1. Lorentz transformations for non-abelian coordinates
Paul Koerber, UBC
Vancouver, January 29, 2005

2. 0. Introduction Coinciding Dp-branes
... 1 2 N 1 2 3 4 N
...
Transversal coordinates become matrix-valued!

3. Contents General coordinate transformations for matrices
hep-th/ , Brecher, Furuuchi, Ling, Van Raamsdonk
Transformation law
Covariant vector field
Invariant actions
Mixing transverse & longitudinal coordinates: Lorentz-boosts
Work in progress, Brecher, PK, Ling, Van Raamsdonk
Static gauge
Search for covariant description
Conclusions/Further research

4. 1. General coordinate invariance for matrices
: matrix-valued D0-brane coordinates
: bulk
Bulk:
, de Boer, Schalm
Impossible!
such that
composition law
naive:

5. Properties: Composition law Basepoint independence
Diagonal matrices abelian limit
Linear transformations: simple
Linear in F

6. Covariant vector field
First step: building block
Abelian case
Non-abelian generalization
construct order by order

7. Existence canonical choice, but not unique Does not always work:
check basepoint independence

8. Invariant actions Second step Integral representation Time= spectator
Integrate
whole space
Scalar

9. Properties:
Every generally covariant action can be written in this form
such that can be covariantized
Many possible invariant actions!

10. 2. Mixing transverse & longitudinal coordinates
Until now: general coordinate invariance for transverse coordinates
Mixing transverse & longitudinal:
harder
Step back: Lorentz boost for D0-branes
Covariant description abelian case
Static gauge
Search for covariant description

11. Covariant description abelian case
10 coordinates treated in the same way
Action:
t weight: -1
Target-space general coordinate invariance
t reparametrization invariance (static gauge )
Derivative corrections possible

12. Non-abelian case Promote to matrices Nature of ? Nature of ?
Static gauge:
must be matrix too
Nature of ?
Not just a matrix function!
Already fails for diagonal matrices:
we want to be independent
Nature of ?
Severe conceptual problems!

13. Static gauge approach: abelian case
Boost:
outside static gauge
Compensating diffeomorphism:

14. Remark
is not invariant
add corrections terms to build:

15. Static gauge approach: non-abelian case
where
Preserve the Lorentz-algebra:
Nested symmetrizations

16. Nested Symmetrizations

17. We find:
We must add a correction:

18. We find (up to 2 commutators):
where

19. Properties transformation law
Unique up to field redefinitions
#d/dt=#X-1
More speculative:
different invariants in the action

20. Covariant objects: currents
Abelian case
D0-brane charge and
current
Static gauge:

21. Calculate moments
Transformation law
becomes:

22. Non-abelian currents Start with: Problem Lorentz covariance:
nested symmetrizations

23. Add correction terms: Solve for
Current conservation=gauge invariance WZ
Do not need to use the trace
Field redefinitions can be absorbed in
Solution is not unique

24. Charge density of dielectric branes
Hashimoto
N D0-branes
Add family of terms

25. Invariant action Avoid partial integration identities
Construct action as a density
with
start with:

26. Result
In particular:

27. Delta function We did not use the trace:
can be used a our `delta function’
We need an object V to produce terms such as:

28. Covariant vector field
Abelian case: complex expression
Non-abelian case: possible
(must satisfy transformation law)

29. Conclusions/Further research
Consistent transformation law: possible!
Invariant actions: possible!
Building blocks: V &
Understand ambiguity
Understand which lowest order terms
All possible invariant actions?
T-duality