Anomalous dimensions of monopole operators in scalar QED_3 with Chern-Simons term
AAnomalous dimensions of monopole operatorsin scalar QED with Chern-Simons term Shai M. Chester Department of Particle Physics and Astrophysics, Weizmann Institute of Science,Rehovot, Israel
Abstract
We study monopole operators with the lowest possible topological charge q = 1 / ) with N complex scalars and Chern-Simons coupling | k | = N . In the large N expansion, monopoleoperators in this theory with spins (cid:96) < O ( √ N ) and associated flavor representations areexpected to have the same scaling dimension to sub-leading order in 1 /N . We use the state-operator correspondence to calculate the scaling dimension to sub-leading order with theresult N − . O (1 /N ), which improves on existing leading order results. We alsocompute the (cid:96) /N term that breaks the degeneracy to sub-leading order for monopoles withspins (cid:96) = O ( √ N ).February 16, 2021 a r X i v : . [ h e p - t h ] F e b ontents q = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3 Scalar thermal Green’s function . . . . . . . . . . . . . . . . . . . . . . . . . 124.4 Matter kernels at q = 1 / CP -preserving terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4.2 CP -violating terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.5 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 Monopole operators are defined in three dimensional Abelian gauge theories as local opera-tors that are charged under the topological global symmetry U (1) top [2, 3], whose conservedcurrent and charge are j µ top = 18 π (cid:15) µνρ F νρ , q = 14 π (cid:90) Σ F , (1.1)where F νρ ≡ ∂ ν A ρ − ∂ ρ A ν is the gauge field strength, Σ is a closed two-dimensional surface,and j µ top is conserved due to the Bianchi identity. In the normalization (1.1), the charge q isrestricted by Dirac quantization to take the values q ∈ Z / q = 1 / with N flavors of complex scalars andnonzero Chern-Simons coupling k to sub-leading order in 1 /N . The action for this theorycan be written as [1, 4] S = N (cid:90) d x (cid:20) g (cid:16)(cid:12)(cid:12) ( ∇ µ − iA µ ) φ i (cid:12)(cid:12) + iλ (cid:0) | φ i | − (cid:1)(cid:17) − iκ π (cid:15) ηνρ A η ∂ ν A ρ (cid:21) , (1.2)where φ i are complex scalars in the fundamental representation of the flavor group SU ( N ), λ is a Lagrange multiplier field that imposes a length constraint, g is a coupling constant, andwe define κ ≡ k/N . When κ = 0, this theory is equivalent to a non-linear sigma model with CP N − target space. The action (1.2) describes a conformal field theory (CFT) providedthat we tune the coupling g = g c for some g c . This CFT can be studied perturbatively inlarge N and k [5–8], where the fluctuations of A µ and λ are suppressed.Monopole operators in scalar QED with k (cid:54) = 0 play a central role in the recently discussedweb of non-supersymmetric dualities [9–22]. For instance, when N = k = 1 the lowestdimension monopole operator with q = 1 / N and k , a more speculative version of this duality states that q = 1 / SU ( N ) N/ − gauge symmetry and N fermions [22]. Monopole operators have also been used to access gauge theories using theconformal bootstrap [23], in which case large N results are crucial to verify the accuracy ofthe non-perturbative bootstrap bounds.As in [1, 3, 4, 24–28], we will compute the scaling dimension of the lowest dimensionmonopole operators using the state-operator correspondence, which identifies the scalingdimensions of monopole operators with charge q with the energies of states in the Hilbertspace on S × R with 4 πq magnetic flux through the S [3]. The ground state energy on S × R can then be computed in the large N and k limit using a saddle point expansion.When k (cid:54) = 0, the Chern-Simons term induces a gauge charge proportional to q , so thatthe naive S × R vacuum must be dressed by charged matter modes. Following [4], we canenforce this dressing by computing the small temperature limit of the thermal free energy on S × S β , where the radius β of S is related to the temperature T by T ≡ β . The dressing isimposed by the saddle point value of the holonomy of the gauge field on S β , which acts like Note that we most closely follow the formulation in [1], which explicitly includes g in the action, and ismost convenient for subleading in 1 /N calculations. In [4], the action is written in the more standard waywithout an explicit g , in which case explicit counterterms would have to be added when going to subleadingorder in 1 /N . q in scalar QED with k (cid:54) = 0 to leading order in 1 /N , as well asthe finite temperature corrections to the thermal free energy to sub-leading order in 1 /N . These finite temperature corrections were used to show that monopole operators with manydifferent spins and flavor symmetry representations had the same energy to sub-leading orderin 1 /N . For instance, for q = 1 /
2, monopoles with spin (cid:96) < O ( √ N ) and an associated spin-dependent flavor irrep are degenerate to sub-leading order. This degeneracy is broken by anenergy splitting term δE (cid:96) ∝ (cid:96) /N , whose coefficient was not evaluated in [4]. For general κ and q , the leading order scaling dimension for these degenerate monopoles could be writtenas an infinite sum that could be evaluated numerically. For the special case | κ | = q + 1 / | k | = (cid:18) | q | + 12 (cid:19) N : ∆ q = 2 | q | ( | q | + 1)(2 | q | + 1)3 N + O ( N ) . (1.3)In particular, for q = 1 / | κ | = 1, we have ∆ / = N + O ( N ), which has intriguingrelations to the aforementioned dualities. For instance, for N = k = 1 the scaling dimension∆ / = 1 is predicted to be exact because the monopole is dual to a free fermion. For large N and k , the leading order result would imply that the scaling dimension of the baryon inthe dual SU ( N ) N/ − gauge theory receives no quantum corrections.The results of this work can be seen as a completion of the scalar QED analysis in [4]for the case q = 1 /
2. To compute the sub-leading correction to ∆ / , given in (4.45), wecompute the Gaussian fluctuations of the gauge and Lagrange multiplier fields around thesaddle point. These results can also be used to compute the coefficient of δE (cid:96) , given in (5.3),which tells us how degenerate the lowest dimension monopoles are.The rest of this paper is organized as follows. In Section 2, we set up our computation.In Section 3, we review the leading order analysis at large N . In Section 4, we computethe 1 /N correction to ∆ / . In Section 5, we compute the coefficient of δE (cid:96) . We end withconcluding remarks in Section 6. Several technical details of our computation are includedin the Appendices. Monopole operators for other Abelian gauge theories with k (cid:54) = 0 were also studied in [3, 4]. For k = 0Abelian gauge theories, monopoles have been studied in fermionic theories [24–26, 29, 30], scalar theories[1, 27, 31–33], and supersymmetric theories [28, 34–38]. Setup
To study scalar QED on S × S β at large N , we rescale the fields in (1.2) and add a conformalmass term to get S = (cid:90) d x (cid:20) √ g g (cid:18)(cid:12)(cid:12) ( ∇ µ − iA µ ) φ i (cid:12)(cid:12) + 14 | φ i | + iλ (cid:0) | φ i | − N (cid:1)(cid:19) − N iκ π (cid:15) µνρ A µ ∂ ν A ρ (cid:21) , (2.1)where κ ≡ k/N and g is the determinant of the metric ds = dθ + sin θdφ + dτ , (2.2)where τ ∈ [ − β/ , β/
2) and S has unit radius. We are interested in computing the thermalfree energy F q in the presence of a magnetic flux (cid:82) dA = 4 πq through S , with q ∈ Z / Z q = e − βF q = (cid:90) (cid:82) S F =4 πq DA exp (cid:20) − N tr log (cid:18) − ( ∇ µ − iA µ ) + 14 + iλ (cid:19) + iN (cid:90) d x (cid:18) κ π (cid:15) µνρ A µ ∂ ν A ρ + √ g g λ (cid:19)(cid:21) . (2.3)We now expand A µ and λ around a saddle point by taking A µ = A µ + a µ , iλ = µ + iσ , (2.4)where a µ and σ are fluctuations around a background A µ = A µ and iλ = µ that satisfiesthe saddle-point conditions δF q [ A µ , λ ] δA µ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ = a µ =0 = δF q [ A µ , λ ] δλ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) σ = a µ =0 = 0 . (2.5)On S × S β with magnetic flux 4 πq , the most general such background is µ constant and A qµ given by A τ = − iα , F θφ dθ ∧ dφ = q sin θdθ ∧ dφ , (2.6) Note that [1] defines µ q = µ . α = iβ − (cid:82) S β A is a real constant called the holonomy of the gauge field. Physically, α corresponds to a chemical potential for the matter fields.Since the integrand in (2.3) is proportional to N , the fluctuations a µ and σ have typicalsize of order 1 / √ N , and so are suppressed at large N . The thermal free energy F q can thenbe expanded at large N as F q = N F (0) q + F (1) q + 1 N F (2) q + . . . , (2.7)where F (0) q comes from evaluating F q at the saddle point and F (1) q comes from the functionaldeterminant of the fluctuations around the saddle point. These terms can furthermore beexpanded at large β to get F (0) q = ∆ (0) q − β S (0) q + O ( e − cβ ) ,F (1) q = ∆ (1) q + 1 β (cid:18)
12 log N + d β + O ( β ) (cid:19) , (2.8)for some integer d , where the temperature independent terms are identified with the scal-ing dimension, the β − terms give the entropy of the degenerate monopole states, and the log ββ term is due to the O ( N − ) splitting of the degenerate monopole spectrum, which is acontinuous spectrum at large N . In the following sections, we will mostly focus on ∆ (0) q and∆ (1) q . For more details on the temperature dependent terms, see [4]. We now briefly review the calculation of the leading order free energy F q = N F (0) q + O ( N ),for more details see [1, 4]. After setting A µ and λ to their saddle point values (2.6) in thefree energy (2.3) and tuning the coupling g to its critical value g = g c , we find F q ( α, µ ) = β − tr log (cid:20) − ( ∇ µ − i A µ ) + 14 + µ (cid:21) − κqα − π g c µ . (3.1)The eigenvalues of the operator (cid:2) − ( ∇ µ − i A µ ) + + µ (cid:3) on S × S β with magnetic flux 4 πq are ( ω n − α ) + λ j , where λ j are the energies of the modes of the theory quantized on S × R : λ j = (cid:112) ( j + 1 / − q + µ , j ∈ { q, q + 1 . . . } , d j = 2 j + 1 , (3.2)5 j are the degeneracies of the modes, and we defined the Matsubara frequencies ω n = πnβ , n ∈ Z . Using this spectrum, we find F (0) q ( α, µ ) = − κqα − π g c µ + β − (cid:88) n ∈ Z (cid:88) j ≥ q d j log (cid:2) ( ω n − iα ) + λ j (cid:3) = − κqα − π g c µ + β − (cid:88) j ≥ q d j log [2 (cosh( βλ j ) − cosh( βα ))] . (3.3)The last two terms are both divergent, but can be evaluated using zeta-function regulariza-tion.Lastly, we solve (2.5) for large β to find the set of possible saddle points values of α and µ .For q = 0, it can be easily checked that α = µ = 0 are saddles provided that g − c = O ( N ),so we can ignore g − c to leading order (see [1] for more details). For q >
0, the α saddle thatgives the lowest real free energy is α ( κ ) = − sgn( κ ) (cid:18) λ q + β − log ξ ξ (cid:19) + O ( e − β ( λ q +1 − λ q ) ) , ξ ≡ q | κ | d q . (3.4)We now plug this α into the µ saddle point equation and zeta-function regularize to get (cid:88) j ≥ q (cid:18) d j λ q ( µ ) − (cid:19) − q + ξd q λ q ( µ ) = 0 . (3.5)We are interested in the special case | κ | = q + 1 /
2, in which case we find the saddle µ = q .Plugging these saddle point values back into (3.3), taking the large β limit, and zeta-functionregularizing, we get (1.3). We will now compute the sub-leading correction F (1) q to the free energy. As shown in (2.8), F (1) q includes β -dependent terms, as well as the β -independent correction ∆ (1) q to the scalingdimension. The β -dependent terms were already calculated in [4], so here we focus on ∆ (1) q .To compute F (1) q , we expand (2.3) to quadratic order in the fluctuations a µ and σ aroundthe saddle point values determined in section 3. The linear fluctuations vanish by definition6f the saddle point, so we are left with a Gaussian integral:exp( − βF (1) q ) = (cid:90) DaDσ exp (cid:20) − N (cid:90) d xd x (cid:48) √ g (cid:112) g (cid:48) (cid:18) a µ ( x ) K µνq ( x, x (cid:48) ) a ν ( x (cid:48) )+ σ ( x ) K σσq ( x, x (cid:48) ) σ ( x (cid:48) ) + 2 σ ( x ) K σνq ( x, x (cid:48) ) a ν ( x (cid:48) ) (cid:19)(cid:21) , (4.1)where K µνq ( x, x (cid:48) ) ≡ N (cid:20) − g (cid:104) J µ ( x ) J ν ( x (cid:48) ) (cid:105) q + 2 g g µν δ ( x − x (cid:48) ) (cid:104) J ( x ) (cid:105) q (cid:21) − iκ π δ ( x, x (cid:48) ) (cid:15) µνρ ∂ (cid:48) ρ ,K σνq ( x, x (cid:48) ) ≡ − iN g (cid:104) J ( x ) J ν ( x (cid:48) ) (cid:105) q , K σσq ( x, x (cid:48) ) ≡ N g (cid:104) J ( x ) J ( x (cid:48) ) (cid:105) q , (4.2)with J µ ≡ i (cid:2) φ ∗ i ( ∇ µ − i A µ ) φ i − φ i ( ∇ µ + i A µ ) φ ∗ i (cid:3) , J ≡ φ ∗ i φ i . (4.3)The subscript q on the expectation values denotes that they are computed under the as-sumption that A µ and λ are non-dynamical and fixed to their saddle point values A µ = A µ and iλ = µ . These expectation values can be evaluated using Wick contractions in terms ofthe scalar thermal Green’s function (cid:104) φ i ( x ) φ ∗ j ( x (cid:48) ) (cid:105) = g δ ij G q ( x, x (cid:48) ) (4.4)to get K µνq, mat ( x, x (cid:48) ) = D µ G q ( x, x (cid:48) ) D ν G q ( x (cid:48) , x ) − G q ( x (cid:48) , x ) D µ D ν G q ( x, x (cid:48) )+ D µ G q ( x (cid:48) , x ) D ν G q ( x, x (cid:48) ) − G q ( x, x (cid:48) ) D µ D ν G q ( x (cid:48) , x )+ 2 g µν δ ( x − x (cid:48) ) G q ( x, x ) ,K µνq, CS ( x, x (cid:48) ) = − iκ π δ ( x, x (cid:48) ) (cid:15) µνρ ∂ (cid:48) ρ ,K σνq ( x, x (cid:48) ) = G q ( x, x (cid:48) ) D ν G q ( x (cid:48) , x ) − G q ( x (cid:48) , x ) D ν G q ( x, x (cid:48) ) ,K σσq ( x, x (cid:48) ) = G q ( x, x (cid:48) ) G q ( x (cid:48) , x ) , (4.5)where D µ = ∂ µ − i A µq ( x ) and D ν = ∂ (cid:48) ν + i A νq ( x (cid:48) ) denote the gauge-covariant derivatives inthe presence of the background gauge fields, and we have separated the matter and Chern-Simons contributions to K µνq ( x, x (cid:48) ). 7he path integral in (4.1) is not yet well defined, because it contains many flat directionsdue to pure gauge modes. Since these pure gauge modes are independent of q , we can removethem by calculating e − βF (1) q /e − βF (1)0 and using the fact that F (1)0 = 0 because it correspondsto the scaling dimension of the unit operator. To perform these Gaussian path integrals, itis convenient to expand the fluctuations in spherical harmonics / Fourier modes: a ( x ) = a E (0) dτ √ πβ + ∞ (cid:88) n = −∞ ∞ (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) (cid:2) a E (cid:96)m ( ω n ) E n(cid:96)m ( x ) + a B (cid:96)m ( ω n ) B n(cid:96)m ( x ) (cid:3) e − iω n τ √ β + dλ ( x ) ,σ ( x ) = ∞ (cid:88) n = −∞ ∞ (cid:88) (cid:96) =0 (cid:96) (cid:88) m = − (cid:96) b (cid:96)m ( ω n ) Y (cid:96)m ( θ, φ ) e − iω n τ √ β , (4.6)where dλ are pure gauge modes and E n(cid:96)m ( x ) and B n(cid:96)m ( x ), together with dτ / (4 πβ ), form anorthonormal basis of polarizations for the one-form a ( x ): E n(cid:96)m ( x ) = (cid:96) ( (cid:96) + 1) Y (cid:96)m dτ − iω n dY (cid:96)m (cid:112) (cid:96) ( (cid:96) + 1) (cid:112) ω n + (cid:96) ( (cid:96) + 1) , B n(cid:96)m ( x ) = ∗ dY (cid:96)m (cid:112) (cid:96) ( (cid:96) + 1) , (4.7)where ∗ is the Hodge dual on S . From now on we will ignore the pure gauge modes dλ because the integral over them cancels between the numerator and denominator in e − βF (1) q /e − βF (1)0 . We can then define the Fourier transform of the kernels (4.2) as K q,(cid:96) ( ω n ) = 4 π j + 1 (cid:90) d x √ ge iω n ( τ − τ (cid:48) ) (cid:96) (cid:88) m = − (cid:96) Y † (cid:96)m ( x ) K σσq ( x, x (cid:48) ) Y (cid:96)m ( x (cid:48) ) E † µ,n(cid:96)m ( x ) K µσq ( x, x (cid:48) ) Y (cid:96)m ( x (cid:48) ) B † µ,n(cid:96)m ( x ) K µσq ( x, x (cid:48) ) Y (cid:96)m ( x (cid:48) ) Y † (cid:96)m ( x ) K σνq ( x, x (cid:48) ) E ν,n(cid:96)m ( x (cid:48) ) E † µ,n(cid:96)m ( x ) K µνq ( x, x (cid:48) ) E ν,n(cid:96)m ( x (cid:48) ) B † µ,n(cid:96)m ( x ) K µνq ( x, x (cid:48) ) E ν,n(cid:96)m ( x (cid:48) ) Y † (cid:96)m ( x ) K σνq ( x, x (cid:48) ) B ν,n(cid:96)m ( x (cid:48) ) E † µ,n(cid:96)m ( x ) K µνq ( x, x (cid:48) ) B ν,n(cid:96)m ( x (cid:48) ) B † µ,n(cid:96)m ( x ) K µνq ( x, x (cid:48) ) B ν,n(cid:96)m ( x (cid:48) ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x (cid:48) =0 , (4.8)where we used S × S β symmetry to eliminate the integral over x (cid:48) , and we will denotethe entries of K q,(cid:96) ( ω n ) by σ, E , B , appropriately. For (cid:96) = 0, B n(cid:96)m ( x ) does not exist, sowe should think of K q, ( ω n ) as a 2 × B entries. If wefurthermore restrict to n (cid:54) = 0 for (cid:96) = 0, then we find that the E terms also vanish. This dY does not exist and Y dτ is pure gauge on on S × R [1], so on S × S β the harmonic E n ( x ) canonly contribute to β -dependent terms, which come from n = 0 [4]. e − βF (1) q /e − βF (1)0 , so we can ignore it and in this case define K q, ( ω n ) ≡ K σσq, ( ω n ) as a scalar. We can now plug in (4.6) and (4.8) into the exponent of(4.1) to get − N b (0) a E (0) † K q, (0) b (0) a E (0) − N (cid:88) n (cid:54) =0 | b ( ω n ) | K σσq, ( ω n ) − N ∞ (cid:88) n = −∞ ∞ (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) b (cid:96)m ( ω n ) a E(cid:96)m ( ω n ) a B(cid:96)m ( ω n ) † K q,(cid:96) ( ω n ) b (cid:96)m ( ω n ) a E(cid:96)m ( ω n ) a B(cid:96)m ( ω n ) . (4.9)As shown in [4], the first term contributes purely to β -depedent terms, and so can be ignoredfor our purposes, while the other terms can be separated in the large β limit into linear in β terms and β -indepedent terms as K q,(cid:96) ( ω n ) = β K q,(cid:96) δ n + (cid:101) K q,(cid:96) ( ω ) (cid:12)(cid:12) ω = ω n + O ( e − ( λ q +1 − λ ) β ) , (4.10)where K q,(cid:96) δ n is a constant matrix, and the β -independent term (cid:101) K q,(cid:96) ( ω ) is approximated bya smooth function of ω ∈ R with exponential precision. For (cid:96) = 0, we similarly define thescalar quantity (cid:101) K σσq, ( ω ) (cid:12)(cid:12) ω = ω n as discussed above.We can now plug (4.9) into e − βF (1) q /e − βF (1)0 , perform the Gaussian integrals, and take thelarge β limit to get F (1) q = (cid:90) dω π (cid:34) log (cid:32) (cid:101) K σσq, ( ω ) (cid:101) K σσ , ( ω ) (cid:33) + ∞ (cid:88) (cid:96) =1 (2 (cid:96) + 1) log det (cid:32) (cid:101) K q,(cid:96), mat ( ω ) + (cid:101) K (cid:96), CS ( ω ) (cid:101) K ,(cid:96), mat ( ω ) + (cid:101) K (cid:96), CS ( ω ) (cid:33)(cid:35) + O ( β − ) , (4.11)where for (cid:96) > ω n has been converted into an integral to exponential precision in β .The quantity written in (4.11) is β -independent and so is identified with ∆ (1) q . For the β -dependent corrections, see [4]. In the subsequent subsections, we will now explicitly computethe kernels appearing in (4.11), and then perform the sum and integral to find ∆ (1) q . To easethe notation, we will no longer write O ( e − β ) next to each expression.9 .1 Chern-Simons kernel We begin by considering the Chern-Simons kernel (cid:101) K (cid:96), CS ( ω ). As we see from (4.5), K µνq, CS ( x, x (cid:48) )is local, and so can be computed without doing any integrals. By plugging K µνq, CS ( x, x (cid:48) ) into(4.8) we find that the only nonzero terms are K BE q,(cid:96), CS ( ω n ) = − iκ j + 1 (cid:96) (cid:88) m = − (cid:96) B † µ,n(cid:96)m (0) (cid:15) µνρ ∂ (cid:48) ρ (cid:104) E ν,n(cid:96)m ( x (cid:48) ) e − iω n τ (cid:48) (cid:105) (cid:12)(cid:12)(cid:12) x (cid:48) =0 (4.12)and K EB q,(cid:96), CS ( ω n ) = K BE q,(cid:96), CS ( ω n ). Since K BE q,(cid:96), CS ( ω n ) is manifestly β -independent, we can define (cid:101) K BE q,(cid:96), CS ( ω ) = K BE q,(cid:96), CS ( ω n ) | ω = ω n by just extending ω n to the real line. We then perform thesum in (4.12) to get (cid:101) K BE q,(cid:96), CS ( ω ) = (cid:101) K EB q,(cid:96), CS ( ω ) = iκ π (cid:112) ω + (cid:96) ( (cid:96) + 1) . (4.13)In a pure Chern-Simons theory, the eigenvalues of K q,(cid:96) ( ω ) would then be ± iκ π (cid:112) ω + (cid:96) ( (cid:96) + 1),which are the expected nonzero eigenvalues for the Chern-Simons operator − iκ π ∗ d on S × S β , as it squares to the vector Laplacian − κ π ∗ d ∗ d that has the nonzero eigenvalue − κ π ( ω + (cid:96) ( (cid:96) + 1)) [39]. q = 0 We next consider the matter kernels (cid:101) K q,(cid:96), mat ( ω ), and start with the simpler case of q = 0.As discussed in Section 3, when q = 0 we have α = µ = 0, so in the large β limit we expectto find the same answers in this case as the S × R kernels computed in [1].In particular, the scalar thermal Green’s function G ( x, x (cid:48) ) on S × S β should be thesame as the Green’s function on S × R up to exponential precision in β . This latter Green’sfunction can be found by conformally mapping the R expression 1 / (4 π | x − x (cid:48) | ), which yields G ( x, x (cid:48) ) = 14 π (cid:112) τ − τ (cid:48) ) − cos γ ) , (4.14)where γ is the angle between the two points on S cos γ = cos θ cos θ (cid:48) + sin θ sin θ (cid:48) cos ( φ − φ (cid:48) ) . (4.15)We can now plug G ( x, x (cid:48) ) into (4.5), take the Fourier transform (4.8) to compute10 ,(cid:96), mat ( ω n ), and send β → ∞ to compute (cid:101) K ,(cid:96), mat ( ω ). For instance, for K σσ ,(cid:96) ( ω n ) we find K σσ ,(cid:96) ( ω n ) = 18 π (cid:90) β/ − β/ dτ (cid:90) π sin θdθ e iω n τ P (cid:96) (cos θ )2(cosh τ − cos θ ) , (4.16)where the Legendre polynomial P (cid:96) (cos θ ) comes from the sum over Y (cid:96)m ( θ, φ ) in (4.8). Wethen perform the θ integral and expand at large τ to find K σσ ,(cid:96) ( ω n ) = 18 π (cid:90) β/ − β/ dτ ∞ (cid:88) p =0 ( p + (cid:96) )!Γ( p + 1 / p !Γ( p + (cid:96) + 3 / e − (2 p + (cid:96) +1) | τ | e iω n τ . (4.17)We can now explicitly take β → ∞ and extend ω n → ω ∈ R to get (cid:101) K σσ ,(cid:96) ( ω ) with exponentialprecision in β . The resulting β -independent integral was performed in [40], and yields (cid:101) K σσ ,(cid:96) ( ω ) ≡ D (cid:96) ( ω ) = (cid:12)(cid:12)(cid:12)(cid:12) Γ (( (cid:96) + 1 + iω ) / (cid:96) + 2 + iω ) / (cid:12)(cid:12)(cid:12)(cid:12) . (4.18)This expression will come up frequently, so we have defined the concise notation D (cid:96) ( ω ). Notethat all the sums and integrals used to compute (cid:101) K σσ ,(cid:96) ( ω ) were free of divergences.If we write down expressions analogous to (4.16) for the other nonzero kernels, K EE ,(cid:96), mat ( ω n )and K BB ,(cid:96), mat ( ω n ), we see that they suffer from two kinds of divergences. In [1], it was shownthat these divergences can be uniquely regularized using gauge invariance and the µ saddlepoint condition. Here we will use a slicker method, whose ultimate justification lies in thatit gives the same answer as [1].The first divergence is due to the term G ( x, x ) that appears in K µνq, mat ( x, x (cid:48) ) as definedin (4.5). From (4.14), we see that G ( x, x ) is polynomially divergent. To analyze thisdivergence, it is useful to rewrite G ( x, x (cid:48) ) as an infinite sum [1]: G ( x, x (cid:48) ) = 14 π ∞ (cid:88) j =0 j + 12 j + 1 P j (cos γ ) e − ( j +1 / | τ − τ (cid:48) | , (4.19)and then set x (cid:48) = x and use zeta-functions to find G ( x, x ) = 14 π ∞ (cid:88) j =0 j + 12 j + 1 = ζ (0 , / π = 0 . (4.20)The second divergence is due to the θ and τ integral in (4.8) over the other terms in K µνq, mat ( x, x (cid:48) ), which diverges polynomially for τ = 0. We can regularize this by first per-11orming the θ integral assuming τ (cid:54) = 0. The resulting function is exponentially damped in τ ,just as we saw explicitly in (4.17), so we can send β → ∞ before performing the τ integralto exponential precision in β . Finally, we perform the τ integral by deforming the contouraround τ = 0, which yields the finite results (cid:101) K BB ,(cid:96), mat ( ω ) = ω + (cid:96) D (cid:96) − ( ω ) , (cid:101) K EE ,(cid:96), mat ( ω ) = ω + (cid:96) ( (cid:96) + 1)2 D (cid:96) ( ω ) . (4.21)The expressions (4.21) match those in [1], which justifies our choice of regularization. We would next like to compute the matter kernels for q (cid:54) = 0, for which we will need thescalar thermal Green’s function G q ( x, x (cid:48) ) in the large β limit. We will focus on the specialcase | κ | = q + 1 /
2, where we will find that G q ( x, x (cid:48) ) can be written in closed form.We can define G q ( x, x (cid:48) ) by (cid:20) − ( ∇ µ − i A µ ) + 14 + µ (cid:21) G q ( x, x (cid:48) ) = δ ( x − x (cid:48) ) . (4.22)In [4], this differential equation was solved for general q and κ to find G q ( x, x (cid:48) ) = e α ( τ − τ (cid:48) ) (cid:88) j,m (cid:20) Y q,jm ( θ, φ ) Y ∗ q,jm ( θ (cid:48) , φ (cid:48) )2 λ j (cid:18) e − λ j | τ − τ (cid:48) | + e − λ j ( τ − τ (cid:48) ) e β ( λ j − α ) − e λ j ( τ − τ (cid:48) ) e β ( λ j + α ) − (cid:19)(cid:21) , (4.23)where Y q,jm ( θ, φ ) are the monopole spherical harmonics introduced in [41, 42]. If we nowplug in the value of α given in (3.4), then to exponential precision in large β we get G q ( x, x (cid:48) ) = e α ( τ − τ (cid:48) ) e − iq Θ (cid:34) q | κ | (2 q + 1) λ q e sgn( κ ) λ q ( τ − τ (cid:48) ) F q,q ( γ ) + (cid:88) j ≥ q e − λ j | τ − τ (cid:48) | λ j F q,j ( γ ) (cid:35) , (4.24) [1] uses a different vector basis to compute the kernels, for the relation to our basis see Appendix B. m following [1] using the definitions F q,j ( γ ) ≡ (cid:114) j + 14 π Y q,j ( − q ) ( γ, ,e i Θ cos( γ/ ≡ cos( θ/
2) cos( θ (cid:48) /
2) + e − i ( φ − φ (cid:48) ) sin( θ/
2) sin( θ (cid:48) / . (4.25)Recall from Section 3 that for the special case | κ | = q + 1 / µ = q , sothat from (3.2) we get λ j = j + 1 /
2. Using this relation, along with the identity F q,j (cos γ ) = 2 j + 14 π (cid:18) γ (cid:19) q P , qj − q (cos γ ) , (4.26)we can write the Green’s function for | κ | = q + 1 / G | κ |− / ( x, x (cid:48) ) = e α ( τ − τ (cid:48) ) π (cid:18) γ (cid:19) q e − iq Θ (cid:104) qe ( q +1 / τ − τ (cid:48) ) sgn κ + (cid:88) j ≥ q e − ( j +1 / | τ − τ (cid:48) | P (0 , q ) j − q (cos γ ) (cid:35) . (4.27)Finally, we sum the Jacobi polynomials P (0 , q ) j − q using their generating function ∞ (cid:88) n =0 P a,bn ( x ) z n = 2 a + b z ( a + b +1) / s − (cid:0) z − / − z / + s (cid:1) − a (cid:0) z − / + z / + s (cid:1) − b ,s ≡ √ z + z − − x , (4.28)to find G | κ |− / ( x, x (cid:48) ) = e α ( τ − τ (cid:48) ) π s (cid:32) e − i Θ cos γ τ − τ (cid:48) + s (cid:33) q + qe ( q +1 / τ − τ (cid:48) ) sgn κ (cid:16) e − i Θ cos γ (cid:17) q ,s = (cid:112) τ − τ (cid:48) ) − γ . (4.29) q = 1 / We will now compute the matter contributions to (cid:101) K q,(cid:96) ( ω ) for the specific case q = 1 / | κ | = 1. 13e find it convenient to separate the Green’s function (4.29) in this case as G / ( x, x (cid:48) ) = e α ( τ − τ (cid:48) ) (cid:16) G ( x, x (cid:48) ) + e ( τ − τ (cid:48) ) sgn κ ˆ G ( x, x (cid:48) ) (cid:17) ,G ( x, x (cid:48) ) ≡ π (cid:112) τ − τ (cid:48) ) − γ (cid:34) e − i Θ cos γ τ − τ (cid:48) + (cid:112) τ − τ (cid:48) ) − γ (cid:35) , ˆ G ( x, x (cid:48) ) ≡ e − i Θ cos γ π , (4.30)where G ( x, x (cid:48) ) is the the same as the κ = 0 Green’s function computed in [1] for µ = q , whileˆ G ( x, x (cid:48) ) is a new contribution due to the nontrivial α that only appears because k (cid:54) = 0. Inthe microscopic description, ˆ G ( x, x (cid:48) ) comes from the modes used to dress the bare monopolein order to cancel the gauge charge induced by k . Both ˆ G ( x, x (cid:48) ) and G ( x, x (cid:48) ) are invariantunder the CP transformation τ − τ (cid:48) → τ (cid:48) − τ , but ˆ G ( x, x (cid:48) ) is multiplied by a phase thatviolates CP .All the terms that appear in the matter kernels (4.5) with q = 1 / G / ( x, x (cid:48) ) and G / ( x (cid:48) , x ), or G / ( x, x ) δ ( x, x (cid:48) ), so the overall phase e α ( τ − τ (cid:48) ) can be neglected as long as we replace the covariant derivative D µ by ˆ D µ ≡ D µ | ( ∇ τ − α ) →∇ τ .The resulting terms can then be divided into three categories. Firstly, there are terms givenby setting G q ( x, x (cid:48) ) → ˆ G ( x, x (cid:48) ) in (4.5), and ignoring the G q ( x, x (cid:48) ) term. These terms areindependent of τ , and so only contribute to the linear in β term K q,(cid:96) defined in (4.10), sowe will not consider them. Secondly, there are terms given by setting G q ( x, x (cid:48) ) → G ( x, x (cid:48) )in (4.5), and so are CP -preserving terms. Lastly, the remaining terms in (4.5) receivecontribution from pairs of G ( x, x (cid:48) ) and ˆ G ( x (cid:48) , x ) or just a single ˆ G ( x (cid:48) , x ), and so are CP -violating terms. In the next couple subsections, we will compute these latter two categoriesseparately. CP -preserving terms We will now compute the CP -preserving kernels (cid:101) K / ,(cid:96),CP ( ω ). Just as we saw with the q = 0kernel in Section 4.2, the function of τ that we get after we perform the θ integral in (4.8) isexponentially damped in τ , so we can send β → ∞ before we do the τ integral in (4.8) withexponentially precision in β .As an illustrative example, let us begin by computing the CP -preserving kernel (cid:101) K σσ / ,(cid:96),CP ( ω ).Replacing G ( x, x (cid:48) ) in (4.5) by G ( x, x (cid:48) ) given in (4.30) and plugging into the β → ∞ limit of144.8) we get (cid:101) K σσ / ,(cid:96),CP ( ω ) = 116 π (cid:90) ∞−∞ dτ (cid:90) π sin θdθe iωτ P (cid:96) (cos θ ) (cid:20) τ − cos θ )+ √ τ − θ − τ (1 + cos θ ) √ τ − θ (cid:35) . (4.31)We already encountered the first term in the brackets in (4.16), so its Fourier transform is D (cid:96) ( ω ) as defined in (4.18). For the second term, we can expand at large τ and perform the θ integral term by term to get( − (cid:96) π (cid:90) ∞−∞ dτ e iωτ ∞ (cid:88) p = (cid:96) +1 ( − p p e − p | τ | , (4.32)and then perform the Fourier transform to get the final answer (cid:101) K σσ / ,(cid:96),CP ( ω ) = D (cid:96) ( ω ) + 12 π ∞ (cid:88) p = (cid:96) +1 ( − p + (cid:96) ω + p . (4.33)As a consistency check, if we send (cid:96) → ∞ we recover the q = 0 kernel (cid:101) K σσ ,(cid:96) ( ω ) = D (cid:96) ( ω ). Notethat all the sums and integrals in this computation were finite. We can perform a similarprocedure for (cid:101) K σ B / ,(cid:96),CP ( ω ), and get a finite but more complicated expression that we give inAppendix A.The other nonzero CP -preserving kernels are (cid:101) K EE / ,(cid:96),CP ( ω ) and (cid:101) K BB / ,(cid:96),CP ( ω ), and theysuffer from the same two kinds of polynomial divergences that we saw for q = 0 in Section4.2: the first from the term G ( x, x ), the second from the θ and τ integral in (4.8) when τ = 0. We will regularize both just as we did in Section 4.2, and justify our regularizationby comparing to the results of [1].To regularize the first divergence, we write G ( x, x ) as the infinite sum given in (4.24) anduse zeta-functions: G ( x, x ) = 14 π ∞ (cid:88) j =1 / j + 12 j + 1 = ζ (0)4 π = − π . (4.34)To regularize the second divergence, we first perform the θ integral assuming τ (cid:54) = 0, andthen the τ integral by deforming the contour around τ = 0. The resulting expressions are15lso quite complicated, and are given in Appendix A. Since these CP preserving kernelswere computed using the κ = 0 Green’s function, we can compare to the κ = 0 resultsof [1]. As we show in Appendix B, our results match those of [1], which justifies our choiceof regularization. CP -violating terms Next, we calculate the CP -violating kernels (cid:101) K / ,(cid:96),NCP ( ω ). These kernels differ from the CP -preserving kernels (cid:101) K / ,(cid:96),CP ( ω ) in two important ways.Firstly, (cid:101) K / ,(cid:96),CP ( ω ) receives contributions from the Green’s function ˆ G ( x, x ) that onlyexists for κ (cid:54) = 0, so we cannot compare our results to the κ = 0 results of [1]. On the otherhand, the entire computation of (cid:101) K / ,(cid:96),CP ( ω ) is finite so there is no regularization ambiguity.In particular, from (4.30) we see that ˆ G ( x, x ) = π , which exactly cancels the regularizedvalue G ( x, x ) = − π so that the total Green’s function G / ( x, x ) vanishes, just as we foundfor q = 0 in (4.20), and as was found for κ = 0 and q (cid:54) = 0 in [1].Secondly, the position space kernels now include both terms that are exponentiallydamped in τ , as well as terms that are independent of τ . If we perform the τ integralin (4.8) and then send β → ∞ , the former terms contribute to (cid:101) K / ,(cid:96),NCP ( ω ) while the latterterms are linear in β and so contribute to K / ,(cid:96) ( ω n ), as defined in (4.10). If we send β → ∞ before we take compute the τ integral, then these linear in β terms will appear as deltafunctions, according to the identity2 πδ ( ω ) = (cid:90) ∞−∞ e iωτ dτ = lim β →∞ (cid:90) β/ − β/ e iω n τ dτ = δ n, lim β →∞ β . (4.35)We should thus identify (cid:101) K / ,(cid:96),NCP ( ω ) with the quantity we get by first sending β → ∞ ,then performing the τ integral, then throwing out any δ ( ω ) that appear.As an illustrative example, let us begin by computing the CP -violating kernel (cid:101) K σσ / ,(cid:96),NCP ( ω ).Collecting the terms in (4.5) that include both ˆ G ( x, x (cid:48) ) and G ( x, x (cid:48) ) and plugging into the β → ∞ limit of (4.8) we get116 π (cid:90) ∞−∞ dτ (cid:90) π sin θdθe iωτ P (cid:96) (cos θ ) cosh τ (cid:20) τ √ τ − θ − (cid:21) . (4.36)16e then expand at large τ and perform the θ integral term by term to get116 π (2 (cid:96) + 1) (cid:90) ∞−∞ dτ e iωτ (cid:80) (cid:96) +2 p = (cid:96) − e − p | τ | p , (cid:96) ≥
21 + (cid:80) p =1 e − p | τ | p , (cid:96) = 11 + e − | τ | , (cid:96) = 0 . (4.37)For (cid:96) = 0 ,
1, the constant terms give delta functions, and so should be thrown out. Takingthe Fourier transform of the remaining terms gives (cid:101) K σσ / ,(cid:96),NCP ( ω ) = 18 π (2 (cid:96) + 1) (cid:96) +2 (cid:88) p = (cid:96) − pω + p . (4.38)As a consistency check, if we send (cid:96) → ∞ this kernel vanishes as expected, as there is noanalogous expression for q = 0. We can perform a similar procedure for other entries inthe symmetric matrix (cid:101) K / ,(cid:96),NCP ( ω ), which are all nonzero. We find similar answers, whichare given in Appendix A. Note that all the kernels are independent of sgn κ , except for (cid:101) K σ E / ,(cid:96),NCP ( ω ) and (cid:101) K EB / ,(cid:96),NCP ( ω ) which are proportional to sgn κ . Now that we have explicit expression for the relevant kernels in (4.17), (4.21), (4.13), (4.33),(4.38), (A.1), and (A.2), we can plug these values into (4.11) to compute the sub-leadingscaling dimension ∆ (1)1 / for | κ | = 1. Note that when we take the eigenvalues of the kernel in(4.11), the sgn κ factors in (A.2) cancel, so that we get the same expression ∆ (1)1 / for κ = ± (1)1 / = 12 (cid:90) dω π ∞ (cid:88) (cid:96) =0 (2 (cid:96) + 1) L (cid:96) ( ω ) . (4.39)As shown in Appendix C, at large ω and (cid:96) the integrand in this expression behaves as L (cid:96) ( ω ) = (cid:18) ω + ( (cid:96) + 1 / (cid:19) (cid:18)
11 + π (cid:19) + . . . , (4.40)which makes the integral (4.39) linearly divergent. The first factor in (C.3) equals the lineardivergence that was found for κ = 0 in [1] if we set µ = q = 1 / (cid:18)
11 + π (cid:19) (cid:90) dω π ∞ (cid:88) (cid:96) =0 (cid:96) + 1) ω + ( (cid:96) + 1 / = (cid:18)
11 + π (cid:19) ∞ (cid:88) (cid:96) =0 (cid:96) + 12 (cid:96) + 1 = ζ (0 , / π = 0 . (4.41)We can then subtract (4.41) from (4.39) to get the expression∆ (1)1 / = 12 (cid:90) dω π ∞ (cid:88) (cid:96) =0 (2 (cid:96) + 1) (cid:20) L (cid:96) ( ω ) − (cid:18) ω + ( (cid:96) + 1 / (cid:19) (cid:18)
11 + π (cid:19)(cid:21) , (4.42)which is no longer linearly divergent.Another way of understanding (4.42) is that the inverse critical coupling g − c , whichwe showed in Section 3 to vanish at leading order, obtains 1 /N corrections. A similarphenomenon was encountered in [43] when computing the thermal free energy of scalarQED in flat space with κ = 0 at sub-leading order in 1 /N , where the shift in the couplingwas found to be δ g − c = N (cid:82) d p (2 π ) p for momentum p . In the large N expansion, one effectof κ (cid:54) = 0 is to shift the momentum p → p (cid:16) κ π (cid:17) [44], so that δ g − c → δ g − c (cid:16) κ π (cid:17) .If we similarly shift the κ = 0 expression computed in [1] we get4 π g c = (cid:90) dω π ∞ (cid:88) (cid:96) =0 (2 (cid:96) + 1) 1 ω + ( (cid:96) + 1 / (cid:20) (cid:18)
11 + π (cid:19) N (cid:21) + O (1 /N ) . (4.43)The 1 /N term in this expression contributes to F (1) q through the second term in (3.3) preciselyas the subtraction implemented in (4.42).Even after the linear divergence has been regularized, there is still a potential logarith-mic divergence in the integral in (4.42). This logarithmic divergence cancels as long as weregularize this integral consistent with conformal symmetry, such as by using the symmetriccutoff ( (cid:96) + 1 / + ω < Λ . (4.44)This can be thought of as preserving rotational invariance on R , as the high energy modesare insensitive to the curvature of the sphere. The cancellation of this divergence is a strongcheck on our results, as the potentially logarithmic divergence in the integrand of (4.42)receives contributions from all the kernels evaluated in the previous sections. With thiscutoff, we can now numerically perform the sum and integral in (4.42) for large Λ, and then18xtrapolate to Λ. We find that the numerics rapidly converge, and yield the final answer∆ (1)1 / ≈ − . . (4.45) The sub-leading correction to the scaling dimension given in (4.45) is expected to apply tomonopoles in different representations of the symmetry group SU ( N ) × U (1) top with differentLorentz spins (cid:96) . As shown in [4] for the case q = 1 / | κ | = 1, this degeneracy applies tomonopoles with spins (cid:96) < O ( √ N ) and the associated SU ( N ) representation R (cid:96) ≡ (cid:124) (cid:123)(cid:122) (cid:125) N/ − (cid:96)N/ (cid:96) (cid:122) (cid:125)(cid:124) (cid:123) · · · · · ·· · · , dim R (cid:96) = (2 (cid:96) + 1) N − N/ (cid:96) − N ξ + (cid:96) + 1 N/ − (cid:96) − N ξ − (cid:96) . (5.1)When (cid:96) = O ( √ N ), a formula for the energy splitting term in (2.8) was also derived basedon the log ββ terms that appear in (2.8), which in our case yields δE (cid:96) N = (cid:96) N (cid:18) √ π (cid:19) (cid:18) − i (cid:19) (cid:101) K − / , (0) − i . (5.2)Since the energy splitting goes like (cid:96) /N , we see that the large N expansion starts to breakdown when (cid:96) = O ( √ N ). Using the expression for (cid:101) K / , (0) calculated in the previoussections, we find δEN = − − π + 63 π − π − π + 18 π (cid:96) N ≈ − . (cid:96) N . (5.3)19
Conclusion
The main result of this paper is that the scaling dimension of q = 1 / (cid:96) < O ( √ N ) and SU ( N ) irrep (5.1) in scalar QED with | κ | = 1 and N scalar flavors is∆ / = N − . O (1 /N ) , (6.1)which is obtained by combining the leading order result [4] given in (1.3) with (4.45). The O (1) correction to ∆ / was obtained by performing a Gaussian integral over the fluctuationsof the gauge field A µ and the Lagrange multiplier λ on S × S β with 2 π magnetic flux as β → ∞ . The scaling dimension (6.1) does not depend on the sign of κ , which is also thecase in the large N computation of non-monopole operators in Abelian gauge theories [45]. When (cid:96) = O ( √ N ), the sub-leading term is affected by a spin-dependent contributiongiven in (5.3). The coefficient − . with κ = 0 and q = 1, except with the positive and much large coefficient . S × S β with 4 πq magnetic flux and | κ | = q + 1 / λ saddle point value µ exactly cancelled the contribution − q of the magnetic flux in the eigenvalue λ (cid:96) (3.2) of the Klein-Gordon operator in this case.The same simplification also occurs for BPS monopoles in N = 2 SQED , because of theLagrange multiplier fields in that case. For this theory, the scaling dimension can be foundexactly for any q and κ using supersymmetric localization and F -maximization [34, 35, 44].The localization result can be expanded in large N , e.g. for q = 1 / κ = 0, to get∆ N =2BPS = N π + O (1 /N ) . (6.2)If we enhance the supersymmetry to N = 4, then ∆ N =4BPS = N exactly. It would be interestingto compute the sub-leading corrections in these cases using the methods of this paper, to see For non-monopole operators, the effect of κ (cid:54) = 0 to sub-leading order in large N is just to shift N → N (cid:113) κ π [44, 45]. N calculation in this work to othernon-perturbative numerical methods. In the k = 0 scalar QED case, the subleading in 1 /N calculation of [1] was found to match finite N quantum Monte Carlo studies in [32, 46, 47]even down to N = 2. For k = 0 fermionic QED , the Monte Carlo estimates in [48, 49]match the subleading in 1 /N estimate in [24] for large N , but differ at small N . It would beinteresting to perform a similar Monte Carlo study for scalar QED with k (cid:54) = 0, and see howthe results compare to the subleading in 1 /N prediction in this work. Such a study couldalso be used to verify the N = 1 duality of [13], which predicts that ∆ / = 1 exactly. Acknowledgments
I thank Ofer Aharony, Silviu Pufu, Mark Mezei, Luca Iliesiu, and Nathan Agmon for helpfuldiscussions, and Ofer Aharony for reviewing the draft. I am supported by the ZuckermanSTEM Leadership Fellowship.
A Matter kernel formulae
In sections 4.4.1 and 4.4.2 we explicitly evaluated the CP -preserving and CP -violating com-ponents of the (cid:101) K σσ / ,(cid:96) ( ω ) kernel, respectively, which we denote as (cid:101) K σσ / ,(cid:96),CP ( ω ) and (cid:101) K σσ,NCP / ,(cid:96) ( ω ).In this appendix we give explicit expression for the other CP -violating and CP -preservingcomponents of the matter kernels. 21or the CP -preserving kernels, we have (cid:101) K BB / ,(cid:96),CP ( ω ) = − π − (cid:96) (cid:96) ( (cid:96) + 1) D (cid:96) ( ω ) + ( − (cid:96) π(cid:96) ( (cid:96) + 1) (cid:96) (cid:88) p =1 ( − p ( (cid:96) ( (cid:96) + 1) − p ) p + ω + ( − (cid:96) (cid:96) ( (cid:96) + 1)4 πω (1 − πω coth( πω )) + ω
16 tanh( πω/ ( − (cid:96) + 14 D ( ω ) + ( − (cid:96) (cid:96) ( (cid:96) + 1) (cid:20) − (cid:96) + (cid:96) / (cid:96) + 2 (cid:96) (cid:21) D ( ω )+ (cid:96) − (cid:88) p =1 ( − (cid:96) + p p + 14 (cid:96) ( (cid:96) + 1) (cid:2) p ( p + 1) − (cid:96) ( (cid:96) + 1)(7 − ( − p + (cid:96) ) (cid:3) D p ( ω ) , (cid:101) K EE / ,(cid:96),CP ( ω ) = (cid:18) ω (cid:96) ( (cid:96) + 1) (cid:19) (cid:34) − π + (cid:96) ( (cid:96) + 1)2 D (cid:96) ( ω ) − (cid:96) (cid:88) p =0 ( − p + (cid:96) p π ( p + ω )+ (cid:100) (cid:96)/ (cid:101)− (cid:88) p =0 ((2 (cid:96) − p − D (cid:96) − − p ( ω ) − ( (cid:96) − − p ) D (cid:96) − − p ( ω ) − ( (cid:96) − p ) D (cid:96) − p ( ω )) , (cid:101) K σ B / ,(cid:96),CP ( ω ) = i ( − (cid:96) (cid:112) (cid:96) ( (cid:96) + 1) + ω (cid:96) ( (cid:96) + 1) (cid:34) (1 − (cid:96) − (cid:96) ) D ( ω ) + (cid:96) − (cid:88) p =1 ( − p (2 p + 1) D p ( ω )+( − (cid:96) (cid:96)D (cid:96) ( ω ) − (cid:96) ( (cid:96) + 1)4 πω (1 − πω coth( πω )) − π (cid:96) (cid:88) p =1 ( − p (cid:96) ( (cid:96) + 1) − p p + ω (cid:35) , (A.1)where D (cid:96) ( ω ) is defined in (4.18). 22or the CP -violating kernels, we have (cid:101) K σ B / ,(cid:96),NCP ( ω ) = i π (2 (cid:96) + 1) (cid:112) (cid:96) ( (cid:96) + 1) (cid:20) − (cid:96) − (cid:96) − + ω + (cid:96) (cid:96) + ω − ( (cid:96) + 1) ( (cid:96) + 1) + ω + (cid:96) ( (cid:96) + 2)( (cid:96) + 2) + ω (cid:21) , (cid:101) K σ E / ,(cid:96),NCP ( ω ) = sgn κ π (2 (cid:96) + 1) (cid:115) ω (cid:96) ( (cid:96) + 1) (cid:20) (cid:96) − (cid:96) − + ω + (cid:96) ( (cid:96) + 2) (cid:96) + ω − (cid:96) − (cid:96) + 1) + ω − (cid:96) ( (cid:96) + 2)( (cid:96) + 2) + ω (cid:21) , (cid:101) K EE / ,(cid:96),NCP ( ω ) = 18 π (cid:18) ω (cid:96) ( (cid:96) + 1) (cid:19) (cid:20) (cid:96) + 1) (cid:18) ( (cid:96) − (cid:96) + 1) ( (cid:96) − + ω + (cid:96) ( (cid:96) + 2) (cid:96) + ω + ( (cid:96) − ( (cid:96) + 1)( (cid:96) + 1) + ω + (cid:96) ( (cid:96) + 2)( (cid:96) + 2) + ω (cid:19)(cid:21) , (cid:101) K BB / ,(cid:96),NCP ( ω ) = 14 π − π (2 (cid:96) + 1) (cid:96) ( (cid:96) + 1) (cid:20) ( (cid:96) − (cid:96) + 1) ( (cid:96) − + ω + (cid:96) (cid:96) + ω + ( (cid:96) + 1) ( (cid:96) + 1) + ω + (cid:96) ( (cid:96) + 2)( (cid:96) + 2) + ω (cid:21) , (cid:101) K EB / ,(cid:96),NCP ( ω ) = sgn( κ ) i (cid:112) (cid:96) ( (cid:96) + 1) + ω π (2 (cid:96) + 1) (cid:96) ( (cid:96) + 1) (cid:20) − ( (cid:96) − (cid:96) + 1) ( (cid:96) − + ω + (cid:96) ( (cid:96) + 2) (cid:96) + ω + ( (cid:96) − (cid:96) + 1) ( (cid:96) + 1) + ω − (cid:96) ( (cid:96) + 2)( (cid:96) + 2) + ω (cid:21) . (A.2) B Comparison to [1]
In this appendix, we will relate the CP -preserving kernels (cid:101) K / ,(cid:96),CP ( ω ) computed in Section4.4.1 to the results of κ = 0 kernels computed in [1], which we use to justify our choice ofregularization for (cid:101) K EE / ,(cid:96),CP ( ω ) and (cid:101) K BE / ,(cid:96),CP ( ω ).In [1], the matter kernels were computed on S × R by expanding the gauge fluctuationin the gauge redundant basis a ( x ) = (cid:90) dω π (cid:34) a E (0) dτ √ π + ∞ (cid:88) (cid:96) =1 (cid:96) (cid:88) m = − (cid:96) (cid:34) a τ(cid:96)m ( ω ) Y (cid:96)m ( x ) dτ + a E(cid:96)m ( ω ) dY ( x ) (cid:96)m (cid:112) (cid:96) ( (cid:96) + 1)+ a B(cid:96)m ( ω ) ∗ dY ( x ) (cid:96)m (cid:112) (cid:96) ( (cid:96) + 1) (cid:35)(cid:35) e − iωτ , (B.1)23o that the matrix of frequency space kernels analogous to (4.11) is a 4 × (cid:101) K σσq,(cid:96) ( ω ) (cid:101) K σBq,(cid:96) ( ω ) (cid:101) K στq,(cid:96) ( ω ) (cid:101) K σEq,(cid:96) ( ω ) − (cid:101) K σB ∗ q,(cid:96) ( ω ) (cid:101) K BBq,(cid:96) ( ω ) (cid:101) K τBq,(cid:96) ( ω ) (cid:101) K BEq,(cid:96) ( ω ) (cid:101) K στ ∗ q,(cid:96) ( ω ) − (cid:101) K τB ∗ q,(cid:96) ( ω ) (cid:101) K ττq,(cid:96) ( ω ) (cid:101) K τEq,(cid:96) ( ω ) (cid:101) K σE ∗ q,(cid:96) ( ω ) − (cid:101) K BE ∗ q,(cid:96) ( ω ) (cid:101) K τE ∗ q,(cid:96) ( ω ) (cid:101) K EEq,(cid:96) ( ω ) , (B.2)where the signs are determined by the reality of the position space kernels (4.5). Since thebasis in (B.1) is gauge redundant, (B.2) has a zero eigenvector that relates the entries as (cid:101) K q,σE(cid:96) ( ω ) = − iω (cid:112) (cid:96) ( (cid:96) + 1) (cid:101) K q,στ(cid:96) ( ω ) , (cid:101) K q,BE(cid:96) ( ω ) = − iω (cid:112) (cid:96) ( (cid:96) + 1) (cid:101) K q,Bτ(cid:96) ( ω ) , (cid:101) K q,τE(cid:96) ( ω ) = − iω (cid:112) (cid:96) ( (cid:96) + 1) (cid:101) K q,ττ(cid:96) ( ω ) , (cid:101) K q,EE(cid:96) ( ω ) = ω (cid:96) ( (cid:96) + 1) (cid:101) K q,ττ(cid:96) ( ω ) . (B.3)By comparing (B.1) and (4.6), we see that the different independent entries in (B.2) arerelated to our (cid:101) K q,(cid:96) ( ω ) as (cid:101) K BB q,(cid:96) ( ω ) = (cid:101) K BBq,(cid:96) ( ω ) , (cid:101) K EE q,(cid:96) ( ω ) = (cid:18) ω (cid:96) ( (cid:96) + 1) (cid:19) (cid:101) K ττq,(cid:96) ( ω ) , (cid:101) K σ E q,(cid:96) ( ω ) = (cid:115) ω (cid:96) ( (cid:96) + 1) (cid:101) K στq,(cid:96) ( ω ) , (cid:101) K BE q,(cid:96) ( ω ) = (cid:115) ω (cid:96) ( (cid:96) + 1) (cid:101) K τBq,(cid:96) ( ω ) , (cid:101) K B σq,(cid:96) ( ω ) = (cid:101) K σ B q,(cid:96) ( ω ) = (cid:101) K σBq,(cid:96) ( ω ) . (B.4)The kernels in (B.2) were computed in [1] for κ = 0, in which case the theory is CP -invariant so (cid:101) K στq,(cid:96) ( ω ), (cid:101) K σEq,(cid:96) ( ω ), (cid:101) K τBq,(cid:96) ( ω ), and (cid:101) K BEq,(cid:96) ( ω ) vanish. For q = 1 / µ , we will denote the remaining kernels as K / ,(cid:96) ( ω ), and they are given by the following24nfinite sums: K σσ / ,(cid:96) ( ω ) = 8 π (cid:96) + 1 ∞ (cid:88) (cid:96) (cid:48) ,(cid:96) (cid:48)(cid:48) =1 / λ (cid:96) (cid:48) + λ (cid:96) (cid:48)(cid:48) λ (cid:96) (cid:48) λ (cid:96) (cid:48)(cid:48) ( ω + ( λ (cid:96) (cid:48) + λ (cid:96) (cid:48)(cid:48) ) ) I D ( (cid:96), (cid:96) (cid:48) , (cid:96) (cid:48)(cid:48) ) , K σB / ,(cid:96) ( ω ) = 8 π i (2 (cid:96) + 1) (cid:112) (cid:96) ( (cid:96) + 1) ∞ (cid:88) (cid:96) (cid:48) ,(cid:96) (cid:48)(cid:48) =1 / λ (cid:96) (cid:48) + λ (cid:96) (cid:48)(cid:48) λ (cid:96) (cid:48) λ (cid:96) (cid:48)(cid:48) ( ω + ( λ (cid:96) (cid:48) + λ (cid:96) (cid:48)(cid:48) ) ) I H ( (cid:96), (cid:96) (cid:48) , (cid:96) (cid:48)(cid:48) ) , K τE / ,(cid:96) ( ω ) = 8 π iω (2 (cid:96) + 1) (cid:112) (cid:96) ( (cid:96) + 1) ∞ (cid:88) (cid:96) (cid:48) ,(cid:96) (cid:48)(cid:48) =1 / ( λ (cid:96) (cid:48) − λ (cid:96) (cid:48)(cid:48) ) [ (cid:96) (cid:48)(cid:48) ( (cid:96) (cid:48)(cid:48) + 1) − (cid:96) (cid:48) ( (cid:96) (cid:48) + 1)]2 λ (cid:96) (cid:48) λ (cid:96) (cid:48)(cid:48) ( ω + ( λ (cid:96) (cid:48) + λ (cid:96) (cid:48)(cid:48) ) ) I D ( (cid:96), (cid:96) (cid:48) , (cid:96) (cid:48)(cid:48) ) , K EE / ,(cid:96) ( ω ) = − π (2 (cid:96) + 1) (cid:96) ( (cid:96) + 1) ∞ (cid:88) (cid:96) (cid:48) =1 / ∞ (cid:88) (cid:96) (cid:48)(cid:48) =1 / ( λ (cid:96) (cid:48) + λ (cid:96) (cid:48)(cid:48) ) [ (cid:96) (cid:48) ( (cid:96) (cid:48) + 1) − (cid:96) (cid:48)(cid:48) ( (cid:96) (cid:48)(cid:48) + 1)] λ (cid:96) (cid:48) λ (cid:96) (cid:48)(cid:48) ( ω + ( λ (cid:96) (cid:48) + λ (cid:96) (cid:48)(cid:48) ) ) I D ( (cid:96), (cid:96) (cid:48) , (cid:96) (cid:48)(cid:48) )+ 2 (cid:96) (cid:48) + 18 πλ (cid:96) (cid:48) (cid:21) + C , K BB / ,(cid:96) ( ω ) = 8 π (2 (cid:96) + 1) (cid:96) ( (cid:96) + 1) ∞ (cid:88) (cid:96) (cid:48) =1 / ∞ (cid:88) (cid:96) (cid:48)(cid:48) =1 / λ (cid:96) (cid:48) + λ (cid:96) (cid:48)(cid:48) λ (cid:96) (cid:48) λ (cid:96) (cid:48)(cid:48) ( ω + ( λ (cid:96) (cid:48) + λ (cid:96) (cid:48)(cid:48) ) ) I B ( (cid:96), (cid:96) (cid:48) , (cid:96) (cid:48)(cid:48) ) + 2 (cid:96) (cid:48) + 18 πλ (cid:96) (cid:48) + C , K ττ / ,(cid:96) ( ω ) = 8 π (2 (cid:96) + 1) ∞ (cid:88) (cid:96) (cid:48) =1 / ∞ (cid:88) (cid:96) (cid:48)(cid:48) =1 / − ( λ (cid:96) (cid:48) + λ (cid:96) (cid:48)(cid:48) )( ω + 4 λ (cid:96) (cid:48) λ (cid:96) (cid:48)(cid:48) )2 λ (cid:96) (cid:48) λ (cid:96) (cid:48)(cid:48) ( ω + ( λ (cid:96) (cid:48) + λ (cid:96) (cid:48)(cid:48) ) ) I D ( (cid:96), (cid:96) (cid:48) , (cid:96) (cid:48)(cid:48) ) + 2 (cid:96) (cid:48) + 14 πλ (cid:96) (cid:48) + C (cid:48) , (B.5)where I D ( (cid:96), (cid:96) (cid:48) , (cid:96) (cid:48)(cid:48) ) = (cid:20) (2 (cid:96) + 1)(2 (cid:96) (cid:48) + 1)(2 (cid:96) (cid:48)(cid:48) + 1)32 π (cid:21) (cid:32) (cid:96) (cid:96) (cid:48) (cid:96) (cid:48)(cid:48) − / / (cid:33) , I H ( (cid:96), (cid:96) (cid:48) , (cid:96) (cid:48)(cid:48) ) = I D ( (cid:96), (cid:96) (cid:48) , (cid:96) (cid:48)(cid:48) ) × (cid:96) ( (cid:96) + 1) − ( (cid:96) (cid:48) − (cid:96) (cid:48)(cid:48) ) for (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) odd (cid:96) ( (cid:96) + 1) − ( (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) + 1) for (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) even I B ( (cid:96), (cid:96) (cid:48) , (cid:96) (cid:48)(cid:48) ) = − I D ( (cid:96), (cid:96) (cid:48) , (cid:96) (cid:48)(cid:48) ) × [( (cid:96) (cid:48) − (cid:96) (cid:48)(cid:48) ) − (cid:96) ( (cid:96) + 1)] for (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) odd[( (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) + 1) − (cid:96) ( (cid:96) + 1)] for (cid:96) + (cid:96) (cid:48) + (cid:96) (cid:48)(cid:48) even. (B.6)The constants C and C (cid:48) parameterize the regularizations required for K EE / ,(cid:96) ( ω ), K BB / ,(cid:96) ( ω ),and K ττ / ,(cid:96) ( ω ), while the rest of the kernels do not require regularization.Since both these K / ,(cid:96) ( ω ) and our (cid:101) K / ,(cid:96),CP ( ω ) are independent of κ , we can relatethem by setting µ = q = 1 / K / ,(cid:96) ( ω ). The total kernel (cid:101) K / ,(cid:96) ( ω ) = K / ,(cid:96) ( ω ) +25 K / ,(cid:96),NCP ( ω ) must satisfy the gauge relations (B.3), which relates the finite term (cid:101) K τE / ,(cid:96) ( ω )to the regularized terms (cid:101) K ττ / ,(cid:96) ( ω ) and (cid:101) K EE / ,(cid:96) ( ω ), which contain the constants C and C (cid:48) . Itcan be checked numerically that this relation is satisfied only if C = − π , C (cid:48) = 14 π . (B.7)The rest of (cid:101) K / ,(cid:96),CP ( ω ) can similarly be numerically matched to K / ,(cid:96) ( ω ), which justifiesour choice of regularization, and is a check on our computation of these terms in general. C Asymptotic expansion
In this appendix we will write the asymptotic formula at large (cid:96) and ω for the integrand in(4.11).The expression for the CP -violating kernels in (A.2) and (4.38), the Chern-Simons kernelin (4.13), and the q = 0 kernels in (4.21) and (4.18) are all simple functions of (cid:96) and ω whose asymptotic expansion is straightforward. The CP -preserving kernels in (A.1) and(4.33) include sums of the form (cid:80) (cid:96)p =1 D p ( ω ) and (cid:80) (cid:96)p =1 f ( p ) p + ω , for some polynomial f ( p ). Theformer can be written in terms of gamma functions using standard identities, while the lattercan be expressed in terms of the polygamma function ψ ( z ), both of which have standard26symptotic formulae. The resulting asymptotic expressions for the kernels are (cid:101) K σσ / ,(cid:96) ( ω ) = (cid:101) K σσ ,(cid:96) ( ω ) + ( (cid:96) + ) − ω π (cid:2) ( (cid:96) + ) + ω (cid:3) + 2( (cid:96) + ) + 25 ω − ω ( (cid:96) + ) π (cid:2) ( (cid:96) + ) + ω (cid:3) + . . . , (cid:101) K σ B / ,(cid:96) ( ω ) = i (cid:96) + (cid:2) ( (cid:96) + ) + ω (cid:3) / + i (cid:96) + ) + ω − ω ( (cid:96) + ) (cid:96) + ) (cid:2) ( (cid:96) + ) + ω (cid:3) / + . . . , (cid:101) K BB / ,(cid:96) ( ω ) = (cid:101) K BB ,(cid:96) ( ω ) + 116 (cid:113) ( (cid:96) + ) + ω − π (cid:2) ( (cid:96) + ) + ω (cid:3) + . . . , (cid:101) K EE / ,(cid:96) ( ω ) = (cid:101) K EE ,(cid:96) ( ω ) + (cid:0) (cid:96) ( (cid:96) + 1) + ω ) (cid:1) (cid:34) (cid:2) ( (cid:96) + ) + ω (cid:3) / + 2( (cid:96) + ) − ω π (cid:2) ( (cid:96) + ) + ω (cid:3) (cid:35) + . . . , (cid:101) K E σ / ,(cid:96) ( ω ) = sgn( κ ) (cid:115) ω (cid:96) ( (cid:96) + 1) (cid:34) ( (cid:96) + ) π (cid:2) ( (cid:96) + ) + ω (cid:3) + 5( (cid:96) + ) − ω − ω ( (cid:96) + ) π (cid:2) ( (cid:96) + ) + ω (cid:3) (cid:35) + . . . , (cid:101) K EB / ,(cid:96) ( ω ) = − sgn( κ ) i (cid:112) (cid:96) ( (cid:96) + 1) + ω (cid:34) − π + (cid:0) ( (cid:96) + ) − ω (cid:1) π (cid:2) ( (cid:96) + ) + ω (cid:3) + 19( (cid:96) + ) + ω − ω ( (cid:96) + ) + 9 ω ( (cid:96) + ) π ( (cid:96) + ) (cid:2) ( (cid:96) + ) + ω (cid:3) (cid:35) + . . . , (C.1)where the q = 0 kernels are all simple functions of (cid:101) K σσ ,(cid:96) ( ω ) = 18 (cid:113) ( (cid:96) + ) + ω + − ( (cid:96) + ) + ω (cid:2) ( (cid:96) + ) + ω (cid:3) / + . . . . (C.2)We can now plug these expressions into (4.39) to get L (cid:96) ( ω ) = 2 π ( π + 64) (cid:2) ( (cid:96) + ) + ω (cid:3) + 4(64 + 3 π )(( (cid:96) + ) − ω ) π (64 + π ) (cid:2) ( (cid:96) + ) + ω (cid:3) / + O (cid:18) (cid:96) + ) + ω ] (cid:19) , (C.3)The first term is the linear divergence that we regularize in (4.41). The second term gives apotential logarithmic divergence, that cancels in the sum and integral in (4.39) as long as weuse a regulator that respects conformal invariance, such as (4.44). The higher order termsare all convergent. 27 eferences [1] E. Dyer, M. Mezei, S. S. Pufu, and S. Sachdev, “Scaling dimensions of monopoleoperators in the CP N b − theory in 2 + 1 dimensions,” JHEP (2015) 037, .[2] A. M. Polyakov, “Compact gauge fields and the infrared catastrophe,” Phys.Lett.
B59 (1975) 82–84.[3] V. Borokhov, A. Kapustin, and X.-k. Wu, “Topological disorder operators inthree-dimensional conformal field theory,”
JHEP (2002) 049, hep-th/0206054 .[4] S. M. Chester, L. V. Iliesiu, M. Mezei, and S. S. Pufu, “Monopole Operators in U (1)Chern-Simons-Matter Theories,” .[5] T. Appelquist and R. D. Pisarski, “High-Temperature Yang-Mills Theories andThree-Dimensional Quantum Chromodynamics,” Phys.Rev.
D23 (1981) 2305.[6] T. W. Appelquist, M. J. Bowick, D. Karabali, and L. Wijewardhana, “SpontaneousChiral Symmetry Breaking in Three-Dimensional QED,”
Phys.Rev.
D33 (1986) 3704.[7] T. Appelquist, D. Nash, and L. Wijewardhana, “Critical behavior in(2 + 1)-dimensional QED,”
Phys.Rev.Lett. (1988) 2575.[8] T. Appelquist and U. W. Heinz, “Three-dimensional O ( N ) theories at largedistances,” Phys.Rev.
D24 (1981) 2169.[9] D. T. Son, “Is the Composite Fermion a Dirac Particle?,”
Phys. Rev. X5 (2015), no. 3031027, .[10] O. Aharony, “Baryons, monopoles and dualities in Chern-Simons-matter theories,” JHEP (2016) 093, .[11] A. Karch and D. Tong, “Particle-Vortex Duality from 3d Bosonization,” Phys. Rev. X6 (2016), no. 3 031043, .[12] J. Murugan and H. Nastase, “Particle-vortex duality in topological insulators andsuperconductors,” .[13] N. Seiberg, T. Senthil, C. Wang, and E. Witten, “A Duality Web in 2+1 Dimensionsand Condensed Matter Physics,” Annals Phys. (2016) 395–433, .2814] P.-S. Hsin and N. Seiberg, “Level/rank Duality and Chern-Simons-Matter Theories,”
JHEP (2016) 095, .[15] D. Radicevic, D. Tong, and C. Turner, “Non-Abelian 3d Bosonization and QuantumHall States,” JHEP (2016) 067, .[16] S. Kachru, M. Mulligan, G. Torroba, and H. Wang, “Bosonization and MirrorSymmetry,” Phys. Rev.
D94 (2016), no. 8 085009, .[17] S. Kachru, M. Mulligan, G. Torroba, and H. Wang, “Nonsupersymmetric dualitiesfrom mirror symmetry,”
Phys. Rev. Lett. (2017), no. 1 011602, .[18] A. Karch, B. Robinson, and D. Tong, “More Abelian Dualities in 2+1 Dimensions,”
JHEP (2017) 017, .[19] M. A. Metlitski, A. Vishwanath, and C. Xu, “Duality and bosonization of (2+1)-dimensional Majorana fermions,” Phys. Rev.
B95 (2017), no. 20 205137, .[20] O. Aharony, F. Benini, P.-S. Hsin, and N. Seiberg, “Chern-Simons-matter dualitieswith SO and U Sp gauge groups,”
JHEP (2017) 072, .[21] F. Benini, P.-S. Hsin, and N. Seiberg, “Comments on Global Symmetries, Anomalies,and Duality in (2+1)d,” .[22] Z. Komargodski and N. Seiberg, “A Symmetry Breaking Scenario for QCD ,” .[23] S. M. Chester and S. S. Pufu, “Towards bootstrapping QED ,” JHEP (2016) 019, .[24] S. S. Pufu, “Anomalous dimensions of monopole operators in three-dimensionalquantum electrodynamics,” Phys.Rev.
D89 (2014), no. 6 065016, .[25] E. Dyer, M. Mezei, and S. S. Pufu, “Monopole Taxonomy in Three-DimensionalConformal Field Theories,” .[26] S. M. Chester, M. Mezei, S. S. Pufu, and I. Yaakov, “Monopole operators from the4 − (cid:15) expansion,” JHEP (2016) 015, .[27] M. A. Metlitski, M. Hermele, T. Senthil, and M. P. Fisher, “Monopoles in CP N − model via the state-operator correspondence,” Phys.Rev.
B78 (2008) 214418, . 2928] V. Borokhov, A. Kapustin, and X.-k. Wu, “Monopole operators and mirror symmetryin three dimensions,”
JHEP (2002) 044, hep-th/0207074 .[29] E. Dupuis, M. B. Paranjape, and W. Witczak-Krempa, “Transition from a Dirac spinliquid to an antiferromagnet: Monopoles in a QED -Gross-Neveu theory,” Phys. Rev.B (Sep, 2019) 094443.[30] E. Dupuis and W. Witczak-Krempa, “Monopole hierarchy in transitions out of a Diracspin liquid,” .[31] G. Murthy and S. Sachdev, “Action of hedgehog instantons in the disordered phase ofthe (2 + 1)-dimensional CP N − model,” Nucl.Phys.
B344 (1990) 557–595.[32] M. S. Block, R. G. Melko, and R. K. Kaul, “Fate of CP N − Fixed Points with qMonopoles,”
Physical Review Letters (Sept., 2013) 137202, .[33] R. K. Kaul and M. Block, “Numerical studies of various Neel-VBS transitions inSU( N ) antiferromagnets,” .[34] F. Benini, C. Closset, and S. Cremonesi, “Chiral flavors and M2-branes at toric CY4singularities,” JHEP (2010) 036, .[35] F. Benini, C. Closset, and S. Cremonesi, “Quantum moduli space of Chern-Simonsquivers, wrapped D6-branes and AdS4/CFT3,”
JHEP (2011) 005, .[36] Y. Imamura and S. Yokoyama, “Index for three dimensional superconformal fieldtheories with general R-charge assignments,”
JHEP (2011) 007, .[37] S. Kim, “The Complete superconformal index for N=6 Chern-Simons theory,” Nucl.Phys.
B821 (2009) 241–284, . [Erratum: Nucl. Phys.B864,884(2012)].[38] O. Aharony, P. Narayan, and T. Sharma, “On monopole operators in supersymmetricChern-Simons-matter theories,” .[39] M. Marino, “Lectures on localization and matrix models in supersymmetricChern-Simons-matter theories,”
J. Phys.
A44 (2011) 463001, .[40] S. S. Pufu and S. Sachdev, “Monopoles in 2 + 1-dimensional conformal field theorieswith global U(1) symmetry,”
JHEP (2013) 127, .3041] T. T. Wu and C. N. Yang, “Dirac Monopole Without Strings: Monopole Harmonics,”
Nucl.Phys.
B107 (1976) 365.[42] T. T. Wu and C. N. Yang, “Some properties of monopole harmonics,”
Phys.Rev.
D16 (1977) 1018–1021.[43] R. K. Kaul and S. Sachdev, “Quantum criticality of U(1) gauge theories withfermionic and bosonic matter in two spatial dimensions,”
Phys.Rev.
B77 (2008)155105, .[44] I. R. Klebanov, S. S. Pufu, S. Sachdev, and B. R. Safdi, “Entanglement entropy of 3-dconformal gauge theories with many flavors,”
JHEP (2012) 036, .[45] J. A. Gracey, “Large N(f) critical exponents for abelian Chern-Simons theory coupledto matter,”
Europhys. Lett. (1993) 651–655.[46] J. Lou, A. W. Sandvik, and N. Kawashima, “Antiferromagnetic to valence-bond-solidtransitions in two-dimensional SU( N ) Heisenberg models with multispin interactions,” Physical Review B (Nov., 2009) 180414, .[47] R. K. Kaul and A. W. Sandvik, “Lattice Model for the SU( N ) N´eel to Valence-BondSolid Quantum Phase Transition at Large N ,” Physical Review Letters (Mar.,2012) 137201, .[48] N. Karthik, “Monopole scaling dimension using Monte-Carlo simulation,”
Phys. Rev.D (2018), no. 7 074513, .[49] N. Karthik and R. Narayanan, “Numerical determination of monopole scalingdimension in parity-invariant three-dimensional noncompact QED,” Phys. Rev. D (2019), no. 5 054514,1908.05500