cAF F −12 12 12 32
cARR 1 −1 5 15
Table 7.2: The different constant multipliers for the shifts of the Chern-Simons terms depending on the type of six-dimensional N = (1,0)-multiplet whose five-dimensional analogue is integrated out. Note that this is the contribution of a single five-dimensional multiplet and not the entire Kaluza-Klein tower. We remark that in order to obtain the right contribution of a tensor multiplet, one must take into account that an (anti-)self-dual tensor in six dimensions reduces to a real five-dimensional tensor and therefore contributes only half the factor of table 7.1.
7.3 Loop-corrected Matchings
Given the explicit expressions of Equation 7.2.1 and Equation 7.2.2, we can compute the loop corrections to the prepotential terms that we have so far not been able to match. First, however, we introduce a bit of notation.
• We write R for a representation of the whole gauge group G, while representations of GnA are referred to asR.
• For a representation R we denote the weights of the whole representation (including U(1)-factors) by w. Weights of only GnA are called w. By the roots α of G (and analogously by the coroots of G) we mean explicitly only the roots of GnA, possibly embedded into the root lattice of G. The set of roots ofG is called Φ(G).
• Expanding the non-Abelian vector fields in the coroot basis of GnA, the charge of a weight w of a representationRunder the Cartan vector field AI is
qwI ≡ hαI∨,wi, (7.3.1)
whereα∨I is the respective coroot. Similarly, we denote the charge of a rootαunderAI by qIα. Together with the charges qm of the representation Runder the Abelian vector fields Am they can be combined into a vector
qIwˆ = (qwI , qm) (7.3.2) and similarly for the roots.
• H(R) is the number of hypermultiplets transforming in a representationR. The com-plete number of involved hypermultiplets is then dim(R)·H(R), where dim(R) is the dimension of the representation R. One similarly defines H(R). Let H(qm, qn) denote
156 CHAPTER 7. THE SIX-DIMENSIONAL EFFECTIVE F-THEORY ACTION the total number of hypermultiplets with U(1)-charges (qm, qn) and proceed likewise for H(qm, qn, qk, ql). Furthermore, we write H(R, qm) for the number of hypermulti-plets transforming in the representation Rand carryingU(1)-chargeqm. An analogous statement holds for H(R, qm, qn). Note that when a hypermultiplet transforms in some representationR in our notation, this actually means that one complex scalar and one symplectic Majorana-Weyl fermion in the multiplet transform in R, while the other complex scalar and fermion transform in the conjugate representation R∗.
• Traces with respect to the representation R are denoted by trR and tr refers to the trace in the fundamental representation.
• We denote the (floored) ratio between the Coulomb branch mass and the Kaluza-Klein mass of a particle corresponding to a weight wby
lw ≡
$|mwCB|
|mKK|
%
=br|w·ζ|c. (7.3.3)
and similarly for W-bosons labeled by rootsα. Here we have introduced the contraction w·ζ ≡ hα∨I,wiζI+qmζm =qIwˆζIˆ (7.3.4) of the weight w of a representation R of the total gauge group G with the vacuum expectation value of the scalars ζIˆ in the vector multiplets. As before, the scalars ζI are the expansion coefficients in the coroot basis of G.
• Finally, we write
sign(w)≡sign(w·ζ) (7.3.5)
and similarly for the roots α.
To compute the actual loop corrections to the Chern-Simons coefficient kΛΣΘ, one must integrate out the (possibly infinite) set of massive fields that are charged with respect to all three vector fieldsAΛ,AΣ, andAΘ. At the end of the previous section we listed all massive fields in the circle-reduced theory that can theoretically contribute. In the following, our task is to identify the correct subset of fields for the Chern-Simons coefficient in question, restrict the sums of Equation 7.2.1 (or Equation 7.2.2) correspondingly and evaluate the resulting expressions using the formulas derived insection E.3. In the following, we will carry out these steps in full detail for the Chern-Simons coefficientsk000 and kI, before we then summarize the results for all the other coefficients insubsection 7.3.2.
7.3.1 Explicit Computation of k000 and k0
The coefficientsk000 andkI are generated entirely by the one-loop corrections to the Chern-Simons coefficients obtained by integrating out the massive states that are still present in the
7.3. LOOP-CORRECTED MATCHINGS 157 circle reduction of section 6.3. To see how this type of computation is performed, we carry it out in detail for these two coefficients. As all the other formulae of subsection 7.3.2 are obtained analogously, the reader should be able to reproduce them on his own.
To compute the loop corrections tok000, one must integrate out all matter states charged under the vector field A0, i.e. every field with non-zero Kaluza-Klein charge. We therefore have to integrate out all the fields mentioned insection 7.2: the hyperinos, the gauginos, the antisymmetric two-tensors, the tensorinos, the gravitino and the two-tensor originating in the six-dimensional gravity multiplet. Usingtable 7.2, we find
k000 =
∞
X
n=−∞
(−n)3
"
3 2+ T
2 +1 2
X
vectors
sign(m)−1 2
X
R
H(R) X
w∈R
sign(m)
#
. (7.3.6) Here the first contribution is the Kaluza-Klein tower of the gravity multiplet, and the second term corresponds to the six-dimensional tensors. The spinors of the third and the fourth term require additional information in order to perform the sums, as their mass terms include a contribution from the Coulomb branch:
m=mCB+n·mKK =qIˆζIˆ+n
r . (7.3.7)
If the states are neutral (as is the case for the neutral hypermultiplets and the vector fields whose zero mode remains massless), then m obviously does not depend on the ζIˆ anymore and the sum can be performed. We point out that in our notation, the nth state in the Kaluza-Klein tower has charge −n. Using Equation E.3.8 to regularize the infinite sum, we find
k000= −1 60
3 2+T
2 +V 2 −H
2
+1 4
X
α∈Φ(G)
l2α(lα+ 1)2−X
R
H(R)X
w∈R
l2w(lw+ 1)2
= 1
120(H−V −T −3) +1 4
X
α∈Φ(G)
l2α(lα+ 1)2−X
R
H(R)X
w∈R
l2w(lw+ 1)2
. (7.3.8)
ComputingkI is very similar, but we nevertheless go through the steps to illustrate how to compute the corrections for states charged not underA0, but a different vector field. Since only states charged with respect to AI contribute, we only need to consider hyperinos from charged hypers and gauginos from the W-bosons ofGnA. We thus have that
kI =
∞
X
n=−∞
"
X
R
H(R) X
w∈R
qIwsign(m)− X
α∈Φ(G)
qIαsign(m)
#
=X
R
H(R) X
w∈R
(2lw+ 1)qwI sign(w)− X
α∈Φ(G)
(2lα+ 1)qIαsign(α), (7.3.9) where we have again inserted the explicit expressions for the mass of the hyperinos and gauginos and usedEquation E.3.4 to evaluate the infinite sum.
158 CHAPTER 7. THE SIX-DIMENSIONAL EFFECTIVE F-THEORY ACTION
7.3.2 Summary of all Loop-Corrected Chern-Simons Terms
Having illustrated explicitly how to compute the loop corrections, we spare the reader the detailed computations for the remaining Chern-Simons coefficients and instead give a compre-hensive summary of all coefficients. We find that the loop-corrected Chern-Simons coefficients for theA∧F ∧F term are
k000 = 1
120(H−V −T −3) + 1 4
X
α∈Φ(G)
l2α(lα+ 1)2−X
R
H(R) X
w∈R
l2w(lw+ 1)2
(7.3.10a) k00I = 1
6
X
α∈Φ(G)
lα(lα+ 1)(2lα+ 1)qαI sign(α) (7.3.10b)
−X
R
H(R)X
w∈R
lw(lw+ 1)(2lw+ 1)qwI sign(w)
k00m =−1 6
X
R
H(R)X
w∈R
lw(lw+ 1)(2lw+ 1)qmsign(w) (7.3.10c) k0IJ = 1
12
X
α∈Φ(G)
(1 + 6lα(lα+ 1))qIαqαJ −X
R
H(R)X
w∈R
(1 + 6lw(lw+ 1))qwI qJw
(7.3.10d) k0Im=−1
12 X
R
H(R)X
w∈R
(1 + 6lw(lw+ 1))qIwqm (7.3.10e)
=−1 2
X
R
H(R)X
w∈R
lw(lw+ 1)qIwqm
k0mn =−1 12
X
R
H(R)X
w∈R
(1 + 6lw(lw+ 1))qmqn (7.3.10f)
kIJ K = 1 2
X
α∈Φ(G)
(2lα+ 1)qαIqJαqKα sign(α)−X
R
H(R) X
w∈R
(2lw+ 1)qwI qJwqwKsign(w)
(7.3.10g) kIJ m =−1
2 X
R
H(R)X
w∈R
(2lw+ 1)qwI qJwqmsign(w) (7.3.10h) kImn =−1
2 X
R
H(R)X
w∈R
(2lw+ 1)qwI qmqnsign(w) (7.3.10i) kmnk =−1
2 X
R
H(R)X
w∈R
(2lw+ 1)qmqnqksign(w). (7.3.10j) To arrive at the second line ofEquation 7.3.10ewe used that the weights of any given repre-sentation all sum up to zero, as we show inAppendix D.
For the higher curvature terms, we determine the loop corrected expressions forkΛto be