Table Of ContentBrown-Het-1368 QMUL-PH-0310
Towards Gauge theory for a class of
commutative and non-associative fuzzy spaces
4
0
0
2
n Sanjaye Ramgoolam
a
J Queen Mary College
2
2 London, E1 4NS
3 [email protected]
v
3
5
1
We discuss gauge theories for commutative but non-associative algebras related to
0
1
the SO(2k + 1) covariant finite dimensional fuzzy 2k-sphere algebras. A consequence
3
0
of non-associativity is that gauge fields and gauge parameters have to be generalized to
/
h
t be functions of coordinates as well as derivatives. The usual gauge fields depending on
-
p
coordinates only are recovered after a partial gauge fixing.The deformation parameter for
e
h
these commutative but non-associative algebras is a scalar of the rotation group. This
:
v
i suggests interesting string-inspired algebraic deformations of spacetime which preserve
X
r Lorentz-invariance.
a
October 2003
1. Introduction
The fuzzy four-sphere [1] has several applications in D-brane physics [2][3]. While
the fuzzy four-sphere has some similarities to the fuzzy two-sphere [4], it is also different
in important ways. The Matrix Algebra of the fuzzy 2k-sphere at finite n is the space
of transformations, End( ), of an irreducible representation, , of Spin(2k + 1). It
n n
R R
contains a truncated spectrum of symmetric traceless representations of SO(2k+1) which
form a truncated spectrum of spherical harmonics, but contains additional representations.
The analysis of the representations in the Matrix Algebras for fuzzy spheres in diverse
dimensions was given in [5]. For even spheres S2k these representations span, at large n,
the space of functions on a higher dimensional coset SO(2k+1)/U(k), which are bundles
overthespheres S2k. TheextradegreesoffreedomintheMatrixalgebracanbeinterpreted
equivalently in terms of non-abelian fields on the spheres [6]. In the case of the fuzzy four
sphere the higher dimensional geometry is SO(5)/U(2) which is one realization of CP3 as
a coset.
Any discussion of field theory on the fuzzy sphere S2k requires a product for the
fields. The configuration space of a scalar field on the fuzzy four-sphere, is the subspace
of End( ) which transforms in the traceless symmetric representations. This vector
n
R
subspace of End( ) admits an obvious product. It is obtained by taking the Matrix
n
R
product followed by a projection back to the space of Matrices transforming as symmetric
traceless representations. The vector space of truncated spherical harmonics equipped
with this product is denoted by (S2k). The product is denoted by m , which can be
n 2
A
viewed as a map from (S2k) (S2k) to (S2k). It is important to distinguish
n n n
A ⊗ A A
End( ) and (S2k), which are the full Matrix algebra and the algebra obtained after
n n
R A
projection, respectively. The product on is non-associative and commutative but the
n
A
non-associativityvanishes atlargen [5][6]. The MatrixalgebraEnd( )contains Matrices
n
R
X transforming in the adjoint of SO(2k+1). It also contains Matrices X transforming
µν µ
in the vector of SO(2k +1). The vector of SO(2k +1) and the adjoint combine into the
adjoint of SO(2k+2). The SO(2k+2) acts by commutators on the whole Matrix algebra.
The projection used in defining the non-associative product on (S2k) commutes
n
A
withSO(2k+1)butdoesnotcommutewithSO(2k+2). HencethegeneratorsofSO(2k+1)
provide derivations on the (S2k) [6]. However derivations transforming in the vector of
n
A
SO(2k + 1) would be very useful in developing gauge theory for the (S2k), where we
n
A
would write a covariant derivative of the form δ iA and use that as a building block for
α α
−
1
the gauge theory, a technique that has found many applications in Matrix Theories [7][8](
see for example [9,10,11,12,13,14] ).
The classical sphere can be described in terms of an algebra generated by the co-
ordinates z of the embedding R2k+1, as a projection following from the constraint
α
2k+1
z z = 1. When we drop the constraint of a constant radius, the algebra of
α=1 α α
Pfunctions on R5 generated by the z admits derivations ∂ which obey ∂ z = δ . These
α α α β αβ
translations can be projected to the tangent space of the sphere to give derivations on the
sphere. The latter P can be written as P = δ Q , where Q is a derivative transverse
α α α α α
−
to the sphere, and can be characterized by its action on the Cartesian coordinates.
The fuzzy sphere algebra contains analogous operators Z which obey a constraint
α
2k+1Z Z = n+2k and form a finite dimensional algebra. It is natural to consider
α=1 α α n
P
at finite n, a finite dimensional algebra generated by the Z obtained by dropping this
α
quadratic constraint, but keeping the structure constants of the algebra. This can be
viewed as a deformation of the algebra of polynomial functions of Z . We call this fi-
α
nite dimensional algebra (R2k+1). Imposing the constraints Z Z = n+2k on this
An α α n
deformed polynomial algebra gives the algebra (S2k). Again Pone can consider ‘deriva-
n
A
tives’ δ defined such that their action on the algebra (R2k+1) at large n is just the
α n
A
action of the generators of the translation group. In general they will not satisfy the
Leibniz rule at finite n.
An approach to gauge theory on the algebra (R2k+1) is to study the deformation of
n
A
Leibniz Rule that is obeyed by δ . This will allow us to obtain the gauge transformation
α
required of A such that the covariant derivative D = δ iA is indeed covariant. We
α α α α
−
can expect the non-associativity to lead to extra terms in the transformation of A . To
α
get to gauge theory on (S2k+1), we would define the projection operators P ,Q and
n α α
A
study their deformed Leibniz rule, and then obtain the gauge transformation rule for A
α
by requiring that P iA is covariant. The strategy of doing non-commutative geometry
α α
−
on the deformed R3 as a way of getting at the non-commutative geometry of the fuzzy S2
is used in [15,16].
The structure constants of the algebra (SD) and (RD+1) show an interesting
n n
A A
simplification at large D. We will study simpler algebras (R2k+1) and (S2k), which
n n
B B
are obtained by choosing 2k + 1 generators among the D + 1 of (RD+1). By keeping
n
A
k fixed and letting D be large we get a simple algebra, whose structure constants can
be easily written down explicitly. This algebra is still commutative/non-associative, and
2
allows us to explore the issues of doing gauge theory in this set-up. We explain the relation
between (S2k+1) and (S2k+1) in section 2.
n n
B A
In section 3, we explore ‘derivatives’ on (R2k+1) which obey δ Z = δ . To com-
n α β αβ
B
pletely define these operators we specify their action on a complete basis of the deformed
polynomial algebra. The action we define reduces in the large n limit to the ordinary
action of infinitesimal translations. Having defined the action of the derivatives on the
space of elements in the algebra, we need to explore the interplay of the derivatives with
the product. The operation δ obeys the Leibniz rule at large n but fails to do so at finite
α
n. The precise deformation of the Leibniz rule is described in section 3.
In section 4 we show that by using a modified product m∗ ( which is also commutative
2
and non-associative ) the derivatives δ do actually obey the Leibniz rule. The modified
α
product is related to the projected Matrix product m by a ‘twisting’. While our use of
2
the term ‘twist’ is partly colloquial, there is some similarity to the ‘Drinfeld twist’ that
appears in quantum groups [17][18]. Applications of the Drinfeld twist in recent literature
related to fuzzy spheres include [19][20][21].
In section 5, using the δ and the product m∗ we discuss an abelian gauge theory
α 2
for the finite deformed polynomial algebra ∗(R2k+1). The transformation law of A ,
n α
B
which guarantees that D is covariant, now has additional terms related to the associator.
α
A detailed description of the associator becomes necessary, some of which is given in
Appendix 2. Appendix 1 is a useful prelude to the discussion of the associator. It gives
the relation between the product m∗ and the product mc which is essentially the classical
2 2
product on polynomials. The discussion of the gauge invariance shows a key consequence
ofthe non-associativity,that gaugefields can pick up derivativedependences under a gauge
transformation by a gauge parameter which depends only on coordinates. Hence we should
enlarge our notion of gauge field to include dependences on derivatives. This naturally
suggests that the gauge parameters should also be generalized to allow dependence on
derivatives. We can then obtain ordinary coordinate dependent gauge fields and their
gauge transformations by a partial gauge fixing. This discussion is somewhat formal since
a complete discussion of gauge theory on a deformation of R2k+1 should include a careful
discussion of the integral. We will postpone discussion of the integral for the deformed
R2k+1 to the future. However the structure of gauge fields, gauge parameters, gauge
fixing in the non-associative context uncovered here carries over to the case of the gauge
theory on ∗(S2k) which we discuss in the next section. One new ingredient needed here
n
B
is the projection of the translations of R2k+1 to the tangent space of the sphere. This
3
requires introduction of new derivatives Q such that the projectors to the tangent space
α
are P = δ Q . The relevant properties of Q are described and the construction of
α α α α
−
abelian gauge theory on ∗(S2k) done in section 5. The discussion of the integral is much
n
B
simpler in this case. Details on the deformed Leibniz rule for Q are given in Appendix
α
3. The final appendix 4, which is not used in the paper, but is an interesting technical
extension of Appendix 1 shows how to write the classical product mc in terms of m , the
2 2
product in (R2k+1)
n
A
We end with conclusions and a discussion of avenues for further research.
2. The Algebra (R2k+1)
n
B
We will define a deformed polynomial algebra (R2k+1) starting from the fuzzy
n
B
sphere algebras (SD). Consider operators in End( ), where is the irrep of
n n n
A R R
Spin(D+1) whose highest weight is n times that of the fundamental spinor.
1
Z = ρ (Γ )ρ (Γ ) ρ (Γ )
µ1µ2···µs ns r1 µ1 r2 µ2 ··· rs µs (2.1)
r16=rX2···6=rs
Each r index can run from 1 to n. The µ indices can take values from 1 to D +1. It is
i
important to note that the maximum value that s can take at finite n is n.
The operators of the form (2.1) contain the symmetric traceless representation corre-
sponding to Young Diagrams of type (s,0, ) where the integers denote the first, second
···
etc. row lengths. By contracting the indices, the operators of fixed l also contain lower
representations, e.g (s 2,0). For example
−
n(n 1)
Z = −
µµ
n2
Xµ
(2.2)
(n 1)(n 2)
Z = − − Z
µµα α
n2
Xµ
We now define (RD+1) we keep the product structure from (SD) but we drop rela-
n n
A A
tions between Z’s with contracted indices and Z’s with fewer indices such as (2.2). Note
that (RD+1) is still finite dimensional because s n, but it a larger algebra that
n
A ≤
(SD). We can obtain (SD) from (RD+1) by imposing the contraction relations.
n n n
A A A
The operators of the form given in (2.1) can be obtained by multiplying Z using the
µ
1
product in (SD). For example
n
A
1
Z Z = Z + δ (2.3)
µ µ µ µ µ µ
1 2 1 2 n 1 2
4
More generally we have
Z = Z Z Z + (1/n) (2.4)
µ1µ2···µs µ1 µ2 ··· µs O
To make clear which product we are using we write
1
m (Z Z ) = Z + δ (2.5)
2 µ µ µ µ µ µ
1 ⊗ 2 1 2 n 1 2
The m denotes the product obtained by taking the Matrix product in End( ) and
2 n
R
projecting out the representations which do not transform in the symmetric traceless rep-
resentations. This kind of product can be used for both (RD+1) and (SD). Since the
n n
A A
higher Z’s are being generated from the Z we can view (RD+1) as a deformation of the
α n
A
algebra of polynomials in the D+1 variables Z . As discussed previously in the context of
α
(SD) [5,6] this product is commutative and non-associative but becomes commutative
n
A
and associative in the large n limit.
Now we look at a deformed R2k+1 subspace of the deformed RD+1 space. We have
generators Z where the µ indices take values from 1 to 2k + 1. This operator is
µ1µ2···µs
symmetric under any permutation of the µ’s. The largest representation of SO(2k + 1)
contained in the set of operators for fixed s is the one associated with Young Diagram of
row lengths (r ,r , ) = (s,0, ) and are symmetric traceless representations. We keep
1 2
··· ···
k fixed and let D be very large. We will get very simple algebras which we denote as
(R2k+1) and (S2k). The relation between (R2k+1) and (S2k) is similar to that
n n n n
B B B B
between (R2k+1) and (S2k+1).
n n
A A
Let us look at some simple examples of the product.
n n
X X = ρ (Γ ) ρ (Γ )
µ µ r µ r µ
1 2 1 1 2 2
rX=1 rX=1
1 2
n n
= ρ (Γ )ρ (Γ )+ ρ (Γ Γ )
r µ r µ r µ µ
1 1 2 2 1 1 2
r16=Xr2=1 r1=Xr2=1 (2.6)
n n
= ρ (Γ )ρ (Γ )+ ρ (δ )
r µ r µ r µ µ
1 1 2 2 1 1 2
r16=Xr2=1 r1=Xr2=1
= X +nδ
µ µ µ µ
1 2 1 2
In the first line we have written the product of two X operators which are both Γ matrices
acting on each of n factors of a symmetrized tensor product of spinors [2]. In the second
line we have separated the double sum into terms where r = r and where they are
1 2
5
different. In other words we are looking separately at terms where the two Γ’s act on
the same tensor factor and when they act on different tensor factors. Where r = r the
1 2
6
expression is symmetric under exchange of µ and µ . When r = r , there is a symmetric
1 2 1 2
part and an antisymmetric part. The antisymmetric part is projected out when we want
to define the product which closes on the fuzzy spherical harmonics [5]. This is why the
third line only keeps δ . Expressing the product in terms of the normalized Z we have
µ µ µ
1 2
1
Z Z = Z + δ (2.7)
µ µ µ µ µ µ
1 2 1 2 n 1 2
By similar means we compute
(n 1)
Z Z = Z + − (δ Z +δ Z ) (2.8)
µ µ µ µ µ µ µ µ µ µ µ µ
1 2 3 1 2 3 n2 1 3 2 2 3 1
The LHS contains sums of the form
= + + (2.9)
rX16=r2Xr3 r16=Xr26=r3 r3=Xr16=r2 r16=Xr2=r3
The three types of sums in (2.9) lead, respectively, to the first, second and third terms on
the RHS of (2.8). The factor of n 1 in the second term, for example, comes from the fact
−
that the r = r sum runs from 1 to n avoiding the single value r . The 1/n2 comes from
1 2 3
normalizations. The relations (2.7) and (2.8) hold both in (R2k+1) and (R2k+1).
n n
B A
Using (2.7) and (2.8) it is easy to see the non-associativity. Indeed
(n 1) 1
(Z Z )Z = Z + − (δ Z +δ Z )+ δ Z
µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ
1 2 3 1 2 3 n2 1 3 2 2 3 1 n 1 2 3
(n 1) 1
Z (Z Z ) = Z + − (δ Z +δ Z )+ δ Z
µ µ µ µ µ µ µ µ µ µ µ µ µ µ µ
1 2 3 1 2 3 n2 1 3 2 1 2 3 n 2 3 1
1
(Z Z )Z Z (Z Z ) = ( δ Z δ Z )
µ µ µ µ µ µ µ µ µ µ µ µ
1 2 3 − 1 2 3 n2 1 2 3 − 2 3 1
(2.10)
We now explain the simplification that arises in (R2k+1) as opposed to (R2k+1).
n n
B A
Consider a product
1
Z Z = ρ (Γ )ρ (Γ ). ρ (Γ )ρ (Γ ) (2.11)
µ µ µ µ r µ r µ r µ r µ
1 2 3 4 n4 1 1 2 2 3 3 4 4
rX16=r2 rX36=r4
In (R2k+1) or (S2k), wewillgettermsoftheformZ wherer = r = r = r .
n n µ µ µ µ 1 2 3 4
A A 1 2 3 4 6 6 6
In addition there will be terms of the form δ Z from terms where we have r = r
µ µ µ µ 1 3
1 3 2 4
6
and r = r . If r = r = r = r , the antisymmetric part of Γ Γ appears in the
2 4 1 3 2 4 µ µ
6 6 6 1 3
Matrix product of Z with Z but transforms as the representation ~r = (2,1,0,...).
µ µ µ µ
1 2 3 4
Hence these are projected out. However when we consider terms coming from r = r
1 3
and r = r and take the operators of the form ρ ([Γ ,Γ ])ρ ([Γ ,Γ ]), these will
2 4 r µ µ r µ µ
1 1 3 2 2 4
include representations of type (2,2,0, ) which are projected out, in addition to those
···
of type (2,0,0 ). The latter come from the trace parts δ . They have coefficients
µ µ
··· 1 2
that go as 1 and hence disappear in the large D limit. We conjecture that all terms
D
of this sort, which transform as symmetric traceless reps, but come from antisymmetric
parts of products in coincident r factors, are subleading in a 1/D expansion. The simple
product on the Z , obtained by dropping all the antisymmetric parts in the coincident
µ1µ2···
r factors ( or according to the conjecture, by going to the large D fixed k limit ) gives
an algebra we denote as (R2k+1). We will not try to give a proof of the conjecture
n
B
that (R2k+1) arises indeed in the large D limit (RD+1) as outlined above. We may
n n
B A
also just study (R2k+1) as an algebra that has many similarities with (R2k+1) but
n n
B A
has simpler structure constants, while sharing the key features of being a commutative
non-associative algebra which approaches the ordinary polynomial algebra of functions on
R2k+1 in the large n limit.
We now give an explicit description of the product on elements Z in the algebra
µ1···µs
(R2k+1) . It will be convenient to set up some definitions as a prelude to a general
n
B
formula. For a set of integers S we will denote by Z an operator of the form (2.1) with
µ(S)
the labels on the µ’s taking values in the set S. For example Z is of the form Z
µ1···µs µ(S)
with the set S being the set of integers ranging from 1 to s. Instead of writing Z we
µ µ µ
1 4 5
can write Z with S being the set of integers 1,4,5 .
µ(S)
{ }
Given two sets of positive integers T and T where T and T have the same number
1 2 1 2
of elements, which we denote as T = T t, we define
1 2
| | | | ≡
˜
δ ( µ(T ) µ(T ) ) = δ( µ µ ) δ( µ µ ) (2.12)
1 2 i1 jσ(1) ··· it jσ(t)
Xσ
The δ’s on the RHS are ordinary Kronecker deltas. The i’s are the integers of the set T
1
with a fixed ordering, which can be taken as i < i < < i . The j’s are the integers
1 2 t
···
of the set T with a similar ordering. The σ in the sum runs over all the permutations in
2
˜
the group of permutations of t elements. The δ is therefore a sum over all the t! ways of
pairing the integers in the set T with those in the set T .
1 2
7
With this notation the general formula for the product can be expressed as
Z Z = δ( T , T )
µ(S ) µ(S ) 1 2
1 2 | | | |
X|T1|X|T2| TX1⊂S1TX2⊂S2 (2.13)
1 (n S S +2 T )!
1 2 1 ˜
−| |−| | | | δ( µ(T ) , µ(T ) ) Z
n2|T1| (n S1 S2 + T1 )! 1 2 µ(S1∪S2\T1∪T2)
−| |−| | | |
We have chosen S and T to be the sets describing two elements of of the form (2.1).
n
B
There is a sum over positive integers T and T which are the cardinalities of subsets T
1 2 1
| | | |
and T of S and T respectively. Given the restriction T = T , the sum over T extends
2 1 2 1
| | | | | |
from 0 to min( S , T ). The factor expressed in terms of factorials comes from the T
1
| | | | | |
summations which can be done after replacing quadratic expressions in Γ’s by δ’s. The
formula expresses the fact that the different terms in the product are obtained by summing
over different ways of picking subsets T and T of S and T which contain the elements
1 2
that lead to δ’s. The set of remaining elements S T T T gives an operator of the
1 2
∪ \ ∪
form (2.1). It is instructive to check that (2.7) and (2.8) are special cases of (2.13). Note
that we should set to zero all Z where S > n. For example this leads to a restriction
µ(S)
| |
on the sum over T requiring it to start at max(0, S + T n). This is not an issue if
1
| | | | | |−
we are doing the 1/n expansion.
3. Deformed derivations on (R2k+1)
n
B
We now define operators δ , which are derivations in the classical limit, by giving
α
their action on the above basis.
s
δ Z = δ Z (3.1)
α µ1µ2···µs αµi µ(1..s\i)
Xi=1
At finite n these are not derivations. Rather they are deformed derivations, which can
described in terms of a co-product, a structure which comes up for example in describing
the action of quantum enveloping algebras on tensor products of representations or on
products of elements of a q-deformed space of functions ( see for example [22,23] )
∞ ( 1)k
∆(δ ) = (δ 1+1 δ ) − δ δ δ δ δ δ
α α ⊗ ⊗ α n2k α1 α2 ··· αk ⊗ α1 α2 ··· αk
Xk=0
∞ ( 1)k
= − δ δ δ δ δ δ δ (3.2)
n2k α α1 α2 ··· αk ⊗ α1 α2 ··· αk
Xk=0
∞ ( 1)k
+ − δ δ δ δ δ δ δ
n2k α1 α2 ··· αk ⊗ α α1 α2 ··· αk
kX=0
8
The leading k = 0 terms just lead to the ordinary Leibniz rule, and the corrections are
subleading as n . These formulae show that co-product is a useful structure for
→ ∞
describing the deformation of Leibniz rule. Another possibility one might consider is to
see if adding 1/n corrections of the type δ (1+ 1 δ δ ) gives a derivative which obeys the
α n2 β β
exact Leibniz rule. This latter possibility does not seem to work.
The co-product (3.2) has the property that
δ .m = m .∆(δ ) (3.3)
α 2 2 α
Another way of expressing this is that
δ (A.B) = m .∆(δ )A B
α 2 α
⊗
∞ ( 1)k
= − δ δ δ δ A . δ δ δ B
n2k α α1 α2 ··· αk α1 α2 ··· αk (3.4)
kX=0
∞ ( 1)k
+ − δ δ δ A . δ δ δ δ B
n2k α1 α2 ··· αk α α1 α2 ··· αk
kX=0
The proof is obtained by evaluating LHS and RHS on arbitrary pair of elements in
the algebra.
It is useful to summarize some properties of δ which act as deformed derivations on
α
the deformed polynomial algebra (R2k+1) They approach ∂ in the large n limit. At
n α
A
finite n they obey [δ ,δ ] = 0 At finite n they transform as a vector under the SO(2k+1)
α β
Lie algebra of derivations that acts on , i.e we have
n
B
[L ,δ ] = δ δ δ δ (3.5)
µν α να µ µα ν
−
These properties can be checked by acting with the LHS ad RHS on a general element of
the algebra.
3.1. A useful identity on binomial coefficients
In proving that the above is the right co-product we find the following to be a useful
identity.
A
(N +1+2A)! (N +2A)!
( 1)k = (3.6)
− (A k)!(N +1+A+k)! A!(N +A)!
kX=0 −
9
