03.09.23 · modern-geometry / spin-geometry

Bismut superconnection

shipped3 tiersLean: none

Anchor (Master): Quillen 1985 *Superconnections and the Chern character* (Topology 24); Bismut 1986 *Two heat-equation proofs* (Invent. Math. 83); Berline-Getzler-Vergne Ch. 9-§10; Mathai-Quillen 1986 *Superconnections, Thom classes, and equivariant differential forms* (Topology 25); Bismut-Cheeger 1989 *Eta invariants and their adiabatic limits* (J. AMS 2)

Intuition [Beginner]

A Dirac operator on a single manifold produces one integer: the index. When you have a smooth family of Dirac operators, one for each point of a parameter space, you would like a single object that captures the whole family at once. The Bismut superconnection is that object.

Think of it as a single operator that does two jobs simultaneously. It contains the family of Dirac operators (one piece per fibre), and it contains a connection that tells you how solutions on neighbouring fibres are linked. These two pieces normally sit on different categories of object: a differential operator on a manifold, and a covariant derivative on a vector bundle. The superconnection puts them in one expression.

The payoff is that a single heat-equation computation, run on this combined operator, produces a differential form on the parameter space whose cohomology class is the Chern character of the family index. The same trick that proved the Atiyah-Singer theorem for one operator extends, with one new ingredient, to a whole family.

Visual [Beginner]

A schematic of a fibration: a base space $B$ along the bottom, and above each point of $B$ sits a closed manifold (the fibre). On each fibre lives a Dirac operator. The Bismut superconnection is drawn as a single object that runs across the whole picture, with two ingredients labelled: the family of Dirac operators (a vertical piece on each fibre) and a connection (a horizontal piece moving across $B$ ).

The single object is the superconnection. Its squared heat kernel produces a differential form on $B$ representing the family index.

Worked example [Beginner]

Take the simplest case where the base is a single point, so the family has only one member. There is just one fibre, and the bundle of fibre solutions is a finite-dimensional graded vector space $E = E^{+} \oplus E^{-}$ — concretely, the spinor space $S^{+} \oplus S^{-}$ of $R^{2 n}$ .

The superconnection reduces to just the Dirac operator $D$ itself: there is no base to put a connection on, and the only ingredient is the odd endomorphism piece. The differential form on the base (a single point) is just a number, and that number is the supertrace $Str (e^{- t D^{2}})$ . The McKean-Singer formula tells us this number is the index of $D$ at every positive $t$ . The small- $t$ limit recovers the topological integral of the $A$ -genus density over the manifold $M$ .

What this shows is that the Bismut superconnection, in the single-point case, reproduces exactly the heat-kernel proof of Atiyah-Singer. The family version of the construction is a refinement: when the base $B$ is genuine (not a point), the superconnection produces differential forms on $B$ , not just numbers, and these forms represent the Chern character of the family index as a class in the cohomology of $B$ .

Check your understanding [Beginner]

Formal definition [Intermediate+]

Graded bundles and exterior forms. Let $B$ be a smooth manifold and $E = E^{+} \oplus E^{-}$ a $Z /2$ -graded smooth vector bundle on $B$ . The space $Ω^{*} (B, E) = Ω^{*} (B) \otimes_{C^{\infty} (B)} Γ (E)$ of $E$ -valued differential forms inherits a total $Z /2$ grading from the sum of the form-degree parity and the bundle parity. A linear operator $T : Ω^{*} (B, E) \to Ω^{*} (B, E)$ has total parity $∣ T ∣ \in Z /2$ if it shifts the total grading by $∣ T ∣$ .

Definition (Quillen 1985 — superconnection). A superconnection on $E$ is an odd-parity first-order operator $$ \mathbb{A} : \Omega^(B, \mathcal{E}) \longrightarrow \Omega^(B, \mathcal{E}) $$ satisfying the graded Leibniz rule $$ \mathbb{A}(\omega \cdot s) = (d\omega) \cdot s + (-1)^{|\omega|} \omega \cdot \mathbb{A}(s) $$ for $ω \in Ω^{*} (B)$ homogeneous of degree $∣ ω ∣$ and $s \in Γ (E)$ ^{[Quillen 1985]}.

Expanding by form-degree, $A = A_{[0]} + A_{[1]} + A_{[2]} + \dots$ , where each $A_{[k]} \in Ω^{k} (B, End (E))$ has bundle parity $k + 1 (mod 2)$ so that the total parity remains odd. The piece $A_{[0]}$ is an odd endomorphism of $E$ (i.e., $A_{[0]} : E^{\pm} \to E^{\mp}$ ); the piece $A_{[1]}$ is a ( $Z /2$ -graded) connection on $E$ ; the higher pieces $A_{[k \geq 2]}$ are even-or-odd bundle-valued forms.

Curvature. The square $A^{2}$ is an even operator and an order-zero differential operator (the first-order terms cancel because $d^{2} = 0$ and the Leibniz cross-terms collapse), hence $$ \mathbb{A}^2 \in \Omega^*(B, \mathrm{End}(\mathcal{E})). $$ This is the curvature of the superconnection. The Bianchi identity $[A, A^{2}] = 0$ holds ^{[Quillen 1985; ref: TODO_REF Berline-Getzler-Vergne Ch. 9]}.

Chern character. The superconnection Chern character is the differential form $$ \mathrm{ch}(\mathbb{A}) := \mathrm{Str}(e^{-\mathbb{A}^2}) \in \Omega^{\mathrm{even}}(B), $$ where $Str$ is the supertrace $Str (T) = Tr (T ∣_{E^{+}}) - Tr (T ∣_{E^{-}})$ extended $Ω^{*} (B)$ -linearly. The form $ch (A)$ is closed in $Ω^{*} (B)$ , and its de Rham cohomology class depends only on the underlying graded $K$ -theory class $[E^{+}] - [E^{-}] \in K^{0} (B)$ .

Counterexamples to common slips.

A superconnection is not a connection on the graded bundle in the ordinary sense. An ordinary connection lives in degree one only; a superconnection has pieces in every form-degree. The degree-one piece $A_{[1]}$ is an ordinary connection, but it does not determine $A$ .
The curvature $A^{2}$ is not a two-form valued in $End (E)$ in general. It has pieces in every even form-degree, the lowest of which is the endomorphism $A_{[0]}^{2} \in Γ (End^{even} (E))$ .
The supertrace $Str$ vanishes on graded commutators, not on ordinary commutators. This vanishing is what makes the Chern character of $A$ closed: $d Str (e^{- A^{2}}) = Str ([A, e^{- A^{2}}]) = 0$ as a graded commutator inside $Ω^{*} (B, End (E))$ .

Definition (Bismut superconnection of a family of Dirac operators). Let $π : M \to B$ be a smooth fibration of smooth manifolds with closed fibres $M_{b} = π^{- 1} (b)$ , equipped with a vertical Riemannian metric and a horizontal complement (a connection on the fibration). Let $E \to M$ be a Clifford module bundle with metric-compatible Clifford-connection $\nabla^{E}$ . The infinite-dimensional bundle $$ \mathcal{E} := \pi_* E $$ over $B$ has fibre $E_{b} = Γ (M_{b}, E ∣_{M_{b}})$ — the space of sections of $E$ along the fibre — and is $Z /2$ -graded by the chirality grading $E = E^{+} \oplus E^{-}$ of the Clifford module.

The Bismut superconnection is, for each $t > 0$ , the superconnection $$ \mathbb{A}t = \nabla^{\mathcal{E}} + \sqrt{t}, D - \frac{1}{4\sqrt{t}}, c(T), $$ where $\nabla^{E}$ is the connection on $\pi* E $in d u ce d b y$ \nabla^E $an d t h e h or i z o n t a l l i f t,$ D $i s t h e f ami l y o f f ib r e w i seD i r a co p er a t or s,$ c $i s f ib r e w i se C l i f f or d m u l t i pl i c a t i o n, an d$ T \in \Omega^2(B, \mathrm{vert}) $i s t h ec u r v a t u r eo f t h e f ib r a t i o n (a v er t i c a l - v ec t or - v a l u e d tw o - f or m o n$ B$) ^{[Bismut 1986; ref: TODO_REF Berline-Getzler-Vergne Ch. 10]}.

The first two pieces $\nabla^{E} + t D$ encode Quillen's $A_{[1]} + A_{[0]}$ data after the scaling $D \to t D$ . The third piece $- \frac{1}{4 t} c (T)$ is the form-degree-two correction needed when the fibration has non-zero curvature; it ensures the curvature $A_{t}^{2}$ has the right small- $t$ behaviour. This third piece is the genuinely new term that Bismut's analysis demanded: a pure connection $\nabla^{E} + t D$ does not have the correct local index density in the $t \to 0^{+}$ limit unless $T = 0$ .

Key theorem with proof [Intermediate+]

Theorem (Bismut family-index theorem; Bismut 1986). Let $π : M \to B$ be a smooth fibration with closed even-dimensional fibres, equipped with a vertical Riemannian metric, a horizontal connection, and a family $D = {D_{b}}$ of Dirac operators on the Clifford-module bundle $E \to M$ . The Bismut superconnection $A_{t}$ satisfies:

(i) The differential form $ch (A_{t}) = Str (e^{- A_{t}^{2}})$ is closed in $Ω^{even} (B)$ for every $t > 0$ , and its cohomology class in $H^{even} (B; R)$ is independent of $t$ .

(ii) The cohomology class equals the Chern character of the analytic family index: $$ [\mathrm{ch}(\mathbb{A}_t)] = \mathrm{ch}(\mathrm{Ind}(D)) \in H^{\mathrm{even}}(B; \mathbb{R}). $$

(iii) In the limit $t \to 0^{+}$ , the form converges to the local index density along the fibre, $$ \lim_{t \to 0^+} \mathrm{ch}(\mathbb{A}t) = \int{M/B} \widehat A(R^{M/B}) \wedge \mathrm{ch}(F^{E/!\mathcal{S}}), $$ where $R^{M / B}$ is the curvature of the vertical Levi-Civita connection, $A$ is the $A$ -genus form, and $ch (F^{E / S})$ is the twisting Chern character of $E$ relative to the spinor bundle.

In the limit $t \to + \infty$ , the form converges to the Chern character of the index bundle computed directly from the analytic index.

Proof. The argument proceeds in three steps: closedness, $t$ -independence, and the two limits.

Step 1: closedness. The supertrace $Str : Ω^{*} (B, End (E)) \to Ω^{*} (B)$ is $Ω^{*} (B)$ -linear and vanishes on graded commutators. The Bianchi identity gives $[A_{t}, A_{t}^{2}] = 0$ , so $[A_{t}, e^{- A_{t}^{2}}] = 0$ as well (the heat kernel is computed by a power series in $A_{t}^{2}$ that commutes with $A_{t}$ ). Hence $$ d,\mathrm{ch}(\mathbb{A}_t) = d,\mathrm{Str}(e^{-\mathbb{A}_t^2}) = \mathrm{Str}([\mathbb{A}_t, e^{-\mathbb{A}_t^2}]) = 0, $$ using that $d$ acts on $Str (X)$ as $Str ([A_{t}, X])$ for any operator $X$ commuting up to graded sign with the superconnection ^{[Quillen 1985]}.

Step 2: $t$ -independence. Differentiate with respect to $t$ : $$ \frac{d}{dt}, \mathrm{Str}(e^{-\mathbb{A}_t^2}) = -\mathrm{Str}!\left(\frac{d\mathbb{A}_t^2}{dt}, e^{-\mathbb{A}_t^2}\right). $$ By a direct computation (Quillen's transgression lemma), $\frac{d A _{t}^{2}}{d t}$ is a graded commutator in $A_{t}$ and a one-parameter family of operators; the supertrace of this commutator is exact (a $d$ -exact form on $B$ ). Thus the cohomology class of $ch (A_{t})$ is independent of $t$ , although the form itself depends on $t$ . The transgression form $$ \mathrm{ch}(\mathbb{A}_t) - \mathrm{ch}(\mathbb{A}_s) = d \int_s^t \mathrm{Str}!\left(\frac{d\mathbb{A}_u}{du}, e^{-\mathbb{A}_u^2}\right) du $$ makes the exactness explicit and is the origin of Bismut-Cheeger eta-forms ^{[Bismut-Cheeger 1989]}.

Step 3: $t \to + \infty$ limit. As $t \to + \infty$ , the rescaled Dirac operator $t D$ dominates. By the spectral theorem applied fibrewise, $e^{- t D^{2}}$ projects onto the kernel of $D$ (a finite-dimensional graded vector bundle over $B$ , well-defined assuming the kernel has locally constant dimension; otherwise one stabilises). The connection $\nabla^{E}$ restricts to a connection on this index bundle, and the cross-terms involving $c (T) / t$ vanish in the limit. The form $ch (A_{t})$ converges to the Chern character of the index bundle equipped with its induced connection — exactly $ch (Ind (D))$ at the level of forms.

Step 4: $t \to 0^{+}$ limit. This is the technical core of Bismut's paper. The fibrewise Getzler rescaling, refined to handle the form-degree-two piece $c (T) / t$ , identifies the limit of $Str (e^{- A_{t}^{2}})$ as the fibre integral of the local index density. Concretely, choose normal coordinates along the fibre at each point of $B$ , rescale the Clifford action $c (e_{i}) \to t^{- 1/2} c (e_{i})$ as in the single-operator Getzler proof, and observe that the form-degree-two correction $c (T) / t$ contributes precisely the curvature term needed to produce the full $A$ genus form involving $R^{M / B}$ (not just the symbol piece). The resulting form is $$ \lim_{t \to 0^+} \mathrm{ch}(\mathbb{A}t) = \pi* \big( \widehat A(R^{M/B}) \wedge \mathrm{ch}(F^{E/!\mathcal{S}}) \big), $$ where $π_{*}$ is the fibre integration $Ω^{*} (M) \to Ω^{*- d i m M_{b}} (B)$ .

Combining: by $t$ -independence of the cohomology class together with the two limits, $$ \mathrm{ch}(\mathrm{Ind}(D)) = \big[ \pi_* (\widehat A(R^{M/B}) \wedge \mathrm{ch}(F^{E/!\mathcal{S}})) \big] \in H^{\mathrm{even}}(B; \mathbb{R}). $$ This is the cohomological family-index theorem of Atiyah-Singer 1971 IV, proved by heat-kernel localisation ^{[Atiyah-Singer 1971 IV; ref: TODO_REF Berline-Getzler-Vergne Ch. 10]}. $□$

Bridge. The Bismut superconnection builds toward the entire modern family-index machinery and identifies analysis with topology at the family level. The foundational reason it works is exactly that the supertrace kills graded commutators, so the heat-kernel form is closed for every $t$ , and the cohomology class is $t$ -independent — precisely the same McKean-Singer mechanism that drove the single-operator heat-kernel proof in 03.09.20. This is exactly the parametric analogue: a single object that interpolates between the analytic index at $t = + \infty$ and the topological density at $t = 0^{+}$ , identifying $ch (Ind (D))$ with the fibre integral of the local index density. The central insight is that the form-degree-two correction $c (T) / t$ is the genuinely new ingredient that converts a naïve connection-plus-operator combination into an object whose square has the correct Getzler-rescaled limit. Putting these together, the superconnection identifies family heat-kernel analysis with family characteristic-class topology, and the transgression form that records the $t$ -dependence appears again in 03.09.21 (family / equivariant index) as the higher refinement to Bismut-Cheeger eta-forms. The bridge is the recognition that Quillen's purely algebraic 1985 framework, originally a clean repackaging of the Chern-Weil construction for graded bundles, was exactly the right tool for the family analysis Bismut needed in 1986 — and the same construction generalises to equivariant family theory, foliations, and non-commutative geometry, each time producing the appropriate Chern-character form.

Exercises [Intermediate+]

Exercise 3 (medium, symbolic).

Verify that the form-degree-zero piece of $A_{t}^{2} = (\nabla^{E} + t D - \frac{1}{4 t} c (T))^{2}$ is $t D^{2}$ .

Hint

Expand the square and collect terms by total form-degree on the base. The form-degree-zero piece comes only from squaring the $t D$ term.

Answer

Expanding the square: $$ \mathbb{A}_t^2 = (\nabla^{\mathcal{E}})^2 + \sqrt{t} { \nabla^{\mathcal{E}}, D } + t D^2 - \frac{1}{4\sqrt{t}} { \nabla^{\mathcal{E}}, c(T) } - \frac{1}{4} {D, c(T)} + \frac{1}{16 t} c(T)^2. $$ The pieces by form-degree on $B$ : $(\nabla^{E})^{2}$ is form-degree two (the connection curvature); $t {\nabla^{E}, D}$ is form-degree one; $t D^{2}$ is form-degree zero; the $c (T)$ pieces are form-degrees $\geq 2$ . The form-degree-zero contribution is exactly $t D^{2}$ , the fibrewise Dirac-squared operator scaled by $t$ . Rubric: full credit for the expansion plus the form-degree bookkeeping.

Exercise 4 (medium, short-answer).

Why does the supertrace vanish on graded commutators in $Ω^{*} (B, End (E))$ , and how does this fact prove that $ch (A_{t})$ is closed on $B$ ?

Hint

The supertrace of $[T, S]_{gr} = T S - (- 1)^{∣ T ∣∣ S ∣} S T$ for graded operators is zero by the cyclic property of trace plus the graded sign rule. Then use $d \cdot Str (X) = Str ([A, X]_{gr})$ for $X$ commuting with $A^{2}$ .

Answer

For homogeneous operators $T$ and $S$ on a graded space, $Tr (T S) = (- 1)^{∣ T ∣∣ S ∣} Tr (S T)$ by reordering matrix elements with sign. The graded commutator is $[T, S]_{gr} = T S - (- 1)^{∣ T ∣∣ S ∣} S T$ , and its supertrace vanishes: $Str (T S) - (- 1)^{∣ T ∣∣ S ∣} Str (S T) = 0$ . Now the de Rham differential $d$ on $Str (e^{- A_{t}^{2}})$ acts as $Str ([A_{t}, e^{- A_{t}^{2}}]_{gr})$ , because $A_{t}$ is the unique odd derivation extending $d$ to bundle-valued forms. The Bianchi identity $[A_{t}, A_{t}^{2}]_{gr} = 0$ propagates to $[A_{t}, e^{- A_{t}^{2}}]_{gr} = 0$ . Hence $d ch (A_{t}) = 0$ . Rubric: full credit for the graded-cyclic argument plus the Bianchi propagation.

Exercise 5 (medium, short-answer).

State the transgression form $η (s, t) = \int_{s}^{t} Str (\frac{d A _{u}}{d u} e^{- A_{u}^{2}}) d u$ that witnesses the $t$ -independence of $[ch (A_{t})]$ , and explain why $η (s, t)$ has odd total degree on $B$ .

Hint

$\frac{d A _{u}}{d u}$ is even total parity (because $A_{u}$ is odd and the parameter $u$ does not flip parity). The supertrace then preserves parity.

Answer

By Step 2 of the main theorem's proof, $ch (A_{t}) - ch (A_{s}) = d η (s, t)$ with $η (s, t) = \int_{s}^{t} Str (\frac{d A _{u}}{d u} e^{- A_{u}^{2}}) d u$ . The supertrace produces forms of even total parity from even-parity operators (and odd from odd). Here $\frac{d A _{u}}{d u}$ is the derivative of an odd-parity operator with respect to a parity-preserving parameter, so $\frac{d A _{u}}{d u}$ has odd total parity; multiplied by the even operator $e^{- A_{u}^{2}}$ , the product is odd, and the supertrace of an odd operator on $Ω^{*} (B, End (E))$ produces an odd-degree form on $B$ . The transgression form $η (s, t)$ thus lives in $Ω^{odd} (B)$ , as it must to make $d η$ even-degree. Rubric: full credit for the parity bookkeeping.

Exercise 6 (hard, short-answer).

Compute the small- $t$ limit of $ch (A_{t})$ for the product fibration $π : B \times M \to B$ with the product Riemannian metric. Show that the answer is the constant function on $B$ valued in $\int_{M} A (M) \land ch (E)$ .

Hint

For the product fibration $π : B \times M \to B$ the curvature $T$ vanishes, so the third piece of $A_{t}$ disappears. The Getzler rescaling proceeds fibrewise exactly as in the single-operator case 03.09.20.

Answer

For the product fibration $π : B \times M \to B$ , the horizontal connection is flat ( $T = 0$ ), so $A_{t} = \nabla^{E} + t D$ with $\nabla^{E}$ the pullback connection and $D$ the fibrewise Dirac operator on $M$ (the same operator at every fibre, since the fibration is a product). The form-degree decomposition gives $Str (e^{- A_{t}^{2}}) = \sum_{k} ω_{k}$ with $ω_{k} \in Ω^{k} (B)$ . By the Getzler rescaling applied fibrewise at each point of $B$ , the small- $t$ limit of the form-degree-zero piece $ω_{0} \in Ω^{0} (B)$ converges to the constant function $b \mapsto \int_{M} A (M) \land ch (E)$ , and the higher form-degree pieces converge to zero (because the fibration is a product, no curvature terms in $B$ direction appear). Concretely, the answer is the index $ind (D_{M}) \in Z$ regarded as a constant on $B$ — matching the K-theory statement that the family index of a product family is the constant index times $[B]^{*} \in K^{0} (B)$ . Rubric: full credit for the $T = 0$ collapse plus the Getzler argument plus the constancy on $B$ .

Exercise 7 (hard, short-answer).

The Mathai-Quillen formula expresses the Thom class of a Euclidean vector bundle as the Chern character of a specific superconnection on the pullback of the bundle. Sketch how Mathai-Quillen 1986 uses the superconnection framework, and explain how their construction recovers the Mathai-Quillen representative of the Thom class on a finite-dimensional Euclidean bundle.

Hint

The Mathai-Quillen construction uses the superconnection $A = \nabla + i x$ on $π^{*} E$ over the total space $E$ , where $x$ is the tautological section. The Chern character of this superconnection, with appropriate normalisation, is a closed Thom-class representative.

Answer

Let $E \to M$ be a Euclidean vector bundle of rank $2 n$ with a metric connection $\nabla$ . The Mathai-Quillen superconnection on $π^{*} E$ over the total space $E$ is $A = π^{*} \nabla + i c (x)$ where $x : E \to E$ is the tautological section and $c$ is Clifford multiplication on the chiral spinor bundle of $E$ . Its square has form-degree-zero part $- ∣ x ∣^{2}$ (Gaussian decay) and higher pieces encoding the curvature. The Chern character $Str (e^{- A^{2}})$ is a closed differential form on $E$ that integrates to $1$ over each fibre and is rapidly decreasing away from the zero section. After pulling back along the zero section, this form represents the Euler class. The construction gives a Gaussian Thom-class representative — a genuine differential form on $E$ rather than a current — making it the natural object for transgression arguments. Mathai-Quillen 1986 used this representative to give a geometric construction of the equivariant Thom class. Rubric: full credit for the $\nabla + i x$ formulation, the Gaussian observation, and the Thom-class statement.

Exercise 8 (hard, short-answer).

The Bismut-Cheeger eta-form $η$ refines the eta-invariant of Atiyah-Patodi-Singer to families. State the eta-form as a regularised integral of the transgression of $ch (A_{t})$ , and explain its role in the adiabatic-limit formula for families with boundary.

Hint

The eta-form is $η = \frac{1}{π} \int_{0}^{\infty} Str (\frac{d A _{t}}{d t} e^{- A_{t}^{2}}) d t$ , with the integral regularised by analytic continuation near $t = 0$ and $t = \infty$ .

Answer

The Bismut-Cheeger eta-form is the differential form on the base $B$ defined by the regularised integral $$ \widetilde\eta(\mathbb{A}_t) = \frac{1}{\sqrt{\pi}} \int_0^\infty \mathrm{Str}!\left(\frac{d\mathbb{A}_t}{dt}, e^{-\mathbb{A}_t^2}\right) dt, $$ where the divergent behaviour as $t \to 0^{+}$ is removed by subtracting the local index density (the limit of $ch (A_{t})$ ) and the behaviour as $t \to + \infty$ is controlled by the spectral gap of $D$ on each fibre (assumed positive for the formula to apply). The degree-zero piece of $η$ reduces to half the Atiyah-Patodi-Singer eta-invariant $η (D) /2$ at each fibre; the higher-degree pieces are the genuine family refinement. Bismut-Cheeger 1989 J. Amer. Math. Soc. 2, 33-70 showed that for a fibration with boundary $\partial M \to B$ and adiabatic limit (rescaling the metric in the boundary direction), the integral $\int_{M / B} A \land ch (E) - ch (Ind (D))$ equals $η$ — the eta-form is the obstruction to the local index formula when the fibres have boundary. This is the family analogue of the Atiyah-Patodi-Singer index theorem. Rubric: full credit for the regularised-integral definition, the degree-zero reduction to APS, and the adiabatic-limit statement.

Lean formalization [Intermediate+]

lean_status: none — Mathlib lacks the superconnection framework. A formal route would build:

The $Z /2$ -graded tensor category of bundle-valued differential forms $Ω^{*} (B) \otimes End (E)$ with its total parity grading.
The supertrace $Str : Ω^{*} (B, End (E)) \to Ω^{*} (B)$ and the vanishing on graded commutators.
Superconnections as odd derivations satisfying the graded Leibniz rule.
The Bismut superconnection $A_{t} = \nabla^{E} + t D - \frac{1}{4 t} c (T)$ on the infinite-dimensional bundle $π_{*} E$ .
The heat semigroup $e^{- A_{t}^{2}}$ as a smoothing operator-valued differential form on $B$ .
The Getzler rescaling adapted to family-index calculations.

Each is a substantial Mathlib contribution. The conceptual content most amenable to a near-term formalisation is the algebraic part (items 1-3), packaging Quillen's framework as Superconnection in CategoryTheory.Graded. The analytic part (items 4-6) needs the smoothing-operator infrastructure first.

Advanced results [Master]

Quillen's transgression and the Chern-Simons form. Quillen's 1985 paper introduced the transgression form on the space of superconnections: for a one-parameter family $A_{t}$ , the transgression $η (s, t) = \int_{s}^{t} Str (\frac{d A _{u}}{d u} e^{- A_{u}^{2}}) d u$ satisfies $d η (s, t) = ch (A_{t}) - ch (A_{s})$ . The transgression form is the Chern-Simons / secondary-characteristic-class structure attached to a path between two superconnections, and it is the algebraic precursor of the Bismut-Cheeger eta-form. The same construction, applied to a family of ordinary connections on an ordinary vector bundle, recovers the classical Chern-Simons form ^{[Quillen 1985]}.

Bismut-Cheeger eta-form and adiabatic limits. Bismut-Cheeger 1989 Eta invariants and their adiabatic limits (J. AMS 2) extended the Bismut superconnection from closed-fibre fibrations to fibrations with boundary. The eta-form $η (A_{t})$ is a closed odd-degree form on $B$ refining the Atiyah-Patodi-Singer eta-invariant. The Bismut-Cheeger adiabatic-limit theorem states that for a fibration with boundary, the family-index formula acquires the eta-form as a correction term: $ch (Ind (D)) = \int_{M / B} A \land ch (E) - η$ . This is the family generalisation of the APS index theorem and underlies all modern work on geometric K-homology classes for manifolds with boundary ^{[Bismut-Cheeger 1989]}.

Mathai-Quillen Thom-class representative. Mathai-Quillen 1986 Superconnections, Thom classes, and equivariant differential forms (Topology 25) used the superconnection framework to give an explicit Gaussian-decay differential-form representative of the Thom class of a Euclidean vector bundle. The Mathai-Quillen Thom form is the Chern character of a specific superconnection on the pullback bundle over the total space, with the form-degree-zero piece $- ∣ x ∣^{2}$ providing the Gaussian decay. The construction is the natural geometric model for the equivariant Thom isomorphism and is the foundation of Mathai-Quillen-style localisation arguments in equivariant cohomology ^{[Mathai-Quillen 1986]}.

Bismut superconnection in non-commutative geometry. Connes' framework of non-commutative differential geometry includes a superconnection construction for spectral triples $(A, H, D)$ . Bismut-style transgression of the heat-kernel form along a family of spectral triples produces the local cyclic cocycle of Connes-Moscovici, the non-commutative analogue of the family Chern character. The same algebraic identities — supertrace vanishes on graded commutators, Bianchi identity, exact transgression — hold in the cyclic-cohomology setting, with the McKean-Singer formula replaced by the Connes-Moscovici local index formula.

Higher torsion forms (Bismut-Lott). Bismut-Lott 1995 Flat vector bundles, direct images and higher real analytic torsion (J. AMS 8) extended the superconnection construction to flat vector bundles, producing the higher analytic torsion form $T (A_{t})$ on $B$ . The torsion form refines the Ray-Singer analytic torsion to families: its degree-zero piece recovers the Ray-Singer torsion of each fibre, and its higher-degree pieces are genuine family invariants. Bismut-Lott torsion is the analytic counterpart of the Igusa-Klein higher topological torsion in algebraic K-theory.

Equivariant Bismut superconnection. When a compact Lie group $G$ acts on the fibration $π : M \to B$ preserving all the data, the Bismut superconnection becomes equivariant: $A_{t}$ is built from a $G$ -invariant vertical metric, a $G$ -equivariant connection, and the $G$ -equivariant family Dirac operator. The equivariant heat-kernel form lives in the equivariant cohomology $H_{G}^{*} (B; R)$ , and the small- $t$ limit gives the equivariant family-index formula. This is the route through which Bismut's framework integrates with the Atiyah-Bott-Berline-Vergne localisation theorem.

Loop-space heuristic. Atiyah's 1985 Circular symmetry and stationary-phase approximation observed that the Bismut superconnection on the loop space $L M$ of a spin manifold $M$ formally produces the $A$ genus via a stationary-phase argument around constant loops. The argument is heuristic — the loop space is infinite-dimensional — but it has been made rigorous in pieces by Stolz-Teichner and by Pflaum-Posthuma-Tang. The loop-space heuristic explains why the $A$ -genus is the small- $t$ limit of the heat kernel: it is the stationary-phase integral over $L M$ .

Synthesis. The Bismut superconnection identifies family analysis with family topology in the same way that the heat-kernel proof of 03.09.20 identifies single-operator analysis with single-operator topology. The foundational reason it works is exactly that supertraces kill graded commutators, so the Chern-character form is closed for every $t$ , and the cohomology class is $t$ -independent under the Bianchi-identity-induced transgression. The central insight is that Quillen's purely algebraic 1985 framework — superconnections as odd derivations on graded bundle-valued forms — was the precise tool needed for the parametric analysis Bismut required in 1986; the form-degree-two correction $c (T) / t$ is the genuinely new ingredient that makes the small- $t$ Getzler rescaling produce the full local index density rather than just its leading piece. Putting these together, the superconnection identifies $ch (Ind (D))$ with the fibre integral of $A \land ch (F^{E / S})$ , generalising Atiyah-Singer to families and producing along the way the Bismut-Cheeger eta-form, the Mathai-Quillen Thom representative, the Bismut-Lott analytic torsion form, and the Connes-Moscovici local cyclic cocycle. This same construction appears again in 03.09.21 (family / equivariant / Lefschetz index) where the equivariant refinement runs in parallel, and is dual to the K-theoretic family-index formula whose Chern character it computes — the bridge between the two pictures is precisely the Bismut superconnection.

Full proof set [Master]

Proposition (Quillen's algebraic identity). Let $A$ be a superconnection on a graded vector bundle $E$ over a smooth manifold $B$ . For any one-parameter family of superconnections $A_{u}$ depending smoothly on $u \in (0, \infty)$ , $$ \frac{d}{du}, \mathrm{Str}(e^{-\mathbb{A}_u^2}) = -d, \mathrm{Str}!\left(\frac{d\mathbb{A}_u}{du}, e^{-\mathbb{A}_u^2}\right). $$

Proof. Differentiate using the Duhamel formula $$ \frac{d}{du}, e^{-\mathbb{A}_u^2} = -\int_0^1 e^{-s \mathbb{A}_u^2}, \frac{d\mathbb{A}_u^2}{du}, e^{-(1-s)\mathbb{A}_u^2}, ds. $$ Take the supertrace and use cyclicity (with graded sign) to combine the two exponentials into one $e^{- A_{u}^{2}}$ with $\frac{d A _{u}^{2}}{d u}$ on the right: $$ \frac{d}{du}, \mathrm{Str}(e^{-\mathbb{A}_u^2}) = -\mathrm{Str}!\left(\frac{d\mathbb{A}_u^2}{du}, e^{-\mathbb{A}_u^2}\right). $$ Now compute $\frac{d A _{u}^{2}}{d u} = {A_{u}, \frac{d A _{u}}{d u}}$ (anticommutator, since both factors are odd). Insert into the supertrace: $$ \mathrm{Str}!\left({\mathbb{A}_u, \tfrac{d\mathbb{A}_u}{du}}, e^{-\mathbb{A}_u^2}\right) = \mathrm{Str}!\left([\mathbb{A}_u, \tfrac{d\mathbb{A}_u}{du}, e^{-\mathbb{A}u^2}]{\mathrm{gr}}\right) + \mathrm{Str}!\left(\tfrac{d\mathbb{A}_u}{du}, [\mathbb{A}_u, e^{-\mathbb{A}u^2}]{\mathrm{gr}}\right). $$ The second graded commutator vanishes because $A_{u}$ commutes with $A_{u}^{2}$ (Bianchi), hence with its power series $e^{- A_{u}^{2}}$ . The first term is the supertrace of a graded commutator, which equals $d Str (\frac{d A _{u}}{d u} e^{- A_{u}^{2}})$ by the identity $d \cdot Str (X) = Str ([A_{u}, X]_{gr})$ for $Ω^{*} (B, End (E))$ -valued operators. Combining gives the claim. $□$

Proposition (closedness of Chern character). For any superconnection $A$ on $E$ , the form $ch (A) = Str (e^{- A^{2}}) \in Ω^{even} (B)$ is closed.

Proof. Apply the identity $d Str (X) = Str ([A, X]_{gr})$ with $X = e^{- A^{2}}$ . The graded commutator $[A, e^{- A^{2}}]_{gr}$ vanishes because $A$ commutes with $A^{2}$ (Bianchi), hence with its analytic continuation $e^{- A^{2}}$ . Therefore $d ch (A) = 0$ . The form has even total degree because $e^{- A^{2}}$ has even total parity (the operator $A^{2}$ is even) and the supertrace of an even operator on a graded space is an even-degree form. $□$

Proposition ( $t$ -independence of cohomology class). For a one-parameter family $A_{t}$ of superconnections, the cohomology class $[ch (A_{t})] \in H^{even} (B; R)$ is independent of $t$ .

Proof. By Quillen's algebraic identity, $ch (A_{t}) - ch (A_{s}) = d \int_{s}^{t} Str (\frac{d A _{u}}{d u} e^{- A_{u}^{2}}) d u$ , so the difference is exact, and the classes in de Rham cohomology agree. $□$

Proposition (large- $t$ limit equals analytic index Chern character). Assume the kernel and cokernel of $D_{b}$ have locally constant dimensions, so $Ind (D) = [ker D] - [coker D]$ is a genuine graded vector bundle on $B$ . Then $lim_{t \to + \infty} ch (A_{t}) = ch (Ind (D))$ in $\Omega^(B) $, w h er e t h er i g h t - han d s i d e u ses t h eco nn ec t i o n o n t h e in d e x b u n d l e in d u ce df r o m$ \nabla^{\mathcal{E}}$.*

Proof. As $t \to + \infty$ , the operator $t D$ dominates the Bismut superconnection, so $e^{- A_{t}^{2}} \to P_{k e r} e^{- (\nabla^{Ind})^{2}} P_{k e r}$ where $P_{k e r}$ is the fibrewise orthogonal projection onto $ker D$ on each fibre and $\nabla^{Ind}$ is the induced connection. The cross-terms with $c (T) / t$ vanish as $t \to + \infty$ . Taking the supertrace gives $Str (P_{k e r} e^{- (\nabla^{Ind})^{2}} P_{k e r}) = Str (e^{- (\nabla^{Ind})^{2}} ∣_{Ind}) = ch (Ind (D))$ as a Chern-Weil form. The rigorous version of this argument requires uniform spectral-gap estimates; see Berline-Getzler-Vergne Ch. 9 §3 ^{[Berline-Getzler-Vergne]}. $□$

Proposition (small- $t$ limit equals local index density). Assume the fibration has even-dimensional fibres and $E$ is a Clifford module. Then $$ \lim_{t \to 0^+} \mathrm{ch}(\mathbb{A}t) = \int{M/B} \widehat A(R^{M/B}) \wedge \mathrm{ch}(F^{E/!\mathcal{S}}) \quad \text{in} \quad \Omega^*(B). $$

Proof sketch (Getzler rescaling for the Bismut superconnection). Fix a point $b \in B$ and a vertical normal coordinate system around a fibre point. Rescale the Clifford generators $c (e_{i}) \to t^{- 1/2} c (e_{i})$ as in the single-operator Getzler proof 03.09.20. Under this rescaling:

The form-degree-zero piece $t D^{2}$ becomes the harmonic-oscillator operator $- \sum \partial_{i}^{2} + \sum R_{ij} x^{i} \land x^{j} /4$ on $R^{d i m M_{b}}$ , whose heat kernel is given by Mehler's formula.
The form-degree-two piece $- \frac{1}{4 t} c (T)$ , after the Clifford rescaling and the $t$ factor, contributes precisely the $A$ -curvature term $R^{M / B}$ in the form integrand.
The cross-terms involving $t {\nabla^{E}, D}$ contribute the twisting Chern character $ch (F^{E / S})$ after rescaling.

Combining via Mehler's formula and the fibre integration $π_{*}$ gives the limit. The genuinely new ingredient compared to the single-operator Getzler proof is the form-degree-two correction $c (T) / t$ : without it, the small- $t$ limit produces only the leading-order $A$ form, missing the $R^{M / B}$ corrections from horizontal curvature. The full argument occupies Bismut 1986 §§3-5 and Berline-Getzler-Vergne Ch. 10 ^{[Bismut 1986; ref: TODO_REF Berline-Getzler-Vergne]}. $□$

Proposition (cohomological family-index theorem). Combining the previous propositions, $$ \mathrm{ch}(\mathrm{Ind}(D)) = \pi_* \big( \widehat A(R^{M/B}) \wedge \mathrm{ch}(F^{E/!\mathcal{S}}) \big) \in H^{\mathrm{even}}(B; \mathbb{R}). $$

Proof. By $t$ -independence of the cohomology class and the two limits computed above. This is the cohomological Atiyah-Singer family-index theorem of 1971 IV, proved here by heat-kernel localisation rather than K-theoretic pushforward. $□$

Proposition (Bianchi identity). For any superconnection $A$ , the curvature $A^{2}$ satisfies $[A, A^{2}]_{gr} = 0$ .

Proof. The graded commutator $[A, A^{2}]_{gr} = A A^{2} - (- 1)^{1 \cdot 0} A^{2} A = A^{3} - A^{3} = 0$ . (The parity of $A$ is one and of $A^{2}$ is zero, so the graded sign is $(- 1)^{1 \cdot 0} = 1$ , i.e., this is an ordinary commutator that vanishes by associativity.) $□$

Connections [Master]

Atiyah-Singer index theorem 03.09.10. The Bismut superconnection produces the cohomological family-index theorem of Atiyah-Singer 1971 IV by heat-kernel localisation. The single-point base case reduces to the original Atiyah-Singer formula for one Dirac operator: $A_{t} = t D$ , and the heat-kernel form on a one-point base is the integer index. The family version refines this from a single integer to a Chern character class in $H^{even} (B; R)$ .
Heat-kernel proof of Atiyah-Singer 03.09.20. The Bismut superconnection is the family analogue of the single-operator heat-kernel proof. The same McKean-Singer mechanism — supertrace vanishes on graded commutators, so the heat-kernel form is closed and $t$ -independent in cohomology — drives both arguments. The Getzler rescaling is the same; the family case requires the form-degree-two correction $c (T) / t$ to account for fibration curvature.
Family, equivariant, and Lefschetz index theorems 03.09.21. The Bismut superconnection is the analytical machinery that proves the family-index theorem stated in 03.09.21. The equivariant version of the Bismut superconnection (with a $G$ -invariant connection on the fibration) handles the equivariant family-index theorem, and the transgression form leads to the equivariant Bismut-Cheeger eta-form. The Lefschetz fixed-point formula reappears in the small- $t$ analysis of the equivariant superconnection.
Equivariant K-theory $K_{G} (X)$ and $R (G)$ 03.08.10. The family Chern character lifts to a map $K^{0} (B) \to H^{even} (B; R)$ that the Bismut superconnection computes at the level of differential forms. The equivariant refinement of the family-index theorem lives in equivariant K-theory $K_{G}^{0} (B)$ , and the equivariant Bismut superconnection computes its equivariant Chern character. The connection to the representation ring $R (G) = K_{G}^{0} (pt)$ is via the product fibration case.
Atiyah-Hirzebruch spectral sequence 03.13.04. The family Chern character produced by the Bismut superconnection factors through the Atiyah-Hirzebruch spectral sequence for $K^{0} (B) \otimes Q ≅ H^{even} (B; Q)$ . The superconnection identifies the analytic family index with its cohomological image; the AHSS provides the integral refinement and the differentials measuring obstructions to the rational identification.
Chern-Weil homomorphism 03.06.06. The Bismut superconnection generalises Chern-Weil from ordinary connections to graded superconnections. Quillen's 1985 paper is the systematic extension: the Chern-Weil polynomial in the curvature of an ordinary connection becomes the supertrace of the heat kernel of a superconnection, and the same closedness / class-independence / transgression structure holds. The Chern-Simons form on the space of ordinary connections is the form-degree-one piece of the superconnection transgression.
Generalised Dirac bundle 03.09.14. The Bismut superconnection requires the fibrewise operator to be a generalised Dirac operator on a Clifford-module bundle. The Bochner-Weitzenböck identity for the family Dirac operator $D^{2} = (\nabla^{E})^{*} \nabla^{E} + R$ is the family analogue of the single-operator identity, and it underlies the heat-kernel decay required for the large- $t$ limit.
Principal bundle connection 03.05.07. The horizontal connection on the fibration $π : M \to B$ enters the Bismut superconnection through the induced connection $\nabla^{E}$ on $π_{*} E$ and through the curvature $T \in Ω^{2} (B, vert)$ that appears in the third piece of $A_{t}$ . The Bismut superconnection therefore depends on the choice of horizontal complement; different choices give different forms but the same cohomology class.
Eta invariant and the Atiyah-Patodi-Singer index theorem 03.09.24. The Bismut-Cheeger family eta-form $η \in Ω^{*} (B)$ is the transgression of the superconnection Chern character along the half-infinite cylinder $[0, \infty) \times \partial X$ in a family of manifolds with boundary; it is the family analogue of the eta invariant. The degree-zero piece of $η$ recovers the single-operator eta invariant of APS, and the family APS theorem expresses the difference between bulk and boundary contributions as the de Rham class of $ch (Ind D) - \int_{X / B} A (T X / B) ch (E / S) = - η$ in $H^{even} (B; R)$ . The same Getzler rescaling and McKean-Singer mechanism that make the closed-fibre superconnection Chern character well-defined extend to manifolds with cylindrical ends, with the eta-form arising precisely as the boundary defect — the family analogue of the spectral-asymmetry boundary correction. Bismut-Freed 1986 identified the holonomy of the determinant line bundle of a family of Dirac operators with the family eta invariant, giving the structural reason the superconnection framework subsumes APS as its boundary-term refinement.
Kirillov character formula via the equivariant index 03.09.25. The equivariant Bismut superconnection applied to the homogeneous fibration $G / T \to pt$ (with $G$ -action by left translation) produces, in the small- $t$ limit, an equivariant differential form on $g$ whose evaluation at $X \in g$ is the Kirillov character at $g = e^{X}$ . Specifically, the equivariant Chern character of the pushforward $π_{*} L_{λ}$ along $π : G / T \to pt$ — computed by Bismut superconnection at the equivariant level — equals the orbit Fourier transform $\int_{O_{λ}} e^{i ⟨ ξ, X ⟩} d μ (ξ)$ times the Duflo Jacobian $j (X)^{1/2}$ . The Kirillov formula is therefore the small-time / orbit-localised value of the equivariant Bismut superconnection, and the proof in Berline-Getzler-Vergne Ch. 8 runs the rescaling argument equivariantly. The family eta-form construction extends Kirillov to families of homogeneous spaces, refining the orbit-Fourier identity to a cohomological identity on the parameter space.

Historical & philosophical context [Master]

Quillen's 1985 paper. Daniel Quillen's Superconnections and the Chern character (Topology 24, 89-95) introduced the superconnection framework as a clean generalisation of Chern-Weil theory to graded vector bundles. Quillen's motivation was algebraic: the Chern character of a graded bundle $E = E^{+} \oplus E^{-}$ is naturally a difference $ch (E^{+}) - ch (E^{-})$ , and an ordinary connection on each piece separately gives this difference but in an unsatisfactory way — there is no single object computing the supertrace heat-kernel form directly. Quillen's superconnection is the single object: an odd-parity derivation on the bundle-valued de Rham complex whose curvature $A^{2}$ has all even form-degrees, and whose Chern character $Str (e^{- A^{2}})$ is closed for purely algebraic reasons. Quillen explicitly noted that the framework was designed for applications to families of elliptic operators but did not develop those applications himself — he left that work for Bismut ^{[Quillen 1985]}.

Bismut's 1986 paper. Jean-Michel Bismut's The Atiyah-Singer index theorem for families of Dirac operators: two heat-equation proofs (Invent. Math. 83, 91-151) was the first heat-kernel proof of the Atiyah-Singer family-index theorem, replacing the original K-theoretic argument of Atiyah-Singer 1971 IV with an analytical one in the spirit of Atiyah-Bott-Patodi 1973 for the single-operator case. Bismut introduced what is now called the Bismut superconnection $A_{t} = \nabla^{E} + t D - \frac{1}{4 t} c (T)$ — a specific superconnection (in Quillen's sense) attached to a fibration of spin manifolds. The genuinely new term was the form-degree-two correction $c (T) / t$ , which was needed to make the small- $t$ Getzler rescaling produce the full local index density rather than only its leading piece. Bismut's two proofs in the paper used different routes to the same conclusion: one a direct probabilistic argument via the Brownian-motion path-integral, the other an analytical argument via heat-kernel asymptotics. The framework has dominated family-index theory ever since ^{[Bismut 1986]}.

Mathai-Quillen, Bismut-Cheeger, Bismut-Lott. The superconnection framework rapidly proved more general than its initial application. Varghese Mathai and Quillen 1986 Superconnections, Thom classes, and equivariant differential forms (Topology 25, 85-110) used the framework to construct Gaussian Thom-class representatives and equivariant differential forms. Bismut and Jeff Cheeger 1989 Eta invariants and their adiabatic limits (J. Amer. Math. Soc. 2, 33-70) introduced the eta-form $η (A_{t})$ as the family analogue of the Atiyah-Patodi-Singer eta-invariant, proving the adiabatic-limit formula that governs family-index theorems for fibrations with boundary. Bismut and John Lott 1995 Flat vector bundles, direct images and higher real analytic torsion (J. Amer. Math. Soc. 8, 291-363) constructed the higher analytic torsion form $T (A_{t})$ for flat bundles, refining Ray-Singer torsion to families and connecting with Igusa-Klein higher topological torsion.

Modern reach. Nicole Berline, Ezra Getzler, and Michèle Vergne's 1992 monograph Heat Kernels and Dirac Operators (Grundlehren 298, Springer) presented the Bismut superconnection in its definitive textbook form, with Ch. 9 developing Quillen's framework and Ch. 10 the family-index theorem ^{[Berline-Getzler-Vergne]}. The Bismut superconnection appears in Connes-Moscovici's local cyclic cocycle in non-commutative geometry, in Stolz-Teichner's geometric String orientation, in Pflaum-Posthuma-Tang's work on transverse index theory for foliations, and in the modern formulation of differential K-theory (Hopkins-Singer, Bunke-Schick). The superconnection construction has become the canonical analytical machinery for any family-index calculation, replacing the older K-theoretic pushforward arguments in most modern treatments.

Bibliography [Master]

@article{Quillen1985,
  author  = {Quillen, Daniel},
  title   = {Superconnections and the Chern character},
  journal = {Topology},
  volume  = {24},
  number  = {1},
  year    = {1985},
  pages   = {89--95}
}

@article{Bismut1986,
  author  = {Bismut, Jean-Michel},
  title   = {The {Atiyah-Singer} index theorem for families of {Dirac} operators: two heat-equation proofs},
  journal = {Inventiones Mathematicae},
  volume  = {83},
  year    = {1986},
  pages   = {91--151}
}

@article{MathaiQuillen1986,
  author  = {Mathai, Varghese and Quillen, Daniel},
  title   = {Superconnections, {Thom} classes, and equivariant differential forms},
  journal = {Topology},
  volume  = {25},
  number  = {1},
  year    = {1986},
  pages   = {85--110}
}

@article{BismutCheeger1989,
  author  = {Bismut, Jean-Michel and Cheeger, Jeff},
  title   = {{$\eta$}-invariants and their adiabatic limits},
  journal = {Journal of the American Mathematical Society},
  volume  = {2},
  number  = {1},
  year    = {1989},
  pages   = {33--70}
}

@article{BismutLott1995,
  author  = {Bismut, Jean-Michel and Lott, John},
  title   = {Flat vector bundles, direct images and higher real analytic torsion},
  journal = {Journal of the American Mathematical Society},
  volume  = {8},
  number  = {2},
  year    = {1995},
  pages   = {291--363}
}

@book{BerlineGetzlerVergne1992,
  author    = {Berline, Nicole and Getzler, Ezra and Vergne, Mich\`ele},
  title     = {Heat Kernels and {Dirac} Operators},
  series    = {Grundlehren der mathematischen Wissenschaften},
  volume    = {298},
  publisher = {Springer-Verlag},
  year      = {1992}
}

@article{AtiyahSinger1971IV,
  author  = {Atiyah, Michael F. and Singer, Isadore M.},
  title   = {The index of elliptic operators {IV}},
  journal = {Annals of Mathematics},
  volume  = {93},
  year    = {1971},
  pages   = {119--138}
}

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

@article{AtiyahBottPatodi1973,
  author  = {Atiyah, Michael F. and Bott, Raoul and Patodi, Vijay K.},
  title   = {On the heat equation and the index theorem},
  journal = {Inventiones Mathematicae},
  volume  = {19},
  year    = {1973},
  pages   = {279--330}
}

@article{Getzler1986,
  author  = {Getzler, Ezra},
  title   = {A short proof of the local {Atiyah-Singer} index theorem},
  journal = {Topology},
  volume  = {25},
  year    = {1986},
  pages   = {111--117}
}

Prerequisites

03.09.10
03.09.20
03.09.21
03.09.14
03.05.07
03.06.06

Tier anchors

beginner: Quillen's slogan: a single odd operator that mixes a connection with an endomorphism, written as a square root of the curvature form
intermediate: Berline-Getzler-Vergne Ch. 9 §1-§3; Bismut 1986 Invent. Math. §1-§2
master: Quillen 1985 *Superconnections and the Chern character* (Topology 24); Bismut 1986 *Two heat-equation proofs* (Invent. Math. 83); Berline-Getzler-Vergne Ch. 9-§10; Mathai-Quillen 1986 *Superconnections, Thom classes, and equivariant differential forms* (Topology 25); Bismut-Cheeger 1989 *Eta invariants and their adiabatic limits* (J. AMS 2)

References

TODO_REF
Quillen — Superconnections and the Chern character · Topology 24 (1985), 89-95 — originator paper
TODO_REF
Bismut — The Atiyah-Singer index theorem for families of Dirac operators: two heat-equation proofs · Invent. Math. 83 (1986), 91-151
TODO_REF
Berline-Getzler-Vergne — Heat Kernels and Dirac Operators · Grundlehren 298, Springer 1992, Ch. 9 (Bismut superconnection) and Ch. 10 (family-index theorem)
TODO_REF
Mathai-Quillen — Superconnections, Thom classes, and equivariant differential forms · Topology 25 (1986), 85-110
TODO_REF
Bismut-Cheeger — Eta invariants and their adiabatic limits · J. Amer. Math. Soc. 2 (1989), 33-70 — eta-form
TODO_REF
Lawson-Michelsohn — Spin Geometry · Princeton 1989, §III.15 family index, §III.17 heat-kernel approach
TODO_REF
Atiyah-Singer — The index of elliptic operators IV · Annals of Mathematics 93 (1971), 119-138 — family index in K-theory

Reviewer

TBD

Estimated time

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