Evidence for 3d bosonization from monopole operators Shai M. Chester Jeerson Physical Laboratory Harvard University Cambridge MA 02138 USA

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Evidence for 3d bosonization from monopole operators

Shai M. Chester

Jeﬀerson Physical Laboratory, Harvard University, Cambridge, MA 02138, USA

Center of Mathematical Sciences and Applications,

Harvard University, Cambridge, MA 02138, USA

Eric Dupuis

D´epartement de physique, Universit´e de Montr´eal, Montr´eal (Qu´ebec), H3C 3J7, Canada

William Witczak-Krempa

D´epartement de physique, Universit´e de Montr´eal, Montr´eal (QC), H3C 3J7, Canada

Centre de Recherches Math´ematiques, Universit´e de Montr´eal; Montr´eal (QC), H3C 3J7, Canada and

Institut Courtois, Universit´e de Montr´eal, Montr´eal (QC), H2V 0B3, Canada

We give evidence for 3d bosonization in Conformal Field Theories (CFTs) by computing monopole

operator scaling dimensions in 2+1 dimensional quantum electrodynamics (QED3) with Chern-

Simons level kand Ncomplex bosons in a large N, k expansion. We ﬁrst consider the k= 0 case,

where we show that scaling dimensions previously computed to subleading order in 1/N can be

extrapolated to N= 1 and matched to O(2) Wilson-Fisher CFT scaling dimensions with around

5% error, which is evidence for particle-vortex duality. We then generalize the subleading calculation

to large N, k and ﬁxed k/N, extrapolate to N=k= 1, and consider monopole operators that are

conjectured to be dual to non-degenerate scalar operators in a theory of a single Dirac fermion. We

ﬁnd matches typically with 1% error or less, which is strong evidence of this so-called ‘seed’ duality

that implies a web of 3d bosonization dualities among CFTs.

Introduction—IR duality is when two quantum ﬁeld

theories that are completely diﬀerent at short distances

(the UV), nonetheless ﬂow to the same conformal ﬁeld

theory (CFT) at long distances (the IR). While dual-

ity is common in two spacetime dimensions, in 3d this

phenomenon is much more rare. For many years, the

only experimentally relevant example in 3d was parti-

cle/vortex duality [1,2], which conjectures that the O(2)

Wilson-Fisher ﬁxed point is dual to 3d quantum electro-

dynamics (QED3) with N= 1 complex bosonic ﬁeld and

k= 0 Chern-Simons level, the so-called abelian Higgs

model. Recently, new dualities were conjectured between

QED3 with various Chern-Simons levels and matter con-

tent [3–5], and these dualities were shown to be part of

a so-called web of dualities that generically relates CFTs

with fermionic matter to bosonic matter [6,7], and is thus

an example of 3d bosonization. This duality web can be

derived from a conjectured ‘seed’ duality, which relates

QED3 with N= 1 boson and k= 1 to the free theory

of a single complex two-component fermion. The central

idea is that the Chern-Simons term eﬀectively attaches

ﬂux (a magnetic instanton or monopole) to the boson,

leading to an additional Berry phase and Fermi statistics

for the boson-ﬂux composite.

QED3 at ﬁnite Nand kis strongly coupled in the IR,

which makes it hard to verify these dualities. While the

conjectured dualities satisfy kinematic consistency checks

such as ’t Hooft anomalies [7], we would ideally like to

check local dynamical observables such as critical expo-

nents (i.e. scaling dimensions). When k= 0, the theory

can be modeled on the lattice [8,9], which found evidence

for particle/vortex duality by comparing the lowest scal-

ing dimensions ∆O

qfrom O(2) lattice studies [10,11] of

operators with charge qunder the U(1) ∼

=O(2) global

symmetry, to lattice estimates ∆Q

qof the dual operator

scaling dimensions in QED3 [52]:

∆O

0= 1.511,∆O

2=.5191,∆O

1= 1.236,∆O

2= 2.109,

∆Q

0= 1.508,∆Q

=.48,∆Q

1= 1.23,∆Q

= 2.15.

(1)

Lattice methods have more diﬃculty when k6= 0 due to

the sign problem, however, so it has been diﬃcult to nu-

merically verify the seed duality for 3d bosonization. In-

stead, the duality has been motivated by an uncontrolled

ﬂow [12,13] from more well established supersymmetric

dualities [14], as well as as an extrapolation to NC= 1 of

3d bosonization for quantum chromodynamics (QCD3)

at large colors NCand k[15], which has been checked at

leading order in 1/NCstarting with [16–18].

Here we give evidence for both particle/vortex dual-

ity, and the seed bosonization duality by computing the

scaling dimension of monopole operators in scalar QED3,

and matching these to the dimensions of the operators in

the dual theories. We will do this by considering QED3

in the limit of large Nscalars, and also large kand ﬁxed

κ≡k/N for the bosonization case, where monopole op-

erator scaling dimensions can be computed in a 1/N ex-

pansion [19,20]. For k= 0, the scaling dimensions have

already been computed to subleading order in [21], so

we will simply extrapolate these results to N= 1 and

compare to scaling dimensions of the critical O(2) model

as computed from the conformal bootstrap [22,23] and

lattice [11]. For nonzero k, we will extend the leading

order calculation in [24] to sub-leading order for general

arXiv:2210.12370v2 [hep-th] 30 Oct 2022

κ, extrapolate to N=κ= 1, and compare to scaling

dimensions of non-degenerate scalar operators in the free

fermion theory [53]. In all cases, we ﬁnd that our pertur-

bative calculation matches the conjectured dualities with

a relative error of just a few percent, as shown in Tables

Iand II.

The rest of this letter is organized as follows. We ﬁrst

introduce monopole operators and discuss our new calcu-

lation of their scaling dimension at large N, k. We then

review how these operators are expected to map to the

dual theories, and compare our new results. We end with

a discussion of our results. Technical details are discussed

in the Appendices.

Monopoles at large N, k—Monopole operators are

deﬁned in three dimensional Abelian gauge theories as

local operators that are charged under the topological

global symmetry U(1)top [20,25], whose conserved cur-

rent and charge are

jµ

top =1

8πµνρFνρ , q =1

4πZΣ

F , (2)

where Fνρ ≡∂νAρ−∂ρAνis the gauge ﬁeld strength with

spacetime index µ= 1,2,3, Σ is a closed two-dimensional

surface, and jµ

top is conserved due to the Bianchi iden-

tity. In the normalization (2), the charge qis restricted

by Dirac quantization to take the values q∈Z/2. As

in [20,21,24,26–34], we will compute the scaling di-

mension of the lowest dimension monopole operators us-

ing the state-operator correspondence, which identiﬁes

the scaling dimensions of monopole operators of charge q

with the energies of states in the Hilbert space on S2×R

with 4πq magnetic ﬂux through the sphere [20]. The

ground state energy on S2×Rcan then be computed

in the large Nand klimit using a saddle point expan-

sion. When k6= 0, the Chern-Simons term induces a

gauge charge proportional to q, so that the naive S2×R

vacuum must be dressed by charged matter modes. Fol-

lowing [24], we can enforce this dressing by computing

the small temperature T≡β−1limit of the thermal free

energy on S2×S1

β, where the saddle point value of the

holonomy of the gauge ﬁeld on S1

βacts like a chemical po-

tential for the matter ﬁelds. This dressing will make the

monopole transform in a nontrivial representation under

the SU (N) ﬂavor symmetry with a nonzero spin for the

SO(3) rotation symmetry.

We begin by writing the conformally invariant action

of QED3 with Ncomplex scalars φion S2×S1

βas [54]

S=Zd3xh√g(∇µ−iAµ)φi2+ (1

4+iλ)|φi|2

−ik

4πµνρAµ∂νAρi,

(3)

where gis the determinant of the metric, λis a Hubbard-

Stratonovich ﬁeld, and i= 1, . . . , N. We are interested in

computing the thermal free energy Fq,κ in the presence

of a magnetic ﬂux RdA = 4πq through S2. We can

integrate out the matter ﬁelds in the path integral on

this background to get

e−βFq,κ =

ZDA exp

−Ntr log1

4+iλ −(∇µ−iAµ)2

+iN Zd3xκ

4πµνρAµ∂νAρ+√gλ,

(4)

where κ=k/N . We now expand Aµand λaround a

saddle point by taking

Aµ=Aµ+aµ, iλ =µ+iσ , (5)

where aµand σare ﬂuctuations around a background

Aµ=Aµand iλ =µthat satisfy

δFq,κ[Aµ, λ]

δAµσ=aµ=0

=δFq,κ[Aµ, λ]

δλ σ=aµ=0

= 0 .(6)

On S2×S1

βwith magnetic ﬂux 4πq, the most general

such background is µconstant and Aq

µgiven by

Aτ=−iα , Fθφdθ ∧dφ =qsin θdθ ∧dφ , (7)

where α=iβ−1RS1

βAis a real constant called the holon-

omy of the gauge ﬁeld.

Since the integrand in (4) is proportional to N, the

thermal free energy Fq,κ can then be expanded at large

Nas

Fq,κ =NF (0)

q,κ +F(1)

q,κ +1

NF(2)

q,κ +. . . , (8)

where F(0)

q,κ comes from evaluating Fq,κ at the saddle

point and F(1)

q,κ comes from the functional determinant of

the quantum ﬂuctuations around the saddle point. The

scaling dimension ∆q,κ is then obtained from the zero

temperature limit as

∆q,κ =N∆(0)

q,κ + ∆(1)

q,κ +. . . , ∆(n)

q,κ ≡lim

β→∞ F(n)

q,κ .(9)

At leading order there are many degenerate monopoles

when k6= 0, due to the diﬀerent ways of dressing the bare

monopole, which can be detected from the O(β−1) terms

in F(0)

q,κ . This leads to degeneracy breaking contributions

to ∆(1)

q,κ that were shown [24] to depend on spin, but

whose explicit form was not worked out in general.

The leading order F(0)

q,κ was computed in [24] for general

k, q by ﬁxing µand αfrom the saddle point equations

(6), setting Aµand λin (4) to their saddle point values

(7), and then doing the resulting mode expansion. We

review the details of this calculation in Appendix A, and

give some of the resulting ∆(0)

q,κ in Tables Iand II. The

sub-leading F(1)

q,κ comes from expanding (4) to quadratic

order in the ﬂuctuations aµand σaround the saddle point

q∆(0)

q,0∆(1)

q,0N= 1 O(2) Error (%)

1/2 0.12459 0.38147 0.50609 0.519130434 2.5

1 0.31110 0.87452 1.1856 1.23648971 4.1

3/2 0.54407 1.4646 2.0087 2.1086(3) 4.7

2 0.81579 2.1388 2.9546 3.11535(73) 5.2

5/2 1.1214 2.8879 4.0093 4.265(6) 5.8

3 1.4575 3.7053 5.1628 5.509(7) 6.3

7/2 1.8217 4.5857 6.4074 6.841(8) 6.3

4 2.2118 5.5249 7.7367 8.278(9) 6.5

9/2 2.6263 6.5194 9.1458 9.796(9) 6.6

5 3.0638 7.5665 10.630 11.399(10) 6.7

TABLE I: Scaling dimensions ∆q,0=N∆(0)

q,0+ ∆(1)

q,0+O(1/N )

for charge qscalar monopole operators in QED3 with N

scalars and k= 0 in a large Nexpansion [21,35] extrapo-

lated to N= 1, compared to values of the dual operators

in the critical O(2) model as computed from the numerical

bootstrap (q≤2) and lattice (q > 2), along with the relative

errors from the comparison.

values to get the Gaussian integral

exp(−βF (1)

q,κ ) =ZDaDσ exp h−N

2Zd3xd3x0

×√gpg0aµ(x)Kµν

q(x, x0)aν(x0)

+σ(x)Kσσ

q(x, x0)σ(x0)+2σ(x)Kσν

q(x, x0)aν(x0)i,

(10)

where the kernels are expressed by expectation values of

the matter ﬁelds for the saddle point values of Aµand λ.

These kernels can be computed in terms of the thermal

Green’s function hφi(x)φ∗

j(x0)i=δi

jGq(x, x0), which was

computed for general q, κ in [24]. We give explicit expres-

sions in Appendix A, where we explain how to use these

kernels to compute the small temperature expansion of

F(1)

q,κ , which yields the sub-leading ∆(1)

q,κ. For κ= 0, this

calculation was performed in [21,35], and we list some

of their results in Table I. For κ6= 0, we have addi-

tional parity breaking contributions to the matter ker-

nels, which makes the calculation much more challenging.

When κ= 1 and q= 1/2, it was shown in [36] that the

Green’s function simpliﬁes, so that matter kernels could

be computed in closed form and used to compute the

scaling dimension. In Appendix A, we extend this cal-

culation to general q, κ using an algorithmic approach,

which yields the scaling dimensions in Table II.

Duality Comparison—We will now extrapolate the

large Nmonopole scaling dimensions to N= 1 and com-

pare to the conjectured dual theories. For k= 0, we

expect the monopole operators of charge qto be dual

to the lowest dimension scalar operators of charge qin

the critical O(2) Wilson-Fisher CFT, where U(1)top is

identiﬁed with the O(2) ﬂavor symmetry. The scaling

dimensions of operators with q= 1/2,1,3/2,2 have been

determined using the conformal bootstrap [22,23], while

higher values of qwere determined with less accuracy us-

q∆(0)

q,1∆(1)

q,1N= 1 Fermion Error (%)

1/2 1 −0.2789 0.7211 1 28

1 2.5833 −0.6312 1.952 2 2.4

3/2 4.5873 −1.052 3.535 4 15

2 6.9380 −1.534 5.404 6 9.9

5/2 9.5904 −2.070 7.521 8 6.0

3 12.514 −2.655 9.859 10 1.4

6 34.727 −7.032 27.70 28 1.1

10 74.141 −14.71 59.43 60 0.95

15 135.67 −26.64 109.03 110 0.88

21 224.23 −43.76 180.5 182 0.84

TABLE II: Scaling dimensions ∆q,1=N∆(0)

q,1+∆(1)

q,1+O(1/N )

for charge qmonopole operators in QED3 with Nscalars and

k/N = 1 in a large N, k expansion extrapolated to N=k= 1,

compared to values of the dual operators in the free fermion

CFT, along with the relative errors from the comparison. We

expect the comparison to be most precise when the operator

is a unique scalar q= 1,3,6, . . . , as denoted in purple.

ing lattice methods [10,11]. We compare these values in

Table I[55], and ﬁnd that the monopole scaling dimen-

sions match their expected duals with just a few percent

relative error, which gradually grows with q. It is remark-

able that the contribution of the quantum correction ∆(1)

q,0

exceeds the leading saddle-point one by a factor of more

than 2. This extends the previous lattice evidence for the

singlet [8] and q= 1/2,1,3/2 [9] monopole scaling dimen-

sions as reviewed in (1). Monopoles with q= 1/2 and

higher Nwere also successfully matched to lattice calcu-

lations in antiferromagnets with SU(N) symmetry that

can be described by an eﬀective CPN−1gauge theory as

in Eq. (3) with k= 0 [21], so the sub-leading compu-

tation seems accurate for general N. Note that all our

large-Nestimates are strictly below the estimates from

other methods, which is also true for the boson-fermion

duality that we now discuss.

We next consider the extrapolation of the large N, k

monopole scaling dimensions to N=k= 1, where the

theory is conjectured to be dual to a single free fermion

ψαwith spinor index α= 1,2. Monopoles of charge q

should be dual to the lowest dimension operator formed

by 2qfermions, where we identify U(1)top with the U(1)

ﬂavor symmetry of the complex fermion [56]. Due to the

antisymmetry of the fermions and the equations of mo-

tion, the lowest operator must sometimes include deriva-

tives, and so there will be multiple such operator with

diﬀerent spins for diﬀerent contractions of the indices.

For instance, while the lowest q= 1/2 operator is the

spin half ψα, and the lowest q= 1 is the spin zero

ψ[αψβ], already at q= 2 the lowest dimension opera-

tors are ψα1ψα2/

∂α3α4ψα5/

∂α6α7ψα8, where the two ways

of contracting the indices give spin zero or two. We can

count the lowest dimension operators of a given qby ex-

panding the S2×Spartition function for the free fermion

in characters of primary operators following [37], which

we do in Appendix B. These operators are unique scalars

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Evidencefor3dbosonizationfrommonopoleoperatorsShaiM.ChesterJeersonPhysicalLaboratory,HarvardUniversity,Cambridge,MA02138,USACenterofMathematicalSciencesandApplications,HarvardUniversity,Cambridge,MA02138,USAEricDupuisDepartementdephysique,UniversitedeMontreal,Montreal(Quebec),H3C3J7,CanadaWil...

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Evidence for 3d bosonization from monopole operators Shai M. Chester Jeerson Physical Laboratory Harvard University Cambridge MA 02138 USA

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