String Theory and M-Theory

5.4 Gauge anomalies and their cancellation 183
the trR 4 terms, since trR 4 cannot be factorized. This requirement gives the
condition 980 + 4(nV − nH − 1) = 0, which simplifies to nH − nV = 244. ✷
EXERCISE 5.10
The type IIA NS5-brane, which is introduced in Chapter 8, has a sixdimensional
world-volume theory with (0, 2) supersymmetry. This means
that both supercharges have the same chirality. As a result, the theory is
chiral, and there is an anomaly associated with it. The resulting anomaly
cannot be canceled by the methods described in this chapter. Instead, the
brane has interactions with fields of the ten-dimensional bulk that lead to an
anomaly-inflow mechanism that cancels the anomaly. Determine the form
of this interaction required for the cancellation.
SOLUTION
The NS5-brane world volume has N = 2 supersymmetry, and the field content
is given by two matter multiplets. The first multiplet contains four
scalars and a right-handed fermion. The second multiplet contains one antiself-dual
tensor, a single scalar and another right-handed fermion. Of these
fields only the fermions and the anti-self-dual tensor are chiral and contribute
to the anomaly. Since the theory is six-dimensional, the anomalies
are characterized by eight-forms. For this problem, it is desirable to keep
track of the overall normalization, which was not relevant in the previous
discussions. For this purpose it is convenient to express the anomalies in
terms of Pontryagin classes. These are defined by the formula
p(R) = det 1 + R
n
= 1 + (λi/2π)
2π
2
.
Thus
p1 = − 1 1
trR2
2 (2π) 2
i=1
and p2 = 1 1
8 (2π) 4
2 2 4
(trR ) − 2trR
and so forth.
Expressed in terms of Pontryagin classes, including the overall normalization
factor, the anomalies are
I (8) 1 2
1/2 = 7p1 − 4p2
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and I (8)
A
1 2
= 16p1 − 112p2 .
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So the total anomaly on the NS5-brane world volume is
I8 = 2I (8) 1 2 1 1
1/2 + I(8)
A = p1 − 4p2 =
192
192 (2π) 4
trR 4 − 1
4 (trR2 ) 2
.