omega^2(M, \Lambda^2 \pi^V). $$ Therefore $$ s_0^ U(V) = \frac{1}{(2\pi)^n} \int^{\mathrm{Ber}} e^{-R^V}. $$ By the Berezin-Pfaffian proposition applied to the $Λ^{2}$ -valued two-form $R^{V}$ (regarded as a skew-symmetric endomorphism of $V$ via the Euclidean metric), the Berezin integral equals the Pfaffian of $R^{V}$ . Normalising by $(2 π)^{n}$ , $$ s_0^* U(V) = \mathrm{Pf}!\left(\frac{R^V}{2\pi}\right). $$ This is the Chern-Weil representative of the Euler class $e (V) \in H^{2 n} (M; R)$ ^{[Mathai-Quillen 1986; ref: TODO_REF Bott-Tu §11]}. $□$

Proposition (universal Thom form on the universal bundle). Let $E S O (2 n) \to B S O (2 n)$ be the universal oriented rank- $2 n$ bundle, equipped with a universal connection $\nabla^{univ}$ . The Mathai-Quillen form $U_{univ} \in Ω_{c v}^{2 n} (E S O (2 n))$ is a universal differential-form representative of the Thom class: for every classifying map $f : M \to B S O (2 n)$ inducing the bundle map $\tilde f : V = f^ E SO(2n) \to E SO(2n) $, t h e M a t hai - Q u i l l e n f or m o f$ V $e q u a l s t h e p u l l ba c k$ U(V) = \tilde f^* U_{\mathrm{univ}}$.*

Proof. The MQ construction depends only on the Euclidean metric and the metric-compatible connection on the bundle. Both pull back: $f^{*}$ of a metric is a metric, and $\tilde{f}^{*}$ of a connection is a connection. The tautological section $v$ of $π^{*} V$ pulls back under the bundle map $\tilde{f}$ to the tautological section of $π^{*} E S O (2 n)$ restricted to the image, and the curvature pulls back via the second-derivative-of-the-connection-form identity. Every ingredient in $U (V)$ is therefore the $\tilde{f}$ -pullback of the corresponding ingredient in $U_{univ}$ , and the Berezin integral commutes with pullback (it is $C^{\infty}$ -linear in the Berezin variable and depends only on the orientation, which pulls back covariantly). Hence $U (V) = \tilde{f}^{*} U_{univ}$ . $□$

Proposition (Mathai-Quillen represents the Thom class). The form $U (V) \in Ω_{c v}^{2 n} (V)$ is a closed form whose cohomology class $[U (V)] \in H_{c v}^{2 n} (V; R)$ equals the Thom class $Th (V)$ .

Proof. Closedness was established in the second proposition. The Thom class is uniquely characterised by the property $π_{*} Th (V) = 1$ in $H^{0} (M)$ (which determines the cohomology class up to torsion that vanishes for real coefficients), and the fibre-integral proposition gives $π_{*} [U (V)] = 1$ . Hence $[U (V)] = Th (V) \in H_{c v}^{2 n} (V; R)$ ^{[Bott-Tu §6]}. The Euler-class identity $s_{0}^{*} Th (V) = e (V)$ then specialises the previous proposition to a cohomological identity, confirming consistency. $□$

Connections [Master]

Bismut superconnection 03.09.23. The Mathai-Quillen formalism is the odd-Clifford analog of the Bismut family superconnection. Both constructions are evaluations of a supertrace of a superconnection exponential — Bismut on the infinite-dimensional bundle of fibre solutions, Mathai-Quillen on the finite-dimensional pullback bundle over the total space. The fermionic Berezin variable in MQ plays the role of the Clifford generators in Bismut, and the bosonic fibre coordinate plays the role of the Brownian-motion path. The same closedness / class-independence argument runs in both cases. Mathai-Quillen 1986 and Bismut 1986 appeared together as twin applications of Quillen's 1985 superconnection framework.
Thom class and global angular form 03.04.09. The Mathai-Quillen form is an explicit smooth refinement of the Bott-Tu Thom representative. The Bott-Tu construction uses the global angular form (a transgression of the volume form on the sphere bundle), producing a Thom representative that is supported in a tubular neighbourhood of the zero section but not Gaussian. The Mathai-Quillen form is supported everywhere with Gaussian decay, making it the canonical representative for transgression arguments and the natural object for infinite-dimensional generalisations.
Thom isomorphism in K-theory 03.08.03. The Mathai-Quillen form gives the differential-form image (under the Chern character) of the K-theoretic Thom class. The K-theoretic Thom class lives in $K_{c v}^{0} (V)$ and corresponds to the spinor bundle $S$ of $V$ (when $V$ is spin or spin $^{c}$ ); the Chern character of $S$ is computed by the Mathai-Quillen form via $ch (S) = A (V)^{- 1} \cdot U (V)$ in differential-form cohomology. The MQ form is therefore the bridge between the K-theoretic and de Rham pictures of the Thom isomorphism.
Chern-Weil homomorphism 03.06.06. The zero-section pullback identity $s_{0}^{*} U (V) = Pf (R^{V} /2 π)$ identifies the Mathai-Quillen form with the Chern-Weil Euler form on the base. The Mathai-Quillen construction therefore upgrades the Chern-Weil homomorphism from a homomorphism $Sym^{*} (so (V)^{*})^{ad} \to H^{*} (M)$ to an explicit form-level lift: every invariant polynomial in the curvature comes from a specific MQ-type integrand on the total space.
Generalised Dirac bundle 03.09.14. The Mathai-Quillen form can be expressed in terms of a generalised Dirac operator on the pullback bundle: the exponent $\nabla v \land \nabla v /2$ encodes the symbol of the fibrewise Dirac operator on $V$ , and the curvature contribution comes from the Bochner-Weitzenböck identity for the pullback connection. This is the route through which Mathai-Quillen connects to the spinor-bundle approach to characteristic classes.
Family, equivariant, and Lefschetz index 03.09.21. The equivariant Mathai-Quillen form lives in the Cartan model of equivariant cohomology and is the equivariant Thom-class representative. Equivariant localisation (Atiyah-Bott-Berline-Vergne) for a Dirac-type operator with a group action computes the index via fibre integration of the equivariant MQ form against the operator's symbol class. The same machinery handles family-index calculations: the equivariant MQ form is the family Thom class restricted to a parametrising space.
Heat-kernel proof of Atiyah-Singer 03.09.20. The local index density in the heat-kernel proof of Atiyah-Singer is the small- $t$ limit of the supertrace of a Bismut-style superconnection, and the Mathai-Quillen form is the finite-dimensional template that the small- $t$ Getzler rescaling reduces to: the Gaussian decay along the fibre comes from the harmonic-oscillator heat kernel of Mehler, the Berezin variable corresponds to the Clifford generators after rescaling, and the curvature piece $R^{V}$ contributes the $A$ -genus.
Atiyah-Singer index theorem 03.09.10. The Mathai-Quillen formula gives the explicit Chern-Weil cocycle for the topological side of the Atiyah-Singer formula: the integrand $A (T M) \land ch (E)$ on the base of a Dirac-type operator can be expressed as a fibre integral of a Mathai-Quillen form on the total space of $T M$ paired with the symbol class of the twisting bundle $E$ . This is the route Atiyah-Jeffrey followed to interpret TQFT path integrals as infinite-dimensional MQ-style Euler-class computations.

Historical & philosophical context [Master]

Mathai-Quillen 1986. Varghese Mathai and Daniel Quillen's Superconnections, Thom classes, and equivariant differential forms (Topology 25, 85-110) was a direct sequel to Quillen's 1985 Superconnections and the Chern character. Where Quillen 1985 had introduced the superconnection framework as an algebraic generalisation of Chern-Weil theory for graded bundles, Mathai-Quillen 1986 applied the framework to a specific question: how does one write down an explicit smooth differential form representing the Thom class of a Euclidean vector bundle, with rapid Gaussian decay along the fibre? The answer was the Mathai-Quillen formula. Quillen had been working on this question independently since the early 1980s; Mathai, then a graduate student, contributed the geometric refinement that turned a formal Berezin-integral expression into a workable formula for explicit characteristic-class computations. The paper also introduced the equivariant version, opening the door to equivariant localisation arguments that have since become central to gauge theory ^{[Mathai-Quillen 1986]}.

Bismut's parallel application. Jean-Michel Bismut's 1986 Invent. Math. 83 paper appeared in the same year and applied Quillen's superconnection framework to a different question — the family-index theorem for Dirac operators — producing what is now called the Bismut superconnection of 03.09.23. The Mathai-Quillen and Bismut papers were independent applications of the same algebraic substrate (Quillen 1985), and they have remained interlocking ever since: Mathai-Quillen for the finite-dimensional Thom class and equivariant cohomology, Bismut for the infinite-dimensional family Chern character ^{[Bismut 1986]}.

Atiyah-Jeffrey 1990 and Witten 1988. Edward Witten's 1988 Topological sigma models (Comm. Math. Phys. 118, 411-449) had already interpreted the path-integral of certain topological field theories as integrals over moduli spaces with Euler-class-like integrands. Michael Atiyah and Lisa Jeffrey 1990 Topological Lagrangians and cohomology (J. Geom. Phys. 7, 119-136) recognised that the Witten integrand was the formal application of the Mathai-Quillen formula to an infinite-dimensional vector bundle, providing a rigorous geometric interpretation of the Donaldson-Witten path-integral as the Euler class of an infinite-dimensional bundle whose zero locus is the moduli space of anti-self-dual connections. The Atiyah-Jeffrey paper became the canonical reference for the Mathai-Quillen approach to TQFT ^{[Atiyah-Jeffrey 1990; ref: TODO_REF Witten 1988]}.

Modern reach. The Mathai-Quillen formalism has become the standard tool for equivariant localisation in gauge theory and supersymmetric quantum field theory. Pestun 2012's calculation of the partition function of $N = 2$ gauge theory on $S^{4}$ , Nekrasov's instanton partition function, and Hori-Vafa's mirror-symmetry-via-localisation framework all rely on the explicit equivariant MQ representative. In differential K-theory (Hopkins-Singer, Bunke-Schick), the MQ form provides the canonical differential refinement of the Thom class. Berline, Getzler, and Vergne's 1992 monograph Heat Kernels and Dirac Operators presented the construction in its definitive textbook form (Ch. 1 §1.6 for the Berezin integral, Ch. 7 §7.7 for the Mathai-Quillen form) ^{[Berline-Getzler-Vergne]}.

Bibliography [Master]

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  number  = {1},
  year    = {1986},
  pages   = {85--110}
}

@article{Quillen1985,
  author  = {Quillen, Daniel},
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}

@article{Bismut1986,
  author  = {Bismut, Jean-Michel},
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}

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}

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}

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  year      = {1992}
}

@book{BottTu1982,
  author    = {Bott, Raoul and Tu, Loring W.},
  title     = {Differential Forms in Algebraic Topology},
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  publisher = {Springer-Verlag},
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}

@article{Kalkman1993,
  author  = {Kalkman, Jaap},
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  journal = {Communications in Mathematical Physics},
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  pages   = {447--463}
}

@article{CordesMooreRamgoolam1995,
  author  = {Cordes, Stefan and Moore, Gregory and Ramgoolam, Sanjaye},
  title   = {Lectures on {2D} {Yang-Mills} theory, equivariant cohomology and topological field theories},
  journal = {Nuclear Physics B Proceedings Supplements},
  volume  = {41},
  year    = {1995},
  pages   = {184--244}
}

@book{LawsonMichelsohn1989,
  author    = {Lawson, H. Blaine and Michelsohn, Marie-Louise},
  title     = {Spin Geometry},
  publisher = {Princeton University Press},
  year      = {1989}
}

Prerequisites

03.09.23
03.09.02
03.08.03
03.04.09
03.06.06
03.05.07
03.09.14

Tier anchors

beginner: a Gaussian-shape differential form that represents the Thom class of a vector bundle and recovers the Euler class after fibre integration
intermediate: Mathai-Quillen 1986 *Superconnections, Thom classes, and equivariant differential forms* (Topology 25) §§4-5; Berline-Getzler-Vergne Ch. 1 §1.6 + Ch. 7 §7.7
master: Mathai-Quillen 1986 *Superconnections, Thom classes, and equivariant differential forms* (Topology 25, originator paper); Bismut 1986 *The Atiyah-Singer index theorem for families of Dirac operators* (Invent. Math. 83, parallel superconnection story); Atiyah-Jeffrey 1990 *Topological Lagrangians and cohomology* (J. Geom. Phys. 7, infinite-dimensional MQ); Witten 1988 *Topological sigma models* (Comm. Math. Phys. 118, TQFT application); Berline-Getzler-Vergne 1992 *Heat Kernels and Dirac Operators* Ch. 1 + Ch. 7 §7.7; Bott-Tu *Differential Forms in Algebraic Topology* §6 + §11 (classical Thom theory)

References

TODO_REF
Mathai-Quillen — Superconnections, Thom classes, and equivariant differential forms · Topology 25 (1986), 85-110 — originator paper, the Mathai-Quillen formula and the explicit Gaussian Thom representative
TODO_REF
Bismut — The Atiyah-Singer index theorem for families of Dirac operators: two heat-equation proofs · Invent. Math. 83 (1986), 91-151 — parallel application of Quillen's superconnection framework
TODO_REF
Quillen — Superconnections and the Chern character · Topology 24 (1985), 89-95 — superconnection framework, the algebraic substrate of the MQ construction
TODO_REF
Atiyah-Jeffrey — Topological Lagrangians and cohomology · J. Geom. Phys. 7 (1990), 119-136 — Mathai-Quillen interpretation of Donaldson-Witten TQFT as an Euler class of an infinite-dimensional bundle
TODO_REF
Witten — Topological sigma models · Comm. Math. Phys. 118 (1988), 411-449 — TQFT application, Mathai-Quillen as path-integral integrand
TODO_REF
Berline-Getzler-Vergne — Heat Kernels and Dirac Operators · Grundlehren 298, Springer 1992, Ch. 1 §1.6 (Berezin integral) and Ch. 7 §7.7 (Mathai-Quillen form)
TODO_REF
Bott-Tu — Differential Forms in Algebraic Topology · GTM 82, Springer 1982, §6 (Thom isomorphism via global angular form) and §11 (Euler class)
TODO_REF
Lawson-Michelsohn — Spin Geometry · Princeton 1989, §III.10 (Thom isomorphism), §III.11 (Euler class and Pfaffian)
TODO_REF
Cordes-Moore-Ramgoolam — Lectures on 2D Yang-Mills, equivariant cohomology and topological field theories · Nucl. Phys. Proc. Suppl. 41 (1995), 184-244 — equivariant Mathai-Quillen in physical TQFT language
TODO_REF
Kalkman — BRST model for equivariant cohomology and representatives for the equivariant Thom class · Comm. Math. Phys. 153 (1993), 447-463 — alternative presentation of the equivariant MQ form

Reviewer

TBD

Estimated time

beginner: 22m
intermediate: 55m
master: 95m