Lorentz transformations for nonabelian coordinates
Paul Koerber, UBC
Vancouver, January 29, 2005
0. Introduction
Coinciding Dp-branes
...
1
2
...
N
U

£ U

£ ¢¢¢£ U


1234 N
U
N
Transversal coordinates become matrix-valued!
Contents
1. General coordinate transformations for
matrices
hep-th/0403289, Brecher, Furuuchi, Ling, Van Raamsdonk



Transformation law
Covariant vector field
Invariant actions
2. Mixing transverse & longitudinal
coordinates: Lorentz-boosts
Work in progress, Brecher, PK, Ling, Van Raamsdonk


Static gauge
Search for covariant description
3. Conclusions/Further research
1. General coordinate invariance for matrices
X i : matrix-valued D0-brane coordinates
i
y : bulk
Bulk:
i
i
y  F 
y
i
i

X 
F; X 
such that

F; 
H; X  
F ±H;X 
composition law
naive:


F; X 
@i : : : @i n F i 
X i : : : X i n
n n
X
Impossible! 0108161, de Boer, Schalm
i
i

X 
F
y; X ; g
y
Properties:
Composition law

F; 
H; X ; g; 
gH  
F ± H ; X ; g
Basepoint independence
@ j
 
F
y; X ¡ y; g
y 
i
@y
Diagonal matrices
abelian limit
Linear transformations: simple
i 
M k l y l ak ; X ; g M i j X j aj
Linear in F
i 
F H ; X ; g i 
F; X ; gi 
H ; X ; g
Covariant vector field
First step: building block
Abelian case
x


vy 
j
j
i
vF 


@
F
y
v
i
y
y
Non-abelian generalization
j
j
i

VF 

@
F
y
V
i
y
y
construct order by order
Existence

Vy
Vyi 
X ; g i 
vy 
x; g; X ; g
canonical choice, but not unique
X
Vy

Does not always work:
check basepoint independence
Invariant actions
Second step
Z
S
Z
dt
q
ddy g
y
L
Vy ; g; R; r R; : : :±
Vy 
Scalar
Time=
spectator
Integrate
whole space
Integral representation
Z
ddk i ki Vyi
±
Vy  q
e
g
y ¼

Properties:
Every generally covariant action can be
written in this form
L such that @² i L 
X ² 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
1. Covariant description abelian case
2. Static gauge
3. Search for covariant description
Covariant description abelian case
10 coordinates treated in the same way
¹
i
x 
¿ 
t
¿; x 
¿
Action:
Z
S¡
s
d¿
dx ¹ dx ¹
d¿ d¿
Z
¡
v
u µ
u dt ¶ 
t
d¿
d¿
Ã
¡
dx i
! 
d¿
t weight: -1
•Target-space general coordinate invariance
• t reparametrization invariance (static gauge
Derivative corrections possible
¿0  t
)
Non-abelian case
T
¿; X i 
¿ to matrices
Promote 
Static gauge: ¿0  T
¿ must be matrix too
Nature of 
T
¿; X i 
¿ ?
Not just a matrix function!
Already fails for diagonal matrices:
Tj 
¿j ; X ji 
¿j  to be independent
we want 
Ã
Nature of
dT dX i
;
d¿ d¿
!
?
Severe conceptual problems!
Static gauge approach:
abelian case
x¹ 
¿ 
t
¿; x i 
¿ 
t ; xi
t 
¿!
Boost:
¿0  t
x 0i 
t  x i 
t ¯ i t
t 0
t  t ¯ i x i 
t
t 06
t
outside static gauge
Compensating diffeomorphism:
ixl
±x i  ¯ i t ¡ ¯ l x
0
i
0
t  t ¡ ¯i x 
t
Remark

x is not invariant

add corrections terms to build:
q
¡


¡ x
Static gauge approach:
non-abelian case
³
iX l
±X i  ¯ i ¡ ¯ l 
 X
´

where 

A : : : A n 
AP 

: : : A P 
n
P n
X
Preserve the Lorentz-algebra:
i ¯ ¡ ¯i
j

±¯; ±¯
X i  ̄

X

j
̄j
Nested symmetrizations
Nested Symmetrizations


A : : : A n 

B : : : B r  

A : : : A n B : : : B r 
³
Xr
 Xn


 
A i ; B k

A j ; B l
i 6 j  
k6
 l 


A : : : A
i : : : A
j : : : A n B : : : B
k : ::B
l :: : Br
¡
Xr
Xn



Bk ; 
A i ; Bl 


 i  k6 l 


A : : : A
i : : : A n B : : : B
k :::B
l : :: Br
:
 O

;
´
´
We find:
i
i
j

±¯
; ±¯
X i  ̄
¯
¡
¯

X
j
̄j

i kl
 ¯ k ¯ 
C
l


j
i
Cij k  ¡ 
Xi ; 
X j ; X k

X
;
X k; X


We must add a correction:
³
´
³
´

i X l  ¯ 
ij l

±X i  ¯ i ¡ ¯ l 
 X

X
C
j
l

We find (up to 2 commutators):
±X i 
µ
h
j t
 j t 
:::
l
j
t
 X


¯ i t ¯ 
 ¡ X
X

E
X
C
¡
E
X
E
X j ; 
X j ; X l 

Xi 
t ij l
t
t
i l

´

Xj 
X j ; 
X
 Xi 
X
;
X j ; X l 

i; X l
j
 j t  j t   j t    
E
X t E
X t E
X t X i X j 
X j ; 
X j ; X l 


³
´
 j t 
j
t












¡ E
X t E
X t 
X i ; X j 

X j ; X l 
X i ; X j 

X j ; X l 
X i; X l

X j ; X j 

µ
 j t 
j t X E j t X ::: 
 E
X
E

X j ; X
X


X j ; X
X i X l ; X j 
t
t
t
j ¡ Xi 
j 
l; X j 


´





X j 
X i ; X j 

X j ; X l ¡ X l 
X i ; X j 

X j ; X j 
i
 j t 
j t   j t   j t    



¡ E
X t E
X t E
X t E
X t X i X j 
X j ; X j 

X j ; X l  O

;
:

¡
where
j

E j t
±t u ¡ X
X

±
u
u
t
Properties transformation law
Unique up to field redefinitions
#d/dt=#X-1
different invariants in the action
More speculative:
³
´
i X l ¯ l 
±X i  ¯ i t ¡ ¯ l 
 X
±l X i
X i
t ¡ ¯ l X l ¡ X i 
t
Covariant objects: currents
Abelian case
Z
Z
C¹ dx ¹ 
dd
y C¹ 
yJ ¹ 
y
½
D0-brane charge and
current
J¹
Z
J¹ 
Ji
¹
dy
d¿±d

y¡ X
¿
d¿
Static gauge:
½ ±d
~¡ ~
y
x
t 
i
d
i
J ± 
y¡ ~
~
x
t y
Calculate moments
i
:::i n 
Á
Z
ddyÁ
y; ty i : : : y i n
Transformation law
~ ¢r ½
~ ¢~
±½
y ; t   ¯~ ¢J 
~
y ; t¡ t ¯
~
y ; t ¡ ¯
~
y @t ½
~
y; t 
~ 
~ ¢r J~
~ ¢~
±J~
y ; t   ¯½
~
y ; t ¡ t ¯
~
y ; t ¡ ¯
~
y @t J~
y; t 
~
becomes:
i
¢¢¢i n   ¯ ¢J 
i
¢¢¢i n nt ¯ 
i
j i
¢¢¢in 
±½
½i ¢¢¢i n ¡ ¯ j @t ½
i
¢¢¢i n 
i
¢¢¢i n 
i ~i ¢¢¢in 
j i
¢¢¢i n 
~ 
±J~
 ¯½
nt ¯ 
J
¡ ¯ j @t J~
Non-abelian currents
Start with:

i ¢¢¢i n 

½

 
X i ¢¢¢X i n 
i 
i ¢¢¢i n 
i i
 
X
X ¢¢¢X i n 

J

Problem Lorentz covariance:
nested symmetrizations
Add correction terms:
X
n

i :::i
Cp  pX i p: : : X i n 

n ¡ p
p p
µ
¶
X
n
i
:::i p
i

i
:::i p
i

i
¢¢¢in 
i 
i p
i n

J



X
C
E
X
:
:
:
X
p
p



n ¡ p
p p
i
¢¢¢in 
½






Solve for Cp; E p
•Current conservation=gauge invariance WZ
d 
i :::i 

i 
i :::i 
Cp  p  pE p   p
dt
•Do not need to use the trace
•Field redefinitions can be absorbed in C
•Solution is not unique
Charge density of dielectric branes
Hashimoto 0401043
N D0-branes
R
Xi 
Li
N
Add family of terms

L i ; L j  i ² i j k L k
Invariant action
Avoid partial integration identities
Construct action as a density
Z
dd
yL 
y ; t  with
~
S
~ ¢~
±L 
y ; t  ¡ t ¯~ ¢r L 
~
y ; t¡ ¯
~
y@
y; t 
~
tL
µq
start with:
µq
±
¶



¡ X

i
:::i n 
L
 

µ
q
¶
d

 ¡ ¯l
Xl 
¡ X
dt
i

¡ XX 
:::X
¶
in
Result
X
0
n
i
:::i p i p

i
¢¢¢i n 
i n
L



C
X
:
:
:
X
p



n ¡ p
p p
0
C i :::i n 
n
q
q
@
i
:::i n 

i
:::i n 
 C
E j 

¡ X


¡ X
n
n
@X
j
q
X
@
@
 Rj 
:::j k 

i
:::i n ;

:::

¡ X


k @X j  @X j k

In particular:
q
q
@
@ @
0

i
 Ri j


C 

¡
X
E


¡
X


i
i @X
j

@X
@
X
q
@ @ @
 ij k

 i

¡
X
R
j
k



@X @X @X
Delta function
We did not use the trace:
L can be used a our `delta function’
We need an object V to produce terms such
as: Z d
¹
º
d
y
V¹ ; Vº 

V ;V 
L
Covariant vector field
Vy¹
Vy¹ dx ¹  
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 & L
 Understand ambiguity
Understand which lowest order terms
All possible invariant actions?
T-duality