Differential Geometry through Supersymmetric Glasses

The following section is included:

• Contents

In 1982, Edward Witten published his seminal paper where he showed that some classical problems of differential geometry and differential topology such as the de Rham complex and Morse theory can be described in a very simple and transparent way using the language of supersymmetric quantum mechanics. However, according to what I could judge from my conversations with mathematicians, the language of supersymmetry has not become yet a common working tool in this branch of mathematics. The experts working in this field prefer to use more traditional methods…

The following section is included:

• Riemannian Geometry
• Differential Forms
• Principal Fiber Bundles and Gauge Fields
• Vielbeins and Tangent Space

The following sections are included:

• Complex Description
  • Complex differential forms
• Real Description
• Examples

The following sections are included:

• Hyper-Kähler Manifolds
  • Examples
• HKT Manifolds
  • Examples

The main idea of this book is that, to study the geometric properties of manifolds, it is very useful to consider the associated dynamics. We will be interested in particular with the dynamics of particles moving along the geodesic trajectories in the manifold. These trajectories realize the shortest way between two points. A geodesic satisfies the equation

The following sections are included:

• Basic Definitions
• Supersymmetric Oscillator
• Electrons in a Magnetic Field
• Witten’s Model
• Extended Supersymmetry
• Classical Supersymmetry

This chapter will provide the reader with the necessary prerequisits to understand the second half of Chap. 8 and also Chap. 14, where we will prove the famous Atiyah-Singer index theorem by supersymmetric methods. It was originally proven by Atiyah and Singer using traditional mathematical tools. But the physical way of deriving this result by analyzing the spectrum of certain SQM systems appears to be both more simple and more beautiful

We have to emphasize that the quantum systems to be considered in this chapter have all a discrete spectrum. Otherwise, the notion of index is ill-defined.…

Supersymmetric Lagrangians (5.46), (5.48), (5.52) constructed in Chap. 5 are invariant up to a total time derivative under the supersymmetry transformations (5.47), (5.49), and (5.53). These are the transformations mixing the ordinary and Grassmann variables, which can be compared to rotations mixing different vector components. In the latter case, we know that the description of a vector as a set of its components is possible, but it is much more convenient to use the vector notation and to write instead of many different scalar equations, a single vector equation, like $\mathbf{B}=\nabla \times \mathbf{A}$ instead of $B_x={\partial }_y{A}_z-{\partial }_z{A}_y$, etc…

The following sections are included:

• Basic Structures
• Euler Characteristic
• Deformations □ Morse Theory □ Quasitorsions

As we know from Chap. 2, there are two ways to describe complex manifolds. One way is to use complex coordinates. Another way is to use real coordinates, but take into account the presence of the complex structure tensor satisfying the integrability conditions (Theorem 2.6). Similarly, there are two possible supersymmetric descriptions.…

In Chaps. 2,3, we defined some special classes of complex manifolds: the Kähler, hyper-Kähler and HKT manifolds. And now we are going to describe these and some other related geometries in the supersymmetric language.

The corresponding sigma models enjoy extended $\mathcal{N}=4$ and $\mathcal{N}=8$ supersymmetries. We consider the $\mathcal{N}=4$ models first.…

A reader who has just finished reading the previous chapter could be somehow confused. A large hip of different models were described, and it might not be immediately clear whether there is a relationship between them, whether this hip has a structure. In this chapter we answer positively to this rhetoric question.…

In Chap. 10 we discussed, among other extended supersymmetric models, the hyper-Kähler and HKT models, but the efficiency and generality of our discussion was limited.

• A hyper-Kähler model was defined as an $\mathcal{N}=2$ model with three extra complex supersymmetries (10.100). We proved that the metric is hyper-Kähler iff the complex structures are quaternionic, but the extra supersymmetries had an external nature; they did not appear as superspace symmetries. As a result, we did not have tools that could allow us to construct the models that do enjoy these extra symmetries.
• We showed that the models with the superspace action (10.67) are HKT, and their metrics are given by the double derivatives of the prepotential, as in (10.70). But all these metrics belonged to a rather special class of HKT metrics, being Obata-flat. The complex structures (I, J, K)_M^N are constant matrices in this case

What physicists call gauge fields, mathematicians call principal G-connections. We gave the mathematical definition of the latter in Chap. 1.3. We discussed systems including gauge fields in several places in this book—in particular, we did so in Chaps. 5,6,9,12. We mentioned there an essential feature of such systems: gauge fields are not arbitrary; certain integrals that involve field densities, like the magnetic flux (6.65), may acquire only a particular set of discrete (quantized) values. However, we did not discuss this remarkable feature (quantization of topological charge) in detail. The time has come to do so.

Gauge fields may live both on non-compact and compact manifolds. For example, the Landau problem discussed in Chap. 5 was defined on ℝ² (and even on ℝ³ if one also takes into account free motion along the direction of the magnetic field). However, in this and the next chapter we will mostly limit our discussion to compact manifolds, where both physics and mathematics are simpler and more interesting in our opinion…

The following sections are included:

• Generalities
• Hirzebruch Signature
• Dirac Index
  • Kähler manifolds
  • Non-Kähler manifolds
• Functional Integral Calculation
  • Non-Abelian Dirac index

The following sections are included:

• Bibliography
• Index