Scalar conformal primary fields in the Brownian loop soup

Prepared for submission to JHEP arXiv:2109.12116v1 [math-ph] 24 Sep 2021 Scalar conformal primary fields in the Brownian loop soup Federico Camiaa,b Valentino F. Foita Alberto Gandolfia Matthew Klebanc a Science Division, New York University Abu Dhabi, Saadiyat Island, Abu Dhabi, United Arab Emirates b Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1111, 1081 HV Amsterdam, The Netherlands c Center for Cosmology and Particle Physics, New York University, 726 Broadway, New York, NY 10003, USA E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: The Brownian loop soup is a conformally invariant statistical ensemble of random loops in two dimensions characterized by an intensity λ > 0, with central charge c = 2λ. Recent progress resulted in an analytic form for the four-point function of a class of scalar conformal primary “layering vertex operators” Oβ with dimensions (∆, ∆), with λ ∆ = 10 (1 − cos β), that compute certain statistical properties of the model. The Virasoro conformal block expansion of the four-point function revealed the existence of a new set of operators with dimensions (∆ + k/3, ∆ + k ′ /3), for all non-negative integers k, k ′ satisfying |k − k ′ | = 0 mod 3. In this paper we introduce the edge counting field E(z) that counts the number of loop boundaries that pass close to the point z. We rigorously prove that the n-point functions of E are well defined and behave as expected for a conformal primary field with dimensions (1/3, 1/3). We analytically compute the four-point function hOβ (z1 )O−β (z2 )E(z3 )E(z4 )i and analyze its conformal block expansion. The operator product expansions of E × E and E × Oβ produce higher-order edge operators with “charge” β and dimensions (∆ + k/3, ∆ + k/3). Hence, we have explicitly identified all scalar primary operators among the new set mentioned above. We also re-compute the central charge by an independent method based on the operator product expansion and find agreement with previous methods. Contents 1 Introduction 1.1 Preliminary definitions 1.2 Summary of the main results 1.3 Structure of the paper 1 2 4 7 2 The edge counting operator 8 3 Correlation functions with a “twist” 10 4 OPE and the edge operator 12 5 A mixed four-point function 15 6 Higher-order and charged edge operators 6.1 Higher-order edge operators 6.2 Charged edge operators 17 17 20 7 The primary operator spectrum 7.1 Virasoro conformal blocks 7.2 The three-point function of the edge operator 23 24 26 8 Central charge 27 9 Conclusions and future work 29 A Proofs 30 1 Introduction The Brownian loop soup (BLS) [1] is an ideal gas of Brownian loops with a distribution chosen so that it is invariant under local conformal transformations. The BLS is implicit in the work of Symanzik [2] on Euclidean quantum field theory, more precisely, in the representation of correlation functions of Euclidean fields in terms of random paths that are locally statistically equivalent to Brownian motion. This representation can be made precise for the Gaussian free field, in which case the random paths are independent of each other and can be generated as a Poisson process. The BLS is closely related not only to Brownian motion and the Gaussian free field but also to the Schramm-Loewner Evolution (SLE) and Conformal Loop Ensembles (CLEs). It provides an interesting and useful link between Brownian motion, field theory, and statistical mechanics. Partly motivated by these connections, as well as by a potential application to –1– cosmology in the form of a conformal field theory for eternal inflation [3], three of the present authors introduced a set of operators that compute properties of the BLS and discovered new families of conformal primary fields depending on a real parameter β [4]. One such λ (1 − cos β) and family are the fields Oβ . These operators have scaling dimensions ∆(β) = 10 are periodic under β → β + 2π, with O0 ≡ O2π = 1 (the identity operator). Their n-point Pn function hOβ1 (z1 ) . . . Oβn (zn )iC in the full plane is identically zero unless j=1 βj = 0 mod 2π, which is reminiscent of the “charge neutrality” or “charge conservation” condition that applies to vertex operators of the free boson [5]. These operators were further studied in [6], where it is shown that the operator product expansion (OPE) Oβi ×Oβj predicts the existence of operators of dimensions (∆ij + k3 , ∆ij + λ k′ ′ ′ 3 ) for all non-negative integers k, k satisfying |k − k | = 0 mod 3, where ∆ij = 10 (1 − cos(βi + βj )). The simplest case is k = k ′ = 1 and βi + βj = 0 mod 2π so that ∆ij = 0 and the dimensions are (1/3, 1/3). These results were derived by exploiting a connection between the BLS and the O(n) model in the limit n → 0. Further generalizations of the layering operators were explored in [7]. While the analysis in [6] demonstrated that new operators must exist and allowed us to compute their dimensions and three-point function coefficients with Oβ , it did not provide a clue as to how they are defined in terms of loops of the BLS loop ensemble. In this paper we introduce a new field E(z) that counts the number of outer boundaries of BLS loops that pass close to z and rigorously prove that its n-point functions are well defined and behave as expected for a primary field. We identify E with the operator of dimensions (1/3, 1/3) discovered in [6], compute the four-point function hOβ (z1 )O−β (z2 )E(z3 )E(z4 )iC, and perform its Virasoro conformal block expansion. This provides further information about three-point function coefficients and the spectrum of primary operators. We further define higher order (k = k ′ > 1) and charged (β 6= 0) generalizations of this operator that can be identified with the operators of dimensions (∆ij + k3 , ∆ij + k3 ). In other words, we identify and explicitly define in terms of the loops all spin-zero primary fields emerging from the Virasoro conformal block expansion derived in [6]. This corpus of results establishes the BLS as a novel conformal field theory (CFT), or class of conformal field theories, with certain unique features (such as the periodicity of the operator dimensions in the charge β). Nevertheless, many aspects of this CFT remain mysterious—among other things, the nature of the operators with non-zero spin, |k−k ′ | = 6 0. The relation of this CFT to other better-known CFTs and its possible role as a model for physical phenomena also remains unclear. 1.1 Preliminary definitions If A is a set of loops in a domain D, the partition function of the BLS restricted to loops from A can be written as ZA = ∞ X λn  n=0 n! µloop D (A) –2– n , (1.1) where λ > 0 is a constant and µloop is a measure on planar loops in D called Brownian loop D measure and defined as Z Z ∞ 1 loop µD := µbr dt dA(z), (1.2) 2 z,t 2πt D 0 where A denotes area and µbr z,t is the complex Brownian bridge measure with starting point 1 z and duration t. ZA can be thought of as the grand canonical partition function of a system of loops with fugacity λ, and the BLS can be shown to be conformally invariant and to have central charge c = 2λ (see [1, 4]). In this paper we will only be concerned with the outer boundaries of Brownian loops. More precisely, given a planar loop γ in C, its outer boundary or “edge” ℓ = ℓ(γ) is the boundary of the unique infinite component of C \ γ. Note that, for any planar loop γ, ℓ(γ) is always a simple closed curve, i.e., a closed loop without self-intersections. Hence, in this paper, we will work with collections L of simple loops ℓ which are the outer boundaries of the loops from a BLS and for us, with a slight abuse of terminology, a BLS will be a collection of simple loops. With these understandings, the λ → 0 limit (interpreted appropriately) reduces to the case of a single self-avoiding loop. There is a unique (up to an overall multiplicative constant) conformally invariant measure on such loops [8], which are also described by the n → 0 limit of the O(n) model. Exploiting this connection allowed us to obtain exact results for certain correlation functions here and in our previous work [6]. Given a simple loop ℓ, let ℓ̄ denote its interior, i.e. the unique bounded simply connected component of C \ ℓ. In other words, a point z belongs to ℓ̄ if ℓ disconnects z from infinity, in which case we write z ∈ ℓ̄. In [4], the authors studied the correlation functions of the layering P operator or field 2 Vβ (z) = exp(iβ ℓ:z∈ℓ̄ σℓ (z)), where σℓ are independent, symmetric, (±1)valued Boolean variables associated to the loops. One difficulty arises immediately due to the scale invariance of the BLS, which implies that the sum at the exponent is infinite with probability one. This difficulty can be overcome by imposing a short-distance cutoff δ > 0 on the diameter of loops (essentially removing from the loop soup all loops with diameter smaller than δ.3 ) As shown in [4], the cutoff δ can be removed by rescaling the cutoff version Vβδ of Vβ by δ −2∆(β) and sending δ → 0. When δ → 0, the n-point correlation functions of δ −2∆(β) Vβδ converge to conformally covariant quantities [4], showing that the limiting field is a scalar conformal primary field with real and positive scaling dimension varying λ ¯ (1 − cos β). This continuously as a periodic function of β, namely as ∆(β) = ∆(β) = 10 limiting field is further studied in [6], where its canonically normalized version is denoted by Oβ .4 The edge field E(z) studied in this paper counts the number of loops ℓ passing within a short-distance ε of the point z. The cutoff and renormalization procedure described in 1 We note that the Brownian loop measure should be interpreted as a measure on “unrooted” loops, that is, loops without a specified starting point. Unrooted loops are equivalence classes of rooted loops. The interested reader is referred to [1] for more details. 2 In this paper we use the terms field and operator interchangeably. 3 An additional infrared cutoff or a “charge neutrality” or “charge conservation” condition may be necessary in some circumstances—we refer the interested reader to [4] for more details. 4 By canonically normalized we mean that the full-plane two-point function hOβ (z)O−β (z ′ )iC = |z − ′ −2∆(β) z| . –3– Section 2 shows that E has well defined n-point functions which are conformally covariant, and that it behaves like a scalar conformal primary with scaling dimension (1/3, 1/3). This scaling dimension can be understood qualitatively as follows. It is known that the fractal dimension of the boundary of a Brownian loop is 4/3 [9]. Fattening the loop’s boundary into a strip5 of width ε, a fractal dimension of 4/3 means that the area of the strip is proportional to ε2/3 . Hence the probability for a loop to come within ε of a given point scales as ε2/3 . Loops that contribute to the two-point function of the edge operator with itself must come close to both points (Figure 1). Therefore the two-point function is proportional to the square of this probability |ε/z12 |4/3 , where the power of z12 follows from invariance under an overall scale transformation (ε, z) → (λε, λz). This dependence on |z12 | is that of a scalar operator with dimension (1/3, 1/3). In Section 6.1 we identify additional scalar fields resulting from combinations of the edge field E with itself that we denote by E (k) and call higher-order edge operators. These fields have holomorphic and anti-holomorphic dimension k3 for all non-negative integers k. In Section 6.2 we discuss “charged” versions of the (higher-order) edge operators resulting from combinations of the edge field with itself and with the layering field Oβ ; we denote (k) these by Eβ and call them charged edge operators. These fields have holomorphic and anti-holomorphic dimension ∆(β) + k3 , with non-negative integer k. The higher-order and charged edge operators complete the list of all scalar primary fields in the conformal block expansion derived in [6]. 1.2 Summary of the main results The domains D considered in this paper are the full (complex) plane C, the upper-half plane H or any domain conformally equivalent to H. In this section and in the rest of the paper, we use h·iD to denote expectation with respect to the BLS in D. The domain will be explicitly present in our notation when we want to emphasize its role; if the domain is not denoted in a particular expression (for example, if we use h·i instead of h·iD or µloop instead of µloop D ), it means that that expression is valid for any of the domains mentioned above. The first group of main results concerns the Brownian loop measure µloop in a domain D D, the n-point functions of the edge operator E, which can be expressed in terms of µloop D , 6 and the relation between E and Oβ . • For any collection of distinct points z1 , . . . , zk ∈ D with k ≥ 2, letting Bε (zj ) denote the disk of radius ε centered at zj , the following limit exists z1 ,...,zk αD := lim ε−2k/3 µloop D (ℓ ∩ Bε (zj ) 6= ∅ ∀j = 1, . . . , k). ε→0 (1.3) z1 ,...,zk Moreover, αD is conformally covariant in the sense that, if D′ is a domain con5 6 Recipes for Wiener sausages in Brownian soups are available on special request. The edge operator is properly defined in Section 2 below. –4– z1 z2 ε Figure 1: A Brownian loop (thin NYU violet line) and its boundary (thick violet line; the interior is shaded). Such a loop would contribute to the two-point function of edge operators inserted at z1 and z2 because the loop comes within ε of both. It would contribute to a layering operator inserted at z1 (but not z2 ) because z1 (but not z2 ) is in the interior of the loop (that is, the loop separates z1 from infinity, but not z2 ). formally equivalent to D and f : D → D′ is a conformal map, then f (z ),...,f (zk ) αD ′ 1  = k Y j=1  z1 ,...,zk |f ′ (zj )|−2/3  αD . (1.4) • The field E formally defined by
ĉ E(z) := √ lim ε−2/3 Nε (z) − hNε (z)i , λ ε→0 (1.5) where Nε (z) counts the number of loops ℓ that come to distance ε of z,7 behaves like a conformal primary field with scaling dimension 2/3. The constant ĉ is chosen so that E is canonically normalized, i.e. hE(z1 )E(z2 )iC = |z1 − z2 |−4/3 . 7 (1.6) We note that Nε (z) is infinite with probability one because of the scale invariance of the BLS, but its centered version Eε (z) := Nε (z) − hNε (z)i has well defined n-point functions—see Lemma 2.1. –5– • More precisely, if D′ is a domain conformally equivalent to D and f : D → D′ is a conformal map, then   n Y |f ′ (zj )|−2/3  hE(z1 ) . . . E(zn )iD . (1.7) hE(f (z1 )) . . . E(f (zn ))iD′ =  j=1 • Letting zjk := zj − zk , we have E hOβ (z1 )O−β (z2 )E(z3 )iC = CO β O−β z12 1 4∆(β) z13 z23 |z12 | 2/3 , (1.8) with three-point structure constant √ E CO = − λ(1 − cos β) β O−β 31/4 √ 27/6 π . 5Γ(1/6)Γ(4/3) (1.9) • The OPE of Oβ × O−β takes the form Oβ (z) × O−β (z ′ )  E (2) E = |z − z ′ |−4∆(β) 1 + CO |z − z ′ |2/3 E(z) + CO |z − z ′ |4/3 E (2) (z) β O−β β O−β
 + o |z − z ′ |4/3 , where 1 is the identity operator and 2  (2) 2 1  E E C . CO = Oβ O−β β O−β 2 (1.10) (1.11) • The mixed full-plane four-point function hOβ (z1 )O−β (z2 )E(z3 )E(z4 )iC has the following explicit expression: hOβ (z1 )O−β (z2 )E(z3 )E(z4 )iC   1 − cos β z4 −4∆(β) 1 + cos β −4/3 2 z3 = |z12 | |z34 | + Ztwist + λ(1 − cos β) α̂z1 |z2 α̂z1 |z2 , 2 2 (1.12) where α̂zzjl |zk ;C = 31/4 √ zjk 27/6 π 5 Γ(1/6)Γ(4/3) zjl zkl 2/3 (1.13) and "   z13 z24 2/3 2 1 2 z12 z34 2 Ztwist = 2 , ; , F − 2 1 3 3 3 z13 z24 z34 z23 z14
 #  6 4Γ 32 z12 z34 2/3 1 2 4 z12 z34 2 − .
2
4 2 F1 − , ; , z13 z24 3 3 3 z13 z24 Γ 34 Γ 31 –6– (1.14) • The OPE of E × E contains the terms E(z) × E(z ′ )   E E (2) |z − z ′ |4/3 E (2) (z) + . . . , = |z − z ′ |−4/3 1 + CEE |z − z ′ |2/3 E(z) + CEE (1.15) where the three-point structure constants are E CEE (2) E CEE √ 1 213/6 31/4 5 π 3/2 Γ =√
3
λ Γ 61 Γ 76 √ = 2. 2 3
(1.16) (1.17) • The OPE of Oβ × E takes the form O E Oβ (z) × E(z ′ ) = COββE |z − z ′ |−2/3 Oβ (z) + COββ E Eβ (z) + . . . (1.18) O E where COββE = CO and β O−β  E COββ E 2 = 1 + cos β . 2 (1.19) • The higher-order edge operators E (k) behave like canonically normalized primary fields. More precisely, for each k ∈ N, D E (1.20) E (k) (z1 )E (k) (z2 ) = |z1 − z2 |−4k/3 . C Moreover, if D′ is a domain conformally equivalent to D and f : D → D′ is a conformal map, then D E E (k1 ) (f (z1 )) . . . E (kn ) (f (zn )) ′   D n D E (1.21) Y |f ′ (zj )|−2kj /3  E (k1 ) (z1 ) . . . E (kn ) (zn ) . = D j=1 • The central charge of the BLS can be independently re-derived to be c = 2λ by computing the two-point function of the stress-tensor hT (z1 )T (z2 )iC = c/2 4 z12 (1.22) from (1.12) by applying the OPEs of E × E and Oβ × O−β . 1.3 Structure of the paper This paper contains both rigorous results and “physics-style” arguments and is written with a mixed audience of mathematicians and physicists in mind. The rigorous results are generally presented as lemmas or theorems in the text; they include explicit expressions for certain correlation functions and the proof that the n-point correlation functions of the –7– edge operator E and of the higher-order edge operators E (k) are conformally covariant. The proofs of most rigorous results are collected in the appendix to avoid breaking the flow of the paper. The results in Sections 2-5 and 6.1 are rigorous except for the use of Eq. (6.19) of [6] in Section 3, the existence of the limit in (4.9) in Section 4, the use of Eq. (52) of [10] and the identification in (5.12) in Section 5. The edge operator E is introduced in Section 2, where its correlation functions are discussed. Section 3 contains the computation of hOβ (z1 )O−β (z2 )E(z3 )iC, including the E structure constant CO . Section 4 contains a derivation of the OPE of Oβ × O−β β O−β and the identification of the edge operator E with the primary operator of dimension (1/3, 1/3) discovered in [6]. Section 5 contains the calculation of the full-plane four-point function hOβ (z1 )O−β (z2 )E(z3 )E(z4 )iC. Higher-order and charged edge operators are introduced in Sections 6.1 and 6.2, respectively, where their correlation functions are discussed. The Virasaoro conformal block expansion resulting from the four-point function hOβ (z1 )O−β (z2 )E(z3 )E(z4 )iC is developed in Section 7.1, while Section 7.2 contains a direct derivation of the full-plane three-point function hE(z1 )E(z2 )E(z3 )iC, including the structure E . Section 8 contains a new derivation of the fact that the central charge of the constant CEE BLS with intensity λ is c = 2λ. 2 The edge counting operator For a domain D ⊆ C, a point z ∈ C, a real number ε > 0, and a collection L of simple loops in D, let nεz (L) denote the number of loops ℓ ∈ L such that ℓ ∩ Bε (z) 6= ∅, where Bε (z) denotes the disk or radius ε centered at z. We define formally the “random variable” Nε (z) = nεz (L) where L is distributed like the collection of outer boundaries ℓ = ℓ(γ) of the loops γ of a Brownian loop soup in D with intensity λ (see Section 1.1). Nε (z) counts the number of loops γ of a Brownian loop soup whose “edge” ℓ (the outer boundary) comes ε−close to z; it is only formally defined because it is infinite with probability one. Nevertheless, we will be interested in the fluctuations of Nε (z) around its infinite mean, which can be formally written as Eε (z) :=Nε (z) − hNε (z)iD =Nε (z) − λµloop D (ℓ ∩ Bε (z) 6= ∅), (2.1) where h·iD denotes expectation with respect to the Brownian loop soup in D (of fixed intensity λ) and µloop is the Brownian loop measure restricted to D, i.e. the unique (up to D a multiplicative constant) conformally invariant measure on simple planar loops [8]. In Lemma A.1 of the appendix we show that, while Eε (z) is only formally defined, its correlation functions hEε (z1 ) . . . Eε (zn )iD are well defined for any collection of points z1 , . . . , zn at distance greater than 2ε from each other, with n ≥ 2. There is a closed-form expression for such correlations in terms of the Brownian loop measure µloop D , as stated in the following lemma, whose proof is presented to the appendix. Lemma 2.1. For any ε > 0 and any collection of distinct points z1 , . . . , zn ∈ D at distance greater than 2ε from each other, with n ≥ 2, let Π denote the set of all partitions of –8– {1, . . . , n} such that each element Il of {I1 , . . . , Ir } ∈ Π has cardinality |Il | ≥ 2; then hEε (z1 ) . . . Eε (zn )iD = X λr {I1 ,...,Ir }∈Π r Y l=1 µloop D (ℓ ∩ Bε (zj ) 6= ∅ ∀j ∈ Il ). (2.2) A central result of this paper is the fact that the field formally defined by ĉ E(z) := √ lim ε−2/3 Eε (z) λ ε→0 (2.3) behaves like a conformal primary field, where the constant ĉ is chosen to ensure that E is canonically normalized, i.e., hE(z1 )E(z2 )iC = |z1 − z2 |−4/3 . (2.4) This result relies crucially on the following lemma, which is interesting in its own right. Lemma 2.2. Let D ⊆ C be either the complex plane C or the upper-half plane H or any domain conformally equivalent to H. For any collection of distinct points z1 , . . . , zk ∈ D with k ≥ 2, the following limit exists: z1 ,...,zk αD := lim ε−2k/3 µloop D (ℓ ∩ Bε (zj ) 6= ∅ ∀j = 1, . . . , k). ε→0 (2.5) z1 ,...,zk Moreover, αD is conformally covariant in the sense that, if D′ is a domain conformally equivalent to D and f : D → D′ is a conformal map, then   k Y f (z ),...,f (zk ) z1 ,...,zk αD ′ 1 = |f ′ (zj )|−2/3  αD . (2.6) j=1 For any collection of points z1 , . . . , zn ∈ D and any subset S = {zj1 , . . . , zjk } of S := αzj1 ,...,zjk . The statement about the operator E defined formally {z1 , . . . , zn }, let αD D in (2.3) is made precise by the following theorem. Theorem 2.3. Let D ⊆ C be either the complex plane C or the upper-half plane H or any domain conformally equivalent to H. For any collection of distinct points z1 , . . . , zn ∈ D with n ≥ 2, the following limit exists: gD (z1 , . . . , zn ) := lim ε−2n/3 hEε (z1 ) . . . Eε (zn )iD . ε→0 (2.7) Moreover, if S = S(z1 , . . . , zn ) denotes the set of all partitions of {z1 , . . . , zn } such that each element Sl of (S1 , . . . , Sr ) ∈ S has cardinality |Sl | ≥ 2, then X S1 Sr λr α D gD (z1 , . . . , zn ) = . . . αD . (2.8) (S1 ,...,Sr )∈S Furthermore, gD (z1 , . . . , zn ) is conformally covariant in the sense that, if D′ is a domain conformally equivalent to D and f : D → D′ is a conformal map, then ! n Y ′ −2/3 gD′ (f (z1 ), . . . , f (zn )) = |f (zk )| gD (z1 , . . . , zn ). (2.9) k=1 –9– Proof. The existence of the limit in (2.7) follows from (2.2) combined with the existence of the limit in (2.5). The expression in (2.8) follows directly from (2.2) and the definition of αz1 ,...,zk (D) in (2.5). The conformal covariance expressed in (2.9) is an immediate consequence of (2.8) and (2.6). Using the notation introduced in (2.3), we will write hE(z1 ) . . . E(zn )iD := ĉn gD (z1 , . . . , zn ), λn/2 (2.10) despite the fact that E is only formally defined. To simplify the notation, we define z1 ,...,zk z1 ,...,zk α̂D := ĉk αD . (2.11) In particular, using this notation, the two-, three- and four-point functions are z1 ,z2 hE(z1 )E(z2 )iD = α̂D 1 z1 ,z2 ,z3 hE(z1 )E(z2 )E(z3 )iD = √ α̂D λ 1 z1 ,z2 ,z3 ,z4 hE(z1 )E(z2 )E(z3 )E(z4 )iD = α̂D λ z1 ,z2 z3 ,z4 z1 ,z3 z2 ,z4 z1 ,z4 z2 ,z3 + α̂D α̂D + α̂D α̂D + α̂D α̂D . 3 (2.12a) (2.12b) (2.12c) Correlation functions with a “twist” In this section we present a simple method to compute certain types of correlation functions involving two vertex layering operators. Later, as an application, we will use this method to show how the edge operator E emerges from the OPE of Oβ × O−β . From now on, we z1 ,...,zk will drop the subscript D from h·iD , µloop and similar expressions when D can be D , αD any domain. To explain the method mentioned above, in the next paragraph we use {·} to denote an unnormilazed sum, that is 1 h·i := {·}, (3.1) Z where Z := {1} denotes the partition function. If we define {·}∗z1 ,z2 ≡ {·}∗z1 ,z2 ;β := {· Oβ (z1 )O−β (z2 )} (3.2) and h·i∗z1 ,z2 ≡ h·i∗z1 ,z2 ;β := {·}∗z1 ,z2 , {1}∗z1 ,z2 (3.3) then we can write {· Oβ (z1 )O−β (z2 )} {1} ∗ {1}z1 ,z2 {·}∗z1 ,z2 = {1} {1}∗z1 ,z2 h· Oβ (z1 )O−β (z2 )i = = hOβ (z1 )O−β (z2 )i h·i∗z1 ,z2 . – 10 – (3.4) This simple formula will be very useful in the rest of the paper thanks to the observation that h·i∗z1 ,z2 is the expectation with respect to the measure µ∗z1 ,z2 ;β ≡ µ∗z1 ,z2 defined by  loop  if ℓ does not separate z1 , z2  µ (ℓ) ∗ iβσ µz1 ,z2 (ℓ) := e ℓ µloop (ℓ) if z1 ∈ ℓ̄, z2 ∈ (3.5) / ℓ̄   e−iβσℓ µloop (ℓ) if z ∈ 1 / ℓ̄, z2 ∈ ℓ̄ where σℓ = ±1 is a symmetric Boolean variable assigned to ℓ. As a first example, to illustrate the use of the method, we calculate ĉ hOβ (z1 )O−β (z2 )E(z3 )i = √ lim ε−2/3 hOβ (z1 )O−β (z2 )Eε (z3 )i λ ε→0 ĉ = √ hOβ (z1 )O−β (z2 )i lim ε−2/3 hEε (z3 )i∗z1 ,z2 . ε→0 λ (3.6) To perform this calculation, we define Nεδ (z) := nεz (Lδ ) and Eεδ (z) := Nεδ (z) − Nεδ (z) , where Lδ is a Brownian loop soup with cutoff δ > 0, obtained by taking the usual Brownian loop soup and removing all loops with diameter (defined to be the largest distance between any two points on the loop) smaller than δ. The random variables Nεδ (z) and Eεδ (z) are well defined because of the cutoffs ε and δ. With these definitions, we have D E∗ E∗ hD D Ei hEε (z3 )i∗z1 ,z2 := lim Eεδ (z3 ) = lim Nεδ (z3 ) − Nεδ (z3 ) δ→0 δ→0 z1 ,z2 z1 ,z2   loop = lim (cos β − 1)λµ (diam(ℓ) > δ, ℓ ∩ Bε (z3 ) 6= ∅, ℓ separates z1 , z2 ) δ→0 = −λ(1 − cos β)µloop (ℓ ∩ Bε (z3 ) 6= ∅, ℓ separates z1 , z2 ). (3.7) The expression above for hEε (z3 )i∗z1 ,z2 follows from the observation that the contributions ∗ to Nεδ (z3 ) z1 ,z2 and Nεδ (z3 ) from loops that do not separate z1 and z2 cancel out, while ∗ the contribution to Nεδ (z3 ) z1 ,z2 from loops that do separate z1 and z2 comes with a factor cos β because of the definition of µ∗z1 ,z2 and the averaging over σℓ = ±1. (Note that {σℓ }ℓ∈L is distributed like a collection of independent, (±1)−valued, symmetric random variables). We conclude that √ hOβ (z1 )O−β (z2 )E(z3 )i = − λ(1 − cos β) α̂zz13 |z2 hOβ (z1 )O−β (z2 )i , (3.8) where α̂zz13 |z2 := ĉ αzz13 |z2 (3.9) αzz13 |z2 ≡ αzz23 |z1 := lim ε−2/3 µloop (ℓ ∩ Bε (z3 ) 6= ∅, ℓ separates z1 , z2 ). (3.10) with ε→0 The existence of the limit in (3.10) follows from the proof of Lemma 2.2. So far our discussion has been completely general and independent of the domain D. If we now specify that D = C and note that the operators Oβ , O−β are canonically normalized hOβ (z1 )O−β (z2 )iC = |z1 − z2 |−4∆(β) , – 11 – (3.11) we get from (3.8) √ hOβ (z1 )O−β (z2 )E(z3 )iC = − λ(1 − cos β) α̂zz13 |z2 ;C|z1 − z2 |−4∆(β) . (3.12) Since (3.12) is a three-point function of primary operators defined on the full plane, its form is fixed by global conformal invariance up to a multiplicative constant (see, for example, the proof of Theorem 4.5 of [4]). In this case, letting zjk := zj − zk , we have E hOβ (z1 )O−β (z2 )E(z3 )iC = CO β O−β 1 z12 4∆(β) z13 z23 |z12 | 2/3 . (3.13) E The coefficient CO , evaluated at β1 = β2 = π, was determined in [6], where it was β O−β called C (1,1) . Comparing (3.12) with (3.13) and using the expression for C (1,1) from Eq. (6.19) of [6] shows that α̂zz13 |z2 ;C = 31/4 √ 27/6 π z12 5 Γ(1/6)Γ(4/3) z13 z23 2/3 . (3.14) Together with (3.12), this implies that, for general values of β, we have the three-point function coefficient √ E CO = − λ(1 − cos β) β O−β 4 31/4 √ 27/6 π . 5 Γ(1/6)Γ(4/3) (3.15) OPE and the edge operator In this section, applying the method presented in the previous section, we show how the edge operator E emerges from the Operator Product Expansion (OPE) of Oβ × O−β . It is shown in [6] that the OPE of the product of two vertex operators Oβi × Oβj contains operators of ′ λ dimensions (∆ij + k3 , ∆ij + k3 ) for non-negative integers k, k ′ , where ∆ij = 10 (1−cos(βi +βj )). 1 1 In what follows, we identify the operator of dimensions ( 3 , 3 ) with the edge operator E. If N δ (z) denotes the number of loops of diameter larger than δ that contain z in their interior, it was shown in [4] that the two-point function D E δ δ ′ Oβ (z)O−β (z ′ ) ∝ lim δ −2∆(β) eiβN (z) e−iβN (z ) δ→0

= lim δ −2∆(β) exp − λ(1 − cos β)µloop ℓ separates z, z ′ , diam(ℓ) > δ δ→0 (4.1) exists. We are interested in the sub-leading behavior of Oβ (z) × O−β (z ′ ) when z ′ → z. The two-point function hOβ (z)O−β (z ′ )i diverges when z ′ → z (see (3.11)), so we normalize Oβ (z)O−β (z ′ ) by its expectation. Taking two distinct points z1 , z2 6= z, z ′ , we compute the four-point function  Oβ (z)O−β (z ′ ) Oβ ′ (z1 )O−β ′ (z2 ) hOβ (z)O−β (z ′ )i  = Oβ ′ (z1 )O−β ′ (z2 ) – 12 – hOβ (z)O−β (z ′ )i∗z1 ,z2 ;β ′ hOβ (z)O−β (z ′ )i . (4.2) The loops that do not separate z1 and z2 contribute equally to hOβ (z)O−β (z ′ )i∗z1 ,z2 and hOβ (z)O−β (z ′ )i, so their contributions cancel out in the ratio on the right hand side. The loops that do separate z1 , z2 contribute differently, as we have already seen in the computation leading to (3.8). An analogous computation using (4.1) gives hOβ (z)O−β (z ′ )i∗z1 ,z2 ;β ′ hOβ (z)O−β (z ′ )i   = exp (1 − cos β ′ )λ(1 − cos β)µloop (ℓ separates z, z ′ and z1 , z2 ) = 1 + (1 − cos β ′ )λ(1 − cos β)µloop (ℓ separates z, z ′ and z1 , z2 ) + O(µloop (ℓ separates z, z ′ and z1 , z2 )2 ), (4.3) as |z − z ′ | → 0. We now let ε = |z − z ′ | and observe that µloop (ℓ separates z, z ′ and z1 , z2 ) = µloop (ℓ ∩ Bε (z) 6= ∅ and ℓ separates z1 , z2 ) − µloop (ℓ ∩ Bε (z) 6= ∅, ℓ does not separate z, z ′ and ℓ separates z1 , z2 ) = µloop (ℓ ∩ Bε (z) 6= ∅ and ℓ separates z1 , z2 )   µloop (ℓ ∩ Bε (z) 6= ∅, ℓ does not separate z, z ′ and ℓ separates z1 , z2 ) , 1− µloop (ℓ ∩ Bε (z) 6= ∅ and ℓ separates z1 , z2 ) (4.4) where µloop (ℓ ∩ Bε (z) 6= ∅ and ℓ separates z1 , z2 ) = O ε3/2 which follows from the proof of Lemma 2.2. Letting
as ε → 0, (4.5) c̃ε ≡ c̃ε (z, z ′ ; z1 , z2 ) := 1 − µloop (ℓ ∩ Bε (z) 6= ∅, ℓ does not separate z, z ′ and ℓ separates z1 , z2 ) µloop (ℓ ∩ Bε (z) 6= ∅ and ℓ separates z1 , z2 ) (4.6) and using (4.4), (4.5) and (3.7), we can write hOβ (z)O−β (z ′ )i∗z1 ,z2 ;β ′ hOβ (z)O−β (z ′ )i = 1 − (1 − cos β) c̃ε hEε (z)i∗z1 ,z2 ;β ′ + o ε2/3 Combining this with (4.2), we obtain * + Oβ (z)O−β (z ′ ) Oβ ′ (z1 )O−β ′ (z2 ) Oβ (z) O−β (z ′ )
as ε → 0. = Oβ ′ (z1 )O−β ′ (z2 ) − (1 − cos β) c̃ε Oβ ′ (z1 )O−β ′ (z2 )Eε (z) + o ε (4.7) (4.8) 2/3
as ε → 0. At this point we make the natural assumption that, as long as the points z, z1 , z2 are distinct, the limit c̃ := lim c̃ε ≡ lim c̃ε (z, z ′ ; z1 , z2 ) (4.9) ′ ′ z →z z →z exists and is independent of the domain and of z, z1 , z2 . This can be justified using arguments analogous to those in the proof of Lemma 2.2. The idea is, essentially, the following. – 13 – One can think in terms of the full scaling limit of critical percolation, as described in the proof of Lemma 2.2. Then one can split the loops separating z1 , z2 and intersecting Bε (z) into excursions from ∂Bε (z) either inside or outside the disk. As explained in the proof of Lemma 2.2, the excursions inside and outside Bε (z) are independent of each other, conditioned on the location on ∂Bε (z) of their starting and ending points. Since the limit in (4.9) is determined only by the behavior of the excursions inside Bε (z), it should not depend on the domain and on z1 , z2 . Using the assumption expressed by (4.9) and the formal definition (2.3) of the edge operator, we can write
Oβ (z)O−β (z ′ ) c̃ √ λ|z − z ′ |2/3 E(z) + o |z − z ′ |2/3 as z ′ → z, (4.10) = 1 − (1 − cos β) ′ hOβ (z)O−β (z )i ĉ where 1 denotes the identity operator. For z away from any boundary and in the limit z ′ → z, using (3.11) this takes the form   √
c̃ ′ ′ −4∆(β) ′ 2/3 ′ 2/3 1 − λ(1 − cos β) |z − z | E(z) + o |z − z | Oβ (z) × O−β (z ) = |z − z | , ĉ (4.11) which shows how the edge operator emerges from the OPE of two layering vertex operators. In order to check for internal consistency, we determine c̃/ĉ. To do this we insert the OPE (4.11) in the three-point function hOβ (z1 )O−β (z2 )E(z3 )iC   √
c̃ −4∆(β) 2/3 2/3 − λ(1 − cos β) hE(z1 )E(z3 )iC |z12 | + o |z12 | = |z12 | . ĉ (4.12) Comparing this with (3.12) and using (3.14) and the fact that E is assumed to be canonically normalized, so that hE(z1 )E(z3 )iC = |z13 |−4/3 , (4.13) we get
c̃ |z13 |−4/3 |z12 |2/3 + o |z12 |2/3 = α̂zz13 |z2 ;C ĉ = 31/4 √ z12 27/6 π 5 Γ(1/6)Γ(4/3) z13 z23 (4.14) 2/3 . Dividing both sides of the equation above by |z12 |2/3 and letting z2 → z1 gives 27/6 π c̃ √ . = ĉ 31/4 5 Γ(1/6)Γ(4/3) (4.15) Based on general principles and on the conformal block expansion performed in [6], the OPE of Oβ × O−β should read
φ Oβ (z) × O−β (z ′ ) = |z − z ′ |−4∆(β) 1 + CO1/3,1/3 |z − z ′ |2/3 φ1/3,1/3 (z) + . . . , β O−β – 14 – (4.16) where φ1/3,1/3 is an operator of dimension (1/3, 1/3). In order to identify φ1/3,1/3 with the φ E edge operator E, we need to identify CO1/3,1/3 with the coefficient CO given in (3.15). β O−β β O−β Comparing (4.16) with (4.11), and using (4.15), this gives √ φ CO1/3,1/3 = − λ(1 − cos β) β O−β 31/4 √ 27/6 π , 5 Γ(1/6)Γ(4/3) (4.17) which indeed coincides with (3.15). 5 A mixed four-point function The method introduced in Section 3 can be used to calculate the mixed four-point function hOβ (z1 )O−β (z2 )E(z3 )E(z4 )i = hOβ (z1 )O−β (z2 )i hE(z3 )E(z4 )i∗z1 ,z2 = λ−1 ĉ2 hOβ (z1 )O−β (z2 )i lim ε−4/3 hEε (z3 )Eε (z4 )i∗z1 ,z2 . ε→0 (5.1) Using the random variables defined just above (3.7), a bit of algebra shows that D E∗ hEε (z3 )Eε (z4 )i∗z1 ,z2 = lim Eεδ (z3 )Eεδ (z4 ) δ→0 z ,z Dh D E ih D1 2 E iE∗ = lim Nεδ (z3 ) − Nεδ (z3 ) Nεδ (z4 ) − Nεδ (z4 ) δ→0 z1 ,z2 h i ∗ D E∗ ih D E∗ Nεδ (z4 ) − Nεδ (z4 ) = lim Nεδ (z3 ) − Nεδ (z3 ) δ→0 hEε (z3 )i∗z1 ,z2 + z1 ,z2 z1 ,z2 hEε (z4 )i∗z1 ,z2 (5.2) z1 ,z2 . Now note that lim δ→0 h D E∗ Nεδ (z3 ) − Nεδ (z3 ) z1 ,z2 ih D E∗ Nεδ (z4 ) − Nεδ (z4 ) z1 ,z2 i ∗ (5.3) z1 ,z2 is exactly analogous to hEε (z3 )Eε (z3 )i, with the measure µloop replaced by µ∗z1 ,z2 . Therefore, combining Lemma 2.1 with (3.5), we have that h D E∗ ih D E∗ i ∗ δ δ δ δ lim Nε (z3 ) − Nε (z3 ) Nε (z4 ) − Nε (z4 ) δ→0 = z1 ,z2 λµ∗z1 ,z2 (ℓ loop = λµ z1 ,z2 z1 ,z2 ∩ Bε (zj ) 6= ∅ for j = 3, 4) (ℓ ∩ Bε (zj ) 6= ∅ for j = 3, 4; ℓ does not separate z1 , z2 ) + λ cos βµ loop (5.4) (ℓ ∩ Bε (zj ) 6= ∅ for j = 3, 4; ℓ separates z1 , z2 ) = λµloop (ℓ ∩ Bε (zj ) 6= ∅ for j = 3, 4) − λ(1 − cos β)µloop (ℓ ∩ Bε (zj ) 6= ∅ for j = 3, 4; ℓ separates z1 , z2 ). Using this and (3.7), we obtain hEε (z3 )Eε (z4 )i∗z1 ,z2 = λµloop (ℓ ∩ Bε (zj ) 6= ∅ for j = 3, 4) − λ(1 − cos β)µloop (ℓ ∩ Bε (zj ) 6= ∅ for j = 3, 4; ℓ separates z1 , z2 ) + λ2 (1 − cos β)2 µloop (ℓ ∩ Bε (z3 ) 6= ∅, ℓ separates z1 , z2 ) · µloop (ℓ ∩ Bε (z4 ) 6= ∅, ℓ separates z1 , z2 ). – 15 – (5.5) Inserting this expression in (5.1) gives where hOβ (z1 )O−β (z2 )E(z3 )E(z4 )i i h ,z4 z4 2 z3 + λ(1 − cos β) α̂ α̂ = hOβ (z1 )O−β (z2 )i α̂z3 ,z4 − (1 − cos β)α̂zz13 |z z1 |z2 z1 |z2 , 2 ,z4 ,z4 := ĉ2 αzz13 |z α̂zz13 |z 2 2 (5.6) (5.7) with ,z4 ,z4 := lim ε−4/3 µloop (ℓ ∩ Bε (zj ) 6= ∅ for j = 3, 4; ℓ separates z1 , z2 ). ≡ αzz23 |z αzz13 |z 1 2 ε→0 (5.8) The existence of the limit in (5.8) follows from the proof of Lemma 2.2. ,z4 ,z4 We note that αzz13 |z ≡ αzz13 |z depends on the domain D. When D = C we can 2 2 ;D z3 ,z4 determine αz1 |z2 in terms of a quantity Ztwist , whose origin and meaning are explained in the next paragraph, and which was computed in [10]. Using Ztwist , the weight can be written as z3 ,z4 − Ztwist α̂C z3 ,z4 α̂z1 |z2 ;C = , (5.9) 2 with 1 z3 ,z4 , (5.10) α̂C = |z3 − z4 |3/4 from (2.12a), (2.4), and where "
6  2  2#  4Γ 32 1 2 4 z13 z24 2/3 2 1 2 2/3 Ztwist = 2 −
2
4 |x| 2 F1 − , ; , x 2 F1 − , ; , x 3 3 3 3 3 3 z34 z23 z14 Γ 34 Γ 31 (5.11) z34 . corresponds to Eq. (52) of [10] with x = zz12 13 z24 In the language of [10], Ztwist is the four-point function of a pair of “2-leg” operators φ0,1 with a pair of “twist” operators φ2,1 8 in the O(n) model in the limit n → 0. The “2-leg” operator φ0,1 (z) forces a self-avoiding loop of the O(n) model to go through z, while a pair of “twist” operators φ2,1 (z1 )φ2,1 (z2 ) acts like Oπ (z1 )O−π (z2 ) in the sense that the weight of each loop that separates z1 and z2 is multiplied by −1. Simmons and Cardy [10] compute this four-point function for the O(n) model for −2 < n < 2, which in the case of n = 0 leads to (5.11). The n = 0 case of the O(n) model corresponds to a self-avoiding loop whose properties are described by µloop , as we will now explain. Strictly speaking, when n = 0 all loops are suppressed, but the inclusion of a pair of 2-leg operators guarantees the presence of at least one loop. Sending n → 0 then singles out the “one loop sector” described by µloop , since all other “sectors” produce a contribution of higher order in n (see the discussion preceding Eq. (49) of [10]). Something analogous happens in the case of the four-point function (5.6). As explained above, the pair of operators Oπ (z1 )O−π (z2 ) acts like φ2,1 (z1 )φ2,1 (z2 ), while the presence of a pair of edge operators guarantees the existence of at least one loop. Since the loop 8 The subscripts label the positions of the operators in the Kac table. – 16 – soup can be thought of as a gas of loops in the grand canonical ensemble with fugacity λ, the four-point function can be written as a sum of contributions from various “sectors” characterized by the number of loops. Because of the normalization of the edge operator, which includes a factor of λ−1/2 , the contribution of the “one loop sector” is of order O(1), while all other contributions are of order O(λ), as one can clearly see from (5.6). As a result, sending λ → 0 in (5.6) singles out the “one loop sector” just like sending n → 0 in the case of the O(n) four-point function calculated by Simmons and Cardy [10]. The two limits can be directly compared because all operators involved are canonically normalized. We can therefore conclude that Ztwist = lim hOπ (z1 )O−π (z2 )E(z3 )E(z4 )iC = λ→0 z3 ,z4 α̂C (5.12) ,z4 − 2α̂zz13 |z . 2 ;C In conclusion, equation (5.6) combined with (5.9)-(5.11) and (3.14) provides an explicit expression for the full-plane mixed four-point function hOβ (z1 )O−β (z2 )E(z3 )E(z4 )iC. 6 Higher-order and charged edge operators We will now extend the analysis of the edge operator E to all spin-zero operators that have non-zero fusion with the vertex operators. We will show that they have holomorphic and anti-holomorphic conformal dimensions (6.1) (∆(β) + k/3, ∆(β) + k/3), λ with ∆(β) = 10 (1−cos β), for any non-negative integer k. They correspond to the operators indicated on the diagonal of Figure 2b. We will first define the operators with β = 0 and dimensions (k/3, k/3) for k ≥ 2, which will be denoted E (k) and will be called higher(k) order edge operators. We will then see that the operators Eβ with dimensions (∆(β) + k/3, ∆(β) + k/3) with β 6= 0 are a product of Oβ with a modified version of E (k) . These will be called charged edge operators. 6.1 Higher-order edge operators Searching for new primary operators, we are guided by their conformal dimensions. For the operators with dimensions (k/3, k/3), it is natural to consider powers of edge operators. However, these are not well defined. Indeed, even if we keep both ε and δ cutoffs, it is
k clear that Eεδ (z) is not the correct starting point because its mean is not zero. A better choice, inspired by Eε(1);δ (z) := Eεδ (z) = Nεδ (z) − λµloop (diam(ℓ) > δ, ℓ ∩ Bε (z) 6= ∅)   ∂ δ loop − λµ (diam(ℓ) > δ, ℓ ∩ Bε (z) 6= ∅) xNε (z) = ∂x – 17 – , x=1 (6.2) is given, for each integer k ≥ 2, by ∂ k δ − λµloop (diam(ℓ) > δ, ℓ ∩ Bε (z) 6= ∅) xNε (z) ∂x x=1   k−1 X
j k = Nε (z) . . . (Nε (z) − (k − j) + 1) λµloop (diam(ℓ) > δ, ℓ ∩ Bε (z) 6= ∅) (−1)j j j=0
k + (−1)k λµloop (diam(ℓ) > δ, ℓ ∩ Bε (z) 6= ∅) . (6.3) Eε(k);δ (z) := This definition is valid in any domain D. Since Nεδ (z) = nεz (Lδ ) (see Section 3 above (3.7) and Appendix A) is a Poisson random variable with parameter λµloop (diam(ℓ) > δ, ℓ ∩ Bε (z) 6= ∅), we have that E D
k−j , Nεδ (z)(Nεδ (z) − 1) . . . (Nεδ (z) − (k − j) + 1) = λµloop (diam(ℓ) > δ, ℓ ∩ Bε (z) 6= ∅) (6.4) D E (k);δ which implies that Eε (z) = 0 for every δ > 0. C With this notation, for each k ≥ 1, we formally define the order k edge operator E (k) (z) := √ ĉk lim ε−2k/3 Eε(k);δ (z). k/2 δ,ε→0 k!λ (6.5) As we will see at the end of this section, the constant in front of the limit is chosen in such a way that E (k) is canonically normalized, i.e., D E (6.6) E (k) (z1 )E (k) (z2 ) = |z1 − z2 |−4k/3 . C For k = 1, we recover the edge operator, i.e., E (1) ≡ E. Definition (6.5) is formal in the sense that E (k) (z) is only well defined within npoint correlation functions. In order to show that E (k) has well-defined n-point functions, we start with an intermediate result, for which we need the following notation. Given a collection of points z1 , . . . , zn and a vector k = (k1 , . . . , kn ), kj ∈ N, we denote by M ≡ M(z1 , . . . , zn ; k1 , . . . , kn ) the collection of all multisets9 M such that (1) the elements S of M are subsets of {z1 , . . . , zn } with |S| > 1, P (2) the multiplicities mM (S) are such that S∈M mM (S)I(zj ∈ S) = kj for each j = 1, . . . , n and each M ∈ M. Condition (2) on the multiplicities essentially says that each point zj has multiplicity exactly kj in each multiset M . Note that M can be empty since conditions (1) and (2) cannot necessarily be satisfied simultaneously for generic choices of the vector k. For a set S, let IS denote the set of indices such that j ∈ IS if and only if zj ∈ S. Then we have the following lemma, proved in the appendix. 9 A multiset is a set whose elements have multiplicity ≥ 1. – 18 – Lemma 6.1. For any n ≥ 2 and δ, ε > 0, for any collection of points z1 , . . . , zn at distance grater than 2ε from each other, with the notation introduced above, we have that * n + * n + Y (kj ) Y (kj );δ Eε (zj ) := lim Eε (zj ) δ→0 j=1  = n Y j=1  kj ! j=1 X Y M ∈M S∈M  mM (S) 1 I(M = 6 ∅), λµloop (ℓ ∩ Bε (zj ) 6= ∅ ∀zj ∈ S) mM (S)! (6.7) where I(M = 6 ∅) denotes the indicator function of the event that M is not empty. The next theorem shows that it is also possible to remove the ε cutoff and demonstrates that the operators E (k) are primaries with dimensions (k/3, k/3) for all non-negative integer k. Theorem 6.2. Let D ⊆ C be either the complex plane C or the upper-half plane H or any domain conformally equivalent to H. With the notation of the previous lemma, for any collection of distinct points z1 , . . . , zn ∈ D with n ≥ 2 and any vector k = (k1 , . . . , kn ) with kj ∈ N such that M is not empty, we have that D E 2 Pn GD (z1 , . . . , zn ; k1 , . . . , kn ) := lim ε− 3 j=1 kj Eε(k1 ) (z1 ) . . . Eε(kn ) (zn ) ε→0 D   n (6.8) Y X Y
m (S) 1 kj ! = λαS M . mM (S)! j=1 M ∈M S∈M Moreover, GD (z1 , . . . , zn ; k1 , . . . , kn ) is conformally invariant in the sense that, if D′ is a domain conformally equivalent to D and f : D → D′ is a conformal map, then GD′ (f (z1 ), . . . , f (zn ); k1 , . . . , kn )   n Y |f ′ (zj )|−2kj /3  GD (z1 , . . . , zn ; k1 , . . . , kn ). = (6.9) j=1 Proof. From the expression for the n-point function in Lemma 6.1, using the fact that P S∈M mM (S)I(zj ∈ S) = kj , for each j = 1, . . . , n and each M ∈ M, we see that * n + Pn Y (kj ) lim ε−2/3 j=1 kj Eβj ;ε (zj ) ε→0 = j=1 n Y kj !  X Y  mM (S) 1 λ lim ε−2|S|/ε µloop (ℓ ∩ Bε (zj ) 6= ∅ ∀zj ∈ S) mM (S)! ε→0 kj !  X Y
m (S) 1 λαS M , mM (S)! j=1 = n Y j=1 M ∈M S∈M M ∈M S∈M – 19 – (6.10) where the last equality follows from Lemma 2.2. Equation (6.9) now follows immediately from the last expression and Lemma 2.2. Using (6.8) and the definition of order k edge operator (6.5), we can now write the correlation of n higher-order edge operators as D E (k1 ) (z1 ) . . . E (kn ) (zn )  = n Y j=1 E = D  λ−kj /2  n Y ĉkj G (z , . . . , zn ; k1 , . . . , kn ) kj /2 D 1 k !λ j j=1 X Y M ∈M S∈M (6.11)
m (S) 1 λα̂S M . mM (S)! In view of (6.9), these n-point functions are manifestly conformally covariant, showing that the higher-order edge operators are conformal primaries. If n = 2 and k1 = k2 = k, it is easy to see that the set M contains a single multiset with only one element S = {z1 , z2 } with multiplicity k. Therefore, specializing (6.11) to this case with D = C gives D E (k) (z1 )E (k) (z2 ) E C = α̂z1 ,z2
k = (hE(z1 )E(z2 )iC)k = |z1 − z2 |−4k/3 , (6.12) which shows that E (k) is canonically normalized. 6.2 Charged edge operators We now apply a “twist” to the (higher-order) edge operators and introduce a new set of operators. A charged edge operator is essentially an edge operator “seen from” the perspective of a measure µ∗z;β ≡ µ∗z defined by µ∗z (ℓ) := ( µloop (ℓ) if z ∈ / ℓ̄ iβσ loop ℓ e µ (ℓ) if z ∈ ℓ̄ (6.13) where σℓ = ±1 is a symmetric Boolean variable assigned to ℓ. This measure, which is similar to that introduced in Section 3, assigns a phase eiβσℓ to each loop covering z. We note that, when taking expectations, one sums over the two possible values of σℓ with equal probability, so that loops ℓ that do not cover z get weight µloop (ℓ), while loops ℓ that cover z get weight cos β µloop (ℓ). With this in mind, for any β ∈ [0, 2π), the simplest charged edge operator with cutoffs δ, ε > 0, corresponding to the “twisted” or “charged” version of (6.2), is defined as (1);δ δ Eβ;ε (z) ≡ Eβ;ε (z)  := Vβδ (z) Nεδ (z) − λ µloop (diam(ℓ) > δ, ℓ ∩ Bε (z) 6= ∅, z ∈ / ℓ) + µloop (ℓ ∩ Bε (z) 6= ∅, z ∈ ℓ) cos β
 , (6.14) – 20 – where Vβδ (z)  := exp iβ X ℓ∈Lδ z∈ℓ̄  (6.15) σℓ , the layering operator with cutoff δ > 0 introduced in [4], induces a phase eiβσℓ for each loop ℓ such that z ∈ ℓ̄, and λ µloop (diam(ℓ) > δ, ℓ ∩ Bε (z) 6= ∅, z ∈ / ℓ) + µloop (ℓ ∩ Bε (z) 6= ∅, z ∈ ℓ) cos β
(6.16) is the expectation of Nεδ (z) under the measure µ∗z . Generalizing this to any k ∈ N, the “twisted” or “charged” version of (6.3) is given by (k);δ Eβ;ε (z) := Vβδ (z) " k−1 X j=0   k (−1) Nε (z) . . . (Nε (z) − (k − j) + 1) j j λ µloop (diam(ℓ) > δ, ℓ ∩ Bε (z) 6= ∅, z ∈ / ℓ) + µloop (ℓ ∩ Bε (z) 6= ∅, z ∈ ℓ) cos β

j + (−1)k λ µloop (diam(ℓ) > δ, ℓ ∩ Bε (z) 6= ∅, z ∈ / ℓ) + µloop (ℓ ∩ Bε (z) 6= ∅, z ∈ ℓ) cos β

k # . (6.17) We now formally define the charged (order k) edge operator (k) Eβ (z) := lim (c′ δ)−2∆(β) δ,ε→0 ĉk (k);δ ε−2k/3 Eβ;ε (z), k/2 k!λ (6.18) where c′ is a normalization constant needed to obtain the canonically normalized operator (0) Oβ from Vβδ , which depends on the domain (see [6]). For completeness, we also define Eβ ≡ (k) Oβ . Unlike their uncharged counterparts, the charged operators Eβ normalized for general β 6= 0. are not canonically As an example, we compute the two-point function of the simplest charged edge operδ (z) as ators, with charge conservation. To that end, we write Eβ;ε  δ Eβ;ε (z) = Vβδ (z) Nεδ (z) − λµloop (diam(ℓ) > ε, ℓ ∩ Bε (z) 6= ∅) + (1 − cos β)λµloop (diam(ℓ) > δ, ℓ ∩ Bε (z) 6= ∅, z ∈ ℓ)  = Vβδ (z)Eεδ (z) + (1 − cos β)λµloop (diam(ℓ) > δ, ℓ ∩ Bε (z) 6= ∅, z ∈ ℓ)Vβδ (z). – 21 – (6.19) Using this expression and the method introduced in Section 3, we have D E E D δ δ δ Eβ;ε (z1 )E−β;ε (z2 ) = Vβδ (z1 )V−β (z2 )Eεδ (z1 )Eεδ (z2 ) E D δ (z2 ) + (1 − cos β)λµloop (diam(ℓ) > δ, ℓ ∩ Bε (z2 ) 6= ∅, z2 ∈ ℓ) Vβδ (z1 )Eεδ (z1 )V−β E D δ + (1 − cos β)λµloop (diam(ℓ) > δ, ℓ ∩ Bε (z1 ) 6= ∅, z1 ∈ ℓ) V−β (z2 )Eεδ (z2 )Vβδ (z1 ) + (1 − cos β)2 λ2 µloop (diam(ℓ) > δ, ℓ ∩ Bε (z1 ) 6= ∅, z1 ∈ ℓ) D E δ µloop (diam(ℓ) > δ, ℓ ∩ Bε (z2 ) 6= ∅, z2 ∈ ℓ) Vβδ (z1 )V−β (z2 ) E∗ D EhD δ = Vβδ (z1 )V−β (z2 ) Eεδ (z1 )Eεδ (z2 ) z1 ,z2 E∗ D loop + (1 − cos β)λµ (diam(ℓ) > δ, ℓ ∩ Bε (z2 ) 6= ∅, z2 ∈ ℓ) Eεδ (z1 ) z ,z E∗1 2 D + (1 − cos β)λµloop (diam(ℓ) > δ, ℓ ∩ Bε (z1 ) 6= ∅, z1 ∈ ℓ) Eεδ (z2 ) z1 ,z2 2 2 loop + (1 − cos β) λ µ (diam(ℓ) > δ, ℓ ∩ Bε (z1 ) 6= ∅, z1 ∈ ℓ) i µloop (diam(ℓ) > δ, ℓ ∩ Bε (z2 ) 6= ∅, z2 ∈ ℓ) . (6.20) After identifying z3 with z1 and z4 with z2 , we can use (3.7) and (5.5) to simplify the above expression. A simple calculation shows that, for for any δ < |z1 − z2 |, D E D Eh δ δ δ (z1 )E−β;ε (z2 ) = Vβδ (z1 )V−β (z2 ) λµloop (ℓ ∩ Bε (zj ) 6= ∅, j = 1, 2) Eβ;ε − (1 − cos β)λµloop (ℓ ∩ Bε (zj ) 6= ∅, j = 1, 2; ℓ separates z1 , z2 ) + λ2 (1 − cos β)2 µloop (ℓ ∩ Bε (z1 ) 6= ∅, z2 ∈ ℓ̄, z1 ∈ / ℓ̄) i µloop (diam(ℓ) > δ, ℓ ∩ Bε (z2 ) 6= ∅, z1 ∈ ℓ̄, z2 ∈ / ℓ̄) . (6.21) Using definition (6.18), we obtain D E δ hEβ (z1 )E−β (z2 )i = lim (ĉ′ δ)−4∆(β) Vβδ (z1 )V−β (z2 ) δ→0 h 2 −4/3 loop ĉ lim ε µ (ℓ ∩ Bε (zj ) 6= ∅, j = 1, 2) ε→0 − (1 − cos β)µloop (ℓ ∩ Bε (zj ) 6= ∅, j = 1, 2; ℓ separates z1 , z2 ) + λ(1 − cos β)2 µloop (ℓ ∩ Bε (z1 ) 6= ∅, z2 ∈ ℓ̄, z1 ∈ / ℓ̄) i / ℓ̄) µloop (ℓ ∩ Bε (z2 ) 6= ∅, z1 ∈ ℓ̄, z2 ∈ h ,z2 = hOβ (z1 )O−β (z2 )i α̂z1 ,z2 − (1 − cos β)α̂zz11 |z 2 + λ(1 − cos β)2 ĉ2 lim ε−4/3 µloop (ℓ ∩ Bε (z1 ) 6= ∅, z2 ∈ ℓ̄, z1 ∈ / ℓ̄) ε→0 i µloop (ℓ ∩ Bε (z2 ) 6= ∅, z1 ∈ ℓ̄, z2 ∈ / ℓ̄) . – 22 – (6.22) At this point, we should note that unfortunately the existence of the limits ,z2 αzz11 |z = lim ε−4/3 µloop (ℓ ∩ Bε (zj ) 6= ∅, j = 1, 2; ℓ separates z1 , z2 ), 2 lim ε ε→0 −2/3 loop ε→0 µ (ℓ ∩ Bε (zj ) 6= ∅, zk ∈ ℓ̄, zj ∈ / ℓ̄) (6.23) does not follow from Lemma 2.2. It is, however, reasonable to assume that they exist. Indeed, in the case of the first limit, observing that lim Ztwist = 0 z3 →z1 z4 →z2 (6.24) and using (5.9) suggests that, in the full plane, 1 z1 ,z2 ,z2 . α̂zz11 |z = α̂C 2 ;C 2 (6.25) The second limit in (6.23) should also exist; moreover, if z α̂Cj (zk ; zj ) := ĉ lim ε−2/3 µloop / ℓ̄) C (ℓ ∩ Bε (zj ) 6= ∅, zk ∈ ℓ̄, zj ∈ ε→0 (6.26) does exist, arguments like those used in the second part of the proof of Lemma 2.2 imply sz (0; z) = s−2/3 α̂z (0; z). Since α̂zj (z ; z ) only depends on |z − z |, that, for any s > 0, α̂C j k C C k j zj −2/3 this would in turn imply that α̂C (zk ; zj ) must take the form const |zj − zk | . If the considerations above are correct, then it follows from (6.22) that hEβ (z1 )E−β (z2 )iC behaves like the correlation function between two conformal primaries of scaling dimension z z ∆(β) + 1/3, as desired. Indeed, we conjecture that, similarly to (6.25), α̂Cj (zk ; zj ) = 21 α̂zkj ;C, which would lead to   1 λ z1 z2 z1 ,z2 hEβ (z1 )E−β (z2 )iC = hOβ (z1 )O−β (z2 )iC (1 + cos β)α̂C + α̂z2 ;Cα̂z1 ;C 2 4 (6.27) −4∆(β)−4/3 ∼ |z1 − z2 | , where the existence and the scaling behavior of z α̂zkj ;C := ĉ lim ε−2/3 µloop C (ℓ ∩ Bε (zj ) 6= ∅, zk ∈ ℓ̄) ε→0 (6.28) follows from the proof of Lemma 2.2. 7 The primary operator spectrum The four-point function of a conformal field theory contains information about the threepoint function coefficients, as well as the spectrum of primary operators. In the following two sections, we perform the Virasoro conformal block expansion of the new four-point function (5.6) in the full plane, and derive the three-point coefficient involving three edge operators through the OPE of the edge operator as an illustration of the conformal block expansion. – 23 – 7.1 Virasoro conformal blocks By a global conformal transformation, one can always map three of the four points of a fourpoint function hA1 (z1 )A2 (z2 )A3 (z3 )A4 (z4 )iC to fixed values, where Aj (zj ) here denotes a generic primary operator evaluated at zj . The remaining dependence is only on the crossz34 ratio x = zz12 and its complex conjugate x̄, which are invariant under global conformal 13 z24 transformations. The following discussion parallels Section 6 of [6]. Following the notation of Section 6.6.4 of [5], it is customary to set z1 = ∞, z2 = 1, z3 = x and z4 = 0. The resulting function ¯ 2∆1 2∆1 z̄1 hA1 (z1 )A2 (1)A3 (x)A4 (0)iC G21 34 (x) := lim z1 z1 →∞ can be decomposed into Virasoro conformal blocks according to X 21 P P 21 (P|x̄). G21 C34 C12 F34 (P|x)F̄34 34 (x) = (7.1) (7.2) P The sum over P runs over all primary operators in the theory, and the CljP are the three-point function coefficients of the operators labeled by l, j, P, that is, −(∆l +∆j −∆P ) −(∆l +∆P −∆j ) −(∆j +∆P −∆l ) z23 z13 ¯ +∆ ¯ −∆ ¯ ) ¯ +∆ ¯ −∆ ¯ ) −(∆ ¯ +∆ ¯ −∆ ¯ ) −(∆ −(∆ z̄12 l j P z̄13 l P j z̄23 j P l , hAl (z1 )Aj (z2 )P(z3 )iC = CljP z12 (7.3) ¯ j are the scaling dimensions of the corresponding fields. where ∆j , ∆ The functions F, F̄ are called Virasoro conformal blocks and are fixed by conformal invariance. Each conformal block can be written as a power series 21 F34 (P|x) = x∆P −∆3 −∆4 ∞ X K=0 FK xK , (7.4) where coefficients FK can be fully determined by the the central charge c, and the conformal dimensions ∆j , ∆P of the five operators involved. F̄ is determined analogously. In the case of (5.6), we obtain 4∆(β) G21 hOβ (z1 )O−β (1)E(x)E(0)iC 34 (x) = lim |z1 | z1 →∞ 4 · 21/3 π 2 (1 − cos β)2 1 + cos β + =λ √

2 2 |1 − x|2/3 2|x|4/3 5 3Γ 16 Γ 43 "
6   2  2# 4Γ 23 1 − cos β 2 1 2 1 2 4 2/3 + . −
2
4 |x| 2 F1 − , ; ; x 2 F1 − , ; ; x 3 3 3 3 3 3 2|x|4/3 |1 − x|2/3 Γ 34 Γ 31 (7.5) The expansion around x = x̄ = 0 allows us to obtain information about the primary operator spectrum and fusion rules of the operators that appear in both the Oβ × O−β and E × E expansions. The hypergeometric functions appearing above are regular around x = 0. The expansion of (7.5) around zero can thus be written G21 34 (x) = |x| −4/3 ∞ X m,n=0 – 24 – am,n xm/3 x̄n/3 . (7.6) 0 1 2 3 4 5 6 7 8 0 9 10 11 12 13 14 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 1 2 3 4 5 6 7 8 9 10 11 12 13 14 (p,p′ ) (p,p′ ) (b) Non-zero COβ O−β (a) Non-zero CEE Figure 2: The non-zero three-point function coefficients are shown. Rows and columns label (p, p′ ). Left: between two edge operators. Right: between two vertex operators. ¯ Using (7.4), this expansion is of the form |x|−4∆E x∆P +k x̄∆P +k̄ , where k, k̄ are non-negative ¯ P can only be multiples of 1/3. This must be integers. Since ∆E = 1/3 we see that ∆P , ∆ equal to (7.2), which can now be written as G21 34 (x) = |x| −4/3 ∞ X (p,p′ ) CEE p,p′ , m,n=0 (p,p′ ) ′ (p) (p ) m/3 n/3 COβ O−β Fm Fn x x̄ . (7.7) By comparing the last two equations, we determine the products of three-point function coefficients at any desired order. Together with the three-point coefficients determined in [6], using [11], we can uniquely determine the coefficients involving edge operators which (p,p′ ) also fuse onto vertex operators. Figure 2 shows the non-zero three-point coefficients CEE which appear in the Virasoro block expansion. The operators appearing in Figure 2a are a subset of those in Figure 2b, and only the operators which fuse onto both sets of operators can be discovered from (7.5). The correct normalization of our operators and four-point function is ensured by (0,0) CEE (0,0) 1 ≡ CEE =1 (7.8) 1 = 1. COβ O−β ≡ CO β O−β Furthermore, we obtain the coefficients (1,1) CEE (2,2) CEE √ 1 4 · 21/6 · 31/4 · 5π 3/2 Γ ≡ =√
3
λ Γ 16 Γ 67 √ E (2) ≡ CEE = 2. E CEE 2 3
(7.9) (7.10) The complexity of these coefficients grows rapidly for larger (p, p′ ). The operator E (2) can be identified with the higher order edge operator of conformal and anti-conformal dimensions 2/3 defined in (6.5). – 25 – By rearranging the operators in the four-point function (7.5), one can easily show that the resulting four-point functions are crossing-symmetric. In particular, by exchanging operators 2 and 4, one may obtain information about the OPE of Oβ × E. The expansion in the cross-ratio in this channel shows logarithmic terms, which indicate the existence of degenerate operators in a logarithmic CFT. The logarithmic properties of the related O(n) model have been studied, for example, in [12]. We do not investigate their relations to the BLS at this point. 21 Nevertheless, one can use G41 32 (x) = G34 (1 − x) to compute the fusion rules for Oβ × E, P and in particular, the squares of three-point function coefficients CO of all primaries P. βE 21 The expansion of G34 (1 − x) analogous to (7.7) allows us to obtain the following operators in the OPE O E Oβ (z) × E(z ′ ) = COββE |z − z ′ |−2/3 Oβ (z) + COββ E Eβ (z) + . . . , (7.11) O E where COββE = CO and β O−β  E COββ E 2 = 1 + cos β . 2 (7.12) The operator Eβ is the k = 1 case of the charged edge operators defined in (6.18), with conformal and anti-conformal dimension ∆(β) + 1/3. 7.2 The three-point function of the edge operator (1,1) E , In this section, we show how to compute the three-point function coefficient CEE ≡ CEE which was derived in the previous section from the conformal block expansion, by applying the OPE of two edge operators. This computation is a special case of the general expansion (7.7), and shows the inner workings of the general method. Using the general expression for the three-point function of a conformal primary operator and (4.13), we have E hE(z1 )E(z2 )E(z3 )iC = CEE |z12 |−2/3 |z13 |−2/3 |z23 |−2/3 E = CEE |z12 |−4/3 |z23 |−2/3 1 + O |z23 |



E = CEE hE(z1 )E(z2 )iC |z23 |−2/3 1 + O |z23 | D h
iE E z23 −2/3 E(z2 ) + O z23 1/3 = E(z1 ) CEE . (7.13) C Additionally, using (5.6) and (5.9) we see that, for β = π, hOπ (z1 )O−π (z2 )E(z3 )E(z4 )iC = |z12 |−4λ/5 Ztwist + 4λ |z12 |−4λ/5 α̂zz13 |z2 ;Cα̂zz14 |z2 ;C. (7.14) The second term on the right hand side is not divergent as z4 → z3 , while we see from (5.11) that limz4 →z3 |z34 |4/3 Ztwist = 1, so that lim |z34 |4/3 hOπ (z1 )O−π (z2 )E(z3 )E(z4 )iC = |z12 |−4λ/5 = hOπ (z1 )O−π (z2 )iC . z4 →z3 – 26 – (7.15) Combining these observations gives the OPE E E(z) × E(z ′ ) = |z − z ′ |−4/3 1 + CEE |z − z ′ |−2/3 E(z) + . . . . (7.16) Plugging this OPE into (5.6) and using (3.13) gives hOβ (z1 )O−β (z2 )E(z3 )E(z4 )iC E hOβ (z1 )O−β (z2 )E(z3 )iC |z34 |−2/3 + O |z34 |1/3 = hOβ (z1 )O−β (z2 )iC |z34 |−4/3 + CEE E E |z12 |−4∆(β) CO = |z12 |−4∆(β) |z34 |−4/3 + CEE β O−β z12 z13 z23 2/3
|z34 |−2/3 + O |z34 |1/3 .
(7.17) For β = π, comparing with (7.14) gives E E CO |z12 |−4λ/5 |z12 |−4λ/5 |z34 |−4/3 + CEE π O−π = |z12 | −4λ/5 Ztwist + 4λ |z12 | −4λ/5 z12 z13 z23 2/3 |z34 |−2/3 + O |z34 |1/3 α̂zz13 |z2 ;Cα̂zz14 |z2 ;C.
(7.18) Using the expression (5.11) for Ztwist , we can write z13 z24 = z23 z14 Ztwist − z12 z23 z14 2/3 2/3 2 2 1 2 |z34 |−4/3 2 F1 − , ; , x 3 3 3
6  2 4Γ 32 1 2 4 |z34 |−2/3 .
2
4 2 F1 − , ; , x 3 3 3 Γ 34 Γ 13  (7.19) Plugging this into (7.18) and observing that lim z4 →z3 z13 z24 z23 z14 2/3 2 F1   2  2 2 1 2 − , ; ,x 3 3 3 (7.20) = 1, shows that E CEE E CO π O−π =− Γ 4Γ
4 2 3
 2 6 1 3 lim 2 F1 − ,
4 3 Γ 31 z3 →z4 2 4 ; ,x 3 3 Finally, using (3.15), after some simplification we obtain √ 1 4 · 21/6 · 31/4 · 5π 3/2 Γ E CEE = √
3
λ Γ 61 Γ 76 =− 2 3
, Γ 4Γ
4 2 3
2 6 3
4 . Γ 13 (7.21) (7.22) which indeed coincides with (7.9). 8 Central charge Given an explicit form of a four-point function of a two dimensional CFT, together with sufficient knowledge of the operator spectrum, one can determine the central charge c of the – 27 – theory. We will now use the previous result (5.6) for the case of the full plane to confirm that c = 2λ in the BLS, as was derived, for instance, in [4]. In every two dimensional CFT, the two-point function of the energy–momentum tensor to leading order is fixed by conformal invariance to be hT (z1 )T (z2 )iC = c/2 4 . z12 (8.1) The energy-momentum tensor can be understood as the level-2 Virasoro descendant of the identity operator I 1 1 (L−2 1)(z) = T (w) = T (z), (8.2) dw 2πi z w−z where the integral is along any contour around the point z, and Ln are the generators of the Virasoro algebra. Its anti-holomorphic counterpart is analogously given by T̄ (z̄) = (L̄−2 1)(z̄). Additionally, the OPE of two primary operators is generally given by (cf. [5], Section 6.6.3) A1 (z + ǫ) × A2 (z) XX ¯ ¯ ¯ P P{k} P{k̄} ∆P −∆1 −∆2 +K ∆ = C12 β12 β̄12 ǫ ǭ P −∆1 −∆2 +K̄ L−k1 . . . L−kN L̄−k̄1 . . . L̄−k̄N̄ P(z), P {k,k̄} (8.3) where CljP are three-point function coefficients, K = P{k} P{k̄} βlj , β̄lj P kj ∈{k} kj with kj ∈ N is the descen- dant level, and are numerical coefficients that depend on the central charge and the conformal dimensions of the involved operators and are fully determined by the Virasoro algebra. The outer sum runs over all primary operators P, and the inner sum is over all subsets {k}, {k̄} of the natural numbers. (This was the basis of the analysis of Section 7.) Since the identity operator has non-zero OPE coefficient for both Oβ × O−β and E × E, we can use (5.6) to obtain the central charge c by identifying the level-2 descendant of the identity. We achieve this by applying the OPE twice to (5.6) and evaluating it in two equivalent ways. First, we expand the expression Oβ (z + ǫ)O−β (z)E(z ′ + ǫ′ )E(z ′ ) (8.4) C analytically around zero for ǫ, ǭ, ǫ′ , ǭ′ . We then identify the term of order (ǫǫ′ )−∆(β)−1/3+2 with the contribution from the algebraic expansion (8.3) at the same order in ǫ, ǫ′ , which is 1{2} 1{2} 1 1 CEE βOβ O−β βEE (ǫǫ′ )−∆(β)−1/3+2 CO β O−β (L−2 1)(z)(L−2 1)(z ′ ) C . (8.5) Generically, one expects contributions like (L−1 A(3,0) )(L−1 A(3,0) ) and A(6,0) A(6,0) to ap′ pear, where A(p,p ) are primary operators of conformal dimensions (p/3, p′ /3). However, the – 28 – previous analysis has shown their relevant three-point coefficients vanish (see e.g. Figure 2a). If the conformal dimensions of a pair of operators are equal, it can be shown that 1{2} βA1 A2 = 2∆A1 /c, where ∆A1 = ∆A2 is the conformal dimension of the operators [5]. We 1 also note that CA denotes the normalization of non-zero two-point functions, which is 1 A2 canonically chosen to be 1. Every quantity in (8.5) has thus been determined. The analytic expansion of (8.4) yields (at the desired order) (ǫǫ′ )−∆(β)−1/3+2 1 1 − cos β . 30 (z − z ′ )4 (8.6) Using (8.1) and (8.2), (8.5) becomes (dropping the powers of ǫ and ǫ′ ) 2∆(β) 2∆E 2 λ 1 − cos β T (z)T (z ′ ) = , c c 3c 10 (z − z ′ )4 (8.7) λ (1 − cos β), ∆E = 1/3. Comparing (8.6) to (8.7) confirms the where we used ∆(β) = 10 result that the BLS with intensity λ has central charge c = 2λ. 9 Conclusions and future work In this work we identified all scalar operators that couple to the layering vertex operators Oβ . This leaves open the question of the nature of the operators with non-zero spin. Perhaps the most interesting is the operator with k = 3, k ′ = 0 and zero charge, which has dimensions (1, 0). This is a (component of a) spin-1 current that should satisfy a conservation law and generate a conserved charge. Understanding the nature and role of this current may greatly clarify the structure of the spectrum of the CFT associated to the BLS. Another question open to investigation is the torus partition function. By further exploiting the connection to the O(n) model it seems possible that this can be computed. If so it would reveal the complete spectrum and degeneracies of the theory (modulo complications resulting from the lack of unitarity of the theory). The theory as we have presented it has a continuous spectrum because the operator dimensions depend on the continuous parameters β. This is reminiscent of the vertex operators of the free boson. There, one can compactify the boson and obtain a discrete spectrum. An analogous procedure seems available here too, where we identify the layering number with itself modulo an integer. If this is indeed self-consistent it would render the spectrum discrete, which has a number of interesting implications that we intend to explore in future work. The largest question is what place this Brownian loop soup conformal field theory has in the spectrum of previously known conformally invariant models. It appears to be a novel, self-consistent, and rich theory in its own right, but its connections with the free field and the O(n) model suggest that it may have ties to other theories that could be exploited to greatly advance our understanding of it. – 29 – Acknowledgments We are grateful to Sylvain Ribault for insightful comments on a draft of the manuscript. The work of M.K. is partially supported by the NSF through the grant PHY-1820814. A Proofs In this section we collect all the proofs that do not appear in the main body of the paper. We first show that the correlations functions hEε (z1 ) . . . Eε (zn )iD are well defined, a necessary step to state Lemma 2.1, proved next in this appendix, and Theorem 2.3. We then provide a proof of Lemma 2.2. We refer to Section 2 for the notation used here, the statements of Lemmas 2.1 and 2.2, as well as the statement and proof of Theorem 2.3. Additionally, we remind the reader of the following definitions from Section 3. For any δ > 0, let Lδ denote a Brownian loop soup in D with intensity λ and cutoff δ > 0, obtained by taking the usual Brownian loop soup and removing all loops with diameter smaller than δ. We define Nεδ (z) ≡ nεz (Lδ ) and Eεδ (z) ≡ Nεδ (z) − hNεδ (z)iD . Note that the random variables Nεδ (z) and Eεδ (z) are well defined because of the cutoffs ε > 0 and δ > 0. The next lemma shows that, if we consider n-point functions of Eεδ for n ≥ 2, the δ cutoff can be removed without the need to renormalize the n-point functions. Lemma A.1. For any collection of points z1 , . . . , zn ∈ D at distance greater than 2ε from each other, with n ≥ 2, the following limit exists: hEε (z1 ) . . . Eε (zn )iD := lim hEεδ (z1 ) . . . Eεδ (zn )iD . δ→0 (A.1) Proof. For each j = 1, . . . , n, we can write Nεδ (zj ) = Mεδ (zj ) + Rεδ (zj ), (A.2) where Mεδ (zj ) := X ℓ∈Lδ Rεδ (zj ) := X ℓ∈Lδ I(ℓ ∩ Bε (zj ) 6= ∅, ℓ ∩ Bε (zk ) = ∅ ∀k 6= j), (A.3) I(ℓ ∩ Bε (zj ) 6= ∅ and ℓ ∩ Bε (zk ) 6= ∅ for at least one k 6= j), (A.4) where I(·) denotes the indicator function. Now consider values of δ < mink,m (|zk − zm | − 2ε) with k, m = 1, . . . , n and m 6= k, then all the loops from L that intersect Bε (zj ) and at least one other disk Bε (zk ) must have diameter larger than δ. Therefore, for δ sufficiently small, Rεδ (zj ) does not depend on δ and we can drop the superscript and write Rε (zj ) instead. – 30 – Defining mδε (zj ) := hMεδ (zj )iD and rε (zj ) := hRε (zj )iD , for values of δ sufficiently small we can write Dh i E E D = Mεδ (z1 ) − mδε (z1 ) + Rε (z1 ) − rε (z1 ) Eεδ (z2 ) . . . Eεδ (zn ) Eεδ (z1 ) . . . Eεδ (zn ) D D Dh i E δ δ δ δ = Mε (z1 ) − mε (z1 ) Eε (z2 ) . . . Eε (zn ) D ED δ δ + [Rε (z1 ) − rε (z1 )] Eε (z2 ) . . . Eε (zn ) . D (A.5) Mεδ (z1 ) is independent of Eεδ (zj ) for all j 6= 1, so Dh i E =0 Mεδ (z1 ) − mδε (z1 ) Eεδ (z2 ) . . . Eεδ (zn ) (A.6) D and D Eεδ (z1 ) . . . Eεδ (zn ) E D E D = [Rε (z1 ) − rε (z1 )] Eεδ (z2 ) . . . Eεδ (zn ) . D Proceeding in the same way for all values of j = 2, . . . , n, we obtain D E = h[Rε (z1 ) − rε (z1 )] . . . [Rε (zn ) − rε (zn )]iD , Eεδ (z1 ) . . . Eεδ (zn ) D (A.7) (A.8) which is independent of δ. Proof of Lemma 2.1. The random variables (Nεδ (z1 ), . . . , Nεδ (zn )) are jointly Poisson. If we let v = (v1 , . . . , vn ) be an n-dimensional vector with components vj = 0 or 1, following [13] we see that their joint distribution is captured by Nεδ (v) := |{ℓ : diam(ℓ) > δ, ℓ ∩ Bε (zj ) 6= ∅ ∀j : vj = 1, ℓ ∩ Bε (zj ) = ∅ ∀j : vj = 0}|, (A.9) where Nεδ (v) is itself a Poisson random variable with parameter λµloop (diam(ℓ) > δ, ℓ ∩ Bε (zj ) 6= ∅ ∀j : vj = 1, ℓ ∩ Bε (zj ) = ∅ ∀j : vj = 0). More precisely, using Theorem 2 of [13], we can write the joint probability generating function of (Nεδ (z1 ), . . . , Nεδ (zn )) as E D δ δ N (z ) ε (zn ) h(x1 , . . . , xn ) := x1 ε 1 , . . . , xN n  X
µloop diam(ℓ) > δ, ℓ ∩ Bε (zj ) 6= ∅ ∀j ∈ I, ℓ ∩ Bε (zj ) = ∅ ∀j ∈ /I = exp λ I subset {1,...,n} |I|≥1    Y ·  xj − 1 . j∈I Letting Dk := ∂ ∂xk (A.10) − λµloop (diam(ℓ) > δ, ℓ ∩ Bε (zk ) 6= ∅), using (6.2) we have D Eεδ (z1 ) . . . Eεδ (zn ) E D = n Y . Dk h(x1 , . . . , xn ) k=1 – 31 – xk =1 (A.11) Using an induction argument, one can show that X µloop I subset {1,...,n} |I|≥1   Y diam(ℓ) > δ, ℓ ∩ Bε (zj ) 6= ∅ ∀j ∈ I, ℓ ∩ Bε (zj ) = ∅ ∀j ∈ / I  x j − 1
X = I subset {1,...,n} j∈I µloop diam(ℓ) > δ, ℓ ∩ Bε (zj ) 6= ∅ ∀j ∈ I |I|≥1
Y j∈I (xj − 1). (A.12) Hence, h h(x1 , . . . , xn ) = exp λ =1+ ∞ X r=1 λr X I subset {1,...,n} X I1 ,...,Ir |I|≥1 r Y l=1 subsets of {1,...,n} µloop diam(ℓ) > δ, ℓ ∩ Bε (zj ) 6= ∅ ∀j ∈ I
Y j∈I (xj − 1) i 
Y 1  loop (xj − 1) , µ diam(ℓ) > δ, ℓ ∩ Bε (zj ) 6= ∅ ∀j ∈ Il m(Il )! j∈Il (A.13) where the second sum is over all unordered collections of subsets of {1, . . . , n} not necessarily distinct (i.e., over multiset), and we have used the fact that the number of ways in which an unordered collection of r elements can be ordered is r! , l=1 m(Il )! (A.14) Qr where m(Il ) is the multiplicity of Il in the multiset. Considering the structure of the last expression, the definition of the differential operator Dk , and the fact that in (A.11) all derivatives ∂x∂ k are evaluated at xk = 1, we can differentiate term by term. it is clear that in the right hand side of (A.11) the only terms that survive are those for which the derivatives saturate the variables xk . Moreover, Lemma A.1 implies that terms of the type µloop (diam(ℓ) > δ, ℓ ∩ Bε (zk )) cannot be present in the right hand side of (A.11) because otherwise the limit δ → 0 wouldn’t exist. (One can reach the same conclusion by looking at (A.13) and observing that terms containing subsets that are single points, i.e. Il = {zk }, disappear when applying Dk .) These considerations single out all partitions Π of {1, . . . , n} whose elements have cardinality at least 2. Therefore, we obtain hEε (z1 ) . . . Eε (zn )iD = lim δ→0 = n Y k=1 Dk h(x1 , . . . , xn ) X {I1 ,...,Ir }∈Π λ r r Y l=1 which concludes the proof. – 32 – xk ≡1 µloop D (ℓ ∩ Bε (zj ) 6= ∅ ∀j ∈ Il ), (A.15) Proof of Lemma 2.2. Consider the full scaling limit of critical percolation in D constructed in [14] and denote it by FD . FD is distributed like CLE6 in D [15]. As explained in [8], the “outer perimeters” of loops from FD are distributed like the outer boundaries of Brownian loops. Hence, there is a close connection between the Brownian loop measure µloop and the collection of loops constructed in [14]. D More precisely, let P denote the distribution of FD and E denote expectation with respect to P. Since FD is conformally invariant, if A is a measurable set of self-avoiding loops and NA is the number of loops Γ from FD such that their outer perimeters ℓ(Γ) are in A, E(NA ) defines a conformally invariant measure on self-avoiding loops. Moreover, since the measure µloop is unique, up to a multiplicative constant, we must have D µloop D (A) = Θ E(NA ), (A.16) where 0 < Θ < ∞ is a constant. Now consider the set of simple loops Sε = {ℓ ∈ D : ℓ ∩ Bε (zj ) 6= ∅ ∀j = 1, . . . , k}. Thanks to the scale invariance of µloop and FD , we can assume without loss of generality D that the points z1 , . . . , zk are at distance much larger than 1 from each other. We write FD ∈ Sε to indicate the event that a configuration from FD contains at least one loop Γ such that ℓ(Γ) ∈ Sε . For each j = 1, . . . , k, consider the annulus Aε,1 (zj ) := B1 (zj ) \ Bε (zj ) centered at zj with outer radius 1 and inner radius ε. Because of our assumption on the distances between the points zj , j = 1, . . . , k, the annuli do not overlap. The configurations from FD for which NSε > 0 (i.e., such that FD ∈ Sε ) are those that contain at least one loop Γ whose outer perimeter ℓ(Γ) intersects Bε (zj ) for each j = 1, . . . , k. They can be split in two groups as described below, where a three-arm event inside Aε,1 (zj ) refers to the presence of a loop Γ such that the annulus Aε,1 (zj ) is crossed from the inside of Bε (zj ) to the outside of B1 (zj ) by two disjoint outer perimeter paths belonging to ℓ(Γ) and by one path within the complement of the unique unbounded component of C \ Γ. (i) Configurations that induce a three-arm event inside Aε,1 (zj ) for each j = 1, . . . , k, for which NSε = 1. (ii) Configuration that induce more than three arms in Aε,1 (zj ) for at least one j = 1, . . . , k, for which NSε ≥ 1. The probability of a three-arm event in Aε,1 (zj ) is ε2/3+o(1) as ε → 0, while the probability to have four or more arms in Aε,1 (zj ) is O(ε5/4 ) as ε → 0; therefore ε−2k/3 E(NSε ) = ε−2k/3 P(FD ∈ Sε and there is a three-arm event in each Aε,1 (zj )) + O(ε7/12 ). (A.17) It follows from the construction of FD in [14], which uses the locality of SLE6 , that a configuration in group (i) can be constructed by first generating independent configurations inside B1 (zj ) for each j = 1, . . . , k, requiring that each induces a three-arm event in Aε,1 (zj ), and then generating a “matching” configuration in D \ ∪kj=1 B1 (zj ). A configuration inside – 33 – B1 (zj ) contains loops and arcs starting and ending on ∂B1 (zj ). Moreover, since Aε,1 (zj ) contains a three-arm event, exactly one outer perimeter arc starting and ending on ∂B1 (zj ) intersects Bε (zj ). Each arc in B1 (zj ) has a pair of endpoints on ∂B1 (zj ). We let Ij denote the collection of endpoints on ∂B1 (zj ), together with the information regarding which endpoints are connected to each other, and we denote by νjε the distribution of Ij , conditioned on the occurrence of a three-arm event. An important observation is that, conditioned on Ij for each j = 1, . . . , k, the configuration in D \ ∪kj=1 B1 (zj ) is independent of the configurations inside B1 (zj ) for j = 1, . . . , k. If we let G denote the event that endpoints on ∂B1 (zj ) are connected in D \ ∪kj=1 B1 (zj ) in such a way that overall the resulting configuration in D is in Sε , this observation allows us to write P(FD ∈ Sε and there is a three-arm event in Aε,1 (zj ) ∀j = 1, . . . , k) = P(FD ∈ Sε | there is a three-arm event in Aε,1 (zj ) ∀j = 1, . . . , k) P(there is a three-arm event in Aε,1 (zj ) ∀j = 1, . . . , k) Z k Y 2k/3+o(1) P(G|I1 , . . . , Ik ) =ε dνjε (Ij ). (A.18) j=1 Combining this with (A.17), we obtain lim ε ε→0 −2k/3 E(NSε ) = lim ε→0 Z P(G|I1 . . . , Ik ) k Y dνjε (Ij ), (A.19) j=1 where P(G|I1 . . . , Ik ) does not depend on ε and νjε is the distribution of endpoints on ∂B1 (zj ) conditioned on the occurrence of a three-arm event in Aε,1 (zj ), or equivalently on the existence of a single outer perimeter arc starting and ending on ∂B1 (zj ) and intersecting Bε (zj ). Now observe that requiring the existence of a single outer perimeter arc that intersects Bε (zj ) and sending ε → 0 is equivalent to centering the disk B1 (zj ) at a typical point10 zj on the outer perimeter of a loop from FD which exits B1 (zj ) and therefore has diameter greater than 1. Therefore, the limit limε→0 νjε exists: it is given by the distribution of endpoints of arcs for a disk of radius 1 centered at a typical point on the outer perimeter of a loop from FD of diameter larger than 1. Equivalently, by scale invariance, it is the distribution of endpoints of arcs on ∂Br (z) for a disk Br (z) centered at a typical point z on the outer perimeter of a loop from FD , with diameter r smaller than the diameter of the loop. Therefore, if we call ν this distribution, from (A.16) and (A.19) we have −2k/3 lim ε−2k/3 µloop E(NSε ) D (Sε ) = Θ lim ε ε→0 ε→0 =Θ Z P(G|I1 , . . . , Ik ) k Y (A.20) dν(Ij ), j=1 proving the existence of the limit in (2.5). 10 Here typical means that it is not a pivotal point, i.e., a point on the outer perimeter of two loops. Pivotal points have a lower fractal dimension. – 34 – In order to prove (2.6), consider a domain D′ conformally equivalent to D and a conformal map f : D → D′ , and let zj′ = f (zj ), sj = |f ′ (zj )| for each j = 1, . . . , k, and Sε′ = {ℓ ∈ D′ : ℓ ∩ Bε (zj′ ) 6= ∅ ∀j = 1, . . . , k}. We are interested in the behavior of z ′ ,...,zk′ αD1′ = lim ε−2k/3 µD′ (Sε′ ) = lim ε−2k/3 µD (ℓ ∩ f −1 (Bε (zj′ )) 6= ∅ ∀j = 1, . . . , k). (A.21) ε→0 ε→0 To evaluate this limit, we will use the fact that µD (ℓ ∩ f −1 (Bε (zj′ )) 6= ∅ ∀j = 1, . . . , k) − µD (ℓ ∩ Bε/sj (zj ) 6= ∅ ∀j = 1, . . . , k) ≤ O(ε4k/3 ). (A.22) To see this, let Arj ,Rj (zj ) = BRj (zj ) \ Brj (zj ) denote the thinnest annulus centered at zj containing the symmetric difference of f −1 (Bε (zj′ )) and Bε/sj (zj ) and note that µD (ℓ ∩ f −1 (Bε (zj′ )) 6= ∅ ∀j = 1, . . . , k) − µD (ℓ ∩ Bε/sj (zj ) 6= ∅ ∀j = 1, . . . , k) ≤ µD (ℓ ∩ BRj (zj ) 6= ∅ and ℓ ∩ Brj (zj ) = ∅ ∀j = 1, . . . , k). (A.23) Since f −1 is analytic and (f −1 (zj′ ))′ = 1/sj , for every w ∈ ∂Bε (zj′ ), |zj − f −1 (w)| = |f −1 (zj′ ) − f −1 (w)| = ε/sj + O(ε2 ), which implies that Rj − rj = O(ε2 ). Therefore one can cover the annulus Arj ,Rj (zj ) with a finite number of overlapping disks of radius of order ε2 . Now, using (A.16), we can bound the right hand side of (A.23) by the probability that, for each annulus Arj ,Rj (zj ), at least one of the covering disks of radius O(ε2 ) is the center of an annulus with outer radius O(1) containing a three-arm event, which is of order (ε2 )2k/3 . Hence, from (A.21), (A.22) and (2.5), we obtain     k  −2/3 k Y Y ′ ′ ε z ,...,z −2/3   µD (ℓ ∩ Bε/s (zj )) 6= ∅ ∀j = 1, . . . , k) lim  sj αD1′ k =  j ε→0 sj j=1 j=1 −2k/3 4k/3 + lim ε O(ε ) ε→0   k Y −2/3  z1 ,...,zk αD , sj = (A.24) j=1 which concludes the proof. Proof of Lemma 6.1 This proof is similar to that of Lemma 2.1. With the notation introduced in the proof of Lemma 2.1, we have that D Eε(k1 );δ (z1 ) . . . Eε(kn );δ (zn ) E D = n Y j=1 k Dj j h(x1 , . . . , xn ) . (A.25) xj ≡1 Considering the structure of (A.13), the definition of the differential operator Dj , and the fact that in (A.25) all derivatives ∂x∂ j are evaluated at xj = 1, it is clear that in the right hand side of (A.25) the only terms that survive are those for which the derivatives saturate the variables xj . Moreover, the structure of (A.13) implies that all terms containing subsets that are single points, i.e. Il = {zj }, disappear when applying Dj . These considerations – 35 – imply that the only non-zero terms are those corresponding to multisets M ∈ M. Note k also that, when ∂x∂ j is applied kj times to h(x1 , . . . , xn ), as prescribed by Dj j it produces a multiplicative factor kj ! for each j = 1, . . . , n. Therefore, if the vector k = (k1 , . . . , kn ) is such that M = ∅, we obtain hEε (z1 ) . . . Eε (zn )iD = lim δ→0 = n Y kj ! j=1  X λ P S∈M n Y j=1 k Dj j h(x1 , . . . , xn ) mM (S) M ∈M xj ≡1 Y S∈M mM (S) 1 µloop , (ℓ ∩ B (z ) = 6 ∅ ∀j ∈ I ) ε j S mM (S)! 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