spin geometry

string geometry

# Contents

## Idea

### General

For $S \to X$ a spinor bundle over a Riemannian manifold $(X,g)$, a Dirac operator on $S$ is an differential operator on (sections of) $S$ whose principal symbol is that of $c \circ d$, where $d$ is the exterior derivative and $c$ is the symbol map.

More abstractly, for $D$ a Dirac operator, its normalization $D(1+ D^2)^{-1/2}$ is a Fredholm operator, hence defines an element in K-homology.

### Origin and role in Physics

The first relativistic Schrödinger type equation found was Klein-Gordon. At first it did not look that K-G equation could be interpreted physically because of negative energy states and other paradoxes. Paul Dirac proposed to take a square root of Laplace operator within the matrix-valued differential operators and obtained a Dirac equation; matrix valued generators involved representations of a Clifford algebra. It also had negative energy solutions, but with half-integer spin interpretation which was appropriate the Pauli exclusion principle together with the Dirac sea picture came at rescue (Klein-Gordon is now also useful with more modern formalisms).

(…)

## Definition

### In components

The tangent bundle of an oriented Riemannian $n$-dimensional manifold $M$ is an $SO(n)$-bundle. Orientation means that the first Stiefel-Whitney class $w_1(M)$ is zero. If $w_2(M)$ is zero than the $SO(n)$ bundle can be lifted to a $Spin(n)$-bundle. A choice of connection on such a $Spin(n)$-bundle is a $Spin$-structure on $M$. There is a standard $n/2$-dimensional representation of $Spin(n)$-group, so called Spin representation, which is depending, if $n$ is odd irreducible, and if $n$ is even it decomposes into the sum of two irreducible representations of equal dimension $S_+$ and $S_-$. Thus we can associate associated bundles to the original $Spin(n)$ bundle $P$ with respect to these representations. Thus we get the spinor bundles $E_\pm := P\times_{Spin(n)} S_\pm\to M$ and $E = E_+\oplus E_-$.

Gamma matrices, which are the representations of the Clifford algebra

$\gamma_a \gamma_b + \gamma_b \gamma_a = -2\delta_{ab} I$
$\gamma_5 = i^{n(n+1)/2}\gamma_1\cdots\gamma_n, \,\,\,\,\gamma^2_5 = I$

thus act on such a space; certain combinations of products of gamma matrices with partial derivatives define a first order Dirac operator $\Gamma(E)\to\Gamma(E_-)$; there are several versions, in mathematics is pretty important the chiral Dirac operator

$\Gamma(M,E_+)\to \Gamma(M,E_-)$

given by local formula

$\sum_a \gamma^a e^\mu_a(x) \nabla_\mu \frac{1+\gamma_5}{2}$

where $e^\mu_a(x)$ are orthonormal frames of tangent vectors and $\nabla_\mu$ is the covariant derivative with respect to the Levi-Civita spin connection. The expression $\frac{1+\gamma_5}{2}$ is the chirality operator.

In Euclidean space the Dirac operator is elliptic, but not in Minkowski space.

The Dirac operator is involved in approaches to the Atiyah-Singer index theorem about the index of an elliptic operator: namely the index can be easier calculated for Dirac operator and the deformation to the Dirac operator does not change the index. An appropriate version of a Dirac operator is a part of a concept of the spectral triple in noncommutative geometry a la Alain Connes.

## Properties

### Index and partition function

###### Proposition

Let $(X,g)$ be a compact Riemannian manifold and $\mathcal{E}$ a smooth super vector bundle and indeed a Clifford module bundle over $X$. Consider a Dirac operator

$D \colon \Gamma(X,\mathcal{E}) \to \Gamma(X, \mathcal{E})$

with components (with respect to the $\mathbb{Z}_2$-grading) to be denoted

$D = \left[ \array{ 0 & D^- \\ D^+ & 0 } \right] \,,$

where $D^- = (D^+)^\ast$. Then $D^+$ is a Fredholm operator and its index is the supertrace of the kernel of $D$, as well as of the heat kernel of $D^2$:

\begin{aligned} ind(D^+) & \coloneqq dim(ker(D^+)) - dim(coker(D^+)) \\ & = dim(ker(D^+)) - dim(ker(D^-)) \\ & = sTr(ker(D)) \\ & = sTr( \exp(-t \, D^2) ) \;\;\; \forall t \gt 0 \end{aligned} \,.

This appears as (Berline-Getzler-Vergne 04, prop. 3.48, prop. 3.50), based on (MacKean-Singer 67).

###### Remark

If one thinks of $D^2$ as the time-evolution Hamiltonian of a system of supersymmetric quantum mechanics with $D$ the supercharge on the worldline, then $ker(D)$ is the space of supersymmetric quantum states, $\exp(-t \, D^2)$ is the Euclidean time evolution operator and its supertrace is the partition function of the system. Hence we have the translation

• index = partition function .

## Examples

$d$partition function in $d$-dimensional QFTsuperchargeindex in cohomology theorygenuslogarithmic coefficients of Hirzebruch series
0push-forward in ordinary cohomology: integration of differential forms
1spinning particleDirac operatorKO-theory indexA-hat genusBernoulli numbers
endpoint of 2d Poisson-Chern-Simons theory stringSpin^c Dirac operator twisted by prequantum line bundlespace of quantum states of boundary phase space/Poisson manifoldTodd genusBernoulli numbers
endpoint of type II superstringSpin^c Dirac operator twisted by Chan-Paton gauge fieldD-brane chargeTodd genusBernoulli numbers
2superstringDirac-Ramond operatorsuperstring partition functionelliptic genus/Witten genusEisenstein series
self-dual stringM5-brane charge

## References

Textbooks include

The relation to index theory is discussed in

based on original articles such as

• H. MacKean, Isadore Singer, Curvature and eigenvalues of the Laplacian, J. Diff. Geom. 1 (1967)
• Michael Atiyah, Raoul Bott, V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279–330.