3.3 Gauged linear sigma models
3.3.6 SR ideal and intersection numbers in the GLSM
Z1,1 Z1,2 Z1,3 Z2,1 Z2,2 Z2,3 Z3,1 Z3,2 Z3,3 X111 X211 X311 C1 C2 C3
U(1)R1 1 1 1 0 0 0 0 0 0 0 0 0 -3 0 0
U(1)R2 0 0 0 1 1 1 0 0 0 0 0 0 0 -3 0
U(1)R3 0 0 0 0 0 0 1 1 1 0 0 0 0 0 -3
U(1)E111 1 0 0 1 0 0 1 0 0 -3 0 0 0 0 0
U(1)E211 0 1 0 1 0 0 1 0 0 0 -3 0 0 0 0
U(1)E311 0 0 1 1 0 0 1 0 0 0 0 -3 0 0 0
Table 3.4: GLSM charge assignment for the a (degenerate) smooth resolution ofT6/Z3.
these xed points remain unresolved. This means that by introducing more exceptional divisors, we have re-introduced unresolved xed points!
However, the zero locusz1,3 =z2,1 =z3,1 = 0already guides us to a solution: We simply introduce another exceptional divisor x311 together with the associated scaling and the associated FI term, see table 3.4. Precisely when this b311 <0, the coordinate x311 has to have a VEV and the exceptional scaling U(1)E311 is broken to a Z3 acting as θ311 in (3.45). However, this time the D term associated with the FI parameter b311 forbids the nine xed points of θ311 when b311 > 0. Thus by introducing three exceptional divisors we have broken the xed point degeneracy from one group containing all 27 xed points into three groups containing 9 xed points each.
If we go on and introduce further exceptional divisors, we will further break the de-generacy into 9 groups of 3 xed points each. This description will have all of its xed points resolved once we have introduced a total of 9exceptional divisors together with their scalings and FI parameters. When going on further and introducing the tenth exceptional divisor, the degeneracy is broken completely, but only 10 out of 27 xed points are resolved. Thus in order to obtain a fully resolved model without remnant xed points, we have to introduce all 27 exceptional divisors. We call this model the maximal resolution model.
Lastly, we have to explain what the meaning of all the induced Z3 actionsθαβγ actually is. In the end, we want to describe a simple T6/Z3 orbifold and not a T6/(Z3)27 orbifold. First, we notice that not all27θαβγ are independent since they can be related by dening linear combinations of charges and by rotating all elds simultaneously in a way similar to what was done in (3.45). Using redenitions of this type, it can be shown that out of the 27only four Z3 actions remain. One of them corresponds to the orbifold action and the other three are used to break the degeneracy of the xed points from 27 to 9 with the rst, to 3 with the second, and to 1 with the the last. All the details of this are worked out in [46], also for the other orbifolds, and we refrain from repeating it here.
3.3 Gauged linear sigma models 57 the StanleyReissner ideal is by far not unique, not even for the T6/Z3 as could have been expected from the discussion in section 3.2.1. Instead, there exist a whole variety of geometric, non-geometric, and mixed phases living in dierent Kähler cones.
We start the discussion in the simple setup of the minimal resolution model. In the example in section 3.2.1, we have seen that intersections like D1,αD2,βD3,γ never form a cone in the toric blowup and hence are part of the StanleyReissner ideal. Indeed, setting z1,1 =z2,1 =z3,1 = 0 in (3.41), we see that we cannot solve the fourth D term constraint in the blowup regime b111 > 0 and thus the intersection number is zero.
Likewise, combinations likeD1,1D1,2D1,3 with all divisors in the same torus, which also do not form a cone after the introduction of the exceptional divisor, are forbidden by the rst three D terms. In contrast, it can be checked that zi,1 = zj,1 = x111 = 0 is allowed by the D terms. And indeed, these divisors do form a cone. In order to nd the intersection numbers, we simply count the number of solutions to the F terms. We obtain three equations of the form zi,23 +zi,33 = 0, each of which has three distinct solution, such that the total intersection number is 27. After introducing all 27 exceptional divisors and thereby breaking the degeneracy completely, we nd the intersection number to be1. Using linear equivalences and the method described above to construct the inherited divisors, all intersections can be calculated.
Note that the discussion is solely based on theDterms. So for example the combination D1,1D1,2D2,1 is allowed by the D terms, which set in particular |z1,3|2 =a in this case.
Nevertheless, setting z1,1 = z1,2 = 0, |z1,3|2 = a does not solve the F terms. In other words, the divisors do intersect in the toric ambient space but not on the CY hypersurface. This illustrates the major complication that is introduced when studying hypersurfaces: one needs to restrict the ambient space data to the hypersurface. This restriction is best formulated via exact sequences. (See (3.19) as an example for the restriction of the tangent bundle; for the restriction of the gauge bundle, we later use the Koszul sequence.)
Let us briey discuss the various phases. For ease of exposition, we assume that all torus Kähler parameters are of the same size, ai = a. As we already know, the case a > 0, b111 < 0 corresponds to the orbifold phase. The case a, b111 < 0 corresponds to the LandauGinzburg orbifold. Furthermore, we have discussed the blowup case a b111 > 0. However, a careful analysis shows that one has to be more careful, as this phase is actually subdivided into four phases, out of which only one corresponds to the toric blowup description we are seeking to describe (blowup I). An overview over the various phases is given in gure 3.7. By analyzing all simultaneous solutions to the D terms, we nd that for0<3b111 < a, we reproduce the SR ideal associated with the toric fan given in gure 3.3. As we increase the ratiob111/a, we leave the Kähler cone of the resolution and enter new ones, where rst (blowup II) the exceptional divisors start to intersect the inherited divisors (thus destroying the local resolution picture of the Z3
singularity). Upon increasing the ratio further, we enter yet other cones (critical blowup + over-blowup) where we obtain a mixed description, as here the ci can be vanishing or non-vanishing. Depending on the situation, the dimensionality of the target space jumps between one and three dimensions. Finally, going to a phase where a < 0, b >0, we end up in a regime where the ci cannot vanish andx111 vanishes identically
geometric
over−blowupsingular
blowup I
orbifold non−
critical blowup
over−blowup blowup II
a b
Figure 3.7: Overview over the various GLSM phases that can be obtained from a sim-plied version of the minimal blowup model of T6/Z3.
(singular over-blowup). This corresponds to a mixed phase as well. Situations like this with topology changing hybrid models and exoop transitions have also been studied in [6668]. In order to avoid such complications, we usually study regions deep inside the orbifold regime (ai 0,bαβγ 0) or the CY regime (ai bαβγ 0).
We want to conclude this section by outlining the resolution procedure for toroidal orbifolds in the GLSM framework:
For each torus T2, choose the description in terms of an elliptic curve that is best-suited for the orbifold action under consideration.
Introduce exceptional divisors and U(1) gaugings that generate the orbifold action when the x's get a VEV and that forbid the xed points when the x's are set to zero. Depending on the intended use, this might be a minimal resolution model, a full resolution model, or even some model that cannot be completely resolved.
Calculate the SR ideal from theDterms in the Kähler cone of the toric resolution.
Read o the linear equivalences from the GLSM charges.
Calculate the intersection numbers form counting the solutions to the F terms and by employing the linear equivalence relations.
Chapter 4
. . . . Anomalies
A
nomalies provide an extremely powerful tool for studying string compactications.This is based on the fact that string theory is, by construction, free of anomalies.
Hence, any theory that is derived from it, has to be free of anomalies as well. The mechanism we are invoking for this argumentation is based on the 't Hooft anomaly matching principle [69], which states that the calculation of anomalies is independent of the energy scale at which the calculation is carried out. Thus anomalies are protected quantities that can be used to connect theories which are dened at dierent scales1. As will be discussed in the following, anomalies are characterized in terms of the massless chiral spectrum of a theory. Since the spectrum computation is accessible in both the supergravity approximation and exact string CFT calculations (cf. sections 2.3 and 2.4), anomalies can be computed and compared in both theories.