3.3 Gauged linear sigma models
3.3.5 CalabiYaus as hypersurfaces in the GLSM
Finally, we are in a position to discuss the compact resolution of orbifold models from the GLSM perspective. The analysis has been carried out in great detail in [46] for the geometry of most orbifold models. We will not repeat the full analysis here and only outline the techniques we need for our studies later. We start by describing a two-torus as an elliptic curve. Then we tune the complex structure (i.e. the coecients in the polynomials dening the torus as a hypersurface) to be compatible with the orbifold action. This can be checked via the Weierstrass ℘ function and its derivative, which establishes a relation between the coordinate u on a double-periodic lattice C/ΛT and the elliptic curve by virtue of the Weierstrass dierential equation. Although there exist various descriptions of elliptic curves which are all equivalent, the dierent complex structures needed for toroidal orbifolds of the type T2/ZN make it convenient for us to choose a specic elliptic curve for a specic ZN orbifold:
Orbifold action Z2 Z3 Z4 Z6
Complex structure free e2πi/3 i e2πi/3 Submanifold CP31111[2,2] CP2111[3] CP2112[4] CP2123[6]
Here we introduced the following notation: The subscripts of CPN give the charges of the weighted projective space and the number(s) in square bracket the degree of the homogeneous polynomial(s) used to dene the hypersurface equation. As we can see, for Z3,4,6, the torus is given as a single hypersurface in some (weighted) projective space while for Z2 the torus arises as the intersection of two degree two hypersurfaces in CP3. Let us exemplify the resolution of T6/Z3.
Example (Resolution of T6/Z3) We start with three copies of CP2111[3] which correspond to T6. This means that we introduce for each CP2 labeled by a = 1,2,3 three coordinatesZa,i, one hypersurface constraint eldCa, and one U(1) gauging. The charge assignment is given in the rst three lines of table 3.2 (let us ignore the last line for the moment). We want to study the geometric regime where ai >0 and thus ca= 0. From the FCa terms, we then obtain the three hypersurface constraints
0 = z1,13 +z1,23 +z31,3+κ1z1,1z1,2z1,3, 0 = z2,13 +z2,23 +z32,3+κ2z2,1z2,2z2,3, 0 = z3,13 +z3,23 +z33,3+κ3z3,1z3,2z3,3.
(3.38)
It can be shown that these are already the most general polynomials of degree 3 since other terms like z1,1z21,2 can be absorbed via eld redenitions. The κi are constants and encode the complex structure τi of the tori. Via the Weierstrass map, it can be checked that for τi =e2πi/3 we getκi = 0. Thus xing the complex structure such that it is compatible with the Z3 orbifold action means that we have to drop the last terms.
3.3 Gauged linear sigma models 53 Z1,1 Z1,2 Z1,3 Z2,1 Z2,2 Z2,3 Z3,1 Z3,2 Z3,3 X111 C1 C2 C3
U(1)R1 1 1 1 0 0 0 0 0 0 0 -3 0 0
U(1)R2 0 0 0 1 1 1 0 0 0 0 0 -3 0
U(1)R3 0 0 0 0 0 0 1 1 1 0 0 0 -3
U(1)E111 1 0 0 1 0 0 1 0 0 -3 0 0 0
Table 3.2: GLSM charge assignment for the minimal resolution ofT6/Z3.
The three D terms read
|z1,1|2+|z1,2|2+|z1,3|2−3|c1|2 =a1,
|z2,1|2+|z2,2|2+|z2,3|2−3|c2|2 =a2,
|z3,1|2+|z3,2|2+|z3,3|2−3|c3|2 =a3.
(3.39)
In the geometric regime ai > 0 we have ci = 0 from the F terms and hence we can drop them from theD terms. In the LG regimeai <0we have ci 6= 0and the residual Z3 generates a LandauGinzburg Z3 orbifold. However, as mentioned before, this is not the orbifold we are after. So let us introduce the orbifold action instead using exceptional divisors. This means we introduce new exceptional elds x and new U(1) charges in order to keep the dimensionality xed. Giving VEVs to the exceptional elds x will remove the corresponding exceptional divisors E and give rise to residual Z3 actions. All this happens in the geometric regime ai > 0. In T6/Z3, there are 27 xed points. We label the position of the xed point in the rst torus by α = 1,2,3, in the second torus by β, and in the third torus by γ. Hence we get 27 exceptional divisors Eαβγ ={xαβγ = 0}, 27 U(1) gauging U(1)Eαβγ, and 27 FI parameters bαβγ on top of the GLSM data needed to describe the T6.
It is instructive to proceed step by step and to rst introduce one exceptional coordinate, sayx111. The GLSM charge assignment for this case is given in the last line of table 3.2.
From this, we obtain in the geometric regime ai >0 the following F terms 0 =z1,13 x111+z31,2+z1,33 ,
0 =z2,13 x111+z32,2+z2,33 , 0 =z3,13 x111+z33,2+z3,33 .
(3.40)
Note that they are almost identical to those of the torus (3.38), except for the appear-ance ofx111 in front of some monomials which is needed for gauge invariance. Also note that the absence of the κizi,1zi,2zi,3 term is crucial for the resolution: with the given charge assignment, such a term would not be gauge invariant. The D terms are now
|z1,1|2+|z1,2|2+|z1,3|2 =a1,
|z2,1|2+|z2,2|2+|z2,3|2 =a2,
|z3,1|2+|z3,2|2+|z3,3|2 =a3,
−3|x111|2 +|z1,1|2+|z2,1|2+|z3,1|2 =b111.
(3.41)
In this case, the rst three D terms are the same as in (3.39). The last D term is precisely the one from the local non-compact resolution (3.37). There the target space was non-compact since the Dterms contain elds of opposite signs, such that an arbitrary large value for the zi could be canceled by a large VEV of x. In this case, however, the rst threeDterms prevent thezi from becoming too large and in this way lead to a compact target space. As before, the geometric orbifold and blowup regime are separated by the value of the Kähler parameter b111. If b111 < 0, the last D term forces xto get a VEV and thus the U(1)E111 is broken to a Z3 acting as
θ111 : (z1,1, z2,1, z3,1)7→(e2πi/3z1,1, e2πi/3z2,1, e2πi/3z3,1). (3.42) This Z3 has xed points at z1,1 = z2,1 = z3,1 = 0. By inserting them into the F term equations (3.40), we nd that the equations can be satised by choosing zi,2 =−e2πik/3zi,3,k = 0,1,2. We thus obtain a total of3·3·3 = 27xed points on the torus, all of which are described by the same zero locus. Note thatx111 6= 0 in the orb-ifold phase and thus the exceptional divisor E111 does not exist on the orbifold. In the geometric blowup regime b >0, it is possible to setx111 = 0, corresponding to the divi-sorE111. In this case, however, the fourth Dterm forbids setting z1,1 =z2,1 =z3,1 = 0, since at least one of these elds has to get a VEV to cancel the FI term, and thus the 27 xed points have been resolved by the same exceptional divisor E111. This is remarkable, since we have only introduced one exceptional divisor instead of27and yet all27singularities are resolved. This model corresponds to a resolution ofT6/Z3 where all 27 exceptional divisors are identical or put dierently, where the same exceptional divisor E111 is glued into all 27 xed points to resolve the singularity simultaneously.
In particular, we only have one FI or Kähler parameter b111 which controls the size of the resolution CP2 that is glued into the 27 xed points. The model corresponds to the blowup of an orbifold where all exceptional divisors are identical and thus have the same size and carry the same gauge ux. Nevertheless, the model is smooth. For this reason, we call it the minimal resolution model. This type of model was e.g. also studied in [65]. Albeit being far from the most general case, these minimal resolu-tion models provide a nice class of traceable examples due to their simplicity, which mainly stems from the fact that the StanleyReissner ideal is much smaller than in the maximal resolution models. For this reason, we will use a resolution model with three exceptional divisors (instead of 27) for the discussion in chapter 7.
Before moving on and introducing more exceptional divisors, let us discuss the corre-spondence between the divisors as introduced in section 3.2.2 and the GLSM coordi-nates: the ordinary divisors Di,σ correspond in the GLSM to {zi,σ = 0}. As already mentioned, the exceptional divisors Eαβγ correspond in the GLSM to {xαβγ = 0}. Lastly, let us turn to the inherited divisors. Remember that these were divisors inher-ited from the torus away from the singularity. Using the Weierstrass map to match the at torus with the elliptic curve description, it can be shown that [46]
R1 ={k1,1z1,13 x111+k1,2z1,23 +k1,3z1,33 = 0} (3.43) and likewise forR2 andR3. We note that the Ri are basically theFCi terms (3.40) that cut out the torus. The coecients ki,σ correspond to the positions of the divisors on
3.3 Gauged linear sigma models 55 Z1,1 Z1,2 Z1,3 Z2,1 Z2,2 Z2,3 Z3,1 Z3,2 Z3,3 X111X211 C1 C2 C3
U(1)R1 1 1 1 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 -3 0
U(1)R3 0 0 0 0 0 0 1 1 1 0 0 0 0 -3
U(1)E111 1 0 0 1 0 0 1 0 0 -3 0 0 0 0
U(1)E211 0 1 0 1 0 0 1 0 0 0 -3 0 0 0
Table 3.3: GLSM charge assignment for a partly singular resolution ofT6/Z3.
the torus (they can be related to thec's appearing in the denition (3.23)). Thus in the GLSM the inheritance of the inherited divisors becomes quite obvious. Furthermore, we see how the three linear equivalent relations involving the same Ri but dierent divisors Di,σ arise, which were used to describe the gluing in the resolution process:
The expressions for the Ri have three dierent terms, each of which has the same charge (the negative of the Ci). Thus each term in the sum will be linearly equivalent to Ri. In this sense, the Ri, which are the inverse of the Ci, dene the gluing of the xed points across the torus. Thus the hypersurface construction that proceeds via the introduction of the Ri and the associated auxiliary polyhedra discussed in section 3.2.2 implicitly uses the hypersurface constraints from theCi, as can be seen from the GLSM.
Let us now consider what happens when we introduce another exceptional divisor, say E211, where the charge assignment is given in table 3.3. This should break the degeneracy which identies the exceptional divisor of all 27 xed points. As we can see, in the orbifold phase ai >0,b111, b211 <0 where bothxhave a VEV, the following two discrete Z3 remnants of U(1)E111 and U(1)E211 are left:
θ111 : (z1,1, z2,1, z3,1)7→(e2πi/3z1,1, e2πi/3z2,1, e2πi/3z3,1),
θ211 : (z1,2, z2,1, z3,1)7→(e2πi/3z1,2, e2πi/3z2,1, e2πi/3z3,1). (3.44) Clearly, these actions have xed points z1,1 =z2,1 =z3,1 = 0and z1,2 =z2,1 =z3,1 = 0, respectively. Proceeding as before and substituting them into theF terms, we nd that the two zero loci have nine dierentF term solutions each. Also as before, substituting the xed points into theDterms shows that in the blowup regimeb111, b211 >0the xed points are forbidden since at least one of thez's has to have a VEV to cancel the FI term.
So the degeneracy of27is broken into sets of9. But so far we have only uncovered 18 of the 27 xed points. To nd the last group of 9 xed points, observe that there is a lin-ear combination of U(1)'s, Ue(1) :=U(1)R1 + 3U(1)R2 + 3U(1)R3 −U(1)E111 −U(1)E211, which is broken by either VEV of x111 or x211 to a discrete Z3 symmetry acting as
θ311 : (z1,3, z2,1, z3,1)7→(e2πi/3z1,3, e2πi/3z2,1, e2πi/3z3,1). (3.45) The xed points of this discrete action are atz1,3 =z2,1 =z3,1 = 0. Inserting these xed points into the F terms, we nd the last group of nine xed points at this zero locus.
A surprise awaits us when inserting the xed point equation into the D terms. Unlike the other cases, there is no D term that forbids setting z1,3 = z2,1 = z3,1 = 0. Thus
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.