Schramm-Loewner evolution with Lie superalgebra symmetry
Shinji Koshida



We propose a generalization of Schramm-Loewner evolution (SLE) that has internal degrees of freedom
described by an affine Lie superalgebra.
We give a general formulation of SLE corresponding to representation theory of an affine Lie superalgebra
whose underlying finite dimensional Lie superalgebra is basic classical type,
and write down stochastic differential equations on internal degrees of freedom in case that
the corresponding affine Lie superalgebra is osp(1|2).
We also demonstrate computation of local martingales associated with the solution from a representation
of osp(1|2).


[
    Jeffrey Hatley
    ?; revised ?; accepted ?. - To be entered by editorial office
=================================================================





§ INTRODUCTION

Schramm-Loewner evolution (SLE) <cit.> is
a stochastic process (g_t)_t≥ 0 that takes its value in a space of formal power series z+C[[z^-1]]
and satisfies the following stochastic differential equation (SDE)

    d/dtg_t(z)=2/g_t(z)-B_t,   g_0(z)=z,

where B_t is the Brownian motion of covariance κ that starts from the origin.
The covariance κ of the Brownian motion parametrizes SLEs,
and we often denote the SLE specified by κ by SLE(κ).
While we regard the solution g_t(z) as a formal power series, it actually becomes a uniformizing map of a simply connected domain in the upper half plane.
More precisely, at each time t, g_t is almost surely a biholomorphic map g_t:H\ K_t→H
for a subset K_t⊂H, called a hull, in the upper half plane.
Moreover the hulls are increasing, i.e., t<s implies K_t⊂ K_s.
This means that SLE describes a stochastic evolution of hulls in the upper half plane.
If we look at this evolution of hulls more closely, we find that it is governed by an evolution of the tip of a slit in the upper half plane.
Thus SLE induces a probability measure on a space of curves in the upper half plane running from the origin to the infinity,
which is called the SLE(κ)-measure.
The SLE(κ)-measure is proved to describe an interface of clusters in several critical systems in two dimensions <cit.>,
which is why SLE is interesting to the field of statistical physics.
There are many literatures<cit.> on SLE from probability theoretical point of view.

Two dimensional critical systems are also studied by means of two dimensional conformal field theory (CFT) <cit.>,
which is distinguished from other quantum field theories due to its symmetry described by infinite dimensional Lie algebras.
A milestone was Cardy's formula <cit.>, later proved by Smirnov <cit.>, which predicts crossing probabilities in the two-dimensional
critical percolation from computation of partition functions for a boundary CFT.

Having two different approaches, SLE and CFT, to two-dimensional critical systems,
one expects that these two notions are related one another in some sense.
The connection between SLE and CFT has been studied under the name of SLE/CFT correspondence <cit.>.
One of significant achievements we follow in this paper is the group theoretical formulation of SLE by Bauer and Bernard <cit.>.
It regards SLE as a random process that is induced from one on an infinite dimensional Lie group that
governs formal coordinate transformations at infinity,
while the same infinite dimensional Lie group acts on a representation of the Virasoro algebra.
This point of view enables one to compute local martingales associated with SLE from a representation of the Virasoro algebra.

There are several directions of generalization of SLE along SLE/CFT correspondence,
including multiple SLE <cit.>, SLE for logarithmic CFT <cit.> and for the N=1 superconformal algebra <cit.>.
CFT with internal degrees of freedom is called Wess-Zumino-Witten (WZW) theory <cit.> and it is associated with
the representation theory of an affine Lie algebra.
Variants of SLE corresponding to WZW theory are also considered in literatures <cit.>.

WZW theory can be generalized to super WZW theory whose internal degrees of freedom are governed by a Lie supergroup,
and there are physics models <cit.> that are expected to be described by super WZW theory.
It is, however, not known whether there is a generalization of SLE that corresponds to super WZW theory,
and this is the issue we address in this paper.

The achievement of this paper is 
derivation of SDEs whose solution can be regarded as a generalization of SLE
corresponding to super WZW theory.
Let us sketch the strategy.
The idea is to generalize the group theoretical formulation of SLE corresponding to WZW theory presented in our previous works<cit.>,
in which the author considered a random process G_t on an infinite dimensional Lie group denoted by Aut_+O⋉ G_+(O),
where Aut_+O governs formal coordinate transformations at infinity,
G is a finite dimensional simple complex Lie group and G_+(O) is a subgroup of the loop group for G.
The analogous object of this infinite dimensional Lie group in super WZW theory seems to be obvious,
it is expected to be obtained by replacing the finite dimensional Lie group G by a finite dimensional Lie supergroup.
We will find, however, that the appropriate analogy forces us to add some Grassmann variables and take an infinite dimensional Lie group
denoted by Aut_+O⋉ G_+(O⊗⋀) as the target space of a random process G_t.
In the previous works<cit.>, the author found that applying the random process G_t on a highest weight vector |v_Λ⟩
of a certain highest weight representation of the corresponding affine Lie algebra,
the resulting random process G_t|v_Λ⟩ on (a formal completion of) the representation is a local martingale
with finely tuned parameters.
In the super case, since we extend the target group by Grassmann variables, correspondingly we have to take the tensor product of
a representation of an affine Lie superalgebra and a Grassmann algebra.
In Sect. <ref>, we will see that for a highest weight vector |v_Λ⟩, a potential local martingale is obtained by

    ∫ dηG_t|v_Λ⟩⊗ (1+η),

where ∫ dη is the Berezin integral over the Grassmann algebra.
After these observations, we can read off a set of SDEs from G_t whose solution is regarded as SLE with internal symmetry
described by an affine Lie superalgebra.
Table <ref> shows the fundamental objects in the group theoretical formulation of SLE.



This paper is organized as follows:
In Sect. <ref>, we recall the notion of affine Lie superalgebras and their representation theory.
In Sect. <ref>, we introduce the infinite dimensional Lie group denoted by Aut_+O⋉ G_+(O⊗⋀) above.
A random process on the infinite dimensional Lie group is proposed in Sect. <ref> as a generalization of the previous works <cit.>.
We focus our attention to the case that the underlying finite dimensional Lie superalgebra is osp(1|2) in Sect. <ref>
and write down SDEs explicitly.
We also show a computation of local martingales associated with a solution from a representation of osp(1|2).
In Sect. <ref>, we make some discussion on the result and directions of future research.



§ ACKNOWLEDGEMENTS

The author is grateful to K. Sakai for leading him to this field of research and suggesting this topic,
and to R. Sato for teaching him much on Lie superalgabras.
This work was supported by a Grant-in-Aid for JSPS Fellows (Grant No. 17J09658).




§ AFFINE LIE SUPERALGEBRAS AND REPRESENTATIONS


In this section, we recall the notion of affine Lie superalgebras and their representation theory.
We refer the readers to books<cit.> for a more detailed exposition.
Let g=g_0̅⊕g_1̅ be a finite dimensional Lie superalgebra of basic classical type except for A(1|1),
and let (·|·) be the nondegenerate invariant even supersymmetric bilinear form on g
normalized so that the square norm of the highest root is 2.
The exception of A(1|1) is not essential and is just for avoiding complexity of description
due to the fact that root spaces of A(1|1) are not one-dimensional.
With these data, the corresponding affine Lie superalgebra is defined by
g=g⊗C[ζ,ζ^-1]⊕CK
with parity

    g_0̅   =g_0̅⊗C[ζ,ζ^-1]⊕CK,   g_1̅   =g_1̅⊗C[ζ,ζ^-1],

and Lie brackets

    [X(m),Y(n)]   =[X,Y](m+n)+m(X|Y)δ_m+n,0K,     [K,g]   ={0},

where we denote X⊗ζ^m for X∈g and m∈Z by X(m).

Let M be a representation of g,
then M becomes a representation of g⊗C[ζ]⊕CK so that
g⊗ζ^0 acts naturally, g⊗C[ζ]ζ acts trivially and
K acts as multiplication by k∈C.
We extend this action to an action of the whole algebra by

    M_k=Ind_g⊗C[ζ]⊕CK^gM=U(g)⊗_U(g⊗C[ζ]⊕CK)M.

By the Poincaé-Birkhoff-Witt theorem, M_k is isomorphic to U(g⊗C[ζ^-1]ζ^-1)⊗ M
as a vector space or a U(g⊗C[ζ^-1]ζ^-1)-module.
The number k introduced above is called the level of the representation.

We fix a Cartan subalgebra h⊂g_0̅.
Let L(Λ) be a finite dimensional highest weight representation of g of highest weight Λ∈h^∗.
Although L(Λ) is irreducible, the induced representation L(Λ)_k is not necessarily irreducible,
and in that case we denote its irreducible quotient by L_k(Λ).

On a representation L_k(Λ) of g, we can define an action of the Virasoro algebra via the (super analogue of) Sugawara construction.
Let {X_a}_a=1^g be a basis of g, each element of which is homogeneous with respect to the Z_2-gradation,
and let {X^a}_a=1^g be its dual basis with respect to (·|·) that is characterized by (X_a|X^b)=δ_ab.
Then the operators

    L_n=1/2(k+h^∨)∑_a=1^g∑_k∈Z(-1)^p(X_a)X_a(n-k)X^a(k),  n∈Z

define an action of the Virasoro algebra of central charge c_k=ksdimg/k+h^∨.
Here h^∨ is the dual Coxeter number of g, p(X) denotes the Z_2-degree of X,
and the normal ordered product is defined by

    A(p)B(q)=
    	
    		A(p)B(q), 	    p≤ q,
    
    		(-1)^p(B)p(A)B(q)A(p),	    p>q.

The superdimension is defined by sdimg=g_0̅-g_1̅.
Notice that the normal ordered product is necessary to make operators well-defined on a representation space L_k(Λ)
of an affine Lie superalgebra.
In our application to SLE, we need a more convenient expression of the Virasoro generators.
Let {J_a}_a=1^g_0̅ be an orthonormal basis of the even part g_0̅ with respect to (·|·).
We also denote the set of odd roots by Δ^odd and the set of positive odd roots by Δ_+^odd.
For a basic classical Lie superalgebra g except for A(1|1), each root space is one dimensional.
Then we normalize a basis E_α of a root space g_α for α∈Δ^odd by
(E_α|E_-α)=1 for α∈Δ_+^odd.
The Virasoro generators in Eq.(<ref>) are also expressed as

    L_n=1/2(k+h^∨)∑_k∈Z[   ∑_a=1^g_0̅J_a(n-k)J_a(k)
       +∑_α∈Δ_+^odd(E_-α(n-k)E_α(k)-E_α(n-k)E_-α(k))].

As has been already noted, the exception of A(1|1) is just for simplification of description,
and we can treat A(1|1) in a parallel way noticing multiplicity in root spaces.




§ TARGET: INFINITE DIMENSIONAL LIE GROUP


In this section, we introduce an infinite dimensional Lie group, which will become the target space of our random process in Sect. <ref>.

Firstly, we define an infinite dimensional Lie group that governs formal coordinate transformations following the book<cit.>.
Let O=C[[z^-1]] be a complete topological C-algebra of formal power series
and D=SpecO be the formal disc attached to infinity.
Here we regard z as a coordinate at the origin.
Then we define an infinite dimensional Lie group AutO by one consisting of continuous automorphisms of O,
each element ρ of which is identified with a formal power series

    ρ(z)=a_1z+a_0+a_-1z^-1+⋯∈ zC[[z^-1]]

such that a_1≠ 0.
The group law of AutO is defined by (ρ∗μ)(z):=μ(ρ(z)) for ρ,μ∈AutO.
A significant subgroup Aut_+O is defined by

    Aut_+O={z+a_0+a_-1z^-1+⋯}

under the identification in Eq.(<ref>).
The Lie algebra of AutO consists of holomorphic vector fields at infinity and is realized as Der_0O=zC[[z^-1]]_z.
Correspondingly, the Lie algebra of Aut_+O is identified with Der_+O=C[[z^-1]]_z.
These Lie algebras are roughly regarded as “halfs" of the Virasoro algebra, and thus seem to act on a representation space of the Virasoro algebra,
but to verify this action, we have to “complete" the representation space.
In Sect. <ref>, we saw that on a highest weight representation L_k(Λ) of an affine Lie superalgebra
one can define an action of the Virasoro algebra via the Sugawara construction.
It is easily shown that this space is diagonalized by the action of L_0 so that L_k(Λ)=⊕_n∈Z_≥ 0L_k(Λ)_h_Λ+n,
where L_k(Λ)_h={v∈ L_k(Λ)|L_0v=hv} is an eigenspace of L_0.
The eigenvalue of the “top component" is given by h_Λ=(Λ|Λ+2ρ)/2(k+h^∨) with ρ being the Weyl vector.
Then the formal completion of L_k(Λ) is defined by L_k(Λ)=∏_n∈Z_≥ 0L_k(Λ)_h_Λ+n,
on which the Lie algebras Der_0O and Der_+O act by the assignment -z^n+1_z↦ L_n.
The action of the Lie algebra Der_+O on L_k(Λ) can be exponentiated to an action of Aut_+O.
If h_Λ is an integer, the action of Der_0O also can be exponentiated to give an action of AutO,
which is, however, not necessary in the present paper.
A detailed description of this action can be seen in the previous paper<cit.>.
We denote this action of Aut_+O by Q:Aut_+O→Aut(L_k(Λ)).

We next introduce a group of internal symmetry.
Since we are considering super WZW theory, the internal symmetry is described by a Lie supergroup,
which is abstractly defined as a group object in the category of supermanifolds <cit.>.
This definition is, however, not convenient for our purpose to construct SLE.
Instead we directly treat a set of super-algebra-valued points of a Lie supergroup.
To present an idea, we let k=k_0̅⊕k_1̅ be a Lie superalgebra
and ⋀=⋀[η_1,⋯,η_n] be a Grassmann algebra generated by n Grassmann variables η_1,⋯,η_n.
The Grassmann envelop <cit.> of k by ⋀ is the even part of k⊗⋀, or explicitly

    k(⋀)=(k_0̅⊗⋀_0̅)⊕ (k_1̅⊗⋀_1̅).

It is a (non-super) Lie algebra with Lie bracket defined by [X⊗η,Y⊗η]=[X,Y]⊗ηη.
We formally exponentiate each element of the Grassmann envelop k(⋀),
and denote a Lie group generated by such objects by K(⋀)=⟨e^X⊗η|$⟩.
Let us apply this construction to our case.

We takek=g⊗C[[ζ^-1]]ζ^-1and⋀=⋀[η_1,η_2].
Then we denote the resulting Lie groupK(⋀)byG_+(O⊗⋀).
The spirit of this notation comes from the observation that we can regard this Lie group as a set of (O⊗⋀)-valued points
of a Lie supergroupGwhose Lie superalgebra is the underlying oneg.
To describe an example of the group law forG_+(O⊗⋀), let us recall the Campbell-Baker-Hausdorff formula:

    exp(A)exp(B)=exp(A+B+1/2[A,B]+1/12([A,[A,B]]+[B,[B,A]])+⋯)

for not necessarily commutative symbolsAandB.
Let us apply this to the case ofA=X(ζ)⊗η_1,B=Y(ζ)⊗η_2forX(ζ), Y(ζ)∈g_1̅⊗C[[ζ^-1]]ζ^-1.
Then they are exponentiated to give elementsexp(X(ζ)⊗η_1)andexp(Y(ζ)⊗η_2)of the groupG_+(O⊗⋀)and the product among them is identified with

    exp(X(ζ)⊗η_1)exp(Y(ζ)⊗η_2) 
       =exp(X(ζ)⊗η_1+Y(ζ)⊗η_2+1/2[X(ζ),Y(ζ)]⊗η_1η_2),

due to the nilpotency of Grassmann variables.
Note that the right hand side defines an element of the groupG_+(O⊗⋀).

We have a natural action ofAut_+Oon the groupG_+(O⊗⋀)by transformation of the loop variableζso thatρ∈Aut_+Otransformsexp(X(ζ)⊗η)toexp(X(ρ(ζ))⊗η).
Thus we have a semi-direct product groupAut_+O⋉G_+(O⊗⋀),
which will be the target group of a random process considered in Sect. <ref>.
Intuitively, this semi-direct product can be considered as one generated by elements inAut_+OandG_+(O⊗⋀)with the following relations being imposed:

    ρexp(X(ζ)⊗η)ρ^-1=exp(X(ρ(ζ))⊗η)

forρ∈Aut_+Oandexp(X(ζ)⊗η)∈G_+(O⊗⋀).

Finally in this section, we see a representation of this infinite dimensional Lie groupAut_+O⋉G_+(O⊗⋀).
As we have already seen, the group of coordinate transformationsAut_+Oacts on the formal completionL_k(Λ)of a representation of the affine Lie superalgebra.
Since the group of internal symmetry involves Grassmann variables, it cannot act on the same space, but acts onL_k(Λ)⊗⋀in a natural way.
Moreover since the action of the Virasoro algebra onL_k(Λ)is defined by the Sugawara construction,
it is compatible with the semi-direct product structure, which verifies the action of the groupAut_+O⋉G_+(O⊗⋀)on a spaceL_k(Λ)⊗⋀.
For example, an element inAut_+Ois represented by an operator likeexp(∑_j<0v_jL_j)forv_j∈C,
which just acts onL_k(Λ).
As another example, an elementexp(X(ζ)⊗η_1)inG_+(O⊗⋀)forX(ζ)∈g_1̅⊗C[[ζ^-1]]ζ^-1acts on a vectorv⊗1∈L_k(Λ)⊗⋀as

    exp(X(ζ)⊗η_1)(v⊗ 1)=v⊗1 +(X(ζ)v)⊗η_1.

One can observe that the right hand side is well-defined inL_k(Λ)⊗⋀.




§ RANDOM PROCESS ON A SYMMETRY GROUP


In the previous Sect. <ref>, we defined an infinite dimensional Lie group denoted byAut_+O⋉G_+(O⊗⋀),
which governs coordinate transformation and internal symmetry.
In this section, we introduce a random process on the infinite dimensional Lie group that induces a generalization of SLE
that possesses internal symmetry described by an affine Lie superalgebra.
Our construction is a natural generalization of one presented in the previous works<cit.> for WZW theory.

We begin with a simple observation.
Let|0⟩be the highest weight vector ofL_k(0), which is called the vacuum.
From the concrete description of the Virasoro generators in Eq.(<ref>),
we have

    L_-2|0⟩=1/2(k+h^∨)(   ∑_a=1^g_0̅J_a(-1)^2
       +∑_α∈Δ_+^odd(E_-α(-1)E_α(-1)-E_α(-1)E_-α(-1)))|0⟩.

From this and the fact that the vacuum vector is translation invariant:L_-1|0⟩=0, we have

    [   -2L_-2+κ/2L_-1^2
       +τ/2(∑_a=1^g_0̅J_a(-1)^2
    	+∑_α∈Δ_+^odd(E_-α(-1)E_α(-1)-E_α(-1)E_-α(-1)))]|0⟩=0

for arbitraryκandτ=2/k+h^∨.
As a generalization of this, given levelk, we assume that we can choose positive numbersκandτso that

    [   -2L_-2+κ/2L_-1^2
       +τ/2(∑_a=1^g_0̅J_a(-1)^2
    	+∑_α∈Δ_+^odd(E_-α(-1)E_α(-1)-E_α(-1)E_-α(-1)))]|v_Λ⟩=0

where|v_Λ⟩is the highest weight vector ofL_k(Λ).
The positivity of the parametersκandτis required since
they later play roles of variances of Brownian motions.
This annihilating operator of a highest weight vector motivates us to consider a random processG_ton 
the infinite dimensional Lie groupAut_+O⋉G_+(O⊗⋀)that satisfies the following SDE:

    G_t^-1dG_t
    	=   [-2L_-2+κ/2L_-1^2
       +τ/2(∑_a=1^g_0̅J_a(-1)^2
       +η_1η_2∑_α∈Δ_+^odd(E_-α(-1)E_α(-1)-E_α(-1)E_-α(-1)))] dt 
       +L_-1dB_t^(0)+∑_a=1^g_0̅J_a(-1)dB_t^(a)
       +∑_α∈Δ_+^odd(η_1E_-α(-1)+η_2E_α(-1))dB_t^(α).

HereB_t^(0),{B_t^(a)}_a=1^g_0̅and{B_t^(α)}_α∈Δ_+^oddare
mutually independent Brownian motions whose covariance areκforB_t^(0)andτfor the others.
Note that for Grassmann variablesη_1andη_2, we have

    (η_1E_-α(-1)+η_2E_α(-1))^2=η_1η_2(E_-α(-1)E_α(-1)-E_α(-1)E_-α(-1)),

thus the SDE in Eq.(<ref>) is in the standard form of one for a random process on a Lie group.
We write the operator in front ofdtin Eq.(<ref>) asΞ(κ,τ),
which acts onL_k(Λ)⊗⋀.
Now we observe that

    ∫ dη_2dη_1Ξ(κ,τ)|v_Λ⟩⊗ (1+η_1η_2) 
       =[-2L_-2+κ/2L_-1^2
       +τ/2(∑_a=1^g_0̅J_a(-1)^2
    	+∑_α∈Δ_+^odd(E_-α(-1)E_α(-1)-E_α(-1)E_-α(-1)))]|v_Λ⟩
       =0,

where∫dη_2dη_1is the Berezin integral over the Grassmann algebra⋀and
the last equality is due to the assumption in Eq.(<ref>).
This implies that the random process

    ∫ dη_2dη_1G_t|v_Λ⟩⊗ (1+η_1η_2)

is a local martingale inL_k(Λ).

To see that the random processG_tonAut_+O⋉G_+(O⊗⋀)actually induces SLE with internal degrees of freedom, we factorize it so that

    G_t=Θ_tQ(ρ_t),

whereρ_tandΘ_tare random processes onAut_+OandG_+(O⊗⋀), respectively.
Under this ansatz, we can find with help of Eq.(<ref>) that
they satisfy the following SDEs:

    Q(ρ_t)^-1dQ(ρ_t)=   (-2L_-2+κ/2L_-1^2)dt +L_-1dB_t^(0), 
    Θ_t^-1dΘ_t=   τ/2[∑_a=1^g_0̅(J_a⊗ρ_t(ζ)^-1)^2
       +η_1η_2∑_α∈Δ_+^odd(E_-α⊗ρ_t(ζ)^-1E_α⊗ρ_t(ζ)^-1
       -E_α⊗ρ_t(ζ)^-1E_-α⊗ρ_t(ζ)^-1)]dt 
       +∑_a=1^g_0̅J_a⊗ρ_t(ζ)^-1dB_t^(a)
       +∑_α∈Δ_+^odd(η_1E_-α⊗ρ_t(ζ)^-1+η_2E_α⊗ρ_t(ζ)^-1) dB_t^(α).

Since the action of the groupAut_+OonL_k(Λ)is defined by assigning
an infinitesimal transformationℓ_n=-z^n+1_zto the Virasoro generatorL_n,
the first equation (<ref>) gives an SDE forρ_tso that

    ρ_t^-1dρ_t=(-2ℓ_-2+κ/2ℓ_-1^2)dt+ℓ_-1dB_t^(0).

When we apply both sides to the variablez, we find that the valueρ_t(z)satisfies the following SDE:

    dρ_t(z)=2/ρ_t(z)dt-dB_t^(0),

of which a solution is equivalent to SLE characterized by the parameterκby settingg_t(z)=ρ_t(z)+B_t^(0).
The SDE (<ref>) is also a generalization of one on internal degrees of freedom 
of SLE corresponding to WZW theory <cit.>.
Thus we can regard a solution of the set of SDEs (<ref>) and (<ref>) as
SLE with internal symmetry described by an affine Lie superalgebra,
and can compute local martingales associated with the solution from a representation-space-valued one in Eq.(<ref>).
Note that this structure is prototypical in SLE/CFT correspondence<cit.>.
Let us remark that the parametersκandτappearing in the annihilating operator (<ref>)
correspond to variances of Brownian motions,
and thus in particular,κcharacterizes the geometry (e.g. fractal dimension) of SLE traces.
For example, each SLE trace is almost surely a simple path ifκ≤4.
The other parameterτalso characterizes the behavior of the random process along the internal degrees of freedom,
while its geometric interpretation such as how the shape of sample paths in the internal space depends onτis unclear so far.




§ EXAMPLE

In this section, we construct a random processG_tintroduced in previous Sect.<ref>
in a concrete way in case thatg=osp(1|2),
and show a computation of local martingale associated with the solution.

The Lie superalgebraosp(1|2)is defined by

    osp(1|2)_0̅   =sl_2=CE⊕CH ⊕CF,    osp(1|2)_1̅   =Ce⊕Cf,

with the Lie superbracket being given by

    [H,e]   =e,     [H,f]   =-f,       
    
    	[E,f]   =-e,     [F,e]   =-f,     [E,e]   =[F,f]=0,
    
    	[e,e]   =E,     [f,f]   =-F,     [e,f]   =H.

The even part is isomorphic tosl_2as a Lie algebra and we omitted the Lie bracket among even elements.
The nondegenerate even invariant supersymmetric bilinear form(·|·)coincides with one ofsl_2on the even part
and is characterized by(e|f)=2.
We fix a Cartan subalgebrah=CH.
The set of roots is{±α,±α/2}, where±αare even roots and±α/2are odd roots.
Together with the normalization with respect to(·|·), we setE_α/2=1/√(2)eandE_-α/2=1/√(2)f.
Then the Sugawara construction in Eq.(<ref>) reads

    L_n=1/2(k+h^∨)∑_k∈Z[    1/2H(n-k)H(k)+E(n-k)F(k)+F(n-k)E(k)
       +1/2(f(n-k)e(k)-e(n-k)f(k))]


We shall construct the random processΘ_tinG_+(O⊗⋀)satisfying Eq.(<ref>) in a concrete way.
We take an orthonormal basis{X_a}_a=1^3of the even part by

    J_1   =1/√(2)H,     J_2   =1/√(2)(E+F),     J_3   =i/√(2)(E-F).

We set an ansatzΘ_t=Θ^1̅_tΘ^0̅_t, with

    Θ^0̅_t   =e^E⊗ x^E_t(ζ)e^H⊗ x^H_t(ζ)e^F⊗ x^F_t(ζ), 
    Θ^1̅_t   =e^η_1L^1_te^η_2L^2_te^η_1η_2L^12_t,

wherex^E_t(ζ), x^H_t(ζ), x^F_t(ζ)areC[[ζ^-1]]ζ^-1-valued random processes,L^1_t, L^2_tareg_1̅⊗C[[ζ^-1]]ζ^-1-valued random processes
andL^12_tis ag_0̅⊗C[[ζ^-1]]ζ^-1-valued random process.
Under this ansatz, the “even" partΘ^0̅_tsatisfies

    (Θ^0̅_t)^-1dΘ^0̅_t
    	=τ/2∑_a=1^g_0̅(J_a⊗ρ_t(ζ)^-1)^2+∑_a=1^g_0̅J_a⊗ρ_t(ζ)^-1dB_t^(a),

which coincides with the SDE on the internal degrees of freedom corresponding tosl_2investigated in the previous paper<cit.>.
There the random processesx^E_t(ζ), x^H_t(ζ), x^F_t(ζ)have been shown to satisfy<cit.>
    dx^E_t(ζ)=   -e^2x^H_t(ζ)/√(2)ρ_t(ζ)dB_t^(2)-ie^2x^H_t(ζ)/√(2)ρ_t(ζ)dB_t^(3), 
    
    	dx^H_t(ζ)=   -τ/2ρ_t(ζ)^2dt
    				-1/√(2)ρ_t(ζ)dB_t^(1)+x^F_t(ζ)/√(2)ρ_t(ζ)dB_t^(2)+ix^F_t(ζ)/√(2)ρ_t(ζ)dB_t^(3), 
    
    	dx^F_t(ζ)=   -√(2)x^F_t(ζ)/ρ_t(ζ)dB_t^(1)-1-x^F_t(ζ)^2/√(2)ρ_t(ζ)dB_t^(2)+i(1+x^F_t(ζ)^2)/√(2)ρ_t(ζ).


To describe the random processesL^1_t,L^2_tandL^12_t, we write them as

    L^1_t   =e⊗ x^1,e_t(ζ)+f⊗ x^1,f_t(ζ), 
    
    	L^2_t   =e⊗ x^2,e_t(ζ)+f⊗ x^2,f_t(ζ), 
    
    	L^12_t   =E⊗ x^12,E_t(ζ)+H⊗ x^12,H_t(ζ)+F⊗ x^12,F_t(ζ),

wherex^i,♮_t(ζ)fori=1,2,♮=e,fandx^12,♮_t(ζ)for♮=E,H,FareC[[ζ^-1]]ζ^-1-valued random processes, the SDEs for which are to be derived.
This task is carried out by substituting these ansatz into Eq.(<ref>), and as a result, we have

    dx^1,e_t(ζ)=   √(2)(e^x^H_t(ζ)+e^-x^H_t(ζ)x^E_t(ζ)x^F_t(ζ))ρ_t(ζ)^-1dB_t^(α/2), 
    
    	dx^1,f_t(ζ)=   -√(2)e^-x^H_t(ζ)x^F_t(ζ)ρ_t(ζ)^-1dB_t^(α/2),
    
    	dx^2,e_t(ζ)=   -√(2)e^-x^H_t(ζ)x^E_t(ζ)ρ_t(ζ)^-1dB_t^(α/2), 
    
    	dx^2,f_t(ζ)=   √(2)e^-x^H_t(ζ)ρ_t(ζ)^-1dB_t^(α/2), 
    
    	dx^12,E_t(ζ)=   τ(1+e^-2x^H_t(ζ)x^E_t(ζ)x^F_t(ζ))x^E_t(ζ)ρ_t(ζ)^-2dt 
       -√(2)x^2,e_t(ζ)(e^x^H_t(ζ)+e^-x^H_t(ζ)x^E_t(ζ)x^F_t(ζ))ρ_t(ζ)^-1dB_t^(α/2), 
    
    	dx^12,H_t(ζ)=   -τ/2(1+2e^-2x^H_t(ζ)x^E_t(ζ)x^F_t(ζ))ρ_t(ζ)^-2dt  
       +√(2)(x^2,e_t(ζ)e^-x^H_t(ζ)x^F_t(ζ)ρ_t(ζ)^-1
       -x^2,f_t(ζ)(e^x^H_t(ζ)+e^-x^H_t(ζ)x^E_t(ζ)x^F_t(ζ)ρ_t(ζ)^-1))dB_t^(α/2).
    
    	dx^12,F_t(ζ)=   -τ e^-2x^H_t(ζ)x^F_t(ζ)ρ_t(ζ)^-2dt  
       -√(2)x^2,f_t(ζ)e^-x^H_t(ζ)x^F_t(ζ)ρ_t(ζ)^-1dB_t^(α/2).


Let us investigate when the annihilating condition Eq.(<ref>) holds forg=osp(1|2).
We already know that the vacuum vector|0⟩gives an example, thus we seek one in a higher spin representation,
but we will see that there is no example of an annihilating operator of the form of Eq.(<ref>) on representations other than the vacuum representation
forg=osp(1|2).
We set

    ψ=(-2L_-1+κ/2L_-1^2+τ/2∑_a=1^g(-1)^p(X_a)X_a(-1)X^a(-1))|v_Λ⟩,

which is a candidate for a null vector.
The vectorψis a null vector if and only ifX(1)ψ=0andX(2)ψ=0for arbitraryX∈g.
These conditions are equivalent to

    ((τ k- τ h^∨-2)X(-1)+κ X(0)L_-1
       +τ∑_a=1^g(-1)^p(X_a)[X,X_a](0)X^a(-1))|v_Λ⟩=0, 
       (κ+τ h^∨-4)X(0)|v_Λ⟩=0.

Other than the vacuum representationΛ=0, the second condition Eq.(<ref>) imposesκ+τh^∨-4=0.
We can see that forg=osp(1|2), the first condition Eq.(<ref>) does not hold.
Indeed, forX=E, Eq.(<ref>) yields

    ((τ k-τ h^∨-2+τ(4+λ))E(-1)+τ/2H(-1))|v_Λ⟩=0,

whereλis defined byH(0)|v_Λ⟩=λ|v_Λ⟩,
but it does not hold unlessτ=0.

As we have seen, an annihilating operator of the form in Eq.(<ref>) exists for the vacuum representation.
Then for arbitraryκ>0andτ=2/k+3/2, the quantity in Eq.(<ref>) is aL_k(0)-valued local martingale.
We can obtain infinitely many local martingales by taking the inner product among the representation-space-valued local martingale and vectors inL_k(0).
Now we show an example: for an current fieldE(z)=∑_n∈ZE(n)z^-n-1,

    ∫ dη_2dη_1⟨0|E(z)G_t|0|⊗⟩(1+η_1η_2)  
       =k(1+x^1,f_t(z)x^2,e_t(z)+2x^12,H_t(z)) x^F_t(z) 
       -k(x^1,f(z)x^2,f_t(z)-x^12,F_t(z))(2 x^H_t(z)+2e^-2x^H_t(z)x^E_t(z) x^F_t(z)) 
       -2kx^1,f_t(z) x^2,f_t(z)+k x^12,F_t(z)

is a local martingale, while its interpretation is not clear so far.




§ DISCUSSION

In this paper, we consider a generalization of SLE that corresponds to super WZW theory.
Our present work is a generalization of previous works <cit.>,
which formulated SLE corresponding to WZW theory appearing in the literature<cit.> along the line of the group theoretical formulation<cit.>.
We began with a random process on an infinite dimensional Lie group that governs coordinate transformations and internal symmetry
and saw that it induces a set of SDEs whose solution is regarded as SLE with internal symmetry described by an affine Lie superalgebra.
Our construction also allows one to compute local martingales associated with the solution from a single representation-space-valued local martingale in Eq.(<ref>)
provided an annihilating condition Eq.(<ref>).

Let us make some discussion on further direction of research.
A significant problem is to find an example of an annihilating operator of a highest weight vector of the form in Eq.(<ref>).
It was shown in this paper that forg=osp(1|2)no example does exist except for one that annihilates the vacuum vector.
Then the problem is whether there is an example for another Lie superalgebra of basic classical type.
In seeking an example of an annihilating operator, one has to notice that
the form of an annihilator Eq.(<ref>) has room of generalization.
Indeed one may make the parameterτdepend on the summation index,
which is realized by using Brownian motions of different covariance along internal degrees of freedom
allowing inhomogeneity in an internal space.
One can also use a termX(-2)forX∈gin an annihilator.
Such a deformation is thought to be inevitable when we twist the Virasoro field by a derivative of a current field in the Sugawara construction
as is checked on the vacuum representation.
Note that current field is not primary with respect to a twisted Virasoro field,
which forces us to change the semi-direct product structure of the target group to formulate SLE.
In conclusion, we suggest that the most general form of an annihilator becomes

    -2L_-2+κ/2L_-1^2+c X(-2)+1/2∑_a=1^g(-1)^p(X_a)τ_aX_a(-1)X^a(-1)

whereκ, τ_a>0,c∈CandX∈g.
It is not obvious whether an annihilating operator of this form exists except for one on the vacuum representation
even for the case ofg=osp(1|2).

We may also look for an annihilating operator of a highest weight vector with degree higher than 2.
It is known that there is a degree 4 operator that annihilates the vacuum vector for the Yang-Lee singularity
and an SLE-type growth process corresponding to this annihilating operator was considered<cit.>.
An analogous annihilating operator of degree 4 was also found for affinesl_2andsl_3,
to which certain SLE-type growth processes with internal symmetry correspond<cit.>.
It may be interesting to find an annihilating operator of a highest weight vector with higher degree for an affine Lie superalgebra
in that it will broaden applicability of the formulation presented in this paper.
Note that assuming existence of an annihilating operator that is realized as at most quadratic in generators of an algebra,
it is possible to write down SDEs associated to the operator.
We also remark that in the case of an affine Lie algebra<cit.>,
an annihilating operator of degree 4 was found with the help of the Frenkel-Kac construction<cit.>,
of which the equivalent to an affine Lie superalgebra is not clear at least to us.

Other possibly interesting Lie superalgebras than basic classical ones we considered in this paper
include the affinization of supercommutative Lie superalgebras.
For such cases, the annihilator conditions will be investigated explicitly.
Construction of random process along internal degrees of freedom and computation of local martingales will also be carried out
for general rank of the underlying Lie superalgeras.
Such examples are super-analogues of SLE corresponding to Heisenberg algebras considered in our previous work<cit.>.

In a literature <cit.>, the author formulated multiple SLEs with internal symmetry corresponding to WZW theories.
It will be interesting to extend the result of the present paper to multiple SLEs.
Note that a formulation of multiple SLEs<cit.> relies on correlation functions,
and local martingales are obtained as correlation functions normalized by partition functions,
which are also correlation functions among boundary fields.
In the formulation of the present paper, we obtain local martingales after integrating Grassmann variables.
We will need the correct analogy of this prescription in the formulation via correlation functions
to extend our result to multiple SLEs.

One of the most important applications of SLE/CFT correspondence is a generalization of Cardy's formula.
Indeed, due to a local martingale that is computed from a representation of the Virasoro algebra,
Cardy's formula was rederived <cit.>.
Thus a generalization of SLE that corresponds to some CFT has potential to propose a generalization of Cardy's formula.
To seek a statistical model that is related to our generalized SLE,
together with a generalization of Cardy's formula, will be a significant direction of future research.
Related to this, we shall comment that a variant of SLE with internal symmetry can be considered as
a complexified Bessel process with internal degrees of freedom 
when we evaluate the coordinatezand the affine parameterζat any point in the upper half plane.
This observation allows one to identify each realization of SLE with internal symmetry
with an evolution of a curve in the upper half plane and coloring of the plane as illustrated in Fig.<ref>.
In this interpretation, the relevant Lie supergroup acts on the coloring reflecting the Lie superalgebra symmetry of the model.
This is expected to serve as a starting point of relating SLE with internal symmetry to physical systems.

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