[Paper Review] The Index Theorem for Homogeneous Differential Operators on Supermanifolds

Dimitry Leites|ArXiv.org|Feb 18, 2002

Advanced Operator Algebra Research8 references3 citations

TL;DR

This paper extends Bott's index theorem for homogeneous elliptic operators on homogeneous spaces to supermanifolds, showing that the refined index of a $G$-invariant differential operator on a supermanifold is determined by the Weyl character formula in the representation ring of the supergroup $G = U(p|q)$. The key result establishes that the index $χ(D) = [\text{Ker} D] - [\text{Coker} D]$ equals the formal induction of the symbol class in $R(H)$, generalizing Bott’s result to supergroups and revealing that atypical representations require the Bernstein–Leites character formula, not the standard Weyl formula.

ABSTRACT

In mid 60s Bott proved that (1) the index theorem for homogeneous, G-invariant, elliptic differential operators acting in the spaces of sections of induced representations of G over G/H reduces to the Weyl character formula and (2) the index of an equivariant elliptic operator does not depend on the operator, but on the representations. Here the same theorem is formulated for the unitary supergroup G=U(p|q). For atypical representations the character formula does not reduce to that for the Lie group underlying the supergroup G and this contradicts a statement of Rempel and Schmitt on index on supermanifolds (Pseudodifferential operators and the index theorem on supermanifolds. Seminar Analysis, 1981/82, 92--131, Akad. Wiss. DDR, Berlin, 1982; id., Pseudodifferential operators and the index theorem on supermanifolds. Math. Nachr. 111 (1983), 153--175).

Motivation & Objective

• To generalize Bott’s index theorem for homogeneous elliptic operators on compact Lie groups to the setting of supermanifolds and supergroups.
• To establish that the refined index of a $G$-invariant differential operator on a supermanifold is determined by the representation-theoretic data of the underlying supergroup $G = U(p|q)$.
• To resolve a contradiction with Rempel and Schmitt’s claim that the index on supermanifolds reduces to that on the underlying manifold by showing it does not for atypical representations.
• To demonstrate that the index theorem on supermanifolds reduces to the super Weyl character formula, not the classical one, especially for atypical representations.
• To provide a foundational framework for extending index theory and Riemann–Roch–Grothendieck theorems to supermanifolds via algebraic supergroup representation theory.

Proposed method

• Adapts Bott’s original proof strategy for homogeneous elliptic operators on $G/H$ to the supergroup setting, focusing on $G = U(p|q)$.
• Replaces the representation ring $R(G)$ of a Lie group with the superrepresentation ring $R(G)$, valued in ${\mathbb{Z}}[\varepsilon]/(\varepsilon^2 - 1)$, where $\varepsilon$ acts as parity reversal via $\varepsilon[M] = [\Pi(M)]$.
• Defines the refined index $\chi(D) = [\text{Ker} D] - [\text{Coker} D]$ in the completed representation ring $\widehat{R}(G)$, with superdimension replacing ordinary dimension.
• Introduces the formal induction map $i_*: R(H) \to \widehat{R}(G)$ using the $G$-invariant pairing $<\cdot, \cdot>_G$ on irreducible $G$-modules.
• Expresses the symbol of a homogeneous operator $D$ as $s(D) = [K] - [L] \in R(H)$, where $K$ and $L$ are the $H$-modules inducing the bundles $E$ and $F$.
• Proves that $\chi(D) = i_*(s(D))$, showing the index is independent of the operator and determined solely by the representation data, generalizing Bott’s result to supergroups.

Experimental results

Research questions

• RQ1 Does Bott’s index theorem for homogeneous elliptic operators on $G/H$ extend to supermanifolds when $G$ is a supergroup?
• RQ2 How does the index of a $G$-invariant differential operator on a supermanifold depend on the representation theory of the supergroup $G = U(p|q)$?
• RQ3 Why does the classical Weyl character formula fail to describe the index for atypical representations in the super setting?
• RQ4 Can the index theorem on supermanifolds be reduced to a character formula, and if so, which one?
• RQ5 Is there a purely algebraic proof of the index theorem on supermanifolds that avoids analytic or geometric assumptions?

Key findings

• The refined index $\chi(D)$ of a $G$-invariant elliptic operator on a supermanifold is equal to the formal induction $i_*(s(D))$ of its symbol class in $R(H)$, generalizing Bott’s result to supergroups.
• For typical representations, the super Weyl character formula reduces to the classical Weyl formula, but for atypical representations, it requires the Bernstein–Leites character formula.
• The index does not depend on the specific operator $D$, but only on the $H$-modules $K$ and $L$ inducing the bundles $E$ and $F$, as guaranteed by the exactness of the sequence $0 \to \text{Ker} D \to E \to F \to \text{Coker} D \to 0$.
• The statement contradicts Rempel and Schmitt’s claim that the index on supermanifolds reduces to that on the underlying manifold, because atypical representations introduce new character-theoretic data not captured by the underlying Lie group.
• The proof relies on the compactness of $G = U(p|q)$, but the author conjectures that the result holds more generally via an algebraic proof independent of analytic methods.
• The ring $R(G)$ of virtual $G$-modules is extended to a ${\mathbb{Z}}[\varepsilon]/(\varepsilon^2 - 1)$-module, with $\varepsilon$ encoding superdimension and parity reversal, enabling the superindex to be well-defined.