\( \mathcal{N}=2 \) Chern-Simons-matter theories without vortices

Journal of High Energy Physics, Jul 2017

We study \( \mathcal{N}=2 \) Chern-Simons-matter theories with gauge group \( {U}_{k_1}(1)\times {U}_{k2}(1) \). We find that, when k 1 + k 2 = 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 mass-deformed ABJM theory with U(1) k × U −k (1) gauge group as a particular case. Similar features are shared by a class of CS-matter theories with gauge group \( {U_k}_{{}_1}(1)\times \cdots \times {U}_{k_N}(1) \) .

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\( \mathcal{N}=2 \) Chern-Simons-matter theories without vortices

JHE Chern-Simons-matter 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 Chern-Simons-matter 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 mass-deformed ABJM theory with U(1)k ×U−k(1) gauge group as a particular case. Similar features are shared by a class of CS-matter theories with gauge group Uk1 (1) × · · · × UkN (1). Chern-Simons 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 non-perturbative 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 two-dimensional 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 finite-dimensional integral. The exact partition function in the different N = 2 theories has, nonetheless, an extremely rich and complicated structure, encapsulating interesting gauge-theory 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 three-dimensional N = 2 supersymmetric Chern-Simonsmatter 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 non-trivial 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 three-sphere. 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 Chern-Simons levels k1, k2 and more general matter content. We show that a similar simplification takes place, both on the three-sphere 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 three-ellipsoid with U(1) × U(1) isometry as in [5]. The three-ellipsoid 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 three-dimensional action contains Chern-Simons 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 Chern-Simons 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 R-charge 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) Chern-Simons-matter 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 non-trivial three-dimensional manifolds with non-contractible one-cycles 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 three-sphere [1]), the exact partition function has the form where Z1c−hilroaolp represents the one-loop 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 sixth-order 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 three-dimensional 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/k-plane, 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 Chern-Simons 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 non-integer 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 Chern-Simons levels for the U(1) × U(1) gauge group are (k1, k2), with general matter content.2 Partition function on the three-sphere. We first consider 2Nf chiral fields with charges (1, −1) and 2Nf chiral fields with charges (−1, 1), all with the same R-charge Δ = 1/2 and masses ±m. The partition function on the three-sphere 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 three-sphere 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 one-loop determinant is the double-sine 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 Chern-Simons levels k1, k2 have been proposed to have an holographic interpretation in terms of AdS4 backgrounds with non-zero Romans mass [30]. – 6 – Then the one-loop determinant for a chiral field of R-charge 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 one-loop factor is given by We consider Nf chiral multiplets φr with R-charge Δ and U(1)×U(1) gauge charges (1, −1) and Na chiral multiplets φ˜s with the same R-charge 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 double-sine 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 one-loop 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 Chern-Simons-matter 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 re-derive this condition from the effective potential. −2k1σ1 − η1 + X |φr|2 − X |φ˜s|2 = 0 , −2k2σ2 − η2 − X |φr|2 + X |φ˜s|2 = 0 , s s ik1D1 − ik2D2 + X |φr|2 + X |φ˜s|2 (σ1 − σ2) = 0 , X |φr|2 + X |φ˜s|2 (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 Chern-Simons 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 non-trivial 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 Chern-Simons 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 |φr|2 + |φ˜s|2 −η0 + (k1 + k2) |φr|2 − |φ˜s|2 2 where sums over r and s are understood (i.e. |φr|2 = PrN=f1 φ¯rφr, |φ˜s|2 = 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 |φr|2 + |φ˜s|2 . 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 sixth-order 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 Chern-Simons-Higgs action for a U(1) gauge group with a sixth-order 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 well-known vortex configurations due to the existence of a non-trivial 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) symmetry-breaking vacua for any sign of η0/(k1 + k2), satisfying hPr |φr|2 − Ps |φ˜s|2i = 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 non-abelian 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) ChernSimons-matter 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 non-trivial partition function (4.10), containing non-perturbative 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 CS-matter 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 no-vortex 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 one-loop 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) CS-matter 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 FPA2013-46570 (MINECO), 2014-SGR-104 (Generalitat de Catalunya) and MDM-2014-0369 of ICCUB (Unidad de Excelencia ‘Mar´ıa de Maeztu’). 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Jorge G. Russo, Fidel A. Schaposnik. \( \mathcal{N}=2 \) Chern-Simons-matter theories without vortices, Journal of High Energy Physics, 2017, 62, DOI: 10.1007/JHEP07(2017)062