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

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

Contents General coordinate transformations for matrices hep-th/0403289, 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

1. General coordinate invariance for matrices : matrix-valued D0-brane coordinates : bulk Bulk: 0108161, de Boer, Schalm Impossible! such that composition law naive:

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

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

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

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

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

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

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

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!

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

Remark is not invariant add corrections terms to build:

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

Nested Symmetrizations

We find: We must add a correction:

We find (up to 2 commutators): where

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

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

Calculate moments Transformation law becomes:

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

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

Charge density of dielectric branes Hashimoto 0401043 N D0-branes Add family of terms

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

Result In particular:

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:

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

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