Tractor bundle explained

In conformal geometry, the tractor bundle is a particular vector bundle constructed on a conformal manifold whose fibres form an effective representation of the conformal group (see associated bundle).

The term tractor is a portmanteau of "Tracy Thomas" and "twistor", the bundle having been introduced first by T. Y. Thomas as an alternative formulation of the Cartan conformal connection,^[1] and later rediscovered within the formalism of local twistors and generalized to projective connections by Michael Eastwood et al. in Tractor bundles can be defined for arbitrary parabolic geometries.^[2]

Conformal manifolds

The tractor bundle for a

n -dimensional conformal manifold M of signature (p,q) is a rank n +2 vector bundle l T\to M equipped with the following data:

• a metric

G: l T ⊗ l T\to R , of signature (p +1 ,q +1 ) ,

• a line subbundle

l X\subset l T ,\nabla , preserving the metric G , and satisfying the nondegeneracy property that, for any local non-vanishing section X of the bundle l X ,$$v\backslash mapsto\; \backslash nabla\_vX\backslash pmod$$is a linear isomorphism at each point from the tangent bundle of M (v\in TM ) to the quotient bundle l X ^{\perp/lX} , where l X ^{\perp} denotes the orthogonal complement of l X in l T relative to the metric G .

Given a tractor bundle, the metrics in the conformal class are given by fixing a local section

X of l X , and defining for v,w\in TM ,$$g\_X(v,w)\; =\; G(\backslash nabla\_vX,\backslash nabla\_wX).$$

To go the other way, and construct a tractor bundle from a conformal structure, requires more work. The tractor bundle is then an associated bundle of the Cartan geometry determined by the conformal structure. The conformal group for a manifold of signature

(p,q) is SO(p +1 ,q +1 ) , and one obtains the tractor bundle (with connection) as the connection induced by the Cartan conformal connection on the bundle associated to the standard representation of the conformal group. Because the fibre of the Cartan conformal bundle is the stabilizer of a null ray, this singles out the line bundle l X .

More explicitly, suppose that

g is a metric on M , with Levi-Civita connection \nabla . The tractor bundle is the space of 2-jets of solutions \sigma to the eigenvalue equation$$(\backslash nabla\_i\backslash nabla\_j\; +\; P\_)\backslash sigma\; =\; \backslash lambda\; g\_$$where P _{ij} is the Schouten tensor. A little work then shows that the sections of the tractor bundle (in a fixed Weyl gauge) can be represented by (n +2 ) -vectors$$U^I=\backslash begin\backslash sigma\backslash \backslash \; \backslash mu^i\backslash \backslash \; \backslash rho\backslash end.$$The connection is $$\backslash nabla\_jU^I=\backslash nabla\_j\backslash begin\backslash sigma\backslash \backslash \; \backslash mu^i\backslash \backslash \; \backslash rho\backslash end=\backslash begin\backslash nabla\_j\backslash sigma-\backslash mu\_j\backslash \backslash \; \backslash nabla\_j\backslash mu^i\; +\; \backslash delta\_j^i\backslash rho\; +\; P\_j^i\backslash sigma\backslash \backslash \; \backslash nabla\_j\backslash rho\; -\; P\_\backslash mu^i\backslash end.$$The metric, on U ^{I=(\sigma \mu} ^{i \rho)} and V ^{J=(\tau \nu} ^{j \alpha)} is:$$G\_U^IV^J\; =\; \backslash mu^i\backslash nu\_i\; +\; \backslash sigma\backslash tau\; +\; \backslash rho\backslash alpha$$The preferred line bundle l X is the span of$$X^I\; =\; \backslash begin0\backslash \backslash 0\backslash \backslash 1\backslash end.$$

Given a change in Weyl gauge

\widehat g _{ij} = e ^{2\gamma} g _{ij} , the components of the tractor bundle change according to the rule$$\backslash begin\backslash widehat\backslash sigma\backslash \backslash \backslash widehat\; \backslash mu^i\backslash \backslash \backslash widehat\backslash rho\backslash end\; =\; \backslash begin\backslash sigma\backslash \backslash \; \backslash mu^i+\backslash gamma^i\backslash sigma\backslash \backslash \; \backslash rho-\backslash gamma\_j\backslash mu^j\; -\; \backslash gamma^2\backslash sigma/2\backslash end$$where \gamma _{i=\nabla} _{i\gamma} , and the inverse metric g ^{ij} has been used in one place to raise the index. Clearly the bundle l X is invariant under the change in gauge, and the connection can be shown to be invariant using the conformal change in the Levi-Civita connection and Schouten tensor.

Projective manifolds

Let

M be a projective manifold of dimension n . Then the tractor bundle is a rank n +1 vector bundle l T , with connection \nabla , on M equipped with the additional data of a line subbundle l X such that, for any non-vanishing local section X of l X , the linear operator$$v\backslash mapsto\; \backslash nabla\_v\; X\backslash pmod$$is a linear isomorphism of the tangent space to l T/ l X .

One recovers an affine connection in the projective class from a section

X of l X by defining$$\backslash nabla\_X\; =\; \backslash nabla\_v\backslash nabla\_wX\; \backslash pmod$$and using the aforementioned isomorphism.

Explicitly, the tractor bundle can be represented in a given affine chart by pairs

(\mu ^{i \rho)} , where the connection is$$\backslash nabla\_j\backslash begin\backslash mu^i\backslash \backslash \; \backslash rho\backslash end\; =\; \backslash begin\backslash nabla\_j\backslash mu^i\; +\; \backslash delta\_j^i\backslash rho\backslash \backslash \; \backslash nabla\_j\backslash rho\; -\; P\_\backslash mu^i\backslash end$$where P _{ij} is the projective Schouten tensor. The preferred subbundle l X is that spanned by X = ( 0 1 ) .

Here the projective Schouten tensor of an affine connection is defined as follows. Define the Riemann tensor in the usual way (indices are abstract)$$(\backslash nabla\_i\backslash nabla\_j-\backslash nabla\_j\backslash nabla\_i)U^\backslash ell\; =\; ^\backslash ell\; U^k.$$Then$$^\backslash ell\; =\; ^\backslash ell\; +\; 2\backslash delta^\backslash ell\_\; +\; \backslash beta\_\backslash delta\_k^\backslash ell$$where the Weyl tensor

{C _{ijk} }^\ell is trace-free, and 2 P _{[ij]} = - \beta _{ij} (by Bianchi).

References

1. Thomas, T. Y., "On conformal differential geometry", Proc. N.A.S. 12 (1926), 352–359; "Conformal tensors", Proc. N.A.S. 18 (1931), 103–189.
2. Čap, A., & Gover, A. (2002). Tractor calculi for parabolic geometries. Transactions of the American Mathematical Society, 354(4), 1511-1548.