Multiple Schramm-Loewner evolutions for conformal field theories with Lie algebra symmetries
We provide multiple Schramm-Loewner evolutions (SLEs) to describe the scaling limit of multiple interfaces in critical lattice models possessing Lie algebra symmetries. The critical behavior of the models is described by Wess-Zumino-Witten (WZW) models. Introducing a multiple Brownian motion on a Lie group as well as that on the real line, we construct the multiple SLE with additional Lie algebra symmetries. The connection between the resultant SLE and the WZW model can be understood via SLE martingales satisfied by the correlation functions in the WZW model. Due to interactions among SLE traces, these Brownian motions have drift terms which are determined by partition functions for the corresponding WZW model. As a concrete example, we apply the formula to the -WZW model. Utilizing the fusion rules in the model, we conjecture that there exists a one-to-one correspondence between the partition functions and the topologically inequivalent configurations of the SLE traces. Furthermore, solving the Knizhnik-Zamolodchikov equation, we exactly compute the probabilities of occurrence for certain configurations (i.e. crossing probabilities) of traces for the triple SLE.
Geometric aspects of critical phenomena are characterized by random fractals such as conformally invariant fluctuations of local order parameters. They have been extensively studied from various different points of view, especially in two dimensions (2D) where the conformal invariance imposes strong constraints on the structure of critical phenomena. Among them, the Schramm-Loewner evolutions (SLEs) , which directly describe geometric aspects of 2D critical phenomena through simple 1D Brownian motions, have brought a renewed interest in the theory of random fractals (see [2, 3, 4, 5, 6, 7, 8, 9] for reviews).
The SLE is a stochastic process defined in the upper half plane . Its evolution is described by the ordinary differential equation
where is a Brownian motion on , starting at the origin (i.e. ), and its expectation value and variance are given by and , respectively. Here is a diffusion coefficient which essentially characterizes the SLE process. The SLE (1.1) has a solution up to the explosion time , i.e. the first time when hits the singularity . Let be the hull at time (see Fig. 1 for a schematic view). Then () is an increasing family of hulls: for . Moreover with at (hydrodynamic normalization) is the unique conformal map uniformizing the complement of the hull in the upper half plane : (see Fig. 2). The image by defines the tip of the growing random curve. More precisely, it is expressed as .
The connection between the SLE (1.1) and conformal field theory (CFT) is well understood [10, 11, 12, 13, 14, 15, 16, 17, 5, 7]. Specifically, it can be accomplished by noticing that CFT correlation functions
are SLE martingales: , where and are boundary condition changing (bcc) operators with conformal weights , which are inserted at the points and , respectively (see [7, 17] or next section for details). Thus one can find the SLE corresponds to the minimal conformal field theory (, are coprime integers satisfying the condition ) where the central charge and the conformal weights of the primary fields are, respectively, given by [18, 19]
Then the diffusion constant and the conformal weight of the boundary field are, respectively, expressed as
Namely the boundary field is degenerate at level two, i.e. possesses a null field at level two.
An extension to the SLE connecting with conformal field theories with Lie algebra symmetries (i.e. WZW models [20, 21, 22]) has been achieved by adding the extra Brownian motion on a (semisimple) group manifold associated with a Lie algebra , where ’s () stand for any representation of the Lie algebra generators [23, 24] (see  for a very different approach). The evolution of this additional stochastic process is defined as
The combination of (1.1) and (1.5) defines a fractal curve living on the Lie group manifold. The SLE martingale , which should be satisfied by the CFT correlation function (1.2) (note that the bcc operators and the operator take their values on ), determines the relation of the SLE defined by (1.1) and (1.5) with the corresponding WZW model. Some extensions to the SLE with other additional symmetries have also been done in [26, 27, 28, 29, 30, 31].
In this paper we generalize the SLE for the system containing multiple random interfaces which possess additional Lie algebra symmetries, according to the theory developed in . Assuming that each SLE interface grows under an independent martingale in the infinitesimal time interval, we extend (1.1) together with (1.5) to the case for the system with multiple interfaces. The evolution (cf. (1.1) for the single case) characterizing the geometric aspect of the interfaces is described by a multiple Brownian motion on the real line, while the evolution (cf. (1.5) for the single case) expressing the algebraic aspect is described by a multiple Brownian motion on the Lie group . Both the Brownian motions, however, have drift terms describing the interaction among the SLE traces. Taking into account the SLE martingale, one finds that these drift terms are determined by the partition function in the corresponding WZW model. Moreover this partition function characterizes the configuration of the SLE traces. As an example, we apply our formula to the SLE for the -WZW models. Utilizing the fusion rules for , we conjecture that there is a one-to-one correspondence between the partition functions and the topologically inequivalent configurations of the SLE traces. Further, we exactly compute the probabilities of occurrence for certain configurations of traces (that correspond to crossing probabilities) for the triple SLE, which can be obtained by solving the Knizhnik-Zamolodchikov (KZ) equation .
The paper is organized as follows. In the subsequent section, we describe some basic notions required in this paper. In section 3, the multiple SLE for conformal field theory with Lie algebra symmetries is formulated. The drift terms of the driving Brownian motions are explicitly determined in section 4. A concrete application of the formula to the SLE for the -WZW model is given in section 5.
In this section, we describe several theoretical foundations required in subsequent sections. In the former part of this section, we introduce SLE martingales from the point of view of statistical mechanics (see  for details). SLE martingales are given by CFT correlation functions involving bcc operators, and hence they become a key to decipher the relation between SLE and CFT. In the latter part, general properties of correlation functions are explained in the case of the WZW model.
2.1 SLE martingales and conformal correlators
To formulate the SLEs correctly describing the behavior of 2D interfaces in critical systems, one must construct SLE martingales in terms of the corresponding statistical systems defined on . Let be the thermal average of an observable defined in , and be the thermal average under a given shape of configuration of (multiple) interfaces, where denotes the shape of configuration with its occurrence probability . Then the thermal average must be given by
where the average should be taken over all the possible configurations. The conditional expectation value is thus time independent (i.e. conserved in mean), and therefore it is a martingale. Here and in what follows, we denote it by .
At the critical point where the conformal invariance is expected in the system, the above observable can be described in terms of the CFT correlation functions. For the situation where the number of the interfaces under consideration is , it reads
where ’s denote the positions of the tips of the interfaces. Note that the CFT correlation functions are defined on the domain removing the hull , i.e. (see Fig. 3). The operators ’s inserted at the positions of the tips denote primary bcc operators with conformal weights : under a local conformal map , a primary field transforms as :
The denominator in (2.2) stands for the CFT partition function with a specific boundary condition fixed by the bcc operators .
Applying the conformal uniformizing map (written with the same symbol as that used for the single SLE (1.1)), we obtain
Here , and is the image of by the map . Note that the Jacobians coming from the conformal map on have been canceled in the numerators and the denominators111 The identity holds for a global conformal map , where the field transforms as . . Now SLE martingales are expressed as the CFT correlation functions on , where the bcc operators are inserted at the points (see Fig. 3).
To proceed further, let us consider the case when the operator is a product of an arbitrary number of primary fields at positions and with conformal weights . By construction, the uniformizing map can be analytically extended to the lower half plane: . Then, the doubling trick can be applied to the CFT correlation functions. The result reads
where we denote that ; ; ; , for . Here () stands for the holomorphic (antiholomorphic) part of the field .
In this paper, we analyze the SLE martingales (2.5) for the system that possesses additional Lie algebra symmetries. Namely we construct the multiple SLE for the WZW models which are one of the most fundamental CFTs with extra Lie algebra symmetries. In this case, the primary fields constructing the SLE martingale (2.5) possess internal degrees of freedom, such as “spin”.
2.2 WZW models and correlation functions
Let us introduce several properties for correlation functions of WZW primary fields to analyze the SLE martingales (2.5). The WZW model is a CFT described by a field taking values in a group manifold associated with a Lie algebra [19, 20, 21, 22]. The model is invariant under
where and denote arbitrary matrices valued in . This invariance gives rise to Noether currents and , which can be written as
where ’s stand for any matrix representation of the generator of , with commutation relations . The parameter is a positive integer referred to as the level. Hereafter we only consider the holomorphic components, as if there were no boundary (cf. (2.5)) . Let be a correlator of -valued fields. Then the infinitesimal transformation leads to the Ward identity
On the other hand, conformal aspects of the WZW model are described by the stress energy tensor :
where denotes the dual Coxeter number of . Note that means the “normal ordering” defined as
The infinitesimal conformal transformation leads to the Ward identity
The current and the stress energy tensor can be expanded in terms of the modes and , respectively. Namely
where and denote, respectively, the generators222Here we use the orthonormal basis in terms of the Killing form: . In that case, the structure constants can be written as , where is antisymmetric in all three indices. of the affine Lie algebra with level , and the generators of the Virasoro algebra:
The central charge of the Virasoro algebra is then given by
By construction (2.9), the Virasoro generators are not independent of the affine generators:
where the normal ordering means that the operator with larger index is placed at the rightmost position.
The primary fields in the WZW model are defined as the fields transforming covariantly with respect to the transformation (2.6): . Together with the conformal covariance (see (2.3)), these properties can be expressed in terms of the operator product expansions (OPE) via the Ward identities (2.11) and (2.8):
where the field takes values in the representation specified by the highest weight , and is the generator in that representation. Furthermore utilizing the field-state correspondence, i.e. , we can translate these properties into
Insertion of the relation (2.15) into the l.h.s. of the first equation in the above yields
In the last equality we used the explicit form of the eigenvalue of the quadratic Casimir. The quantity denotes the Weyl vector, i.e. the sum of the fundamental weights , where is the rank of . Comparing the r.h.s. in the first equation in (2.17) with the above, we find
Since the Virasoro generators are expressed in terms of affine generators as (2.15), the other arbitrary states are of the form with positive integers. Let and be the descendent fields corresponding to the states and , respectively. Combining the OPE (2.16) and the mode expansions (2.12), one finds that the correlation functions and , where , satisfy the following equations:
This constraint together with the global conformal invariance (or equivalently -invariance) fix the structure of the two- and three-points correlation functions.
To close this section, let us derive a crucial equation called Knizhnik-Zamolodchikov (KZ) equation  satisfied by the correlation functions of the WZW primary fields. The constraint stems from the fact that the Virasoro generators are not independent of the affine generators as in (2.15). For , we have . Then a null state is given by
3 Multiple SLEs for WZW models
Now we generalize the SLE (1.1) and (1.5) for the system containing multiple random interfaces with additional Lie algebra symmetries. In the infinitesimal time interval, we expect that each SLE interface grows under an independent martingale. Let us discuss the case where the number of the interfaces is . Then the uniformizing map with the hydrodynamic normalization may be of the form  (see also [33, 34, 35, 36] for other approaches).
where ’s mean infinitesimal time intervals satisfying the condition . The random processes (), which play a role as driving forces for the growth of interfaces, should be written as the Itô stochastic differential equations:
where is an -valued Brownian motion whose expectation value and variance are, respectively, given by
Namely prescribes the growth rate of each interface. The quantity denotes a drift term proportional to , which comes from interactions among interfaces, and will be determined later by the SLE martingale.
To extend (3.1) to evolutions with Lie algebra symmetries, we define a stochastic process () living on a Lie group manifold , where is written as
Here is an -valued Brownian motion with
and stands for a drift term proportional to , which will also be determined later. For , we must define
4 Drift terms and SLE martingales
The remaining problem is to determine the drift terms appearing in the driving forces (3.2) and in the Brownian motion (3.5). To achieve it, we must evaluate the variation of the SLE martingale involving WZW primary fields. For the WZW models, the SLE martingale is written as
where and (see section 2.1 for details). Note that the primary fields and constructing the correlation function take values in the representation specified by the highest weights and , respectively.
The drift terms are determined by the condition which makes to be a martingale. To simplify the notations we sometimes omit the index , and (e.g. , , , , etc.), if there is no ambiguity. Let , and be the denominator, numerator and Jacobian factor of (4.1), respectively. First we explicitly compute . A simple manipulation leads to
where in the last equality, we applied the geometric component of the SLE (3.1). Thus we obtain the variation of the Jacobian factor:
Next we shall calculate . The Itô derivative of the bulk fields is given by
The above two relations together with (4.3) give the form of . Explicitly it reads
where we define
and (resp. ) is given by subtracting the first sum depending on from the r.h.s. of the first (resp. second) equation in (4.7). Note that the operator (resp. ) characteristically appears in the correlation functions between a descendent field (resp. ) and some composite primary field (see (2.20)). To derive (4.6), we have also applied the identity , which comes from the translation invariance of the correlation functions (see the explanation below (2.23)).
In completely the same manner, can also be evaluated:
By construction, we must set () for the conformal weights of the bcc operators. Furthermore, due to the constraint between the conformal weight and the representation (2.19), all the bcc operators except for must take values in the representation specified by , or its conjugate (note that holds). The conformal weight for the bcc operator inserted at is determined by fusion rules. Thanks to this restriction, we can determine, in principle, the relation of the parameters , and the weight of the bcc operators by considering the case that there is only a single interface () as in [23, 24]. We see this in the next subsection.
4.1 Single case ()
For the single SLE (set and ), one sees due to the translation invariance of the two-point correlation functions (it can also be obtained directly from (4.8) by setting ). Therefore the SLE martingale gives a constraint to the correlation function , i.e. . From the relation (4.6) and the definitions (3.2) and (3.5), one easily see that this restriction leads to , and
Namely any drift terms do not show up in the driving forces (3.2) and (3.5), as expected. Moreover the constraint (4.9) indicates that the bcc operators must have null states at level 2. Namely is a martingale if and only if the bcc operators have the following null states at level 2:
Here and are, respectively, the Virasoro and affine generators (2.13). This is equivalent to the conditions and , which respectively lead to
This necessary and sufficient condition is rather involved. For some simple cases (e.g. ), however, we can directly solve the above equations by acting the generator . (Note that more elegant procedure utilizing the KZ-equation has been developed in .)
For later convenience, here we explicitly write down the results for the -WZW model. The central charge (2.14) and the conformal weight (2.19) of the bcc operator in the spin- representation are, respectively, written as
where denotes the fundamental weight, and we used . From the direct evaluation of (4.11) for case, one finds that the conditions in (4.11) are valid only for the case that the bcc operator carries spin-1/2 () . Then [23, 24]
For , and can not be specified, and only the relation is imposed. This case, however, corresponds to a CFT (cf. (1.4)), and therefore we shall set .
4.2 Multiple case ()
Now we identified the bcc operators . Namely they have null states at level 2 and must satisfy the condition (4.10). Utilizing the field-state correspondence, one finds that the first sums in (4.6) and (4.8) vanish.
Thus the drift term of the Itô derivative of the CFT correlation function
is explicitly given by