SelectSciPost Phys. 7, 056 (2019)

Exceptional Chern-Simons-Matter dualities
Clay Córdova1? , Po-Shen Hsin 2† and Kantaro Ohmori 1‡
1 School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, USA
2 Walter Burke Institute for Theoretical Physics,
California Institute of Technology, Pasadena, CA, USA? claycordova@ias.edu, † phsin@caltech.edu, ‡ kantaro@ias.edu,

Abstract
We use conformal embeddings involving exceptional affine Kac-Moody algebras to derive new dualities of three-dimensional topological field theories. These generalize thefamiliar level-rank duality of Chern-Simons theories based on classical gauge groups tothe setting of exceptional gauge groups. For instance, one duality sequence we discuss is(E N )1 ↔ SU(9− N)−1 . Others such as SO(3)8 ↔ PSU(3)−6 , are dualities among theories
with classical gauge groups that arise due to their embedding into an exceptional chiralalgebra. We apply these equivalences between topological field theories to conjecture
new boson-boson Chern-Simons-matter dualities. We also use them to determine candidate phase diagrams of time-reversal invariant G2 gauge theory coupled to either anadjoint fermion, or two fundamental fermions.
Copyright C. Córdova et al.
This work is licensed under the Creative Commons
Attribution 4.0 International License.
Published by the SciPost Foundation.

Received 31-07-2019
Accepted 18-10-2019
Check forupdatesPublished 28-10-2019
doi:10.21468/SciPostPhys.7.4.056

Contents
1 Introduction

1

2 Duality of Chern-Simons Theories
2.1 Conformal Embeddings and Chiral Algebra Dualities
2.2 Exceptional Chern-Simons Dualities
2.3 Examples

4
4
5
7

3 Chern-Simons-Matter Dualities
3.1 S p(3) ↔ SU(3) or U(1)3.2 Time-Reversal Invariant F4 Theory
3.3 E6 ↔ G2
3.4 EN ↔ SU or U(1)

9
9
10
10
11

4 Phase Diagrams of (G2 )0 with Fermions
4.1 (G2 )0 with a single adjoint fermion
4.2 (G2 )0 with two fundamental fermions

12
13
14

A Potential for Higgsing

15

References

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Introduction

In this paper we derive new dualities of three-dimensional Chern-Simons theories, and use
them to propose new dualities of Chern-Simons-matter theories. Our results for TQFTs generalize the familiar level-rank dualities of Chern-Simons theories with classical gauge groups
which are of the form:
SU(N )K ↔ U(K)−N ,−N ,U(N )K,K±N ↔ U(K)−N ,−N ∓K ,SO(N )K ↔ SO(K)−N ,O(N )1K,K−1+L ↔ O(K)1−N ,−N +1+L ,S p(N )K ↔ S p(K)−N ,O(N )0K,K ↔ S pin(K)−N .
(1.1)
In the case of unitary gauge groups, these dualities were discussed in [1–6]. The dualities for
SO and S p gauge groups were derived in [7]. The O(N ) and S pin(N ) results were derived
in [8].1 To these, we will contribute the following identities:2(E8 )1 ↔ trivial ↔ SU(9)1 /Z3 ,SU(6)1 ↔ (E7 )1 × SU(3)−1 ,SU(5)1 ↔ SU(5)−1 ,(E7 )1 ↔ SU(2)−1 ↔ SU(8)1 /Z2 ,SU(2)3 ↔ (E7 )1 × (F4 )−1 ,SO(3)8 ↔ PSU(3)−6 ,(E6 )1 ↔ SU(3)−1 ↔ PS p(4)1 ,SU(2)7 ↔ (E7 )1 × (G2 )−2 ,(G2 )1 ↔ (F4 )−1 .

(1.3)
An important feature of the dualities (1.3) (unlike some of the entries in (1.1) [6]) is that they
are equivalences among bosonic TQFTs, i.e. they do not require a choice of spin structure. We
can also combine (1.3) with some of the results of (1.1) to show that(G2 )1 ←→ U(2)3,1 ,(G2 )2 ←→ U(2)−7,−1 .

(1.4)

One way to summarize some of the dualities in (1.3) is via the equivalence(EN )1 ←→ SU(9 − N )−1 .

(1.5)

In particular, as discussed in section 2 this naturally extends to all 0 ≤ N ≤ 8 with a suitable
identification of the group EN for small N .
The typical path to proving such dualities of Chern-Simons theories is to establish theequivalence of their corresponding chiral algebras. For connected and simply connected gauge
groups, these are standard Kac-Moody current algebras (for a review see e.g. [9]). Gauging
additional discrete symmetries, or reducing the gauge group by a central quotient modifies thechiral algebra respectively by an orbifold or an extension [10, 11].
One way to prove that two chiral algebras are equivalent is to start from a chiral algebraof a larger group at unit level (realized for a classical group for instance by suitable free 2dfermions.) We then decompose the initial algebra under a conformally embedded productchiral algebra, with each factor related to a given Chern-Simons theory (see e.g. [9]). In
section 2, we briefly review this procedure in the case of classical groups, and then we follow it
for conformal embeddings of exceptional Kac-Moody algebras (EN )1 , (F4 )1 , and (G2 )1 studiedin [12–15].
The outcome of our analysis are the dualities (1.3) and (1.4). In each case, we track therelative gravitational Chern-Simons counterterm across the duality. For some of these dualities,
1
For some groups in (1.1), a careful definition of the Chern-Simons theory requires multiple levels. In theunitary group case the SU(N ) and U(1) subalgebras can have separate integral levels [6] and we set

U(N )K,L ≡ (SU(N )K × U(1)N L ) /ZN .

(1.2)

Meanwhile for the orthogonal group there are two additional discrete levels indicating the action for the gauged
reflection. In the notation above these are a superscript valued mod two and a second subscript valued modeight [8].
2
Our level conventions are SO(3)K ≡ SU(2)2K /Z2 , and PSU(3)K ≡ SU(3)K /Z3 .

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SciPost Phys. 7, 056 (2019)(G2 )0 + ψadj
U(2) 13 , 1 + Ψfund + 7 CSgrav
20
U(2)− 13 ,− 1 + Ψfund− 7 CSgrav

2

2

2

χ̄
m=0

m → −∞(G2 )−2 − 7 CSgravlU(2)7,1 + 9 CSgrav

U(2)6,0 + 5 CSgravlU(2)−6,0 − 4 CSgravm→∞
U(2)6,0 + 4 CSgravlU(2)−6,0 − 5 CSgrav(G2 )2 + 7 CSgravlU(2)−7,−1 − 9 CSgrav

Figure 1: A conjectural phase diagram of (G2 )0 coupled to an adjoint Majorana
fermion as a function of the fermion mass m. When m = 0 the theory is time-reversal
invariant, and has N = 1 supersymmetry. At large |m| the IR is described by a TQFT
visible semiclassically. As |m| is reduced there is a transition to a quantum phase.
This transition is weakly coupled in dual variables with a U(2) gauge field and a fundamental fermion. At m = 0 there is also massless fermion χ̄ which is the Goldstino
arising from the expected spontaneous breaking of supersymmetry [30]. In eachphase, we also indicate the coefficient of the gravitational Chern-Simons term which
can be used as a consistency check on the phase diagram (see section 4).
we also describe in some detail the spins of the anyons as well as the map of lines between thetwo presentations of the theory.
In section 3 we apply our topological field theory analysis to conjecture several new bosonboson Chern-Simons-matter dualities.3 They take the form:(EN )1 + N f Φfund←→

SU(9 + N f − N )−1 + N f Φ0fund ,(1.6)

where in the above 1 ≤ N ≤ 8, N f ≤ N , and the meaning of the fundamental representationof EN will be clarified below. As well as:(E6 )1 + Φ27←→(G2 )−1 + Φ07 ,S p(3)1 + Φ140←→

(G2 )−1 × SU(3)−1 + Φ03 ,←→(F4 )−1 + Φ026 ,(F4 )1 + Φ26(1.7)

where in the last line above, the content of the duality is that the IR theory has emergent
time-reversal symmetry.
Our analysis parallels that of [6–8, 16, 17], which analyzed boson-fermion dualities forclassical groups, extending previous large N analysis in [18–20], connections to related supersymmetric dualities in [21–23], and recent work on particle vortex duality [24–27] buildingon the foundational results in [28, 29]. In each case of (1.6)-(1.7), relevant deformations of
the proposed dualities yield equivalences among TQFTs which can be verified from our earlier
results (1.3).
As a final application of our TQFT results, in section 4 we study the phase diagram of (G2 )0
coupled to either an adjoint Majorana fermion, or two Majorana fermions in the fundamental
7. Our results for the adjoint Majorana case are illustrated in Figure 1.
Phase diagrams of this sort have been investigated for classical groups in [8, 31–34]. As afunction of the fermion masses there are semi-classical phases which are visible when |m| islarge. As |m| is reduced there is a phase transition to a new quantum phase which is not obvious
3

Using known boson-fermion dualities, we can also obtain fermionic duals of some of these theories.

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SciPost Phys. 7, 056 (2019)from the UV Lagrangian. In our models, the transitions to the quantum phases are weakly
coupled in dual variables with U(2) gauge groups which are motivated by the TQFT dualities(1.4). The transition to the quantum phase is mediated by a U(2) fundamental fermion dialing
through zero mass.
In the case of (G2 )0 coupled to an adjoint fermion, the quantum phase we discuss includes
the special point m = 0 where the theory has 3d N = 1 supersymmetry. Our results thus
join a growing body of recent literature on such models [35–39]. For (G2 )0 coupled to two
fundamental fermions, the theory enjoys a U(1) global symmetry and in the quantum phase
this symmetry is spontaneously broken leading to a periodic scalar at long distances as in[32, 34, 40–43].

2

Duality of Chern-Simons Theories

2.1

Conformal Embeddings and Chiral Algebra Dualities

A conformal embedding of a 2d chiral algebra is the decomposition of a chiral algebra into
smaller chiral algebras while respecting the conformal symmetry (see e.g. [9]). Such decompositions give rise to equivalences among chiral algebras and hence dualities of the corresponding
Chern-Simons theories [11].
In general, one starts with a affine Kac-Moody algebra at unit level and decomposes it
into a finite product of other Kac-Moody algebras.4 Relatedly one obtains a decomposition ofrepresentations of the initial chiral algebra into modules of the product. To obtain a duality,
it is essential that each chiral algebra factor acts faithfully and that all modules appear in thedecomposition of some representation. To account for selection rules, one must often extend
the chiral algebra. This is equivalent to performing a quotient on the gauge group in the
Chern-Simons theory [10, 11].
Let us illustrate this idea briefly in several examples. The first is the decomposition of
the chiral algebra SU(N K)1 into SU(N )K × SU(K)N (see e.g. [44, 45]). All representations of
SU(N )K and SU(K)N appear in the decomposition. Each representation of SU(N )K is pairedwith a representation of the coset SU(N K)1 /SU(K)N . This gives a duality between chiralalgebras. Using the map between cosets and Chern-Simons theories from [11] we find theChern-Simons duality [45]SU(N )K←→

SU(N K)1 × SU(K)−N
.
ZK

(2.1)

This is an equivalence between bosonic TQFTs and does not require a spin structure. The
quotient denotes a gauging of the diagonal ZK one-form symmetry [46] (for a recent review
see e.g. [47]), and it corresponds in 2d to extending the chiral algebras [10,11]. This basic 2dchiral algebra duality can be generalized to a rich set of level-rank dualities discussed in [6].
Another example is the decomposition of the chiral algebra S pin(N K)1 , which represents
N K real fermions in 2d [8, 44, 48].
S pin(N K)1 ⊃ S pin(N )K × S pin(K)N .

(2.2)

Depending on N and K, not every highest weight representation of the subalgebra S pin(N )K
can appear in the decomposition of representations of S pin(N K)1 . If the representations of
the subalgebra that appear only consist of tensorial representations (i.e. those that can be built
from the vector representation), then we extend the chiral algebra to SO(N )K by including
4

If the initial algebra is not at unit level, no such finite product decompositions exist [12, 13].

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SciPost Phys. 7, 056 (2019)the simple current corresponding to the Z2 center in S pin(N ) that gives rise to the quotient
SO(N ) = S pin(N )/Z2 . The conformal embedding then implies a duality between S pin(N )K(or SO(N )K if it is extended) and the coset S pin(N K)1 /S pin(K)N or S pin(N K)1 /SO(K)N
depending on the representations of S pin(K) that appear in the decomposition. This gives
rise to dualities of the corresponding Chern-Simons theories [7, 8] which are stated in (1.1).
In the rest of the section we will apply the above method to the conformal embeddings into
the exceptional chiral algebras (EN )1 , (F4 )1 and (G2 )1 which were studied in [12–15]. Theyare:(E8 )1 ⊃ SU(9)1 ,(E8 )1 ⊃ SU(2)16 × SU(3)6 ,(E8 )1 ⊃ (E7 )1 × SU(2)1 ,(E8 )1 ⊃ SU(5)1 × SU(5)1 ,(E8 )1 ⊃ (E6 )1 × SU(3)1 ,(E8 )1 ⊃ (G2 )1 × (F4 )1 ,(E7 )1 ⊃ SU(8)1 ,(E7 )1 ⊃ S pin(12)1 × SU(2)1 ,(E7 )1 ⊃ SU(6)1 × SU(3)1 ,(E7 )1 ⊃ SU(2)3 × (F4 )1 ,(E7 )1 ⊃ (G2 )1 × S p(3)1 ,(E7 )1 ⊃ SU(2)7 × (G2 )2 ,(E6 )1 ⊃ S p(4)1 ,(E6 )1 ⊃ SU(6)1 × SU(2)1 ,(E6 )1 ⊃ SU(3)2 × (G2 )1 ,(F4 )1 ⊃ SU(3)2 × SU(3)1 ,(F4 )1 ⊃ S p(3)1 × SU(2)1 ,(F4 )1 ⊃ (G2 )1 × SU(2)8 ,(G2 )1 ⊃ SU(2)3 × SU(2)1 .

(2.3)
The decompositions for representations under these conformal embedding are given in [13].5

2.2

Exceptional Chern-Simons Dualities

From the conformal embeddings (2.3) and the corresponding decomposition of the representations [13] we find the following dualities between non-spin Chern-Simons theories(E8 )1←→

trivial − 16CSgrav←→

SU(9)1 /Z3 ,(E7 )1←→

SU(2)−1 − 16CSgrav←→

SU(8)1 /Z2 ,(E6 )1←→

SU(3)−1 − 16CSgrav←→

PS p(4)1 ,SU(5)1←→

SU(5)−1 − 16CSgrav ,SU(6)1←→(E7 )1 × SU(3)−1 ,SO(3)8←→

PSU(3)−6 − 16CSgrav ,(G2 )1←→(F4 )−1 − 16CSgrav ,SU(2)3←→(E7 )1 × (F4 )−1 ,SU(2)7←→(E7 )1 × (G2 )−2 .

(2.5)

In the above we have included the relative value of the gravitational Chern-Simons term CSgrav
R
across each duality. Our normalization is such that CSgrav = π 4d Â(R). When the coefficientis a multiple of 16 (as in all cases above), this counterterm does not require a choice of spin
structure. Hence (−16)CSgrav defines an invertible non-spin TQFT with framing anomaly c = 8
that has only the trivial line, such as the (E8 )1 Chern-Simons theory.6
5

In the above list we include only the conformal embeddings where we can identify all the chiral algebras
yielding the representations that appear in the decompositions. Examples of conformal embeddings not listed
above are:(E6 )1 ⊃ SU(3)9 , (E6 )1 ⊃ (G2 )3 , (E7 )1 ⊃ SU(3)21 , (E8 )1 ⊃ Sp(2)12 .
(2.4)

In general, the invertible spin TQFT SO(L)1 has partition function exp −i LCSgrav (see e.g. Appendix B in [27]).
R
The theory has framing anomaly c = L/2. Since 8 4d Â(R) ∈ Z on any closed orientable four-manifold, for L ∈ 16Z
the gravitational Chern-Simons term does not require a spin structure.
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Let us comment on some aspects of these dualities:• The theories (EN )1 and SU(N )1 are Abelian TQFTs.7 Thus, the dualities in (2.5) that involve only (EN )1 and SU(N )1 can be alternatively obtained from the defining properties
of Abelian TQFTs. Namely, an Abelian TQFT is specified by the fusion rules, the spin of
the lines, and the framing anomaly (for a review see e.g. [49]).
• The duality SU(5)1 ↔ SU(5)−1 − 16CSgrav implies the theory SU(5)1 is time-reversal
invariant as a non-spin TQFT. Using SU(5)1 ↔ SO(5)2 ∼= S p(2)2 /Z2 , this also follows
from the time-reversal symmetry of S p(2)2 [7]. This duality was also discussed in [50].
• The last three dualities in (2.5) give examples of a phenomenon recently discussed in[47]. Specifically, they present factorizations of TQFTs into the product of a “minimal
Abelian TQFT” [47] and another TQFT. The factorization is a consequence of the oneform global symmetry [47].
One way to summarize the first five dualities in (2.5) is8(EN )1←→

SU(9 − N )−1 − 16CSgrav ,(2.6)

where E5 ∼= S pin(10), E4 ∼= SU(5), and E3 ∼= SU(3) × SU(2). This extends to integers
90 ≤ N ≤ 8 provided that we define(E2 )1 ≡ U(2)1,7 ,(E1 )1 ≡SU(2)1 ×(Z4 )14,
Z2

e1 )1 ≡ U(1)8 ,(E(E0 )1 ≡ (Z3 )2 .

(2.7)

In the above, we use the convention that (Zn )k is the Zn topological gauge theory that canknbe expressed using U(1) × U(1) gauge fields x, y with the action 4πx d x + 2π
x d y. Moreover,H
H
the Z2 quotient in (E1 )1 on (Z4 )14 uses the line exp(3i x + 2i y). Finally, the two theoriese1 )1 are both dual to SU(8)−1 (and hence dual to each other). We list them both
(E1 )1 and ( Ehere for later application to dualities involving matter in section 3.
Apart from our unusually careful attention to the global form of the group, the above
definition of EN for small N are standard and natural from e.g. the perspective of Higgsing.10
This will play a crucial role in our discussion of Chern-Simons-matter dualities in section 3.
We can also use the level-rank duality [6] to dualize the right-hand side of (2.6) to obtain an
equality of spin TQFTs
spin TQFTs:(EN )1←→

U(1)−N +9 − 2(N − 1)CSgrav ,(2.8)

where on both sides we tensor with the theory {1, ψ} (ψ is the transparent spin 1/2 line) withzero framing anomaly to turn them into spin TQFTs.11
Finally, let us remark that the last two dualities in (2.5) imply that(F4 )−1 ←→ SU(2)3 × (E7 )−1 /Z2 ,(G2 )−2 ←→ SU(2)7 × (E7 )−1 /Z2 .
(2.9)
7They can be described as Abelian Chern-Simons theories with K-matrices given by their Cartan matrices.
The duality (2.6) with N = 5 is a special case of the duality Spin(L)1 ↔ Spin(16 − L)−1 − 16CSgrav using
SU(4) ∼
= S pin(6), and we include it in (2.6) for completeness. This duality for general L can be derived from theseries of conformal embeddings S pin(L)1 × S pin(16 − L)1 ⊂ Spin(16)1 ⊂ (E8 )1 [12]. This “16 periodicity” for the
TQFT described by S pin(L)1 is also discussed in [51].
9
Since the theories (EN )1 and SU(9 − N )−1 are Abelian TQFTs, the dualities can be proven from the fusion rules
and the spin of the lines.
10
For instance they also occur in the discussion of five-dimensional superconformal field theories [52, 53]. The
disconnected parts of E1 and E0 also exist in the global symmetries of the corresponding superconformal field
theories, which are the subgroups preserving the (p, q) brane webs engineering these theories.
11
For N = 1 the above duality is also valid between non-spin TQFTs [7, 54].
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Using (F4 )−1 ↔ (G2 )1 + 16CSgrav , (E7 )1 ↔ SU(2)−1 − 16CSgrav in (2.5) and SU(2)1 ↔ U(1)2
we can further simplify them into the non-spin dualities(G2 )1 ←→ U(2)3,1 ,(G2 )2 ←→ U(2)−7,−1 − 16CSgrav .

(2.10)

We will make use of these TQFT dualities in section 4.

2.3

ExamplesThe map of representations/lines across the dualities (2.5) can be obtained from the decomposition of modules in the conformal embeddings [13]. Here we discuss some of them.
Consider the duality(EN )1←→

SU(9 − N )−1 − 16CSgrav ,(2.11)

where for 5 ≥ N ≥ 0 the left-hand side is given in (2.7). The lines on the left-hand side
are labelled by the Dynkin labels λ1 , λ2 , · · · , λn , while the lines on the right-hand side arelabelled by the antisymmetric tensor representation with r indices with r = 0, 1, · · · , (8 − N )mod (9 − N ).
• N = 8. The highest weight representations in (E8 )1 satisfy
2λ1 + 3λ2 + 4λ3 + 5λ4 + 6λ5 + 4λ6 + 2λ7 + 3λ8 ≤ 1 ,(2.12)and thus the Chern-Simons theory only has the trivial line, which maps to the trivial lineon the other side of the duality.
• N = 7. The highest weight representations in (E7 )1 satisfy
2λ1 + 3λ2 + 4λ3 + 3λ4 + 2λ5 + λ6 + 2λ7 ≤ 1 ,(2.13)and thus there are two lines, with λ6 = 0, 1 and all other Dynkin labels vanishing. The
two lines map to r = 0, 1 on the right-hand side of the duality. The lines have spinh[r] = 34 r 2 mod 1.
• N = 6. The highest weight representations in (E6 )1 satisfy
λ1 + 2λ2 + 3λ3 + 2λ4 + λ5 + 2λ6 ≤ 1 ,(2.14)and thus there are three lines, with (λ1 , λ5 ) = (0, 0), (0, 1), (1, 0) and all other Dynkinlabels vanishing. The three lines map to r = 0, 1, 2 on the right-hand side of the duality.
The lines have spin h[r] = 23 r 2 mod 1.
• N = 5. The highest weight representations in (E5 )1 ∼= S pin(10)1 satisfy
λ1 + 2λ2 + 2λ3 + λ4 + λ5 ≤ 1 ,(2.15)and thus there are 4 lines, with (λ1 , λ4 , λ5 ) = (0, 0, 0), (0, 1, 0), (1, 0, 0), (0, 0, 1) and allother Dynkin labels vanishing. The three lines map to r = 0, 1, 2, 3 on the right-handside of the duality. The lines have spin h[r] = 58 r 2 mod 1.
• N = 4. The highest weight representations in (E4 )1 ∼= SU(5)1 are antisymmetric tensor
representations with r = 0, 1, 2, 3, 4 indices. There are 5 lines, and they map to therepresentations in SU(5)−1 byr −→ r 0 = 2r .
(2.16)
This map has inverse r = 3r 0 . The lines have spins h[r] = 25 r 2 mod 1.
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SciPost Phys. 7, 056 (2019)• N = 3. The lines in (E3 )1 ∼= SU(3)1 × SU(2)1 theory can be labelled by antisymmetrictensor representations in SU(3) and SU(2) with r1 and r2 indices, with r1 = 0, 1, 2 andr2 = 0, 1. There are 6 lines, and they map to the lines in SU(6)1 theory as r-index
antisymmetric tensor representations with r = 2r1 + 3r2 mod 6. The lines have spins
7 2
h[r] = 12
r mod 1.
• N = 2. The lines in (E2 )1 ≡ U(2)1,7 can be labelled by SU(2) isospin zero and U(1)charge Q = 2q = 0, 2, 4, 6, 8, 10, 12, and thus there are 7 lines. They map to the antisymmetric tensor representations of SU(7) with r = 3q (mod 7) indices. The lines havespins h[r] = 47 r 2 mod 1.
• N = 1. The lines in (E1 )1 ≡ SU(2)1 × (Z4 )14 /Z2 can be labelled by powers p of theHbasic magnetic line exp(i y) in (Z4 )14 , which has order 8. They map to the antisymmetric tensor representations of SU(8) with p (mod 8) indices. The lines have spins
9 2
p mod 1.
h[p] = 16• N = 0.H The lines in (E0 )1 ≡ (Z3 )2 can be labelled by powers p of the basic magnetic line
exp(i y), which has order 9. They map to the antisymmetric tensor representations of
SU(9) with r = 4p (mod 9) indices. The spins are h[r] = 95 r 2 mod 1.
Consider the duality
SO(3)8←→

PSU(3)−6 − 16CSgrav .

(2.17)The representations on the right have SU(2) isospin j = 0, 1, 2, 3, 4± where the subscript is
associated with the Z2 quotient of SO(3)8 = SU(2)16 /Z2 . There are 6 lines, and they map to
the PSU(3)−6 theory as the lines with SU(3) Dynkin labels(λ1 , λ2 ) = (0, 0), (2, 2)0 , (3, 0), (1, 1), (2, 2)1 , (2, 2)2 ,(2.18)

where the subscript denotes the 3 copies associated with the Z3 quotient in PSU(3)6 =SU(3)6 /Z3 . The lines have spins h = 0, 19 , 13 , 23 , 19 , 19 mod 1. The theory has S3 0-form symmetry that permutes the 3 copies associated with the Z3 quotient.12
Consider the duality(G2 )1 ←→ (F4 )−1 − 16CSgrav .
(2.20)The highest weigh representations for (G2 )1 satisfy 2λ1 + λ2 ≤ 1 and thus there are two lines,
with λ2 = 0, 1 and λ1 = 0. The highest weight representations for (F4 )1 satisfy
2λ01 + 3λ02 + 2λ03 + λ04 ≤ 1, and thus the theory also has two lines, with λ04 = 0, 1 and allother Dynkin labels vanish. The map of the duality is then λ04 = λ2 . The lines have spins
h = 0, 52 mod 1.
Consider the duality(G2 )1 ←→ U(2)3,1 .
(2.21)
12

The left-hand side of the duality (2.17) has another description [7]SO(3)8 ←→S pin(24)1 × Spin(8)−3Spin(8)1 × Spin(8)−3←→− 16CSgrav ,Z2 × Z2
Z2 × Z2(2.19)

where the Z2 ×Z2 quotient uses the diagonal center, and the second duality uses Spin(24)1 ↔ Spin(8)1 −16CSgrav .
The TQFT consists of the lines in S pin(8)−3 that are invariant under the Z2 × Z2 one-form symmetry. The lines in
S pin(8)−3 that generate the one-form symmetry have Dynkin labels (0, 0, 0, 0), (3, 0, 0, 0), (0, 0, 0, 3) and (0, 0, 3, 0),
which form closed orbits under the S pin(8) triality. Thus the triality is a symmetry of the TQFT, and it reproduces
the 0-form permutation symmetry.

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The representations on the right-hand side can be labelled by zero U(1) charge and SU(2)isospin j = 0, 1. These two lines map to the lines λ2 = 0, 1 in (G2 )1 . We remark that thedualities (2.20),(2.21) provide different Chern-Simons theory descriptions for the Fibonacci
Anyons (see e.g. [55–57]), namely, the only non-trivial line τ in the theory has the fusion rule
τ × τ = 1 + τ.
Consider the duality(G2 )2←→U(2)−7,−1 − 16CSgrav .

(2.22)The highest weight representations in (G2 )2 satisfy 2λ1 + λ2 ≤ 2 and thus there are four lines,
with (λ1 , λ2 ) = (0, 0), (0, 2), (0, 1), (1, 0). They map to the lines on the right-hand side of U(1)charge zero and SU(2) isospin j = 0, 1, 2, 3. The lines have spins h = 0, 97 , 13 , 23 mod 1.

3

Chern-Simons-Matter Dualities

In this section we conjecture new Chern-Simons-matter dualities. They enjoy the consistency
check that under relevant deformations, they flow to the rigorous dualities of pure ChernSimons theories discussed in section 2. In particular, this also implies that all anomalies ofglobal symmetries match across our dualities. We can also verify that the gravitational ChernSimons counterterms in the dualities are reproduced in a non-trivial way from integrating out
massive fermions
The Chern-Simons-matter theories that participate in the dualities have small Chern-Simonslevels and small gauge groups. Thus the dualities discussed here do not admit an obvious
semi-classical limit, unlike the dualities proposed for other gauge groups such as in [6, 7]. (Itis perhaps possible that they can be obtained from supersymmetric dualities as in [21–23].)
Throughout, we assume that theories involving scalars are driven to fixed points by a suitable potential. Moreover we often have need of various potentials achieving desired patternsof Higgsing. The general existence of these potentials is discussed in Appendix A.
We adopt the convention that Φ is a complex scalar, and Ψ is a complex (Dirac) fermion.
We also set m L,R to be the relevant deformation parameter on the left and right-hand side ofthe dualities.
The dualities proposed below involve complex scalars or Dirac fermions, and the theories
have (at least) a U(1) global symmetry. In particular we assume that all scalar potentials
respect the global phase rotation Φ → e iα Φ. We check that the dualities are consistent underrelevant deformations preserving this symmetry.

3.1

S p(3) ↔ SU(3) or U(1)We propose that the following theories are dual (IR equivalent):
S p(3)1 + Φ140←→

(G2 )−1 × SU(3)−1 + Φ03 − 16CSgrav←→(G2 )−1 × U(1) 5 + Ψ − 11CSgrav ,
2

(3.1)

where the second line uses the SU/U duality discussed in [6] with Ψ a Dirac fermion of charge
one.
Note that S p(3) has a U(3) subgroup and the index of the SU(3) subgroup is two. Under
U(3) the 140 decomposes as
140 → 13 ⊕ 1−3 ⊕ 6−1 ⊕ 6̄1 .

9

(3.2)

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We assume that on the left-hand side of (3.1) the potential is such that for one sign of therelevant operator an SU(3) singlet gets an expectation value. (The unbroken Z3 in U(1) is infact part of the SU(3)).
With this setup we can now investigate the massive phases. The fixed point has a relevant
operator described by the mass terms in the dual descriptions. The potentials in the scalar
theories are such that the mass deformations lead to the following pure Chern-Simons dualitiesin (2.5):
m2L > 0 ↔ m2R < 0 :S p(3)1 ←→ (G2 )−1 × SU(2)−1 − 16CSgrav ,m2L < 0 ↔ m2R > 0 :SU(3)2 ←→ (G2 )−1 × SU(3)−1 − 16CSgrav .

(3.3)

Let us explain in more detail how to obtain the dualities (3.3) from (2.5). To compare the
top line of (3.3) with (2.5), we use the two spin dualities S p(3)1 ↔ S p(1)−3 − 12CSgrav ∼=SU(2)−3 − 12CSgrav , SU(2)−1 ∼= S p(1)−1 ↔ S p(1)1 + 4CSgrav ∼= SU(2)1 + 4CSgrav [7].13 The
second duality in (3.3) is equivalent to (G2 )−1 being the non-Abelian sector of SU(3)2 , namely,(G2 )−1 ↔ (SU(3)2 × SU(3)1 ) /Z3 + 16CSgrav [47]. The later follows from (2.10) and the two
spin dualities U(2)−3,−1 ↔ U(3)2,−1 + 10CSgrav , U(1)−3 ↔ SU(3)1 + 6CSgrav [6].

3.2Time-Reversal Invariant F4 Theory

We conjecture the IR duality(F4 )1 + Φ26←→(F4 )−1 + Φ026 − 16CSgrav .

(3.4)

Observe that the two theories above are equivalent up to an action of time-reversal symmetry.
Thus the claim in the duality is that the IR theory is time-reversal invariant.
Note that F4 has a G2 × SO(3) subgroup of index (1,4). Under this subgroup, the 26
decomposes as
26 → (7, 3) ⊕ (1, 5) .
(3.5)
We choose the potential on both sides such that for one sign of the relevant operator, a fieldin the (1, 5) gets a generic expectation value.14 This Higgses F4 to G2 (there is no additional
discrete subgroup by our assumption of a generic expectation value.)In the gapped phases we find dualities from (2.5):

3.3

m2L > 0 ↔ m2R < 0 :(F4 )1 ←→ (G2 )−1 − 16CSgrav ,m2L < 0 ↔ m2R > 0 :(G2 )1 ←→ (F4 )−1 − 16CSgrav .

(3.6)

E6 ↔ G2

We conjecture the IR duality(E6 )1 + Φ27←→(G2 )−1 + Φ07 − 16CSgrav .

(3.7)

Note that E6 has an F4 subgroup of index 1. Under this subgroup, the 27 decomposes as
27 → 1 ⊕ 26 .
13

(3.8)

We also use the fact that a spin duality implies a non-spin duality when the sectors other than the factorized {1, ψ} are non-spin TQFTs, and their framing anomaly differs by a multiple of 8 (i.e. the coefficients of thegravitational Chern-Simons term on the two sides differ by a multiple of 16) [54].
14
By generic, we mean that Φ and Φ̄ do not commute as matrices in the 5 (the symmetric traceless tensor).

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We choose the potential such that for one sign of the relevant operator, a field in the 1 getsan expectation value. We also use the fact that G2 has an SU(3) subgroup of index 1. Under
SU(3) the 7 decomposes as
7 → 1 ⊕ 3 ⊕ 3̄ .
(3.9)
We chose the potential such that for one sign of the relevant operator, a field in the 1 gets an
expectation value.
In the gapped phases we find the following dualities from (2.5):

3.4

m2L > 0 ↔ m2R < 0 :(E6 )1 ←→ SU(3)−1 − 16CSgrav ,m2L < 0 ↔ m2R > 0 :(F4 )1 ←→ (G2 )−1 − 16CSgrav .

(3.10)EN ↔ SU or U(1)We conjecture the IR duality(EN )1 + Φfund

SU(10 − N )−1 + Φ0fund − 16CSgrav ,←→(3.11)

where the allowed range of N is N ≥ 3 (it will be extended below), and the definition of thegroup EN for general N is as specified in (2.7). Meanwhile, the definition of the fundamental
representation of EN as follows:
E8 : 248 ,E7 : 56 ,E4 ∼= SU(5) : 10 ,E6 : 27 ,E5 ∼= S pin(10) : 16 ,(3.12)E3 ∼= SU(3) × SU(2) : (3, 2) .

As we will see below, the essential property of this definition of the fundamental of EN is thata generic expectation value of such a fundamental can Higgs EN to EN −1 .
The duality sequence (3.11) can be extended to smaller N as follows. For N = 2 we have
two dualities:(E2 )1 ∼= U(2)1,7 + 23

(E2 )1 ∼= U(2)1,7 + 14

SU(8)−1 + Φ0fund − 16CSgrav .
(3.13)
e1 (but none for E1 ):15
Meanwhile for N = 1, there is an analogous duality involving E←→e1 )1 ∼
(E
= U(1)8 + 13←→←→

SU(9)−1 + Φ0fund − 16CSgrav .

(3.14)In all of the dualities above, for one sign of the relevant operator, the scalar Φ on the lefthand side does not condense but the scalar Φ0 condenses and Higgses SU(10−N ) to SU(9−N ).
This reproduces the duality (2.6). For the other sign of the relevant operator, the scalar Φ
condenses but the scalar Φ0 does not. The left-hand side becomes a pure Chern-Simons theory,
and we will check whether the theory reduces to (EN −1 )1 after Higgsing.
• For N = 8 the representation 248 decomposes under E8 ⊃ (E7 × SU(2))/Z2 as
248 → (1, 3) ⊕ (56, 2) ⊕ (133, 1) .

(3.15)

Thus a generic expectation value for (1, 3) breaks the group to E7 . (Here we assume
that real and imaginary parts of the vev are not aligned so that SU(2) is broken.)
15

This is similar to the pattern of relevant deformations of five-dimensional superconformal field theories with
EN global symmetry [52, 53].

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SciPost Phys. 7, 056 (2019)• For N = 7 the representation 56 decomposes under E7 → (E6 × U(1))/Z3 as
56 → 13 ⊕ 1−3 ⊕ 27−1 ⊕ 271 .

(3.16)

Thus condensing 13 breaks the group to E6 .
• For N = 6 the representation 27 decomposes under E6 → (S pin(10) × U(1))/Z4 as
27 → 1−4 ⊕ 102 ⊕ 16−1 .

(3.17)

Thus condensing 1−4 breaks the group to S pin(10) = E5 .16• For N = 5 the representation 16 decomposes under S pin(10) → (SU(5) × U(1))/Z5 as
16 → 1−5 ⊕ 5̄3 ⊕ 10−1 .

(3.18)

Thus condensing 1−5 breaks the group to SU(5) = E4 .
• For N = 4 the representation 10 decomposes under SU(5) → (SU(3)×SU(2)×U(1))/Z6
as
10 → (1, 1)−6 ⊕ (3̄, 1)4 ⊕ (3, 2)−1 .
(3.19)
Thus condensing (1, 1)−6 breaks the group to SU(3) × SU(2) = E3 .
• For N = 3 condensing the scalar in (3, 2) Higgses the gauge group to U(2). To see thelevel, one can consider the Cartan subgroup embedded in SU(2) × SU(3) asdiag(e iα , e−iα )×diag(e iβ , e iγ , e−iβ−iγ ) with α, β, γ ∈ R/(2πZ). Then the SU(2)1 ×SU(3)1
Chern-Simons term becomes U(1)2 × (U(1)2 × U(1)6 ) /Z2 . After condensing the scalar
it becomes U(1)2 × U(1)14 /Z2 , where U(1)2 is embedded in SU(2) from the original
SU(3) gauge group. Thus the Chern-Simons term after Higgsing is U(2)1,7 .
• For the first duality in the N = 2 case, condensing the scalar of charge 4 Higgses
SU(2)1 ×U(1)14(E2 )1 ∼to SU(2)1 × (Z4 )14 /Z2 ∼= U(2)1,7 ∼== (E1 )1 as expected. For theZ2second duality, condensing 23 Higgses U(2) down to a U(1) subgroup embedded inside
Cartan of U(2). To see the level, we parametrize the Caratan of U(2) as diag(e iα+iβ , e iα−iβ ).
The condensation forces 3α = β. Since the Chern-Simons term for U(2)1,7 isπ
4 (Tr(udu) + 3 Tr(u)d Tr(u)) with U(2) gauge field u, by restricting it to the U(1) sube1 )1 .
group specified by 3α = β we get U(1)8 ∼= (Ee1 )1 ∼• For N = 1, condensing the scalar of charge 3 Higgses ( E= U(1)8 down to(Z3 )8 ∼= (Z3 )2 ∼= (E0 )1 . (Here we use the fact that the level of Z3 , defined below (2.7),
is periodic mod 6 as a non-spin TQFT by the field redefinition y → y + x.)
We can generalize the dualities above to allow N f scalars for N f ≤ N ,(EN )1 + N f Φfund←→

SU(9 + N f − N )−1 + N f Φ0fund − 16CSgrav .

(3.20)

e1 and use the duality (3.14). We can
For the case with N = 1, in the left hand side we use E
also dualize the right-hand side to U(1)−N +9+N f /2 theory with N f fermions [6] and obtain thefollowing boson-fermion duality(EN )1 + N f Φfund ←→ U(1)−N +9+N f /2 + N f Ψfund − 2 N − N f /2 − 1 CSgrav . (3.21)
The right-hand side has a U(1) magnetic global symmetry, which is dual to the phase rotationof the complex scalar field in the EN gauge theory.
16

Note that although the UV E6 theory has the same matter content as that in Section 3.3 the potentials aredifferent. Thus we do not expect that the theories discussed here and in Section 3.3 are dual.

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Phase Diagrams of (G2 )0 with Fermions

In this section we propose phase diagrams for theories involving G2 gauge fields coupled withfermions. In the first subsection we study (G2 )0 gauge theory with a Majorana fermion in theadjoint representation, while the second subsection we study (G2 )0 gauge theory with two
Majorana fermions in the fundamental 7 representation.

4.1 (G2 )0 with a single adjoint fermion
Here we consider the theory (G2 )0 with a Majorana adjoint ψadj . We propose a phase diagramof this theory as a function of the mass m of the adjoint. Our analysis is similar to that of[8, 32–34]. The phase structure is motivated by the TQFT duality (2.10).
When the mass of the adjoint Majorana fermion is zero, we conjecture the theory flows toa quantum phase that consists of three sectors:• A Majorana fermion χ̄ which is the Goldstino for spontaneously broken N = 1 supersymmetry [30].
• A TQFT which is SO(3)3 = SU(2)6 /Z2 Chern-Simons theory.
• A real compact scalar ϕ ∼ ϕ + 2π denoted below by the S 1 sigma model.
Actually, the real scalar ϕ and the TQFT SO(3)3 are coupled and the precise description of thephase is the Goldstino and:SU(2)6 × S 1= U(2)6,0 .
(4.1)
Z2
Above, the Z2 quotient means gauging the diagonal Z2 one-form symmetry of SU(2)6 and theU(1) one-form symmetry of the real scalar theory generated by the current ?dϕ.
Note that the S 1 sigma model is time-reversal invariant, and SO(3)3 is also time-reversal
invariant by level-rank duality [7]. Therefore, the theory (4.1) is also time-reversal invariant
as expected from the time-reversal symmetry of the UV theory. We return to this point below.
As we move away from the point m = 0, supersymmetry is broken and the Goldstinoacquires a mass leaving only (4.1) as dynamical degrees of freedom at long distances. As |m|is increased the quantum phase ends at two phase transition points that have descriptions as
U(2) Chern-Simons theory with one fundamental fermion. The dual descriptions have nonanomalous U(1) magnetic symmetry, which is an emergent symmetry in the G2 theory. Thissymmetry is spontaneously broken in the quantum phase with the free scalar the corresponding
Nambu-Goldstone boson. Finally, after passing through these phases transitions we arriveat U(2) topological field theories which are dual by (2.10) to the expected (G2 )±2 visible
semiclassically. Figure 1 depicts the proposed phase structure.
There are several consistency checks on this phase diagram:• Time-reversal symmetry and anomaly. The theory at m = 0 is T invariant and therefore
has a time-reversal anomaly ν which is an integer modulo 16 [58–62]. This anomaly
must match between the UV and IR. In the UV we find νU V = 14 (the number of
UV fermions). Meanwhile, the proposed IR phase is T invariant by level-rank duality
SO(3)3 ↔ SO(3)−3 , and moreover this theory has ν = ±3 [63]. Combining with theν = 1 theory of the free Goldstino χ we find ν IR = −2 which agrees with νU V modulo
16.
• Gravitational Chern-Simons counterterm. In the UV it is clear that the theory at large
positive m and large negative mass differ by 14 units of gravitational Chern-Simonscounterterm: 14 CSgrav . This difference by the counterterm should be reproduced by
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the proposed IR phase diagram. If we set the amount of the counterterm in the UV
theory to be zero as a reference, we get 7 CSgrav when m → ∞, and −7 CSgrav whenm → −∞. The amount of the counterterm in other phases and other duality frames
are also indicated in Figure 1. When the mass crosses each quantum transition point
U(2)± 13 ,± 1 + Ψfund , the theory gets an additional 4 CSgrav . While, in the middle phase,22
we use the time-reversal level-rank duality
U(2)6,0 ↔ U(2)−6,0 − 9 CSgrav(4.2)

discussed above to track the coefficient of CSgrav . One can thus verify that the gravitational counterterm works out consistently from Figure 1.17

4.2

(G2 )0 with two fundamental fermions

Let us also study the (G2 )0 gauge theory with two Majorana fermions ψ7i in the fundamental
representation 7 of G2 . This theory has two mass parameters m1 and m2 for each fermion,
and a U(1) global symmetry rotating the two fermions when m1 = m2 . We set the amount of
the gravitational counterterm in the UV to be zero as a reference.
We first consider the case where the masses are restricted to m1 = m2 , and subsequently
generalize. We find that for small mass there is a quantum phase where the U(1) global
symmetry is spontaneously broken. This is similar to the analysis for classical groups in [31].
When the masses are large and positive m1 = m2 → ∞, we have the TQFT (G2 )1 TQFT,
and when m1 = m2 → −∞, we find the TQFT (G2 )−1 . We can use the duality (2.10) to go to
the U(2)±3,±1 description. We propose that between these two phases there is a quantum phase
SU(2) ×S 1

2
described by the U(2)2,0 =theory. The S 1 valued scalar field should be identified withZ2
the Goldstone boson for the U(1) symmetry.18 From the semi-classical phase to the quantum
phase, we expect a transition described by U(2)∓ 5 ,∓ 1 with a complex fermion Ψfund in the22fundamental representation. The U(1) global symmetry in UV should be identified with themonopole symmetry in the U(2) description. The phase structure is summarized in Figure 2.
As in the previous subsection, we perform the following consistency checks:• Time-reversal symmetry: The UV theory at m1 = m2 = 0 is T invariant. It also enjoys adiscrete Z2 symmetry C which exchanges the two fermions. The time-reversal symmetry
C T of the UV theory is preserved provided m1 = −m2 and acts trivially on the system
when |mi | → ∞ (this is because there the system itself is trivial in the IR, see below).19
Therefore we conclude that C T has vanishing ν.
In the IR, the proposed phase U(2)2,0 =SU(2)2 ×S 1
is also time-reversal invariant since
Z2

SU(2)

both the S 1 scalar and Z2 2 = SO(3)1 TQFT, (which is actually a trivial TQFT), aretime-reversal invariant. We refer to this time-reversal symmetry as T IR ; it has ν = 0. It
is thus consistent to expect that the UV symmetry C T flows at long distances to the IR
symmetry T IR .20• Gravitational Chern-Simons counterterm: From the UV description, the jump of thecounterterm when the mass goes from m1 = m2 → −∞ to m1 = m2 → ∞ is 14 CSgrav .
To verify this, we assume that the mass of the Goldstino χ̄ is negatively related to the UV mass m.
For this analysis that follows to be correct, it is important that the Z2 subgroup of the UV U(1) symmetry is not
spontaneously broken in the quantum phase. Instead, we identify this unbroken Z2 0-form symmetry of the UV
17
18

SU(2) ×S 1

2.
with the Z2 0-form symmetry that arises from the Z2 1-form gauging in the IR, which defines the quotient
Z2
19
The symmetry T maps mi → −mi while C exchanges the mi . Hence C T is a symmetry if m1 = −m2 .
20
The UV theory also has the anomalous T symmetry with ν=14. However it is possible that this is matched in
the IR by mixing with the 1-form symmetry of the massless scalar (see e.g. [34, 64, 65]).

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m1 = m2 = m(G2 )0 + 2ψ7
U(2)− 5 ,− 1 + Ψfund − 5 CSgrav
20U(2) 5 , 1 + Ψfund+ 5 CSgrav22 2

m → −∞(G2 )−1 − 7 CSgravlU(2)−3,−1 − 7 CSgravm→∞
U(2)2,0 + 3 CSgravlU(2)−2,0 − 3 CSgrav(G2 )1 + 7 CSgravlU(2)3,1 + 7 CSgrav

Figure 2: A conjectural phase diagram of (G2 )0 gauge theory with 2 fermions in thefundamental representation where m1 = m2 . In the center is a quantum phase where
the U(1) global symmetry is spontaneously broken.
One can verify that the same amount of counterterm is generated when the system goes
through the two transition points into the quantum phase from Figure 2.
Next we would like to generalize the phase diagram into where m1 6= m2 . Assuming
Figure 2 we further deduce:• If we set m1 = −m2 and take the limit of m1 = −m2 → ±∞, in the theory becomes(G2 )0 Yang-Mills theory. In particular it is confined and trivially gapped in the IR andthe coefficient of the gravitational Chern-Simons term is zero there.
• If we add a small U(1) breaking mass term m1 −m2 to the quantum phase U(2)2,0 +3 CSgrav ,SU(2)the S 1 valued Goldstone scalar gets mass, and we are left with Z2 2 + 3 CSgrav =SO(3)1 + 3 CSgrav = 0 CSgrav , namely the trivial theory with zero amount of gravitational
Chern-Simons counterterm.
Therefore, it is consistent to smoothly connect the phase m1 = −m2 = ±∞ and the phasewith small m1 = −m2 .
Thus we propose the phase diagram for (G2 )0 + 2ψ7 with general masses as Figure 3. On
the m1 → ±∞ line, we reduce to (G2 )± 1 + ψ7 . On the line, we expect a transition point2
where the fermion mass changes sign, between the semiclassical (G2 )±1 and (G2 )0 phases. If
we assume that this transition point continues to exist when |m2 | is reduced until we intersect
the line of U(1) global symmetry, we get the diagram Figure 3. If this diagram is correct, themonopole operator of the theory U(2)± 5 ,± 1 + Ψfund should be relevant, and it triggers a flow22
to (G2 )± 1 + ψ7 with a tuned mass for Ψfund .
2

Acknowledgements
We thank N. Seiberg for discussions. C.C. is supported by DOE grant de-sc0009988. P.-S.H. issupported by the U.S. Department of Energy, Office of Science, Office of High Energy Physics,
under Award Number DE-SC0011632, and by the Simons Foundation through the Simons
Investigator Award. K.O. is supported by NSF Grant PHY-1606531 and the Paul Dirac fund.

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U(2) 5 , 1 + Ψfund + 5 CSgrav
2 2(G2 )1 + 7 CSgrav(G2 )0ltrivial0
U(2)− 5 ,− 1 + Ψfund− 5 CSgrav
22

SU(2)2 ×S 1
+ 3 CSgrav
Z2(G2 )− 1 + ψ7 − 72 CSgrav
2(G2 ) 1 + ψ7 + 72 CSgrav
2(G2 )0ltrivial(G2 )−1 − 7 CSgrav
−∞
−∞

∞

m1

Figure 3: A conjectural phase diagram for the (G2 )0 gauge theory with two real
fermions 2ψ7 in the fundamental representation of G2 . The diagonal line wherem1 = m2 is identical to Figure 2. As explained there, on the solid line in the middle,
SU(2) ×S 1

2
the theory flows to, and at the black and white stars the theory is dual toZ2
U(2)± 5 ,± 1 + Ψfund . On the solid and dashed doubled lines we expect that the theory2
2
flows to a gapless theory dual to (G2 )± 1 + ψ7 , and intersects with the m1 = m2 line2
at the black and white stars.

A

Potential for Higgsing

Consider a real scalar field φ taking value in an orthogonal representation R of a compact
gauge group G. In the main text we often need to find a one-parameter family of potentials
V (φ; m2 ) such that with m2 → +∞ the scalar is massive and decouples, and with m2 → −∞the gauge group G is Higgsed down to a subgroup H. It is necessary for such a V to exist thatthe representation R includes a vector whose stabilizer is H. Below we show that actually theconverse is also true.
Let v be a vector in R with stabilizer H and norm |v| = 1. Denote the G-orbit of v by O(v).
The potential V (φ; m2 ) with the desired property can be constructed as

2 2 2
2

max y∈O(v)∪O(−v) (|φ + m y| ) m > 0
V (φ; m2 ) =

|φ|4

min
2 2 2
y∈O (v)∪O (−v) (|φ + m y| )m2 = 0

(A.1)

m <0.
2

When m2 > 0, for any vector φ and y either |φ + m2 y|2 or |φ − m2 y|2 is greater than|m2 y|2 (= m4 ). Therefore, φ = 0 is the minimum point of V (φ, m2 > 0). Moreover, as m2
goes to infinity, the mass of φ also goes to infinity. On the other hand, when m2 < 0, V has
minimum 0 if and only if φ/m2 ∈ O(v) ∪ O(−v). Any element of O(v) ∪ O(−v) has stabilizer
H, so classically the gauge group is Higgsed down to H. Moreover, when m2  0, φ gets a16

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large vev, and therefore the classical analysis is reliable. The potential V (φ; m2 ) constructedabove is not necessarily smooth, but it should be possible find a nearby smooth potential ifneeded.
When m2 is close to zero, we cannot say anything concrete about the quantum theory. The
purpose of this appendix is to explicitly show that the semiclassical phase with G-TQFT and
H-TQFT are continuously connected by the potential deformation, and therefore we expect
some phase transition connecting them.
For a complex scalar Φ valued in a unitary representation, a similar V can be written as

2 2 2
2

max y∈∪α O(e iα v) (|Φ + m y| ) m > 0V (Φ; m2 ) = |Φ|4
(A.2)
m2 = 0

min
2 2 2
2
m <0.
iα (|Φ + m y| )y∈∪α O (e v)This potential preserves the U(1) symmetry rotating Φ to e iα Φ.

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[2] E. J. Mlawer, S. G. Naculich, H. A. Riggs and H. J. Schnitzer, Group-level duality of WZW
fusion coefficients and Chern-Simons link observables, Nucl. Phys. B 352, 863 (1991),
doi:10.1016/0550-3213(91)90110-J.
[3] E. Witten, The Verlinde algebra and the cohomology of the Grassmannian, arXiv:hepth/9312104.
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