SUSY QM as point-particle limit of Superstrings

Question

There is a pretty clear resemblance between the Lagrangians for SUSY QM (1-dim susy sigma model) and various superstring theories (2-dim susy sigma models).

Again intuitively, one should expect the point particle theory to emerge as a zero-length/infinite string tension limit of the string theory.

How precisely is SUSY QM a limit of supersymmetric string theory?

From the nLab, we have

This deformed supersymmetric quantum mechanics arises as the point-particle limit of the type II superstring regarded as quantum mechanics on the smooth loop space (the string’s Wheeler superspace), a relation that is stated more explicitly in

Edward Witten, p. 92-94 in: Global anomalies in string theory, in: W. Bardeen and A. White (eds.) Symposium on Anomalies, Geometry, Topology, World Scientific (1985) 61-99 [pdf, spire:214913] and then in

Edward Witten, The Index Of The Dirac Operator In Loop Space, in: Elliptic Curves and Modular Forms in Algebraic Topology, Lecture Notes in Mathematics 1326, Springer (1988) 161-181 [doi:10.1007/BFb0078045, spire]

What is the precise statement and where does it occur in these two cited works?

Answer 1

"How precisely is SUSY QM a limit of supersymmetric string theory?"

The large volume limit of the partition function of the superstring is an index of the charge-operator in the super vertex operator algebra called the Dirac-Ramond operator on the target $M$. Basicly a more general version of the A-hat-genus. This power series is an elliptic genus called the Witten genus and equivalent to the Witten index.

In supersymetric quantum mechanics for sigma-models with target a spin-manifold, the Witten index is, by the Atiyah-Singer index theorem, equivalent to the index of the Dirac operator.

This fueled the idea that the large volume limit of the superstring can be described by some equivariant version of the A-hat-genus. More precisely, an $S^1$-equivariant index of the Dirac operator. It is known that such indices are closely related to the characters of certain vertex operator algebras. The correctness of this assumption is an open problem as far as I am aware. While the mathematical foundation of string theory is subject to ongoing research, the Witten genus is more generally reproduced by topological modular forms. Therefore, a pricise statement is to my knowledge currently unavailable until we know the full picture.

As a site note, again to my knowledge the understanding of the relation between the Dirac operator in the Standard Model from the point of view of Connes' "spectral action" and "non-commutative geometry" and a higher version of such construction involving the Dirac-Ramon operator suffers basicly from the same lack of understanding about non-perturbative superstring theory.