\( \mathcal{N}=2 \) ChernSimonsmatter theories without vortices
JHE
ChernSimonsmatter theories without vortices
Jorge G. Russo 0 1 2
Fidel A. Schaposnik 0 1 2
0 Mart ́ı Franqu`es , 1, 08028 Barcelona , Spain
1 Universitat de Barcelona
2 Pg. Lluis Companys , 23, 08010 Barcelona , Spain
We study N = 2 ChernSimonsmatter theories with gauge group Uk1 (1) × Uk2 (1). We find that, when k1 + k2 = 0, the partition function computed by localization dramatically simplifies and collapses to a single term. We show that the same condition prevents the theory from having supersymmetric vortex configurations. The theories include massdeformed ABJM theory with U(1)k ×U−k(1) gauge group as a particular case. Similar features are shared by a class of CSmatter theories with gauge group Uk1 (1) × · · · × UkN (1).
ChernSimons Theories; Extended Supersymmetry; Matrix Models; Super

HJEP07(21)6
1 Introduction
2
3
4
5
1
N = 2 supersymmetric gauge theories on the ellipsoid
U(1) × U(1) ABJM theory with FI and mass deformations
More general U(1) × U(1) model
Flat space analysis
Introduction
ular, the partition function on the squashed sphere Sb3, computed in [5], can be expressed
as infinite sums where each term contains the product of the vortex times the antivortex
partition function [6]. This structure also appeared in N = 2 superconformal indices [
7, 8
]
and general properties underlying this decomposition in “holomorphic blocks” were further
explored in [9–11].
The physical origin of the nonperturbative terms, and its connection to vortices, can
be understood in a more direct way if one implements the method of localization by adding
a different deformation term to the path integral, in such a way that the classical
supersymmetric configurations contributing to the partition function are precisely vortices at
the north pole and antivortices at the south pole of Sb3 [12, 13]. This alternative
localization, called “Higgs branch localization”, was first discovered in [14, 15] in the context
of twodimensional N = (2, 2) gauge theories on S2. Many other remarkable phenomena
appeared in related works, in particular, mirror duality exchanging vortex loop and Wilson
loop operators [16, 17] (see [18] for a recent review and a more complete list of references).
Supersymmetric localization reduces the problem of computing a highly complicated
functional integral to a far much simpler finitedimensional integral. The exact partition
function in the different N = 2 theories has, nonetheless, an extremely rich and complicated
structure, encapsulating interesting gaugetheory phenomena in an exact formula. The
integrals can be computed by residues, leading to long expressions representing the sum over
vortex and antivortex partition functions described above. A natural question is whether
there are cases where this extremely complicated structure simplifies. In this note we
– 1 –
identify one example where a huge simplification occurs and disclose the physical origin of
such simplification. We consider threedimensional N = 2 supersymmetric
ChernSimonsmatter gauge theories on the squashed sphere Sb3 with gauge group Uk1 (1) × Uk2 (1), with
matter charged under both gauge groups. We will find a peculiar phenomenon. For generic
parameters, the theory contains vortices and antivortices associated with north and south
poles of Sb3, with the expected partition function factorizing in terms of holomorphic blocks.
However, when the couplings satisfy a certain condition, supersymmetric vortex
configurations are no longer possible: the theory then contains a unique, topologically trivial vacuum
and the partition function reduces to a single term (yet with highly nontrivial dependence
on the couplings). We will also discuss this phenomenon in terms of the effective potential.
The paper is organized as follows. In section 2 we give a brief review of
threedimensional N
= 2 gauge theories on S3, with a focus on theories with U(1) × U(1)
b
gauge group. In section 3 we consider U(1)k × U(1)−k ABJM theory with mass and
FayetIliopoulos (FI) deformations and compute the partition function on the threesphere. We
show that the integrals can be carried out in a straightforward way, leading to a very
simple compact formula for the partition function. In section 4 we consider a U(1) × U(1)
gauge theory with arbitrary ChernSimons levels k1, k2 and more general matter content.
We show that a similar simplification takes place, both on the threesphere S3 and on the
ellipsoid Sb3, provided the parameters satisfy a certain constraint. Finally, in section 5, we
study supersymmetric vortex configurations in flat space for the general model of section
4 and show that all vortices disappear when the same condition on the parameters is
imposed. We also show that, for any arbitrary parameters not satisfying this condition, the
theory has vortices with an action compatible with the vortex counting parameter that one
derives from the partition function on the ellipsoid.
2
N
= 2 supersymmetric gauge theories on the ellipsoid
We consider the threeellipsoid with U(1) × U(1) isometry as in [5]. The threeellipsoid is
defined by the hypersurface
x20 + x12 + x22 + x32 = 1 ,
ds2 = ℓ2(dx02 + dx12) + ℓ˜2(dx22 + dx32) .
(2.1)
(2.2)
(2.3)
(2.4)
with metric
Introducing coordinates
the metric takes the form
– 2 –
(x0, x1, x2, x3) = (cos θ cos ϕ2, cos θ sin ϕ2, sin θ cos ϕ1, sin θ sin ϕ1) ,
ds2 = r
2 f (θ)2dθ2 + b2 sin2 θdϕ12 + b−2 cos2 θdϕ22 ,
b ≡
qℓ˜/ℓ ,
r ≡
pℓℓ˜,
f (θ) ≡
pb−2 sin2 θ + b2 cos2 θ .
group U(1) × U(1) and chiral matter.
(A1, σ1, λ1, λ¯1, D1) and (A2, σ2, λ2, λ¯2, D2).
Here we shall study N = 2 supersymmetric gauge theories on this space, with gauge
The theories thus have two vector multiplets
The threedimensional action contains ChernSimons terms for each U(1) gauge group,
i.e.
SCS[k] = i
k Z
4π
A ∧ dA − i 4π
k Z
d3x√g(−λ¯λ + 2Dσ) ,
with general ChernSimons levels k1, k2.
N = 2 background vector multiplets ((A˜a)μ, σ˜a, λ˜a, λ˜a, D˜ a), a = 1, 2. One gets
¯
The FI deformations can be constructed as usual by coupling the vector multiplets to
SFI =
i Z
4π
d3x√g(D˜ aσa + σ˜aDa),
a = 1, 2 .
Chiral matter may couple to both vector multiplets V1 = (A1, σ1, λ1, λ¯1, D1), V2 =
(A2, σ2, λ2, λ¯2, D2), with some given charges q1, q2, so that the covariant derivative is
¯
Dμφ = ∂μφ − iq1(A1)μφ − iq2(A2)μφ. Defining a vector multiplet Vˆ = q1V1 + q2V2, with
components Vˆ = (Aˆ, σˆ, λˆ, λˆ, Dˆ ), the action for a chiral multiplet of Rcharge Δ is then
given by (we follow the conventions of [12])
Smatter =
Z
d3x√g Dμφ¯Dμφ − iψ¯γμDμψ +
Δ(2 − Δ) φ¯φ + iφ¯Dˆ φ −
2Δ − 1 ψ¯ψ
2rf
+iψ¯σˆψ + iψ¯λˆφ − iφ¯λˆ¯ψ + φ¯σˆ2φ +
rf
i(2Δ − 1) φ¯σˆφ + F¯F ,
(2.7)
where γμ are the Pauli matrices. An N = 2 preserving mass deformation can be added in
the usual way by coupling the chiral fields to vector multiplets associated with the flavor
symmetry. Real masses mi then correspond to the expectation values of the scalar fields
of these background vector multiplets.
We will consider models with Nf chiral multiplets having the same charges q1, q2
the abelian case on Sb3 do not need to be quantized).1
and Na chiral multiplets with charges −q1, −q2, with q1q2 6= 0 (the case q1q2 = 0 leads
to a decoupled U(1) ChernSimonsmatter theory, which has already been studied in the
literature, see e.g. [
33, 34
].). For these models, it is convenient to normalize the gauge fields
by setting e.g. q1 = 1, q2 = −1. This normalization rescales the CS levels k1, k2 (which in
We will need the supersymmetric transformations for the fermions, which are as follows
(2.5)
(2.6)
HJEP07(21)6
(2.8)
(2.9)
and
δλ =
δλ¯ =
1
1
2 ǫμνρF νρ
− ∂μσ γμǫ − iDǫ − rf
2 ǫμνρF νρ + ∂μσ γμǫ¯ + iDǫ¯ +
i
i
rf
σǫ ,
σǫ¯,
δψ = −γμǫDμφ − ǫσˆφ − rf
δψ¯ = −γμǫ¯Dμφ¯ − ǫ¯σˆφ¯ − rf
ǫφ + iǫ¯F ,
iΔ ǫ¯φ¯ + iǫF¯ .
iΔ
1In nontrivial threedimensional manifolds with noncontractible onecycles the normalization of the
,
,
requires that the background fields appearing in the FI deformations also satisfy Dˆ a = − σrˆfa .
Integrating over θ, the FI terms localize to 2πiηaσa, where ηa, a = 1, 2 represent constant
parameters related to the values of the background fields.
For the present theory, using the rules derived [5] (generalizing the formula for the
partition function on the threesphere [1]), the exact partition function has the form
where Z1c−hilroaolp represents the oneloop determinant coming from the matter sector.
3
U(1) × U(1) ABJM theory with FI and mass deformations
The first model is inspired by ABJM theory [19]. Specifically, the U(1) × U(1) model
contains CS actions with opposite levels. There are two chiral multiplets with Δ = 1/2,
gauge charges (1, −1) and mass parameters ±m and two antichiral multiplets with the
same masses and gauge charges (−1, 1). In addition, we shall also include a FI term for the
diagonal U(1). One can anticipate that this theory will be particularly simple, since for the
abelian U(1) × U(1) ABJM theory the sixthorder potential vanishes [19], leaving only the
mass deformations and therefore a theory of two chiral and two antichiral free superfields.
For the theory on the threedimensional ellipsoid (2.4), the action of the model is
defined by
S = (SCS[k] + SFI[η])1 + (SCS[−k] + SFI[η])2 + Smatter ,
where the different terms have been defined above.
We start with the simplest case where b = 1, corresponding to the sphere limit of the
ellipsoid. In this case, f (θ) = 1. In the next section we will generalize the formulas for a
model with arbitrary Δ and b parameters. The partition function is given by
Z =
Z
dσ1dσ2 cosh π(σ1 − σ2 + m) cosh π(σ1 − σ2 − m)
.
e−iπk(σ12−σ22)+2πiη(σ1+σ2)
This is the same expression for the mass/FI deformed ABJM partition function given in [2]
particularized to N = 1. Now we introduce new integration variables:
The partition function becomes
σ+ =
σ1 + σ2
2
,
σ− =
σ1 − σ2
2
(3.1)
(3.2)
(3.3)
(3.4)
– 4 –
Z
Therefore, the partition function has the compact form
δ(kσ− − η)
cosh π(2σ− + m) cosh π(2σ− − m)
1
k cosh π( 2kη + m) cosh π( 2kη − m)
Note that the expansion in powers of 1/k corresponds to the perturbative expansion. It
has a finite radius of convergence 1/k0, determined by the first zero of cosh π(2η/k ± m)
in the complex 1/kplane, i.e.
1
k0
=
1
2η
m ± 2
i
This is in contradistinction with the behavior of the weak coupling perturbation series in
more general N = 2 supersymmetric gauge theories, which is asymptotic [20–22].
For integer k, in some cases the partition function on S3 has a finite number of terms
(see [
23–28
] for many examples). This can be illustrated by U(1) N = 2 ChernSimons
theory with a FI deformation, coupled to a pair of massless chiral fields of Δ = 1/2 and
opposite gauge charges. The partition function is given by
integration contour, since the integrand does not decay exponentially on a large semicircle.
It is convenient to go to the “dual” representation by writing
1
cosh πσ
=
Z
dτ
e2πiτσ
cosh(πτ )
Computing the Gaussian integral over σ, and shifting τ → τ − η, we find
This is a Mordell integral [29] (see [
26, 28
] for explicit examples in the context of N = 2
CS theories). For integer k, the integral can be computed by choosing an appropriate
rectangular contour, leading to a finite sum [26]
Z = − e2iπx
− 1
2eiπ(x−k/4)
k−1
k n=0
r −i X e− ikπ (x− k2 −n)2 + ie2iπx
!
with x ≡ −iη − 1/2. For noninteger k, the integral gives rise to an infinite sum which can
be expressed in terms of θ functions [29].
On the other hand, on the ellipsoid, the partition function with k1 + k2 6= 0 contains
an infinite series of terms and they represent vortex contributions as in [6, 12, 13, 16, 17].
We discuss the ellipsoid partition function in the next section.
– 5 –
Introducing new integration variables as in (3.3), the partition function takes the form
Z
Z = 2
e−iπ(k1+k2)σ−2−iπ(k1+k2)σ+2−2πi(k1−k2)σ+σ−+2πi(η+σ++η−σ−)
(cosh π(2σ− + m) cosh π(2σ− − m))Nf
, (4.2)
We now consider the specific model with parameters satisfying the relation
As a result, the σ+2 term in the exponent of (4.2) cancels out and the integral over σ+
gives a Dirac delta function. If one considers chiral multiplets with generic gauge charges
the relation that eliminates the σ+2 term from the exponent is
(q1, q2) and (−q1, −q2) –thus maintaining the original normalization for the gauge fields–
In this section we consider a more general model where the ChernSimons levels for the
U(1) × U(1) gauge group are (k1, k2), with general matter content.2
Partition function on the threesphere.
We first consider 2Nf chiral fields with
charges (1, −1) and 2Nf chiral fields with charges (−1, 1), all with the same Rcharge
Δ = 1/2 and masses ±m. The partition function on the threesphere is now given by
Z
(cosh π(σ1 − σ2 + m) cosh π(σ1 − σ2 − m))Nf .
(4.1)
case discussed earlier, where it has now been extended to more flavors and to the case
η1 6= η2. The final expression for the partition function on the threesphere is
Z =
1
k1
e
iπ η+k1η−
cosh π( ηk+1 + m) cosh π( ηk+1 − m)
Nf .
Partition function on the ellipsoid. The calculation is similar, but now the basic
building block in the oneloop determinant is the doublesine function sb. It is defined by
sb(x) =
∞
Y
k,n=0
kb + nb−1 + Q/2 − ix
kb + nb−1 + Q/2 + ix
,
Q = b + b−1 .
2Deformations of ABJM theory to general ChernSimons levels k1, k2 have been proposed to have an
holographic interpretation in terms of AdS4 backgrounds with nonzero Romans mass [30].
– 6 –
Then the oneloop determinant for a chiral field of Rcharge Δ, gauge charges (q1, q2) and
mass m is given by
Z1c−hilroaolp = sb
iQ
2
(1 − Δ) − q1σ1 − q2σ2 + m .
(4.7)
PsN=a1 m˜s = 0. Thus the total oneloop factor is given by
We consider Nf chiral multiplets φr with Rcharge Δ and U(1)×U(1) gauge charges (1, −1)
and Na chiral multiplets φ˜s with the same Rcharge Δ and opposite gauge charges (−1, 1).
In addition, with add mass deformation parameters mr, m˜s satisfying PrN=f1 mr = 0,
HJEP07(21)6
Z1m−altotoepr(σ−, Δ; mi) = Y sb
(1 − Δ) − 2σ− + mr
(1 − Δ) + 2σ− + m˜s
Nf
r=1
iQ
2
Na
Y sb
s=1
iQ
2
For k1 + k2 = 0, the partition function on the ellipsoid is given by
Z = 2
Z dσ+dσ− e−4πik1σ+σ−+2πi(η+σ++η−σ−)Z1m−altotoepr(σ−, Δ; mi) ,
Thus
Z =
with k ≡ k1.
1
k
 
eiπ η+kη− YNf sb
r=1
k=1
w1 = eiπb2 ,
w2 = eiπ/b2 .
The doublesine function can be written in another form, which is useful to study the
limit b → 0 (or, alternatively, b → ∞), where the ellipsoid degenerates to R2
×
S1:
sb(x) = e− iπ2x2 ∞
Y
1 − w1−(2k+1)e−2πbx −1
1 − w2−(2k+1)e−2πx/b −1
,
In the present case, the partition function (4.10) contains contributions proportional to
e−2πbx
∼ e− 2πkb (η+±kmi) ,
e−2πx/b
∼ e− 2bπk (η+±kmi)
It would be interesting to understand the physical origin of these contributions. As shown
in the next section, these cannot be vortex contributions because the theory does not
have supersymmetric vortices. Indeed, the theory only admits the trivial vacuum with all
φr = φ˜s = 0 (see section 5).
The key point that allows one to explicitly carry out the two integrations in (4.2) is that,
upon imposing (4.3), the integrand depends on one of the two integration variables, σ+, only
in the exponent, with linear dependence. The oneloop determinant does not depend on
σ+, since the chiral matter only couples to the vector multiplet V1 −V2. As discussed below,
the underlying physical reason of the simplicity of these theories is that these are precisely
the cases where the theory does not have vortex configurations associated with north and
iQ
2
(1 − Δ) − k
η+ + mr
Na
Y sb
s=1
iQ
2
(1 − Δ) +
k
η+ + m˜s , (4.10)
(4.9)
(4.11)
(4.12)
south poles of the ellipsoid. It is worth noting that the simplicity of these theories is not
related to possible enhancement of supersymmetries, that arises only for special matter
content. In particular, if Nf 6= Na, the theory always has N = 2 supersymmetry.
In more general 3d models where (4.3) is not satisfied, the partition function on the
ellipsoid is given in terms of infinite sums where each term represents a contribution from
supersymmetric vortex configurations [6]. This is evident in the “Higgs branch
localization” [12, 13], where another deformation term is added. The localized field configuration is
then given in terms of vortex numbers. In the Coulomb branch localization, the equivalent
result is obtained by computing the integrals by residue integration [6, 10, 12, 13].
As an example, we may consider the case where q2 = 0. In this case, one has U(1)k1
ChernSimonsmatter plus a decoupled pure U(1)k2 CS sector without matter. The
nontrivial part in the partition function comes from the first sector. It is a particular case of
the partition functions considered in [13] for U(N ) CS theory coupled to Nf
fundamentals and Na antifundamentals. In our case, Nf , Na correspond to the number of chiral
multiplets with charge q1 and −q1, respectively. The partition function is then given by
particularizing (2.75) of [13] to N = 1. One obtains an expression for Z as a product of
vortex and antivortex partition functions Zv, Zav. In particular, in the case with only FI
mass deformation, one finds an expression of the form
Zv = X e−2πb−1η1nzv(n)(b) ,
Zav = X e−2πbη1nza(nv)(b) ,
(4.13)
∞
n=0
∞
n=0
where n is identified with the absolute value of the vortex topological charge. We see
that the vortex and antivortex actions have the expected linear dependence with the FI
parameter and linear dependence with the topological charges [6].
5
Flat space analysis
Here we will show that vortex configurations disappear precisely in the case when the
partition function reduces to a single term due to the condition (4.3),
In flat spacetime, the FI term is
The part of the action involving D1, D2, σ1, σ2 is
The equations for D1, D2 give
S′ =
Z
d3x
− 2ik1D1σ1 − 2ik2D2σ2 − i(η1D1 + η2D2)
+i(D1 − D2)φ¯φ + (σ1 − σ2)2φ¯φ .
− 2k1σ1 − η1 + φ¯φ = 0 ,
−2k2σ2 − η2 − φ¯φ = 0 .
It follows that 2k1σ1 + 2k2σ2 = −η1 − η2 = −η+ = const.
– 8 –
(5.1)
(5.2)
(5.3)
The equations of motion for σ1, σ2 give
Therefore, k1D1 = −k2D2. When there are several copies of scalar fields φr, r = 1, . . . , Nf ,
with charges (1, −1), and φ˜s, s = 1, . . . , Na, with charges (−1, 1), then the above equations
generalize as follows
We look for supersymmetric configurations. In the flat limit, the supersymmetric
transformations (2.8), (2.9) become
(5.5)
(5.6)
(5.7)
Recall Dμφ = (∂μ − i(A1)μ + i(A2)μ) φ. We must impose δλ1 = δλ2 = δψr = δψ˜s = 0.
Considering the equation k1δλ1 + k2δλ2 = 0, we deduce that
i.e.
The scalar field φ couples to the gauge field
A vortex solution φ = f (r)einϕ implies a circulation for Aˆ and, by Stokes theorem, a flux
F12[Aˆ] ∝ n 6= 0. However, this is impossible if Aˆ is proportional to A˜, since F12[A˜] = 0.
These gauge fields are proportional to each other when
which is nothing but the same condition (4.3) that leads to a simple partition function with
a single term. In the next subsection we will rederive this condition from the effective
potential.
−2k1σ1 − η1 + X φr2 − X φ˜s2 = 0 ,
−2k2σ2 − η2 − X φr2 + X φ˜s2 = 0 ,
s
s
ik1D1 −
ik2D2 +
X φr2 + X φ˜s2 (σ1 − σ2) = 0 ,
X φr2 + X φ˜s2 (σ1 − σ2) = 0 .
r
r
s
s
δλ1 =
δλ2 =
2 ǫμνρF νρ[A1] − ∂μσ1 γμǫ − iD1ǫ ,
2 ǫμνρF νρ[A2] − ∂μσ2 γμǫ − iD2ǫ ,
δψ = −γμǫDμφ − ǫ(σ1 − σ2)φ + iǫ¯F .
k1F νρ[A1] = −k2F νρ[A2] ,
F νρ[A˜] ≡ 0 ,
˜
A ≡ k1A1 + k2A2 .
1
1
r
r
Aˆ = A1 − A2 .
k1 + k2 = 0 ,
– 9 –
(Bμ, λB, λ¯B, σB, DB) the action becomes
The resulting theory with k1 + k2 = 0 can be cast in a familiar form. Introducing
new vector multiplets VA = V1 − V2 ≡ (Aμ, λA, λ¯A, σA, DA) and VB = V1 + V2 ≡
S = ik
where the matter action is given by (2.7) by replacing Vˆ by VA. This is nothing but a BF
ChernSimons model with matter coupled to only one of the two gauge fields. One can
directly see that there are no supersymmetric vortex solutions. The equations of motion
HJEP07(21)6
of Bμ set
Fμν [A] = 0 .
Considering now the supersymmetric variation δψ = 0, one finds that preserving 1/2 of
the supersymmetries requires
(D1 + iD2)φ = 0
or
(D1 − iD2)φ = 0 .
In either case, by (5.10), a solution with nontrivial topological phase, φ = f (r)einϕ, implies
H dxiAi = 2πn, in contradiction with (5.9). Thus the topological charge must be zero.
Standard vortex solutions in the general case.
Let us consider the general model
with arbitrary ChernSimons levels k1, k2. From (5.3), (5.4) one can express D1, D2, σ1, σ2
in terms of φ¯φ. Substituting the solution into the action, the bosonic part of the Euclidean
action takes the form (recall that normalization of gauge fields is conventional)
Z
Sbos = ik1
A1 ∧ dA1 + ik2
A2 ∧ dA2 +
Z
Z
d3x Dμ[Aˆ]φ¯Dμ[Aˆ]φ + Veff (φ) , (5.11)
Dμ[Aˆ]φ = ∂μφ − iAˆμφ ,
Aˆ = A1 − A2 ,
where
Veff =
1
4k12k22 φ¯φ
η0 ≡ k2η1 − k1η2 .
−η0 + (k1 + k2) φ¯φ 2
,
When there are several copies of scalar fields φr, φ˜s with charges (1, −1) and (−1, 1),
r = 1, . . . , Nf , s = 1, . . . , Na, the potential becomes
Veff =
1
4k12k22 φr2 + φ˜s2
−η0 + (k1 + k2) φr2 − φ˜s2
2
where sums over r and s are understood (i.e. φr2 = PrN=f1 φ¯rφr, φ˜s2 = PsN=a1 φ¯˜sφ˜s).
We now see the physical origin of the absence of vortices when k1 + k2 = 0: the
potential becomes
η
2
Veff → 4k12k22 φr2 + φ˜s2 .
0
(5.9)
(5.10)
(5.12)
(5.13)
(5.14)
This potential has only the trivial vacuum φr = φ˜s = 0. In the particular case of ABJM
sixthorder potential vanishes, leaving only the mass deformations.3
theory [19], this just reflects the familiar feature that in the abelian U(1) × U(1) case the
Let us introduce the gauge field
B = A1 + A2 .
The part of the action containing the vector bosons takes the form
Now Bμ can be integrated out by its equation of motion. One obtains
(k1 + k2)dB + (k1 − k2)dAˆ = 0 .
In the special case when k1 + k2 = 0, we recover the condition found above that the flux
dAˆ = 0. This is the theory with no vortex configurations. For k1 + k2 6= 0, we can solve
the above equation for Bμ and find the bosonic (Euclidean) action
S = ik˜ Z
A ∧ dAˆ +
ˆ
Z
d3x
Dμ[Aˆ]φ¯Dμ[Aˆ]φ +
41k˜2 φ¯φ
φ¯φ − k1 + k2
η0
2!
,
i Z
4
+
Z
d3xDμ[Aˆ]φ¯Dμ[Aˆ]φ .
S =
(k1 + k2)Aˆ ∧ dAˆ + (k1 + k2)B ∧ dB + 2(k1 − k2)B ∧ dAˆ
where
k˜ =
k1k2
(k1 + k2)
,
We recognize the ChernSimonsHiggs action for a U(1) gauge group with a sixthorder
Higgs potential. This theory has been studied extensively in the literature [
33, 34
].
Indeed, the potential is the same effective potential that arises from N = 2 supersymmetric
U(1) CS theory coupled to chiral matter –with the precise overall coefficient required by
supersymmetry [
35
] (for recent discussions, see [
36, 37
] and references therein). The theory
has wellknown vortex configurations due to the existence of a nontrivial U(1)
symmetrybreaking vacuum φ¯φ = η0/(k1 + k2), provided η0/(k1 + k2) > 0. More generally, in the
presence of scalar fields φr, φ˜s, there are U(1) symmetrybreaking vacua for any sign of
η0/(k1 + k2), satisfying hPr φr2 − Ps φ˜s2i = η0/(k1 + k2).
× S1β, the Euclidean action for a vortex configuration of vortex number n is
given by [
33, 34
]
S = 2πβ
η0
k1 + k2
n .
From a more physical perspective, one can see why vortices are forbidden in the theory with
k1 + k2 = 0. In the limit k1 + k2 → 0, the action of a vortex is infinity. The result (5.19)
may be compared with the vortex action obtained from the ellipsoid partition function
of the U(1) CS model discussed above, S = 2πη1n/b or S = 2πbη1n for b → 0, ∞. The
3Studies of vortex configurations in nonabelian U(N ) × U(N ) ABJM theory can be found in [31, 32].
(5.17)
(5.18)
(5.19)
effective potentials are the same, with the parameter η1 identified with η0/(k1 + k2).4 Thus
the vortex actions agree with the identification β → b or β → 1/b (this is of course the
observation in [6, 12, 13], now adapted to our context).
In conclusion, the theory with k1 + k2 6= 0 is essentially equivalent to a U(1)
ChernSimonsmatter theory plus a decoupled U(1) pure CS sector. This theory has vortices. The
theory with k1 + k2 = 0 is special: it does not have vortices, nonetheless it has a nontrivial
partition function (4.10), containing nonperturbative contributions in the FI coupling. It
would be very interesting to clarify the origin of such contributions and to have a physical
understanding of the structure of (4.10).
Finally, let us consider CSmatter theories with gauge group Uk1 (1) × · · · × UkN (1) and
Nf chiral multiplets with the same charges (q1, . . . , qN ). Then a straightforward
generalization of the above discussion gives the potential
(5.20)
(5.21)
(5.22)
where
Nf
1
,
Nf
r=1
η0 ≡
XN qaηa
It follows that the potential simplifies when c = 0. In this case the potential becomes
quadratic and the theory does not have vortices. In particular, if all charges qa are different
from zero we can normalize the vector fields by setting qa = ±1. Then the novortex
condition becomes
N
X
plifies. The partition function on Sb3 is given by
Clearly, the same properties hold if supersymmetric mass deformations are added
(contributing as mr2φ¯rφr to the bosonic potential (5.20)).
Like in the U(1) × U(1) case, when c = 0 the partition function Z dramatically
simdN σ e−iπ Pa kaσa2+2πi Pa ηaσa Z1−loop σˆ; Δ, b, mr ,
(5.23)
Since the oneloop determinant depends on σa only through the combination σˆ, it is
convenient to introduce new integration variables σ1, . . . , σN−1, σˆ. Then the integrations over
~σ ≡ (σ1, . . . , σN−1) only involve a Gaussian factor exp(~σT.M.~σ + V~ .~σ), where M is an
(N − 1) × (N − 1) matrix. However, when c = 0, the determinant of M vanishes. Therefore
4Another way to see this is by restoring the dependence on the original gauge charges q1, q2, by rescaling
k1 → k1/q12, k2 → k2/q22, η1 → η1/q1, η2 → −η2/q2. Then S = 2πβ k2q1η1+k1q2η2 n. The U(1) CSmatter
theory is then obtained for (q1, q2) = (1, 0), again giving S = 2πβη1n.
k1q22+k2q12
M has (at least) one zero eigenvalue. As a result, one can perform N − 2 Gaussian
integrations and the remaining integration over the eigenvector σ˜ with vanishing eigenvalue yields
a delta function, of the type δ σˆ −σˆ0(ηa, ka, qa) . Thus the complete integral can be carried
out explicitly, just as in the U(1) × U(1) case, giving rise to a compact expression. For
example, setting all qa = 1, under the condition (5.22), one finds a delta function setting
σˆ → σˆ0 ≡
(−q1, . . . , −qN ) with similar results.
Acknowledgments
One can also extend the model by adding Na chiral multiplets with opposite charges
Z1−loop σˆ0; Δ, b, mr .
(5.24)
HJEP07(21)6
We are grateful to M. Honda for useful discussions. J.G.R. would like to thank the
Department of Physics, FCEN of Universidad de Buenos Aires, for hospitality during the
course of this work. He also acknowledges financial support from projects FPA201346570
(MINECO), 2014SGR104 (Generalitat de Catalunya) and MDM20140369 of ICCUB
(Unidad de Excelencia ‘Mar´ıa de Maeztu’). F.A.S. acknowledges financial support from
ANPCyT, CICBA, CONICET.
Open Access.
This article is distributed under the terms of the Creative Commons
Attribution License (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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