The Brownian loop soup stress-energy tensor

The Brownian loop soup (BLS) is a conformally invariant statistical ensemble of random loops in two dimensions characterized by an intensity $\lambda>0$. Recently, we constructed families of operators in the BLS and showed that they transform as conformal primary operators. In this paper we provide an explicit expression for the BLS stress-energy tensor and compute its operator product expansion with other operators. Our results are consistent with the conformal Ward identities and our previous result that the central charge is $c = 2 \lambda$. In the case of domains with boundary we identify a boundary operator that has properties consistent with the boundary stress-energy tensor. We show that this operator generates local deformations of the boundary and that it is related to a boundary operator that induces a Brownian excursion starting or ending at its insertion point.


Introduction
The Brownian loop soup (BLS) [1] is an ideal gas of Brownian loops in two dimensions with a distribution chosen so that it is invariant under local conformal transformations. The distribution depends on a single parameter, the intensity λ > 0. The BLS is implicit in the work of Symanzik [2] on Euclidean quantum field theory, more precisely, in the representation of the partition function 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. Viewed this way, the BLS consists of 2λ Gaussian free fields (a generally non-integer number).
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. Inspired and partly motivated by these connections, in [3][4][5] we introduced and studied operators that compute properties of the BLS, discovering new families of conformal primary fields and analyzing the operator content of the emerging conformal field theory.
In this paper, we derive the bulk stress-energy tensor of the theory as well as the boundary stress-energy tensor in the upper half-plane. We also analyze the relation of the boundary stress-energy tensor to domain perturbations and boundary condition changing operators.

Preliminary definitions
The Brownian Loop Soup (BLS) is an ideal gas of planar loops. If A is a set of loops, the partition function of the BLS restricted to loops from A can be written as where λ > 0 is a constant and µ loop is a measure on planar loops called Brownian loop measure and defined as where A denotes area and µ br z,t is the complex Brownian bridge measure with starting point z and duration t. 1 Z A 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 [1,3].
As in [5], in this paper we will only be concerned with the of outer boundaries of Brownian loops. More precisely, given a planar loop γ in C, its outer boundary = (γ) is the boundary of the unique infinite component of C \ γ. Note that, for any planar loop γ, (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 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.
Given a simple loop , let¯ denote its interior, i.e., the unique, bounded, simply connected component of C \ . In other words, a point z belongs to¯ if disconnects z from infinity, in which case we write z ∈¯ . In [3], the authors studied the correlation functions of the layering operator 2 where σ 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 3 (essentially removing from the loop soup all loops with diameter smaller than δ), which produces a cutoff version V δ β of V β . The cutoff δ can be removed by rescaling V δ β by δ −2∆(β) , with ∆(β) = λ 10 (1 − cos β), and sending δ → 0. When δ → 0, the n-point correlation functions of δ −2∆(β) V δ β converge to conformally covariant quantities [3], showing that the limiting field is a scalar conformal primary field with real and positive conformal dimensions varying continuously as periodic functions of β, namely as This limiting field is further studied in [4], where its canonically normalized version is denoted by O β . 4 For a domain D with a boundary ∂D and a point ζ ∈ ∂D, the boundary field B δ ε (ζ) studied in this paper counts the number of loops with diameter at least δ passing within a short-distance ε of ζ. This is a boundary version of the edge field studied in [5] and discussed in Section 1.2 below. As in [5], the cutoffs δ, ε > 0 are necessary to keep B δ ε (ζ) from being infinite or identically zero. We show that they can be removed (i.e. sent to zero) by placing the field B δ ε inside n-point correlation functions with n ≥ 2 and renormalizing it by an appropriate power of ε.

The edge counting operator
For the reader's convenience, in this section we provide a brief introduction to the edge counting operator introduced in [5]. Some of the properties of the edge counting operator will be essential to the analysis carried out in this paper.
The edge counting operator E(z), for z in the interior of a domain D, is defined in [5] as the limit (1.6) Here N δ ε (z) counts the number of loops γ with diameter 5 diam(γ) = diam( ) larger than δ and whose "edge" (the outer boundary) comes ε−close to z. The angle brackets · D denote expectation with respect to the Brownian loop soup in D (of fixed intensity λ), µ loop D is the Brownian loop measure µ loop restricted to D, i.e. the unique (up to a multiplicative constant) conformally invariant measure on simple planar loops [6], and B ε (z) is the disk of radius ε centered at z.
The subtraction of the mean in (1.6) is needed because the mean is divergent in the limit δ → 0, due to the scale invariance of the BLS. With this, the field E ε is well defined in 4 By canonically normalized we mean that O β (z)O −β (z ) C = |z − z | −4∆(β) . 5 The diameter of a loop is the largest distance between any two point on the loop. the sense of correlation functions. More precisely, it is shown in [5] that, for any collection of points z 1 , . . . , z n ∈ D at distance greater than 2ε from each other, with n ≥ 2, the following limit exists: (1.7) The exponent 2/3 used in (1.5) is a consequence of the fact that the µ loop D -measure of "macroscopic" loops coming to distance ε of z in the interior of D tends to zero as ε 2/3 when ε → 0. This can be understood using a deep connection [6] between the Brownian loop measure µ loop D and a conformally invariant model of non-simple loops, called CLE 6 , which emerges from the scaling limit of critical two-dimensional percolation [7]. Using this connection, the exponent 2/3 is directly related to the probability of a three-arm event in the bulk in two-dimensional percolation [8,9].
The field E is only formally defined by (1.5) because it cannot be evaluated pointwise, as is customary in (Euclidean) quantum field theory. However, it is shown in [5] that E has well-defined n-point functions and behaves like a conformal primary field with dimensions (1/3, 1/3). The constantĉ and the multiplicative factorĉ √ λ are chosen so that the field is canonically normalized, that is, (1.8) More generally, the n-point functions of the edge operator can be expressed as where S = S(z 1 , . . . , z n ) denotes the set of all partitions of {z 1 , . . . , z n } such that each element S l of (S 1 , . . . , S r ) ∈ S has cardinality |S l | ≥ 2 and, for any subset S l = {z j 1 , . . . , z j k } of {z 1 , . . . , z n }, It is shown in [5] that, if D ⊆ C is either the complex plane C or the upper-half plane H or any domain conformally equivalent to H, for any collection of distinct points z 1 , . . . , z k ∈ D with k ≥ 2, the above limit exists and has the following property. If D is a domain conformally equivalent to D and f : D → D is a conformal map, then (1.11) This, combined with (1.9), implies that (1.12) 1.3 Summary of the main results and structure of the paper Section 2. We provide integral expressions of the bulk stress-energy tensor T that can be used to compute certain correlation functions involving T . We verify that the Ward identities involving these correlation functions are satisfied, confirming the validity of our expressions for T . Section 3. We study a boundary version of the edge counting operator E (1.5), namely where ζ ∈ ∂D. We show that, when D is the upper half-plane H and x 1 , x 2 ∈ R, (1.14) We identify a new operator, formally whose behavior is consistent with the role of boundary stress-energy tensor. We derive part of the operator product expansion of T × T (3.35) and check the Ward identity (3.16).
Section 4. We link the operator T to local deformations in the boundary of the domain and to the insertion of a pair of operators that generate a Brownian excursion in the domain (with the insertion points as starting and ending points of the excursion) and appear to behave like boundary condition changing operators.

The bulk stress-energy tensor
In this section we show how to express the bulk stress-energy tensor T in the BLS in terms of the edge counting operator E and in terms of the vertex layering operator O. From these expressions, we compute two examples of correlation functions of primary fields with T and show that they satisfy the conformal Ward identities, confirming the validity of our expressions for T . The strategy of this section is inspired by [10,11]. In particular, the techniques of [11], where the authors derive the central charge of the O(n → 0) model, have been used in [5] to determine the central charge of the BLS.

Identification
The holomorphic and anti-holomorphic components of the stress-energy tensor can be understood as the generators of conformal transformations. The holomorphic component of the stress-energy tensor in a two-dimensional CFT can generally be understood as the level-2 descendant of the identity operator The anti-holomorphic componentT is equivalently given byL −2 . Their conformal dimensions are (2, 0) and (0, 2) for the holomorphic and anti-holomorphic component, respectively.
Our strategy is to identify the stress-energy tensor starting from the operator product expansion (OPE) of two primary operators A 1 and A 2 (see Section 6.6.3 of [12]) where the outer sum runs over all primary operators P, and the inner sum is over all collections of indices (the descendant levels) are numerical coefficients fully determined by the Virasoro algebra (they depend on the central charge and the conformal dimensions of the operators involved), and C P 12 are three-point function coefficients of the theory. If we choose A 1 and A 2 so that (2.2) contains the identity operator then the stress tensor will appear as a descendent. In particular, the identity is contained in the OPE since the integral in (2.4) singles out the term of the expansion in the right hand side of (2.3) that corresponds to the stress-energy tensor. As shown in Figure 2, the loops that contribute to the integral in (2.4) are those that come close to z and that intersect the circle of radius ρ centered at z. This provides a geometric interpretation for the stress-energy tensor.T is obtained by replacing e −2iθ with e 2iθ . As a consequence of conformal invariance, correlation functions involving the stressenergy tensor in the bulk obey the conformal Ward identities where A j denotes a generic primary operator with conformal dimension ∆ j . In the next section, we test (2.4) using (2.5).

Correlation functions
We now use the full-plane mixed four-point function .) The four-point function was computed in [5] and can be written as and corresponds to Eq. (52) of [13] with x = z 12 z 34 z 13 z 24 . We can then apply (2.4) to (2.6) using (2.7) by expandingα z+ρe iθ z 1 |z 2 ;C in powers of ρ. Observe that the terms independent of θ (e.g.,α z,z+ρe iθ C = ρ −3/4 ) do not contribute because of the integral 2π 0 dθ e −2iθ , while the terms containing powers of ρ greater than 2/3 give zero in the limit ρ → 0. This gives (2.9) Using the fact that ρ −2/3 Z twist (z 1 , z 2 ; z, z + ρe iθ ) is analytic and expanding it in powers of ρ, performing the integral, and sending ρ → 0, we obtain In a very similar manner, one can obtain T (z)E(z 3 )E(z 4 ) C using the fact that the identity block is also contained in the OPE of O β (z + ) × O −β (z). The term containing T (z) appears at order O( −2∆(b)+2¯ −2∆(b) ) (the anti-holomorphic componentT appears at order O( −2∆(β)¯ −2∆(β)+2 )), but in this case there are other contributions to the OPE at the same order. The relevant operators are conformal primaries with dimensions (1, 0) and (2, 0), and we denote them J and W , respectively. They are shown in Figure 1b, which is obtained using results from [4]. More precisely, in the case of two layering vertex operators, (2.2) gives can be computed 6 as explained in [4], which gives (2.12) Based on these considerations, one can write (2.13) Applying (2.13), (2.6) and (2.7), and using the fact that JEE C = W EE C = 0, as 6 In [4] we found that C J = 0 for generic β1, β2. In the special case β1 + β2 = 0 mod 2π the coefficient is given in (2.12).
can be seen from Figure 1a, we obtain This is the correct form of the conformal Ward identity in the bulk, i.e. (2.5), with N = 2, A j = E and ∆ j = 1/3.
We point out that using (2.13) in which has been derived in [4], and performing a similar analysis that takes into account the contributions from J and W from (2.11), leads again to (2.10), as expected.

(2.16)
This is indeed the correct form of the two-point function of stress-energy tensors in a theory with central charge c = 2λ.

The boundary stress-energy tensor
In this section we introduce a boundary version of the edge operator discussed in Section 1.2 and study some of its properties. In particular, we show that it is essentially the boundary stress-energy tensor (up to a factor of √ λ).

The boundary edge counting operator and its correlation functions
The same analysis briefly discussed in the Section 1.2 can be carried out for a point ζ on the boundary ∂D of a domain D, where D is either a finite domain with a sufficiently smooth boundary or the upper half plane H. In this case, using the connection with critical percolation and CLE 6 , the relevant probability is that of a three-arm event at the boundary, which gives an exponent 2 instead of 2/3. This is consistent with the well-known fact that in a renormalizable quantum field theory operators at the boundary require a different renormalization from those in the bulk [14] (producing, in general, a different set of conformal dimensions) and leads to the following (formal) definition of boundary edge counting operator whereĉ b is chosen so that B is canonically normalized when D = H, that is, for points x 1 , x 2 on the real line, At the end of Section 3.2, we will see thatĉ b = 1. Definition (3.1) is convenient because it uses the edge counting operator analyzed in [5], but we could have equivalently defined B by counting the loops crossing an infinitesimal slit attached to the boundary of the domain. This alternative definition, reminiscent of the definitions of the operators discussed in [10,15], would not change our conclusions.
The existence of the n-point functions of the field B follows from the same arguments used in [5]. First of all, we note that the proof that the limit in (1.7) exists extends trivially to the case in which (some of) the points z j are moved to the boundary of the domain (see Lemma A.1 of [5]). This means that exists for all x 1 , . . . , x n ∈ R with n ≥ 2, which allows us to state the following result.
Proposition 3.1. For any collection of distinct points x 1 , . . . , x n on the real line, with n ≥ 2, the following limit exists: where S = S(x 1 , . . . , x n ) denotes the set of all partitions of {x 1 , . . . , x n } such that each element S l of (S 1 , . . . , S r ) ∈ S has cardinality |S l | ≥ 2 and, for any subset Proof. The proposition is analogous to Theorem 2.3 of [5] and its proof follows directly from the arguments used in the proofs of Lemmas 2.1 and 2.2 of [5].
The two-point function B(x 1 )B(x 2 ) H can be calculated exactly, as follows. For any s > 0, the scale invariance of the BLS implies that (3.7) Since B(x 1 )B(x 2 ) H can only depend on |x 1 − x 2 |, this means that and we can chooseĉ b so that the proportionality constant is 1. (As mentioned above, it will turn out thatĉ b = 1.)

Identification of the boundary stress-energy tensor
We now present the main results of this section, which allow us to identify √ λB with the boundary stress-energy tensor of the theory. To this end, we recall the formal definition of layering vertex operator in the upper-half plane [4] O β (z) := lim where ∆(β) =∆(β) = λ 10 (1 − cos β), σ is a symmetric (±1)-valued random variable associated to loop , and L δ is a BLS from which all loops of diameter smaller than δ have been removed, z ∈¯ means that z is in the interior of the domain bounded by (it is disconnected from infinity by );ĉ H is chosen so that (3.10) Equation (3.10) is obtained in [4], while the existence and conformal covariance properties of more general n-point functions are proved in [3]. Essentially, when a layering operator is inserted at z, each loop such that z ∈¯ picks up a phase e iβσ . We note that the layering vertex operators are scalar operators.
Proposition 3.2. For any choice of points z 1 , z 2 ∈ H, we have that

12)
and 2 F 1 is the hypergeometric function.
Proof. Using (3.6) and (3.7) from [5] applied to the upper half-plane and with z 3 = 0 and the exponent 2/3 replaced by 2, we obtain an analog of (3.8) therein, namely (3.13) In order to evaluate the limit in (3.13), we use the fact that where µ bub H is the measure on Brownian bubbles in H, pinned at the origin, defined in [1], and the equality follows from Theorem 1 of [1].
As explained in Remark 7.1 of [16] and in [17], µ bub H is closely related to the SLE 8/3  Since µ SLE 8/3 is obtained by multiplying the law of an SLE 8/3 ε-bubble (i.e., an SLE 8/3 excursion in H starting at the origin and ending at ε) by ε −2 and passing to the limit, the proof is concluded by using (3.2) and (3.3) from Proposition 1 of [17] to evaluate the three terms in the last line of the last equation.
The importance of Proposition 3.2 lies in the fact that it allows us to identify the boundary stress-energy tensor of our CFT. The ingredients for this identification are • the Ward identity at the presence of a boundary (see Eq. (24) of [14]) where T denotes the boundary stress-energy tensor with x ∈ R, • the expression (see Eq. (2.2) of [4]) where 3 F 2 is the hypergeometric function. 7 Plugging (3.17) into the right hand side of (3.16), sending x → 0, and comparing with (3.11) shows that the result equals . By translation invariance, this means that, on the real line, we can make the identification T = √ λ c b B. According to (3.2), for any two points x 1 , x 2 ∈ R, T would then have the two-point function but since the stress-energy tensor T satisfying (3.16) is assumed to be normalized in such a way that where c denotes the central charge (see, e.g., Eq. (8) of [14]), from the relation c = 2λ valid for the BLS with intensity λ, we conclude thatĉ b = 1 and determine the boundary stress-energy tensor to be In other words, the discussion above leads to the following result.

Operator product expansion
In this section we discuss the OPE of T × T . In order to do that, we first express the mixed four-point function B(x 1 )B(x 2 )Õ β (z 1 )Õ −β (z 2 ) H in terms of µ loop H -weights. For that purpose, it is useful to introduce some additional notation. We define and where the existence of both limits follows from the proof of Lemma 2.2 of [5].
Proposition 3.4. For any choice of points z 1 , z 2 ∈ H and x 1 , x 2 ∈ R, with the notation of Proposition 3.2, we have that Proof. The result follows from carrying out the analysis leading to (5.6) in [5] to the upper half-plane, with the exponents 2/3 replaced by 2. The last line of (3.23) follows from Proposition 3.1, which allows us to make the identification B( s > 0, the scale invariance of the BLS implies that where 1 denotes the identity operator. The identification T = √ λ B, made in the previous section, gives the OPE (3.29) which is consistent with T being the stress-energy tensor of a CFT with central charge c = 2λ.
In order to guess the next term in the OPE of T × T , note that .

(3.30)
Observing that is finite, (3.26) suggests that, as ε → 0, converges to a function f (x 1 , x 2 ) such that lim |x 1 −x 2 |→0 |x 1 − x 2 | 2 f (x 1 , x 2 ) = const. If we make this assumption, we are led to the conclusion that which implies the OPEs and The general form of the OPE of the stress energy tensor with itself (see, e.g., Eq. (8) of [14]) is consistent with (3.35) and the fact that the central charge of the theory is 2λ, and implies that const = 2.
one hand and between P H;0,x (Γ 0,x separates z 1 , z 2 ) . (4.7) This leads to two interesting conclusions. On the one hand, using (3.2) and (3.3) from Proposition 1 of [17], and lettingΓ 0,x denote the interior of the loop formed by the concatenation of Γ 0,x with [0, x], we have where G and σ are the same as in Proposition 3.2. Plugging this expression into (4.7) and using Proposition 3.2 (remembering thatĉ b = 1 and T = √ λB) gives showing the appearance of the boundary stress-energy tensor.
On the other hand, it follows from the analysis in Section VI of [13] that 9 with ∆ = 5/8 and c = 2λ is consistent with an OPE for primary operators with scaling dimension 5/8, with structure constant C T φ 1,2 ,φ 1,2 = 1 for all λ, see (2.2). It was argued in [4] that the n → 0 limit of the O(n) model, which describes single self-avoiding loops, is closely related to the the BLS in the λ → 0 limit. We saw additional evidence for this identification in this section, where we identified the boundary changing operator Φ with the O(n) model operator φ 1,2 and presented evidence that this identification holds for finite λ.
It is curious to note that a similar identification can be made for T . Its scaling dimension, ∆ = 2, coincides with that of a "2-leg" boundary operator in the O(0) model, namely ∆(κ) = (8−κ)/κ for κ = 8/3. Geometrically, T counts the number of loops coming close to its insertion point on the boundary (see Figure 3). This geometric interpretation coincides with that of the 2-leg operator, which acts as a sink or source of two self-avoiding random walks at its insertion point that avoid each other and the boundary of the domain. Moreover, although the ensemble of loops of the BLS contains an infinite number of loops, if one considers the set of points x on the real line such that the semi-disk of radius ε centered at x intersects a "macroscopic" loop, its fractal dimension goes to 0 as ε → 0, regardless of the value of λ. This allows us to conjecture that T can be identified with the 2-leg operator for all λ. Additionally, it gives an explanation for why the scaling dimension of B (and T ) is independent of λ.
The discussion in this section can in principle be extended to operators other thañ O β and to the case in which multiple pairs of boundary condition changing operators are inserted at the boundary.

Conclusions and future perspectives
For conformal field theories associated to the BLS, we have provided integral expressions of the bulk stress-energy tensor T based on the OPE of E × E and of O β × O −β . These expressions can be used to compute certain correlation functions involving T itself, and we have done this for two specific examples, verifying that the corresponding Ward identities are satisfied. It would be interesting to extend this analysis to J, the dimension (1, 0) current that also occurs in these OPEs (see Figure 1, position (3,0)). Furthermore, we have identified a new operator T whose properties are consistent with those of a boundary stress-energy tensor. The operator T is essentially a boundary version of the edge counting operator introduced in [5], and is reminiscent of the boundary stressenergy tensor discussed in [10,15]. A full verification that T is the boundary stress-energy tensor, for example by verifying all Ward identities involving T , is an interesting open problem.
We have shown that the insertion of T is linked to local deformations of the boundary of the domain. This is in the spirit of the discussion at the end of Section 6 of [6], where it is argued that the measure on simple loops induced by µ loop is well suited to the study of local deformations of the complex structure of Riemann surfaces. It would indeed be interesting to study the CFTs arising from the BLS on Riemann surfaces, in particular in the case of the torus.
The operator T can also be linked to the insertion of a pair of boundary operators that generate a Brownian excursion between the insertion points. These operators appear to behave like boundary condition changing operators with scaling dimension ∆ = 5/8, consistent with ∆(κ) = 6−κ 2κ for κ = 8/3. The appearance of the value κ = 8/3 is not surprising since the outer boundary of a Brownian loop is locally distributed like an SLE 8/3 curve. It would also be interesting to further explore the properties of these putative boundary condition changing operators.