5.4 The F-theory Spectrum
5.4.3 Matter
Having outlined the procedure to obtain the gauge group and its generators we can compute the matter content. The charged matter content is generically given by fiber singularities of codimension two in the base, as opposed to the uncharged part.
12Note that the divisors do not intersect the [KB].
Charged matter: Codimension 2 singularities
In a similar spirit as the gauge groups, we can identify loci of charged matter in F-theory as a singularity of codimension two. These loci are generically obtained when we intersect multiple divisors in the base Bnthat already give codimension one singularities and see how the vanishing order of the discriminant
∆increases. Similarly to the codimension one case, the elliptic curve factorizes at codimension two and contributes additional fiber componentsci. These curves corresponds to the weights of a given representation and we can compute theirDynkin labelsλiby intersecting with the Cartan divisorsDi
λi=c·Di, (5.58)
and obtain their U(1) charge completely analogous by intersecting with the Shioda map:
qn=c·Sn−c·S0+
N
X
I=1
(SmcGiI)(CG−1I)i j(DGjI ·c). (5.59) Note that the charges are generically quantized but the sum that involves the inverse Cartan matrix con-tributes factors of 1/rank(GI).
The matter loci of purely U(1) charged matter are a bit harder to find than their non-Abelian counter parts because there is no codimension one locus to begin with where we could start from and observe where the singularity enhances further. However the existence of a rational pointslthat has coordinates (1,A,B) in the Weierstrass form implies its factorization [86]
0=−(y−B)(y+B)+(x−A)(x2+Ax+C), (5.60) which then gives a relation for f,gand∆by expanding and comparing coefficients in (5.24)
f =C−A2, (5.61)
g=B2−AC, (5.62)
∆ =16· B2
27B2−54AC
+(C+2A2)2(4C−A2)
. (5.63)
We find for the Weierstrass equation to be singular, the constraints
B=(C+2A2)=0, (5.64)
have to hold with vanishing order of∆to be two. This corresponds to anA1singularity (compare with Table5.3) and is indeed codimension two in the base.
The complete intersection (5.64) is in general a reducible variety in the base. The main complication is that certain non-generic loci that fulfill these constraints are also included in other more complicated ones. Mathematically we have to decompose these loci by making a prime ideal decomposition. A given component we denote by I(k) and the vanishing locus asV(I(k)). After having found all loci by using algebra programs such asSingular, we analyze the fiber for all componentsI(k)and compute the charges of the associated matter curves.
To obtain the multiplicities of the states we have to distinguish the cases of a six and to four dimensional compactification. The four dimensional case we discuss in Chapter7. Six dimensions are special as there the chirality is fixed [33] and a codimension two locus in the base is simply a point. Hence in order to find the multiplicities of a matter curve defined by the vanishing of a polynomial, or more in
5.4 The F-theory Spectrum
general the idealI(k), we have to count the amount of solutions to that constraint.
Lets for example consider a matter locus that is defined by the intersection of the two idealsV(I(1)∩I(2)).
If this variety contains simpler constraints, such as s1 = s2 = 0 which however are different matter curves, we have to subtract those solutions as they correspond to matter multiplicities of other states.
In general it can happen that the simpler solutions s1 = s2 = 0 are included n-times in the variety V(I(1)∩I(2)). In such a case we have to be subtract exactlyn-times the corresponding points. The factor ncan be easily computed using theresultantof the polynomial systemI(1)=0 andI(2)=0 whens1and s2are viewed as variables in those polynomials. More details concerning the resultant techniques can be found in [75].
Lets apply those computations again concretely in the F11 example: By specifying s9 = 0 we first go to the SU(3) locus and then we also set s5 = 0 and go to codimension two. By inspecting (5.50) and comparing with (5.3) we see that the singularity enhances to an SU(4) factor and we expect to find triplet states. This can be explicitly seen by confirming the factorization of the elliptic curve in its cubic form on that locus to
pF11|s9=s5=0=u·e2·e3·p2, (5.65) with p2 a quadratic polynomial. The situation of how the elliptic curve enhances further is depicted in Figure5.7. Now we can see, how the polynomialq2splits up into the curves p2ande3. We choose
Figure 5.7: The enhancement of the SU(3) gauge locus ats9 =0 to the SU(4) matter point ats5 =0. Note how we can use the zero section to track the affine node that splits up over the enhanced locus.
p2=c1as the matter curve and by subsequently adding roots of the non-Abelian group we can complete the full representation. The Dynkin labelsλi(cj)=cj·Di can be readily obtain from Fig.5.7where we can read offthe intersection properties.
Curve Dynkin label
c1 (0,1)
c1+D2 (1,−1) c1+D1+D2 (−1,0)
The U(1) charges we obtain similarly by intersecting the curve with the Shioda map5.55:
qc1 =c1·σ1=c| {z }1·S1
=0
−c| {z }1·S0
=0
+1 3
c1·D1
| {z }
=0
+2c| {z }1·D2
=1
= 2
3. (5.66)
To summarize we got a (3,1)2/3representation, with the multiplicity simply given as the product of the two bundle classes of the base in which the sectionss9 ands5have transformed obtained from Table 5.2: #
(3,1)2/3
= S9·(2[KB−1]− S7). Equivalently we could have chosen the curvee3 as the matter curve resulting in the conjugated representation which still gives the the state in the 6D theory.
Uncharged matter: Adjoints and moduli
The last two kinds of states do not come from codimension two singularities. The first ones are mul-tiplets in the adjoint representation. In [87] it was shown that over a divisorSI in a two dimensional base B2 that supports a non-Abelian groupGI the quantization of the M2-brane moduli space gives additionalgI hyper multiplets charged in the adjoint representation. Here gI denotes thegenusof the divisorSI given by
gI =1+ 1
2SGI · SGI +[KB], (5.67)
where we tookSGI as the divisor classes in the base and · denotes their intersection. These adjoint hypermultiplets come along with the vector multiplets of the gauge symmetry which makes them a codimension one phenomenon.
The last type of matter we want to discuss are the geometric moduli of the compactification and are neutral singlets. We start by noting that the rank of thetotalgauge group of the fibrationYn+1is given by the Hodge numbers
Rank(GY)=h(1,1)(Y)−h(1,1)(B)−1. (5.68) This formula simply counts the amount ofh(1,1)forms that come from the fiber minus its overall volume that is not physical in the F-theory limit. The Euler number of a threefold is given by
χ=2
h(1,1)−h(2,1)
. (5.69)
The neutral singlets are then essentially given by the complex structure moduli that can be related to the geometric properties as
Hneutral=h(2,1)(Y)+1=h(1,1)(B)+2+rank(GY)− 1
2χ(Y). (5.70)
Finally we also haveT tensor multiplets in our construction. The tensor multiplets are supported along cycles that are only in the baseBand hence their multiplicity is given by
T =h(1,1)(B)−1, (5.71)
whereas we have to subtract the overall volume of the base, as it is supported by an uncharged hyper-multiplet. The knowledge of all the matter and their multiplicities is essential in the cancellation of the anomalies by the Green-Schwarz mechanism that we review in the next section.