String Theory and M-Theory

242 T-duality and D-branes
matrix, so this identification is not so straightforward anymore. In the absence
of the four-form electric field, the preferred configurations that minimize
the potential have [Φ i , Φ j ] = 0. This allows one to define a moduli
space on which these matrices are simultaneously diagonal. One can interpret
the diagonal entries as characterizing the positions of the N D-branes.
The pattern of U(N) symmetry breaking is encoded in the degeneracies of
these positions.
In the presence of the four-form flux, the Φ i no longer commute at the
extrema of the potential, and so the classical interpretation of the D-brane
positions breaks down. There is an irreducible fuzziness in the description of
their positions. One can say that the mean-square value of the ith coordinate
(averaged over all N D-branes) is given by
〈(X i ) 2 〉 = 1
N (2πα′ ) 2 Tr[(Φ i ) 2 ]. (6.145)
Summing over the three coordinates gives a “fuzzy sphere” whose radius
R squared is the sum of three such terms. Substituting the ground-state
solution gives
R 2 = (πα ′ f) 2 (N 2 − 1). (6.146)
For large N the sphere becomes less fuzzy, and the radius is approximately
R = πα ′ fN. Specifically, the uncertainty δR is proportional to 1/N. So
the radius is proportional to the strength of the electric field and the number
of D0-branes. If one used a reducible representation of SU(2) instead,
one would find a set of concentric fuzzy spheres, one for each irreducible
component. However, such solutions are energetically disfavored.
The fuzzy sphere has an alternative interpretation as a spherical D2-brane
with N dissolved D0-branes. For large N this can be analyzed using the
abelian D2-brane theory. The total D2-brane charge is zero, though there is
a nonzero D2-charge electric dipole moment, which couples to the four-form
electric field. The previous results can be reproduced, at least for large N,
in this picture.
EXERCISES
EXERCISE 6.9
Expand (6.106) to quartic order in k and show that the quadratic term gives
the Maxwell action (6.111).