QUANTUM CHIRAL RINGS IN FOUR DIMENSIONAL N = 1 ADJOINT SQCD
6.3 Classical chiral rings of Kutasov model GeneralitiesGeneralities
In this subsection we mainly focus on the massless model with superpotential (6.2).
We will briefly comment on its relation to the mass deformed counterpart at the end.
From the Lagrangian of the theory we know that the correspondingD-term equation reads
[Φ†,Φ]+(QiQ†i −Qe†˜iQei˜)=0, (6.12) while theF-term constraint is
Φk =0, (6.13)
soΦis nilpotent4with degree k. The nilpotent matrix always has degree no bigger than its order, so for simplicity we only discuss k ≤ Nc in this paper5. The only nilpotent matrix which is diagonalizable is zero matrix; others can only be put into Jordan normal form:
Φ=
©
« J1
J2 . . .
Jn
ª
®
®
®
®
®
®
®
®
®
¬
, (6.14)
where the Jordan block Jiis
Ji =
©
« λi 1
λi 1 . . . 1
λi
ª
®
®
®
®
®
®
®
®
®
¬
. (6.15)
The nilpotency implies thatλ1 =λ2 = · · ·= λn= 0. A Jordan block Jiis uniquely determined by its orderNi. Thus a nilpotent matrix can be labelled by a partition of Nc,[N1,N2, . . . ,Nn], characterizing the size of Jordan blocks : N1+N2+· · ·+Nn= Nc
with k ≥ N1 ≥ N2 ≥ · · · ≥ Nn. We use the symbolY as a Young tableau withi-th
4This is not true forSU(Nc)theories, where a traceless condition should be imposed. This additional constraint makesΦeither diagonalizable or nilpotent. See [203, 204] for more details.
5Strictly speaking, k = Nc case is in fact a double trace superpotential, as TrXNc+1 is not independent.
row of length Ni. It is a Young tableau for partition of Nc into integers no larger thank.
For nilpotent matrix, we always have
TrΦj =0, j > 0, (6.16)
which means that classically the vevs of Casimir operatorsuj in (6.8a) are always zero. Note this does not mean uj = 0 in the chiral ring6. In the meantime, the vevs of generalized glueballsrj are in general proportional to the strong coupling scaleΛ2Nc−Nf, and are constrained by fermionic statistics. Since they can only be formulated using adjoint Φ and vector superfield Wα as in (6.8), the constraints are exactly the same as that in [222] and we will not include them in current analysis. Therefore, modulo generalized glueballs and photinos, the classical chiral ring ofU(Nc)Kutasov model is a quotient ring of the polynomial ring generated by generalized mesons and Casimir operators:
RNc,Nf,k = C h
u1,u2, . . . ,uk−1,vf
0,f˜,vf
0,f˜, . . . ,vf
k−1,f˜
i
/S(u1,u2, . . . ,uk−1,v0,v1, . . . ,vk−1). (6.17) The constraintS(u1,u2, . . . ,uk−1,v0,v1, . . . ,vk−1)is hard to compute in general. A powerful tool that helps is from computational algebraic geometry. To be more specific, classically we can form a quotient ring using microscopic fields:
Rmicro =C h
Qeα
˜
f,Qαf,Φαβ i
/SF, (6.18)
where SF comes from F-term equations of the superpotential. We do not have to consider the D-term once we complexify the gauge group [228]. The vacuum is parameterized by gauge invariant data, c.f. equation (6.10). The natural map arising from composing microscopic field into gauge invariant ones extends to a map between rings:
ψ :C h
uk,vk,f
˜ f
i
→ Rmicro. (6.19)
Then by definition
S= kerψ. (6.20)
Computation of this kernel is standard in the theory of Gröbner basis [229, 230].
This method has already been adopted in understanding the vacua and computing
6In mathematical language, the two coordinate ring may define the same classical algebraic varieties, but they do not define the same scheme.
Hilbert series of the vacuum moduli, see e.g. [231, 232]. In section 6.3, we will explicitly see how this works.
The above algebraic construction is quite abstract. We now turn to concrete de- scription in terms of the moduli space of vacua. As we already know, the Coulomb branch vev hΦi is parametrized by Young tableau[N1,N2, . . . ,Nn]. There are two cases to consider:
(1) When all Ni = 1. The D-term equation becomes that of SQCD with funda- mental matter, and there is nontrivial Higgs branch. For k Nf > Nc+1, at the root of the Higgs branch the theory is conjectured to be in non-abelian Coulomb phase [202].
(2) Ni > 1 for somei. Since nontrivial Jordan block does not commute with its conjugate, in general the vevs of quark superfields hQiand hQ˜iare not zero.
We will call it the mixed branch.
In (6.16) we see the vevs of gauge invariant Casimir operators are always zero.
However the above two cases reveal there are distinct branches in the vacuum moduli.
Then the natural question is how can one distinguish between them. Classically, we might tell which branch we are in by looking at the flat directions of generalized mesons. In the branch[1,1, . . . ,1]onlyv0is nontrivial, but for other branches more non-trivial generalized mesons appear. However, we will not use such descriptions because such flat directions receive quantum corrections.
Alternatively one can try to study the branch when the deformation (6.4a) is turned on. Moreover we require the deformation is sufficiently generic andg0,0 in (6.4a).
It is not hard to see that now Φ must be diagonalizable, with entries the roots of polynomial
We0(z)=
k
Õ
n=0
gnzn=
k
Ö
j=1
(z−aj). (6.21)
Then the Coulomb branch vev hΦiis labelled by integers s1 ≥ s2 ≥ · · · ≥ sk, the number of each root of (6.21). Therefore we can label this in in terms of another Young diagramY0: [s1,s2, . . . ,sk], the partition ofNcinto no more thankintegers7.
It is a standard fact that
Y0=YD, (6.22)
7We use an underline to remind the reader that they are Young tableau for mass deformed theory.
whereYDis the dual Young diagram ofY. This is also frequently used in the literature as the mapping between nilpotent element and semisimple element. Careful readers now may worry that the mapping is not one-to-one; one can permute the roots {ai} corresponding to the integer {si}. However, there is a natural way to make this mapping one-to-one, due to the fact that their semi-classical unbroken gauge group for a given set of si are uniquely fixed regardless of permutation of roots:
U(s1) ×U(s2) × · · ·U(sk). Therefore we may define our map from a nilpotenthΦi to the image taking the rank of unbroken subgroup ofU(Nc). In figure 6.1 we give an example of the correspondence of the Young diagrams.
(a) (b)
Figure 6.1: The deformation of nilpotent matrix in the groupU(10)C 'GL(10). In (a) the nilpotent matrix is labelled byY = [3,3,2,1,1], while the deformed matrix is given byY0=[5,3,2], with low energy gauge groupU(5) ×U(3) ×U(2).
This identification is more robust than the previous one in the sense that patterns of unbroken gauge group are rigid against quantum corrections. We will see that it is indeed the case in section 6.4.
As we have seen that the deformation (6.4a) is important to distinguish between different branches, it is illustrative to summarize what the vacua look like if the full deformation (6.3) is turned on [209]. In this case, the vacua consist of Coulomb branch (pseudo-confining branch) and Higgs branch. For Coulomb branch, we have hΦi =diag(a1, . . . ,a1,a2, . . . ,a2, . . . ,ak, . . .ak), hQef˜i= hQfi= 0. (6.23)
For Higgs branch we have
hΦi=diag(b,a1, . . . ,a1,a2, . . . ,a2, . . . ,ak, . . .ak), (6.24a) hQeβ
˜
fi= hQβfi= 0, β= 2,3, . . .Nc, (6.24b)
Qf
1 l+1
Õ
n=1
(n−1)bn−2mf,nf˜
!
Qe1f˜+We0(b)= 0, (6.24c) where bis the root of B(z) = det
h mff˜(z)i
= 0. Similar reasoning to that of [209]
reveals that rootbcan only appear in hΦionce. The solution can also be elegantly packaged as
M(z)=−
l Nf
Õ
I=1
rIWe0(bI) z−bI
1 2πi
∮
bI
1
m(x)dx, (6.25)
whererI =0,1 is the number ofbI in the diagonal ofhΦi. This solution of classical Higgs branch will be important in section 6.4.
Example: U(2)theory withk = 2
Having discussed generalities, it is time to get refreshed by a couple of examples.
In this subsection we will be illustrating the case Nc = 2, k = 2 with Nf = 1,2.
We have two choices of Young tableau for hΦi: [1,1] or[2]. Upon deformations by (6.4a),[1,1]corresponds to the dual vacua[2] where the gauge group remains unbroken asU(2), but[2]corresponds to the dual vacua[1,1]where gauge group is broken down toU(1)2. For[1,1]branch,v1= 0 but it is nonzero for[2]. SinceΦ2 vanishes, one concludes thatvj =0 for j ≥ 2. Therefore we know classically,
R2,Nf,2 =C[u1,v0,v1] /S(u1,v0,v1). (6.26) Next we turn to the classical relationS. A nice computer program that produces the kernel of the mapψ in (6.19) is Macaulay 2[233, 234]. In the following we list the relationsS(u1,v0,v1)for Nf = 1,2:
•Nf =1.
R2,1,2= C[u1,v0,v1] /hu3
1,u2
1v1,u1v2
1,u2
1v0−2u1v1i. (6.27) Notice thatu1is nilpotent in the chiral ring; the classical relation implies thatu1 =0 as an algebraic variety, and the rest constraints in the relations are trivially satisfied.
Sov0,v1take arbitrary complex values.