Supersymmetric Quantum Mechanics and Topology Supersymmetric quantum mechanical models are computed by the path integral approach. In the limit, the integrals localize to the zero modes. This allows us to perform the index computations exactly because of supersymmetric localization, and we will show how the geometry of target space enters the physics of sigma models resulting in the relationship between the supersymmetric model and the geometry of the target space in the form of topological invariants. Explicit computation details are given for the Euler characteristics of the target manifold and the index of Dirac operator for the model on a spin manifold.
1. Introduction
Supersymmetry is a quantum mechanical space-time symmetry which induces transformations between bosons and fermions. The generators of this symmetry are spinors which are anticommuting (fermionic) variables rather than the ordinary commuting (bosonic) variables; hence their algebra involves anticommutators instead of commutators. A unified framework consisting of bosons and fermions thus became possible, both combined in the same supersymmetric multiplet [1]. It is overwhelmingly accepted that supersymmetry is an essential feature of any unified theory as it not only provides a unified ground for bosons and fermions but is also helpful in reducing ultraviolet divergences. It was discovered by Gel’fand and Likhtman [2], Ramond [3], Neveu and Schwarz [4], and later by a few physicists [1, 5]. Whether Supersymmetry (SUSY) is actually realized in nature or not is still not clear; however, it has provided powerful mathematical tools and enormous amount of insights have been obtained [6]. For example, SUSY could be used to unify the space-time and internal symmetries of the S-matrix avoiding the no-go theorem of Coleman and Mandula [7], imposing local gauge invariance to SUSY which gives rise to supergravity [8, 9]. In such theories, locally gauged SUSY gives rise to Einstein’s general theory of relativity, which highlights that the local SUSY theories give a natural framework for the unification of gravity and other fundamental forces.
Supersymmetric quantum mechanics was originally developed by Witten [10], as a toy model to test the breaking of supersymmetry. In answering the same question, SUSY was also studied in the simplest case of SUSY QM by Cooper and Freedman [11]. In a later paper, the so-called “Witten Index” was proposed by Witten [12], which is a topological invariant and it essentially provides a tool to study the SUSY breaking nonperturbatively. A year later, Bender et al. [13] proposed a new critical index to study SUSY breaking in a lattice regulated system nonperturbatively. In its early days, SUSY QM was studied as a test to check the SUSY breaking nonperturbatively.
Later, when people started to explore further aspects of SUSY QM, it was realized that this was a field of research worthy of further exploration in its own right. The introduction of the topological index by Witten [12] attracted a lot of attention from the physics community and people started to study different topological aspects of SUSY QM.
Witten Index was extensively explored and it was shown that the index exhibited anomalies in certain theories with discrete and continuous spectra [14–18]. Using SUSY QM, proofs of Atiyah-Singer Index theorem were given [19–21]. A link between SUSY QM and stochastic differential equations was investigated in [22], which was used to prove algorithms about stochastic quantization; Salomonson and van Holten were the first to give a path integral formulation of SUSY QM [23]. The ideas from SUSY QM were extended to study higher dimensional systems and systems with many particles to implement such ideas to problems in different branches of physics, for example, condensed matter physics, atomic physics, and statistical physics [24–29]. Another interesting application is [30], in which the low energy dynamics of -monopoles in supersymmetric Yang-Mills theory are determined by supersymmetric quantum mechanics based on the moduli space of static monopole solutions.
There are also situations where SUSY QM arises naturally, for example, in the semiclassical quantization of instanton solitons in field theory. In the classical limit, the dynamics can often be described in terms of motion on the moduli space of the instanton solitons. Semiclassical effects are then described by quantum mechanics on the moduli space. In a supersymmetric theory, soliton solutions generally preserve half the supersymmetries of the parent theory and these are inherited by the quantum mechanical system. Complying with this, Hollowood and Kingaby in [31] show that a simple modification of SUSY QM involving the mass term for half the fermions naturally leads to a derivation of the integral formula for the genus, which is a quantity that interpolated between the Euler characteristic and arithmetic genus.
The research work in the direction of using supersymmetry to exploit topology occurred in phases: first one started in early 80s with the work of Witten [10, 32], Álvarez-Gaumé [33], and Friedan and Windey [34] and the later phase starting from late 80s and early nineties is still going on. A couple of major breakthroughs in the second phase were due to Witten: in [35], Jone’s polynomials for knot invariants which were understood quantum field theoretically, and, in [36], Donaldson’s invariants for four manifolds. Supersymmetric localization is a powerful technique to achieve exact results in quantum field theories. A recent development using supersymmetric localization technique is the exact computation of the entropy of black holes by a topologically twisted index of ABJM theory [37]. SUSY QM also has important applications in mathematical physics, as in providing simple proof of index theorems which establishes connection between topological properties of differentiable manifolds to local properties.
This review gives a basic introduction to supersymmetric quantum mechanics and later it establishes SUSY QM’s relevance to the index theorem. We will consider a couple of problems in dimensions, that is, supersymmetric quantum mechanics, by using supersymmetric path integrals, to illustrate the relationship between physics of the supersymmetric model and geometry of the background space which is some manifold in the form of Euler characteristic of this manifold . Furthermore, for a manifold admitting spin structure, we study a more refined model which yields the index of Dirac operator. Both the Euler characteristic of a manifold and the index of Dirac operator are the Witten indices of the appropriate supersymmetric quantum mechanical systems. Put differently, we will reveal the connection between supersymmetry and index theorem by path integrals.
The organization of this paper is as follows: Section 2 is an introduction to the calculus of Grassmann variables and their properties. Section 3 is an introduction to the Gaussian integrals, for both commuting (bosonic) and anticommuting (fermionic) variables including some basic examples. Section 4 involves the study of supersymmetric sigma models on both flat and curved space. Section 5 is the summary and conclusion.
