Table Of ContentPrepared for submission to JHEP Imperial-TP-17-AH-01
Coulomb Branch for A-type Balanced Quivers in 3d
7
1 N = 4 gauge theories
0
2
n
a
J
3
1
Gong Cheng∗ Amihay Hanany∗∗ Yabo Li∗ Yidi Zhao∗
h] ∗Department ofModernPhysics, UniversityofScienceandTechnology ofChina,Hefei,Anhui230026, China
t ∗∗TheoreticalPhysicsGroup,TheBlackettLaboratory,ImperialCollegeLondon,PrinceConsortRoad,London,
-
p SW7 2AZ, UK
e
E-mail: [email protected], [email protected],
h
[ [email protected], [email protected]
1
v Abstract: We study Coulomb branch moduli spaces of a class of three dimensional N = 4 gauge
5 theories whose quiver satisfies the balance condition. The Coulomb branch is described by dressed
2
monopole operators which can be counted using the Monopole formula. We mainly focus on A-type
8
3 quivers in this paper, using Hilbert Series to study their moduli spaces, and present the interesting
0 pattern which emerges. All of these balanced A-type quiver gauge theories can be realized on brane
.
1 intervals in Type IIB string theory, where mirror symmetry acts by exchanging the five branes and
0 induces an equivalence between Coulomb branch and Higgs branch of mirror pairs. For each theory,
7
weexplicitlydiscussthegaugeinvariantgeneratorsontheHiggsbranchandtherelationstheysatisfy.
1
v: Finally, some analysis on D4 balanced quivers also presents an interesting structure of their moduli
spaces.
i
X
r
a
Contents
1 Introduction 2
2 Mirror symmetry 3
3 Hilbert Series and Plethystic Logarithm 4
4 Three kinds of relations 5
4.1 Trace relation 5
4.2 Matrix relation 7
4.3 Meson relation 8
5 A -type quiver 9
2
5.1 General pattern 9
5.2 Why no meson relation? 10
5.3 Examples 10
5.3.1 Quiver [2]-(2)-(2)-[2] 10
5.3.2 Quiver [3]-(3)-(3)-[3] 12
5.4 Why matrix relation terminates? 14
5.4.1 Quiver [3]-(3)-(3)-[3] 14
5.4.2 Quiver [2]-(3)-(4)-[5] 15
5.4.3 Quiver [1]-(3)-(5)-[7] 15
5.4.4 Quiver (3)-(6)-[9] 15
5.4.5 General case 16
6 General A -type quiver: an attempt of complete description 16
n
6.1 General description 16
6.2 One generator case 16
6.3 General case 17
6.3.1 Bare pattern: Quiver 242-246 17
6.3.2 Modified pattern: Quiver 232-234 19
6.4 Do we have predictability? 20
7 Balanced D-type quiver 21
7.1 2211 21
7.2 2322 22
A Details of calculations 25
A.1 Quiver [2]-(2)-(2)-[2] 25
A.2 Quiver [3]-(3)-(3)-[3] 26
A.3 Quiver 245-246 29
A.4 Quiver 232-234 29
A.5 Quiver 343-345 30
A.6 Quiver [3]-(3)-(3)-(3)-[3] 31
A.7 Higgs branch of [SO(8)]−C −SO(2) 31
2
– 1 –
B Results of some A -type quiver 32
n
1 Introduction
An infinite class of 3d gauge theories with N = 4 supersymmetry can be realized by using the
specific brane and orientifold configurations in M-theory and Type IIB string theory. They can be
described by quiver diagrams which encode the essential information like gauge symmetry as well as
the representation under which the fields in the theory transform. In this article, we consider the
quivers with an unitary gauge group on each gauge node.
The balance of a gauge node U(N ) in a simply laced ADE Series quiver is defined as [7]:
i
BalanceADE(i)=−2N + N (1.1)
i j
X
j∈adjacentnodes
If all gauge nodes in a quiver have a balance of zero, the quiver is termed balanced.
We mainly focus on A-type balanced quivers and study their Coulomb branch moduli space by
finding all the generators and relations. Those generators and relations are present in the form of
matrices composed of quarks in the mirror theory Higgs branch. We analyze the pattern of the
relations and try to make a general description.
The moduli space of the theory can be viewed as the complex varieties described by the chiral
ring of holomorphic functions. One can read the information of the moduli space from its Hilbert
Series which enumerates the gauge invariant BPS operators of chiral ring. Recently, a new efficient
technique was proposed in [5] to calculate the series, i.e. the monopole formula:
HS(t,z)= zJ(m)t∆(m)P (t,m) (1.2)
G
X
m∈Λ(Gˆ)/W(Gˆ)
∆(m) is the conformaldimension or R-charge of monopole operators,which was calculated using
radial quantization in [2]. We quote the result here:
n
1
∆(m)= |ρ (m)|− |α(m)| (1.3)
i
2
X X X
i=1ρi∈Ri α∈∆+
m denotes the magnetic charge which takes value in the quotient of dual weight lattice by Weyl
group. J(m) is the topological charge, and is counted by fugacity z. P is the generating function
G
of Casimir invariants which plays the role of dressing factor. The first term of conformal dimension
formula (1.3) comes from the hypermultiplets that transform in representation ρ . The second term
i
accounts for the vector multiplets, and ∆ is the set of positive rootsof the gaugegroup. The details
+
of monopole formula can be found in [9] and [5].
It was discussed in [9] the method of dividing weight lattice into fans in order to sum up infinite
terms. We found an algorithm that generalizes this method to high dimensions. This allows the
computation for quivers with many gauge nodes.
The outline of this paper is as follows. In section 2 and 3 we review the method for finding the
mirrorandreadingtherelationsfromHilbertSeries. Insection4weintroducethreekindsofrelations
of A-type balanced quiver. In section 5 we begin the analysis of A balanced quiver and introduce
2
– 2 –
the tool we adopt. In section 6 we use this method to study the general A -type balanced quiver and
n
make prediction of some A -type quiver where the HS is difficult to calculate. Finally, we put some
4
results of D-type balanced quiver in the section 7.
2 Mirror symmetry
The A-type quiver gauge theory can be realized in the brane configuration discussed in [10]. Specif-
ically, it is probed by observer living in the D3 brane which moves in the interval of 5-branes, and
thereforeis(2+1)dimensional. Eachofthese configurationscouldbe describedby apairofpartitions
(ρ,σ), and was denoted by Tσ(G) in [4].
ρ
At the fixed point of these SYM theories, an action called mirror symmetry [11] exchanges the
HiggsbranchandCoulombbranchofamirrorpair. Generally,inthe3dN =4gaugetheory,thesetwo
branches are both hyper-Ka¨hler spaces. The Higgs branch is protected from quantum correction and
therefore the classical description is enough, while Coulomb branch does receive quantum correction
and is described by dressed monopole operators.
Mirror symmetry acts on brane configurationby exchanging D5 brane and NS5 brane, with their
linking number also exchanged. The linking number is conserved quantity of each 5-brane in brane
transitions. The definition is:
1
L = (r−l)+(L−R)
NS
2
(2.1)
1
L = (r−l)+(L−R)
D
2
For eachNS5 (D5) brane, r denotes the number ofD5 (NS5) branes to the rightof this NS5 (D5)
brane, l denotes the number of D5 (NS5) branes to the left of NS5 (D5) brane. The second term is
the net number of D3 brane ending on each 5-brane, from left minus from right.
When all the D5 branes are to the one side of all the NS5 branes, we only need to count for the
net number of D3 branes to keep track of the linking numbers. In these configuration, (σ , σ ,...,σ )
1 2 l
corresponds to the net number of D3 branes on each D5 brane from interior to exterior, and (ρ , ρ ,
1 2
...,ρl′ ) corresponds to the net number of D3 branes on each NS5 brane from interior to exterior.
Given the data of ρ and σ, one can recover the quiver diagram in the following way:
Suppose that ρ=(ρ1,...,ρl′) and σ =(σ1,...,σl) are two partitions of N, which satisfies:
l l′
σ1 ≥...≥σl >0, ρ1 ≥...≥ρl′ >0, σi = ρi =N (2.2)
X X
i=1 i=1
The quiver diagram for Tσ(SU(N)) is as follows:
ρ
N1 N2 Nl′−1
M1 M2 Ml′−1
Figure 1. General A-typeQuiver
– 3 –
l and l′ are the length of partition σ and ρ, ˆl is the length of transpose of σ.
M =σT −σT , withσT =0fori≥ˆl+1
j j j+1 i
l′ ˆl (2.3)
N = ρ − σT
j k i
X X
k=j+1 i=j+1
We can get ρ and σ using the above equations. Then finding the mirror can easily be done by
exchanging ρ and σ, corresponding to exchanging of the linking number of 5-branes.
3 Hilbert Series and Plethystic Logarithm
The Hilbert Series (HS) is a partitionfunction counting BPSoperatorsin the chiralring accordingto
their chargesin different symmetry groups. For example, a (dressed) monopole operator is character-
ized by its topologicalchargeJ(m) andconformaldimension∆(m), and appearsin HS in the formof
zJ(m)t∆(m). The generatorsofthe chiralring canbe convenientlyreadfromthe PlethysticLogarithm
in [1].
ThePlethysticLogarithm(PL)isthereverseoperationofPlethysticExponential,whichgenerates
the symmetric products of different orders. It can be calculated iteratively if we know the expansion
of Hilbert Series.
For example, the unrefined Hilbert Series of quiver (1)-(2)-[3] is
HS =1+8t+35t2+111t3+... (3.1)
(1)−(2)−[3]
Then we try to find whose symmetric products give this expression. From the first order term,
we know that PL=8t+O(t2), but sym2[8t]=36t2. So we need a term −t2 in the PL to compensate
and get the correct 35t2 in the HS. So we know that PL=8t−t2+O(t3). One can do this order by
order and finally gets PL=8t−t2−t3.
The refined version of this process gives the Plethystic Logarithm of refined Hilbert Series as:
1 1 1
PL=(2+z +z + + +z z + )t−t2−t3 (3.2)
1 2 z z 1 2 z z
1 2 1 2
The firstseveralpositivetermsaregenerators. Theirtopologicalchargesconstitute the weightsof
adjoint representation of SU(3) group, the enhanced topological symmetry in the original theory and
flavour symmetry in the mirror theory perspective. The weights are measured by the basis of simple
roots, i.e. z and z correspond to the two simple roots of SU(3). Therefore, in order to write it in
1 2
the standardform of SU(3) characters,one need to do a transformationusing Cartanmatrix in [5] to
shift the basis from simple roots to fundamental weights in Weyl chamber. In our subsequent results
of Hilbert Series or PL, we will denote the character using Dynkin label. So in this example, it is
represented as PL=[11]t−t2−t3.
The negativeterms representrelations,andthey transformin singletrepresentation. These infor-
mation helps people finding the concrete form of relations.
Insummary,wefirstcalculatetheHilbertSeriesusingmonopoleformula,andthentakePlethystic
Logarithmofitandlearnhowtherelationstransformundertheenhancedsymmetry. Finally,wecome
to the Higgs branch of the mirror theory, which do not receive quantum correction, and make use of
F-term to find the concrete form of relations.
– 4 –
4
u d
R1 r1 r2
L l l
1 2 1 3 1 2 2 1
Figure 2. Mirror quiver1-2-3-2-1
4 Three kinds of relations
AgeneralA-typequiversatisfiesthreekindsofrelations,i.e. matrixrelation,tracerelationandmeson
relation. The first two types of relations, matrix and trace relations, only rely on the shape of mirror
quiver.
Suppose we have k generators:
M =du, M =dr l u, ...... , M =dr r ...l l u (4.1)
1 2 1 1 k 1 2 2 1
The relations they satisfy can be read from F-terms:
ud+r l +L R =0
1 1 1 1
r l −l r =0
2 2 1 1
l r =0 (4.2)
2 2
R L −L R =0
1 1 2 2
R L =0
2 2
r andl arethe edgesofshortarmofmirrorquiver,R andLarethose oflongarmofmirrorquiver(as
shown in the above figure). Using the F-term relations, we can turn all the r l to r l . For example,
i i 1 1
dr r l l u = dr l r l u. In the following several parts, we will discuss these three kinds of relations
1 2 2 1 1 1 1 1
separately.
4.1 Trace relation
Proposition: the Nth order trace relation is
1
Tr(Σ M M ...M )=0 (4.3)
λm i1 i2 ir
λ
We denote λ′ ={i′,i′...i′} as all the possible partition of N (not ordered) that satisfies i′ +i′ +
1 2 r 1 2
···+i′ =N, with i′ ≤k, where k is the number of generators. Then λ={i ,i ...i } is the quotient
r s 1 2 r
of λ′ by cyclic group. e.g. λ′ = {1,2,3} is identified with λ′ = {2,3,1} to give λ = {1,2,3}. m is
λ
the power of monomial Mi1Mi2...Mir when Mi1Mi2...Mir can be written as (Mi1Mi2...Mis)mλ,
for some s<r.
This relation was conjectured in [8].
To see an example, consider the 6th order trace relation with 2 generators. It is
1 1 1
Tr( M 6+ M 3+M 2M 2+ (M M )2+M M 4)=0 (4.4)
1 2 2 1 2 1 2 1
6 3 2
The proof of the proposition is as follows.
– 5 –
Proof: Degree i generator:
M =d r ...r l ...l u =d (r l )i−1u (4.5)
i 1 1 i i 1 1 1 1 1 1
We consider trace of product of generators in degree N:
Tr(M M M ...)
i j k
=Tr((r l )i−1u d (r l )j−1u d (r l )k−1u d ...u d )
1 1 1 1 1 1 1 1 1 1 1 1 1 1
(4.6)
=Tr((r l )i−1(−L R −r l )(r l )j−1(−L R −r l )(r l )k−1
1 1 1 1 1 1 1 1 1 1 1 1 1 1
(−L R −r l )...(−L R −r l ))
1 1 1 1 1 1 1 1
That is to say,
Tr(M...)=Tr(combinations of (−L R −r l ) and r l ) (4.7)
1 1 1 1 1 1
We add up all the possible degree N terms of this form, and name it S:
S =Tr(r l ...r l )+...+Tr((−L R −r l )...(−L R −r l )) (4.8)
1 1 1 1 1 1 1 1 1 1 1 1
S can also be written by combination of traces of products of generators:
m [ Tr( m M )
S =NTr(M )+N δ(N − s ) i=1 sσ(i)
N i M(TrQ( m M ))
1≤s1...≤Xsm≤N−1 Xi=1 σX∈Sm Qi=1 sσ(i)
+ r δ(r− l s )[ Tr[( lj=1Msρ(j))N/r] (4.9)
1≤r≤XN2,r|N 1≤s1.X..≤sm≤r Xj=1 j ρX∈SlM(TrQ(Qlj=1Msρ(j))N/r)
+Tr(r l r l ...r l )
1 1 1 1 1 1
where denotes the summation over σ ∈S such that m M is not an operator with an
integerPpoσw∈Semr greater than 1, and M(Tr( m Mm )) denoteQs ti=he1 musσlt(ii)plicity of Tr( m M ).
c i=1 sσ(i) i=1 sσ(i)
According to the F-terms, the last terQm equals to 0. And now we are able to proveQthat S equals
to 0.
S is the traceofthe sumofallpossible termscombining (−L R −r l )andr l . We expandthe
1 1 1 1 1 1
parenthesis, then it’s a polynomial of L R and r l .
1 1 1 1
We first consider terms without L R . According to F-terms, these terms equal to 0.
1 1
Now we consider terms with one L R . All these terms are in same configuration, because of the
1 1
rotation invariance of trace. (# of rotations of the configuration) = N.
N
(# of terms)=(# of rotations of the configuration)· (−1)k−1Ck−1
N−1
X (4.10)
k=1
=N(1−1)N−1 =0
Then we consider terms with two consecutive L R . In this case we also have (# of rotations of
1 1
the configuration) = N.
N
(# of terms)=(# of rotations of the configuration)· (−1)k−2Ck−2
N−2
X (4.11)
k=2
=N(1−1)N−2 =0
– 6 –
Similarly, we cancompute the number ofterms inarbitraryconfiguration(withn L R ), and find
1 1
out it’s 0:
N
(# of terms)=(# of rotations of the configuration)· (−1)k−nCk−n
N−n
X (4.12)
k=n
=(# of rotations of the configuration)·(1−1)N−n =0
Finally,therearesometermswhicharepurelytheproductsofN L R ,butasourfirstcase,they
1 1
equal to 0 because of the F-terms.
Now we know that each term of S equals to 0, then S = 0. Therefore we conclude that:
m [ Tr( m M )
0=NTr(M )+N δ(N − s ) i=1 sσ(i)
N i M(TrQ( m M ))
1≤s1...≤Xsm≤N−1 Xi=1 σX∈Sm Qi=1 sσ(i) (4.13)
l [ Tr[( l M )N/r]
+ r δ(r− s ) j=1 sρ(j)
1≤r≤XN2,r|N 1≤s1.X..≤sm≤r Xj=1 j ρX∈SlM(TrQ(Qlj=1Msρ(j))N/r)
Let’s have a quick check for the simplest case, N=2.
Tr(r l (−L R −r l )+(−L R −r l )r l +(−L R −r l )(−L R
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
−r l )+r l r l ) (4.14)
1 1 1 1 1 1
=Tr(−L R −L R +0·r l r l −0·L R r l )=0
1 1 1 1 1 1 1 1 1 1 1 1
And,
Tr(M )=Tr(r l (−L R −r l ))
2 1 1 1 1 1 1
(4.15)
Tr(M 2)=Tr((−L R −r l )(−L R −r l ))
1 1 1 1 1 1 1 1 1
Therefore we have:
2Tr(M )+Tr(M 2)=0 (4.16)
2 1
4.2 Matrix relation
The Nth order matrix relation is
Σλ′Mi′Mi′ ...Mi′ =0 (4.17)
1 2 r
Again λ′ = {i′,i′...i′} is one of all the possible partitions of N (not ordered), with i ≤ k. This
1 2 r s
time we do not quotient it by the cyclic group.
Proposition: Suppose the length of the long arm of mirror quiver is l, there will be a matrix
relation of the above form in each order starting from (l+1)th order.
Proof: Suppose we have k generators,and we want to prove nth (n>k) order matrix relation.
First we enlarge the generators {M ,M ,...,M } to a larger set {M ,M ,...,M }. For s > k,
1 2 k 1 2 n
let
M =dr l r l ...r l u=0. (4.18)
s 1 1 1 1 1 1
Note that the product of r l ’s, in which the number of r l exceeds the length of short arm of
1 1 1 1
mirror quiver, is zero. We claim that:
Σλ′Mi′Mi′ ...Mi′ =(−1)n−1dL1R1L1R1...L1R1u (4.19)
1 2 r
– 7 –
where λ′ ={i′,i′...i′} is a partition of n, with i ≤n.
1 2 r r
Observe that
Σλ′Mi′Mi′ ...Mi′ =d(combinations of r1l1 and −(r1l1+L1R1))u (4.20)
1 2 r
For example
M 4+M M 2+M M M +M 2M +M M +M M +M 2+M
1 2 1 1 2 1 1 2 3 1 1 3 2 4
=d(r l r l r l −(r l +L R )r l r l −r l (r l +L R )r l −r l r l (r l
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
(4.21)
+L R )+(r l +L R )(r l +L R )r l +(r l +L R )r l (r l +L R )
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
+r l (r l +L R )(r l +L R )−(r l +L R )(r l +L R )(r l +L R ))u
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
There should be 2n terms. The strategy is to find 2n−1 pairs and pair them up to turn the
rightmost term to L R :
1 1
d(−r l r l +r l (r l +L R )−(r l +L R )(r l +L R )+(r l
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
(4.22)
+L R )r l )L R u
1 1 1 1 1 1
Then repeat the process to find 2n−2 pairs and pair them up to get:
d(r l −(r l +L R ))L R L R u (4.23)
1 1 1 1 1 1 1 1 1 1
and finally get:
−d(L R L R L R )u (4.24)
1 1 1 1 1 1
WhenthenumberofL R ’sexceedsthelengthoflongarmofmirrorquiver,thetermdL R L R ...L R u
1 1 1 1 1 1 1 1
vanishes. This finishes the proof of the proposition.
4.3 Meson relation
Ingeneral,mesonrelationisreferredtoastherelationwhichinvolvestheanti-symmetrizationofupper
andlowerindicesseparately. Forexample,forA thesecondmesonrelation(M ) [j(M ) l]transforms
3 1 [i 2 k]
in[010]×[010]=[020]+[101]+1. ForA ,ittransformsin[0100]×[0010]=[0110]+[1001]+1. Ingeneral,
4
the two indices of M are in the [10...0] and [0...01]respectively. Thus, the kth anti-symmetrizationof
lower indices give [00...1...0]. The kth anti-symmetrization of up indices is [0...1...00].
k k
If M =M =|M{,z(M} ) [j(M ) l] =M [jM l] is just the 2 by 2 minor of M,|an{dz i}ts vanishing
1 2 1 [i 2 k] [i k]
means rank(M) ≤ 1. We take A -type quiver [2]-(2)-(2)-(2)-[2] as an example. Its mirror has the
3
following form.
4
u d
r
1
l
1 2 1 1
Figure 3. Quiver1-2-1
The generators are M =du and M =dr l u.
1 2 1 1
– 8 –
Proposition: (M ) [j(M ) l] =0
1 [i 2 k]
Proof:
(M ) j =d αu jδ β,i,j =1,2,3,4;α,β =1,2;
1 i i β α
(M ) l =d γu lQ κ,where Q κ =(r l ) κ =(r ) 1(l ) κ
2 k k κ γ γ 1 1 γ 1 γ 1 1
So
(M ) [j(M ) l] =d αd γu [ju l]δ βQ κ
1 [i 2 k] [i k] β κ α γ
=d [αd γ]u ju lδ βQ κ (4.25)
i k [β κ] α γ
=d αd γu ju lδ [βQ κ]
i k β κ [α γ]
and
δ [βQ κ] =ǫ ǫαγδ βQ κ
[α γ] βκ α γ
=δ γQ κ (4.26)
κ γ
=Tr(Q)
Using F-term, we have Tr(Q) = Tr(r l ) = Tr(l r ) = 0. So (M ) [j(M ) l] = 0. Generator M
1 1 1 1 1 [i 2 k] 2
is of rank 1, so (M ) [j(M ) l] =0. In sum, there are two meson relations: (M ) [j(M ) l] =0 and
2 [i 2 k] 1 [i 2 k]
(M ) [j(M ) l] =0.
2 [i 2 k]
Inadditiontothiskindofmesonrelationwiththeanti-symmetrizationoftwomatrices,otherme-
sonrelationsinvolvingmore matrices alsoexist. For example,we considerquiver {{12221},{01010}}1
whose mirror is Figure 4
6
u d
2
Figure 4. Quiver2-[6]
The generator M has rank 2, so one would expect a relation M [jM lM q] = 0. It transforms
[i k p]
in Λ3[10000]×Λ3[00001]=[00200]+[01010]+[10001]+[00000]. This relation emerges at A quiver,
5
because for lower rank quiver like A , Λ3[1000]×Λ3[0001] = [0010]×[0100] = Λ2[1000]×Λ2[0001].
4
In general, for every A -type quiver with n odd, a meson that transforms in a new representation
n
emerges.
For higher degree generators, the meson relation can be proved iteratively (just as we get rid of
outermost d and u and leave the anti-symmetrized inner part, one can continue to do this to get rid
of the outer part).
5 A -type quiver
2
5.1 General pattern
The A -type quiver has a very clear pattern.2 First, from the Hilbert Series, one conjectures that
2
there are only trace relations (in singlet representation) and matrix relations (in adjoint represen-
1Meaning of this symbol: the first parenthesis contains the gauge nodes, and the second parenthesis contains the
flavournodes atthecorrespondingposition.
2SeeAppendixBforthePLandrelationsofsomeA2-typequivers.
– 9 –
