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C ( S [ Introduction h n admdistributions random and = ] 00,6J7 82C22. 60J67, 60D05, Z 1 N akdbudr ons(MBPs) points boundary marked 1 + |∇ h | 2 , H rvnb h Dyson the by driven H ihbudr condition boundary with e on D tion-valued nue by induced e hog the through ned CP is BCCPs iiylaw bility ,H D, .When ). a conformal nal N ndary ormal distribution manners. o nceptually, ≥ eral of ) 3. s M13]. tiating und 2 MAKOTOKATORIANDSHINJIKOSHIDA the GFF. Such defined random metric formulates the Liouville quantum gravity (LQG) [DS11], a model of two-dimensional quantum gravity, the original idea of which was given by Polyakov [Pol81a,Pol81b]. In the other one, we consider a flow line of the vector field eih/χ, where we identify the two dimensional domain with one in the complex plane. Compared to the LQG, this type of geometry is often called the imaginary geometry [MS16a,MS16b,MS16c,MS17]. Owing to the recent studies [Dub09, MS16a, She16], it has been clarified that both of these models of random geometry are closely related to the theory of Schramm–Loewner evolution (SLE). SLE was first introduced in [Sch00] as a candidate for a stochastic model that describes a cluster interface in a two-dimensional critical lattice model at a scaling limit, relying on the stochastic analogue of the Loewner theory [L¨ow23, KSS68]. After its introduction, many authors studied SLE extensively revealing its prop- erties [RS05] and the relation to lattice models [Smi01, CDCH+14]. Useful expo- sitions of SLE can be found in [Wer04, Law09b]. The relation between SLE and the LQG or the imaginary geometry is formu- lated by considering couplings of SLEs and GFFs. Roughly speaking, a GFF perturbed by an appropriate harmonic function is shown to be stationary under the operation of cutting the domain along an SLE path. Based on this fact, it can be argued [Dub09,MS16a,She16] that an instance of GFF determines an SLE path compatibly to the LQG or the imaginary geometry. Then it seems natural to ask how the coupling between an SLE and a GFF can be extended to the case of multiple SLE and how it can be interpreted in the context of the LQG or the imaginary geometry. In the present paper, we address this issue by considering the conformal welding problem for a quantum surface with multiple marked boundary points and the flow line problem for an imaginary surface with multiple boundary condition changing points. Consequently, we will find that the multiple SLE driven by a time change of the Dyson model plays an analogous role as the one that an SLE plays in the original works [She16,MS16a]. The present paper is organized as follows: In the following Sect. 2, we make preliminaries on GFF and multiple SLE that are needed in the remaining part of the paper. In Sect. 3, with a short review on the classical matter on the LQG and the imaginary geometry, we formulate our problems, the conformal welding problem and the flow line problem. We also present the main results Theorems 3.10 and 3.12 there, which are proved in the succeeding Sects. 4 and 5, respectively. In the final Sect. 6, we discuss related topics and future directions. In Appendix A, we give a construction of the spaces γ and γ,N+1 of pre-quantum surfaces and ones with marked boundary points as orbifoldsS S and investigate their structures in detail. In Appendix B, we summarize the approach in [BBK05] that formulated multiple SLE in relation to CFT and the analogous way of defining the reverse flow of multiple SLE. CONFORMAL WELDING PROBLEM 3
Acknowledgments. The authors would like to thank Takuya Murayama, Ikkei Hotta and Daiya Yamauchi for useful discussion. The authors are also grate- ful to Sebastian Schleissinger for his valuable comments on the first version of the manuscript and Kalle Kyt¨ol¨afor discussions on multiple SLEs. The present work was partially supported by the Research Institute for Mathematical Sciences (RIMS), a Joint Usage/Research Center located in Kyoto University. The authors thank Naotaka Kajino, Takashi Kumagai, and Daisuke Shiraishi for organizing the very fruitful workshop, “RIMS Research Project: Gaussian Free Fields and Related Topics”, held in 18-21 September 2018 at RIMS. MK was supported by the Grant-in-Aid for Scientific Research (C) (No.26400405), (C) (No.19K03674), (B) (No.18H01124), and (S) (No.16H06338) of Japan Society for the Promotion of Science (JSPS). SK was supported by the Grant-in-Aid for JSPS Fellows (No. 17J09658, No. 19J01279).
2. Preliminaries This section is devoted to preliminaries on GFF and multiple SLE. In Subsect. 2.1, we review the notion of GFF both under Dirichlet boundary condition and free boundary condition, and in Subsect. 2.2, we fix a framework of multiple SLE starting from a Loewner chain for a family of multiple slit domains. 2.1. Gaussian free field. Dirichlet boundary case. Let D ( C be a simply connected domain and let Dir W (D) be the Hilbert space completion of the space C0∞(D) of smooth functions supported in D with respect to the Dirichlet inner product 1 (2.1) (f,g) = ( f)(z) ( g)(z)dµ(z), f,g C0∞(D), ∇ 2π ∇ · ∇ ∈ ZD √ 1 where µ is the Lebesgue measure on D C; dµ(z) = 2− dzdz. A GFF with ⊂ Dir Dir Dirichlet boundary condition on D is defined as an isometry HD : W (D) 2 Dir Dir Dir Dir Dir Dir → L (ΩD , D , PD ) where (ΩD , D , PD ) is a probability space such that each Dir F Dir FDir (HD , ρ) := HD (ρ), ρ W (D), is a mean-zero Gaussian random vari- able [She07].∇ One can construct∈ such an isometry relying on the Bochner-Minlos theorem that is an analogue of Bochner’s theorem applicable to the case when the source Hilbert space is infinite dimensional [Hid80, Chapter 3]. It is also Dir known that, in this construction, the sigma field D is generated by the image Dir Dir Dir F of W (D) under HD , i.e., HD is full. Let us denote by C0∞(D)′ the space of distributions with test functions in C0∞(D) with respect to the topology of W Dir(D). From a construction of a Dirichlet boundary GFF, for each ω ΩDir, the assignment ∈ D Dir Dir HD (ω): C0∞(D) C, ρ (HD , ρ) (ω) → 7→ ∇ Dir is continuous so that HD (ω) C0∞(D)′. Therefore a GFF under Dirichlet bound- ∈ Dir Dir ary condition can be regarded as a random distribution H : Ω C∞(D)′. D D → 0 4 MAKOTOKATORIANDSHINJIKOSHIDA
Dir Dir 1 For ρ C0∞(D), we define (HD , ρ) := 2π(HD , ( ∆)− ρ) . Then we have ∈ − ∇ Dir Dir Dir E[(HD , ρ1)(HD , ρ2)] = ρ1(z)GD (z,w)ρ2(w)d(µ µ)(z,w), D D ⊗ Z × Dir ρ1, ρ2 C0∞(D), where GD (z,w) is the Green function on D with Dirichlet bound- ∈ Dir ary condition. For example, when D is the upper half plane H, GH (z,w) = log z w + log z w . When we regard the GFF as a random distribution on − | − | | − | Dir D, it is reasonable to symbolically express (H , ρ) for ρ C∞(D) as D ∈ 0 Dir Dir (HD , ρ)= HD (z)ρ(z)dµ(z). ZD We understand the object HDir( )= HDir( ,ω), ω ΩDir, in this sense. D · D · ∈ D Free boundary case. A GFF with free boundary condition is defined in a similar way, but on a different space of test functions. Let C∞(D) be the space of smooth functions on D whose gradients are compactly supported∇ in D and whose total- masses are zero: fdµ = 0, f C∞(D). Then we denote the Hilbert space com- ∈ ∇ pletion of C∞(D) with respect to the Dirichlet inner product defined by the same formula as (2.1)∇ byR W Fr(D). A GFF with free boundary condition is defined as an Fr Fr 2 Fr Fr Fr Fr Fr Fr isometry HD : W (D) L (ΩD , D , PD ), where (ΩD , D , PD ) is a probability → F F Fr Fr space. This can also be regarded as a random distribution HD : ΩD C∞(D)′, → ∇ where C∞(D)′ denote the space of distributions with test functions in C∞(D). A significant∇ difference of the free boundary GFF from the Dirichlet boundary∇ one Fr lies in the fact that the GFF with free boundary condition HD is regarded as a random distribution on D modulo additive constants. Since, in this paper, we only treat simply connected domains, it suffices to present the GFF with free boundary condition on H (see [She16,Ber16,QW18]). Let ρ1, ρ2 C∞(H) be test functions. Fr ∈ ∇ Then (HH , ρi), i =1, 2 are mean-zero Gaussian variables with covariance
Fr Fr Fr E[(HH , ρ1)(HH , ρ2)] = ρ1(z)GH (z,w)ρ2(w)d(µ µ)(z,w), H H ⊗ Z × where GFr(z,w)= log z w log z w . H − | − | − | − | 2.2. Multiple SLE. We begin with recalling the result in [RS17] that dealt with the multiple-slit version of the Loewner theory [L¨ow23, KSS68]. Let η(i) : (0, ) H, i = 1,...,N, be non-intersectiong curves in H anchored on R: (i)∞ → (i) η N (i) η0 := limt 0 η (t) R, i = 1,...,N. We set Ht := H η (0, t] . Then, → i=1 ∈ \ η at each time t (0, ), there is a unique conformal mappingS gt : Ht H under the hydrodynamic∈ normalization∞ (Fig. 2.1): → η C(Ht ) 2 g (z)= z + + O( z − ), z . t z | | →∞ η η The constant C(Ht ) is called the half plane capacity of Ht . Notice that, by η changing the parametrization of the curves, we can take C(Ht )=2Nt. We call such a parametrization of curves a standard parametrization. CONFORMAL WELDING PROBLEM 5
Theorem 2.1 (Alternative expression of [RS17, Theorem 1.1]). Let η(i) : (0, ) H, i =1,...,N, be non-intersectiong curves in H anchored on R with a standard∞ → parametrization. There exists a unique set of continuous driving functions Xt = (1) (N) N (Xt ,...,Xt ) R , t [0, ), such that the family of conformal mappings g : Hη H solves∈ the multiple∈ ∞ Loewner equation: t t → d N 2 (2.2) g (z)= , t 0, g (z)= z H, dt t (i) ≥ 0 ∈ i=1 gt(z) Xt X − i.e., gt t 0 is the Loewner chain driven by Xt : t 0 . Moreover, the driving { } ≥ (i) { ≥ } functions are determined by Xt = limz η(i)(t),z Hη gt(z), i =1,...,N. → ∈ t Hη H t η(2) η(N) η(1) gt
(1) (2) (N) X(1) X(2) X(N) η0 η0 η0 t t t
η Figure 2.1. Uniformization of the domain Ht by the Loewner chain. By virtue of the above theorem, an N-tuple of random slits in H anchored on R N is converted to a stochastic process Xt R : t 0 . We assume that it solves a system of stochastic differential equations{ ∈ (SDEs)≥ } (2.3) dX(i) = √κdB(i) + F (i)(X )dt, i =1,...,N, t 0, t t t ≥ (i) N where Bt : t 0 i=1 are mutually independent one-dimensional standard Brow- { ≥ } (i) N nian motions and F (x) i=1 are suitable functions of x = (x1,...,xN ) not ex- plicitly dependent on{ t, i.e.,} X : t 0 are of the Markovian type (see [IW89, Eq. { t ≥ } (2.11) in Chapter IV]). Then the solution gt t 0 is just the multiple SLE con- sidered in [BBK05, Gra07]. While, in [BBK05],{ } ≥ the set of driving processes was derived from a single auxiliary function in relation to CFT as is summarized in Appendix B, here we do not assume any CFT origin of the multiple SLE. We write SLEκ if we need to specify the parameter κ.
3. Formulation of problems and results In this section, we formulate the problems we address in this paper, the confor- mal welding problem and the flow line problem, and describe the results. These two problems are, in fact, very similar in the sense that the only difference is 6 MAKOTOKATORIANDSHINJIKOSHIDA
the boundary condition of a GFF and solved analogously by means of multiple SLE, but have different origins in the random geometry. Thus we present them separately.
3.1. Conformal welding problem.
Liouville quantum gravity. GFF plays a relevant role in constructing the LQG [DS11] (see also [RV17,DKRV16,HRV18]). Following the original idea by Polyakov γHFr(z) [Pol81a,Pol81b], it is expected that the object e D dµ(z) is the desired random area measure on D, where γ (0, 2). This does not work, however, because each Fr Fr∈ Fr realization of HD , h( ) = HD ( ,ω), ω ΩD , is not a function but a distribution on D, thus its exponentiation· has· to be∈ verified in some sense. This difficulty is overcome by a certain regularization. Let us fix a realization h C∞(D)′ and let ∈ ∇ hǫ(z) be the mean value of h on the circle ∂Bǫ(z) of radius ǫ centered at z D assuming that the distance from z to the boundary is larger than ǫ. Then∈ the required area measure is obtained by
2 dµγ(z) := lim ǫγ /2eγhǫ(z)dµ(z), z D. h ǫ 0 → ∈ In a similar way, one can construct a linear measure on the boundary
2 dνγ (x) := lim ǫγ /4eγhǫ(x)/2dν(x), x ∂D, h ǫ 0 → ∈
where ν is the Lebesgue measure on the boundary, while, in this case, hǫ(x) is the average over the semi-circle centered at x ∂D of radius ǫ included in D. Let D˜ (C be another simply connected∈ domain, and ψ : D˜ D be a conformal equivalence. Then an area measure is induced on D˜ by pulling→ back the measure γ ˜ γ µh on D. That is, for a measurable set A D, its area is computed as ψ∗µh(A) := γ ⊂ µh(ψ(A)). When we closely look at the pulled-back measure, we find that it can also be realized as µγ built from a distribution h˜ on D˜. Indeed, by changing h˜ integration variables, the area of A D˜ with respect to the pulled-back measure γ ⊂ ψ∗µh becomes
2 2 γ /2 γhǫ(z) γ /2 γ(h ψ)ǫ(w) 2 lim ǫ e dµ(z) = lim ( ψ′(w) ǫ) e ◦ ψ′(w) dµ(w), ǫ 0 ǫ 0 | | | | → Zψ(A) → ZA dψ where ψ′(w) = dw (w). Note that, in the right hand side, in which the integral is taken over A D˜, the regularization parameter ǫ has to be rescaled by ψ′(w) . ⊂ | | This implies that if we introduce a distribution on D˜ by h˜ = h ψ + Q log ψ′ γ2 2 γ ◦ | | with the parametrization Q = ( 2 + 2)/γ = γ + 2 , then the corresponding area measure µγ agrees with the pulled-back measure ψ µγ. It can be verified that h˜ ∗ h the boundary measure also behaves correctly: νγ(I)= νγ (ψ(I)) for a measurable h˜ h I ∂D˜. Motivated by this, we make the following definition: ⊂ CONFORMAL WELDING PROBLEM 7
Definition 3.1. Let γ (0, 2). Pairs (D , h ) and (D , h ) of simply connected ∈ 1 1 2 2 domains Di ( C and distributions hi C∞(Di)′, i = 1, 2, are said to be γ- equivalent if there exists a conformal equivalence∈ ∇ ψ : D D such that 1 → 2 h = h ψ + Q log ψ′ 1 2 ◦ | | 2 γ holds, where Q = γ + 2 . Definition 3.2 (Pre-quantum surface). Let γ (0, 2). A γ-pre-quantum surface is an γ-equivalence class of pairs (D, h) of simply∈ connected domains D (C and distributions h C∞(D)′. We denote the γ-equivalence class of (D, h) by [D, h]γ and write the collection∈ ∇ of γ-pre-quantum surfaces as
γ := [D, h]γ D (C, h C∞(D)′ . S { | ∈ ∇ } We will give the construction of γ as an orbifold in Appendix A. The quantization of γ-pre-quantumS surfaces is carried out by randomizing them: Definition 3.3 (Quantum surface). Let γ (0, 2). A γ-quantum surface is a probability measure on . Equivalently, a γ∈-quantum surface is a collection of Sγ pairs (D,HD), where D (C is a simply connected domain and HD is a C∞(D)′- valued random field subject to the condition that, for all simply connected domains∇ D1,D2 (C and conformal equivalences ψ : D1 D2, the equality in probability law → (law) (3.1) H = H ψ + Q log ψ′ D1 D2 ◦ | | 2 γ holds, where Q = γ + 2 . We write this collection as [D,HD]γ.
Remark 3.4. If one has a pair (D,HD) of a simply connected domain D (C and a C∞(D)′-valued random field HD, then it uniquely extends to a γ-quantum surface ∇ [D,HD]γ . Example 3.5. A relevant example of a γ-quantum surface arises from the free Fr boundary GFF. A pair (D,HD ) of a simply connected domain D ( C and the Fr free boundary GFF HD on D defines a quantum surface. Indeed, the assignment Fr Fr ω [D,HD (ω)]γ gives an γ-valued random field on ΩD and induces a probability measure7→ on . S Sγ Quantum surface with marked boundary points. In order to formulate and address the conformal welding problem, we define a quantum surface with marked boundary points, which has also been introduced by an earlier literature [DMS14]. It is a refined version of a quantum surface decorated by data of boundary points. Let D (C be a simply connected domain. For N Z 0, we define the configura- tion space for N + 1 ordered boundary points of D∈as≥
< N+1 x =x , i=j Conf (∂D)= (x ,...,x ) (∂D) i6 j 6 , N+1 1 N+1 ∈ |aligned counterclockwise n o where ∂D = D D is the boundary of D. \ 8 MAKOTOKATORIANDSHINJIKOSHIDA
(1) (N+1) Definition 3.6. Let γ (0, 2), N Z 0 and let (D, hD, (xD ,...,xD )) be a triple consisting of ∈ ∈ ≥ A simply connected domain D (C. • A distribution hD C∞(D)′. • ∈ ∇ (1) (N+1) < An (N+1)-tuple of ordered boundary points (xD ,...,xD ) ConfN+1(∂D). • (1) (N+1) (1) (N+1) ∈ Triples (D1, hD1 , (xD1 ,...,xD1 )) and (D2, hD2 , (xD2 ,...,xD2 )) are said to be γ-equivalent if there exists a conformal equivalence ψ : D1 D2 such that (i) (i) → ψ(xD1 ) = xD2 , i = 1,...,N + 1 and the following identity among distributions holds:
hD1 = hD2 ψ + Q log ψ′ , 2 γ ◦ | | where Q = γ + 2 . Definition 3.7 (Pre-quantum surface with marked boundary points). Let γ ∈ (0, 2), N Z 0. A γ-pre-quantum surface with N +1 marked boundary points ∈ ≥ (1) (N+1) is a γ-equivalence class of triples (D, hD, (xD ,...,xD )) of simply connected domains D ( C, distributions hD C∞(D)′, and ordered boundary points (1) (N+1) < ∈ ∇ (1) (N+1) (xD ,...,xD ) ConfN+1(∂D). We denote the γ-equivalence class of (D, hD, (xD ,...,xD )) (1) ∈ (N+1) as [D, hD, (xD ,...,xD )]γ and write the collection of γ-pre-quantum surfaces with N + 1 marked boundary points as . Sγ,N+1 We will give the construction of as an orbifold in Appendix A. Sγ,N+1 Definition 3.8 (Quantum surface with marked boundary points). Let γ (0, 2) ∈ and N Z 0. A γ-quantum surface with N + 1 marked boundary points is ∈ ≥ a probability measure on γ,N+1. Equivalently, a γ-quantum surface with N + S (1) (N+1) 1 marked boundary points is a collection of triples (D,HD, (XD ,...,XD )), (1) (N+1) where D ( C is a simply connected domain and (HD, (XD ,...,XD )) is a < (C∞(D)′ ConfN+1(∂D))-valued random field subject to the condition that, for all simply∇ connected× domains D ,D (C and conformal equivalences ψ : D D , 1 2 1 → 2 (1) (N+1) (law) 1 (1) 1 (N+1) (HD , (X ,...,X )) = (HD ψ +Q log ψ′ , (ψ− (X ),...,ψ− (X ))) 1 D1 D1 2 ◦ | | D2 D2 2 γ (1) (N+1) holds, where Q = γ + 2 . We write this collection as [D,HD, (XD ,...,XD )]γ . The relevant example of γ-quantum surfaces with marked boundary points in the present paper is of the standard type defined as follows: We consider the following space:
^Rot < (3.2) γ,N+1(H) := C∞(H)′ ConfN (R), S ∇ × < N where ConfN (R) := (x1,...,xN ) R x1 < < xN . Another presentation of this space contained{ in Appendix A∈ will| motivate··· the superscript} “Rot”. Suppose that a probability space Conf< (R), , P is given. It is equivalent to a random N FN N N-point configuration X = (X ,...,X ) on R defined by X : Conf< (R) R; 1 N i N (x ,...,x ) x , i = 1,...,N. Let α = (α , ,α ) be an N-tuple of→ real 1 N 7→ i 1 ··· N CONFORMAL WELDING PROBLEM 9
< numbers. For given N points x =(x1,...,xN ) ConfN (R), we define a function on H ∈ N x (3.3) u ,α(z)= α log z x . H i | − i| i=1 X Here z is the standard coordinate on H embedded in C. Then the assignment X x (3.4) (H ,α, X) : ΩFr Conf< (R) (ω, x) (HFr(ω)+ u ,α, x) ^Rot (H) H H × N ∋ 7→ H H ∈ Sγ,N+1 ^Rot Fr Fr < gives an γ,N+1(H)-valued random field on ΩH,N := ΩH ConfN (R) equipped with S Fr × the product probability measure PH PN . We denote the probability measure on induced along the surjection ⊗ Sγ,N+1 ^Rot (3.5) π∞ : ։ , (h, x) [H, h, (x, )] γ,N+1 Sγ,N+1 Sγ,N+1 7→ ∞ γ X,α by Pγ,N+1. N Definition 3.9 (Standard type). Let α = (α1,...,αN ) R and let X = < ∈ (X1,...,XN ) be a ConfN (R)-valued random variable. The associated probability X measure P ,α on constructed above is called a γ-quantum surface with γ,N+1 Sγ,N+1 N + 1 marked boundary points of the (X,α)-standard type. Equivalently, a γ- quantum surface with N +1 marked boundary points of the (X,α)-standard type X X X is a collection of the form [H,H ,α, (X, )] , where H ,α = HFr + u ,α. H ∞ γ H H H Fr By definition, the random variable X and the random field HH are independent. X,α They are combined when constructing the probability measure Pγ,N+1 giving rise X,α Fr X,α to a C∞(H)′-valued random field HH = HH + uH depending on the random ∇ < boundary points X taking values in ConfN (R). x,α Note that the function uH defined by Eq. (3.3) is harmonic with logarithmic singularities at the (N + 1)-st point as well as at points xi R, i = 1,...,N. ∞ ∈ x,α Indeed, introducing a coordinate w =1/z vanishing at , we see that uH (1/w) N ∞ ∼ i= αi log w as w 0. − In the rest| of| the present→ paper, we will abbreviate a γ-(pre-)quantum surface withP N + 1 marked boundary points as a γ-(pre-)QS-(N + 1)-MBPs. Now we propose the conformal welding problem for a γ-QS-(N + 1)-MBPs of the (X,α)-standard type with N Z 1 as follows: ∈ ≥ X,α Problem 1. Let Pγ,N+1, γ (0, 2), N Z 1, be a γ-QS-(N + 1)-MBPs of the ∈ ∈ ≥ ^Rot X,α (X,α)-standard type. Then for each realization (h, x) γ,N+1(H)of(HH , X), find a member ∈ S N H η(i)(0, 1], h,˜ η(1)(1),...η(N)(1), ∞ i=1 ! / [ (i) in the equivalence class [H, h, (x, )]γ such that η : (0, 1] H, i =1,...,N are non-intersecting slits in H satisfying∞ the following conditions:→ 10 MAKOTOKATORIANDSHINJIKOSHIDA
(i) (i) 1. The slits are anchored on the real axis: η0 := limt 0 η (t) R, i = 1,...,N. → ∈ 2. The slits are seams: νγ (η(i)(0, t] )= νγ (η(i)(0, t] ), t (0, 1], i =1,...,N. h˜ L h˜ R ∈ (i) (i) Here η (0, t]L (resp. η (0, t]R) is the boundary segment lying on the left (resp. right) of the slit η(i)(0, t]. To explain its geometric meaning, suppose that Problem 1 was solved. For x,α X,α N (i) each realization HH (ω) of HH , let ψ : H H i=1 η (0, 1] be a conformal + → \ + equivalence. Write zi− and zi for the points on the real axis such that zi− < xi < zi (i) S and ψ(zi±)= η0 , i =1,...,N. Then the conformal mapping ψ glues the intervals + γ [z−, xi] and [xi, z ] by means of the boundary measure ν x,α (see Fig. 3.1). i i HH (ω)
x,α (H, H H (ω)) (1) η η(2)
ψ η(N)
x1 x2 xN (1) (2) (N) η0 η0 η0
Figure 3.1. The conformal welding problem.
(i) N As a solution to Problem 1, a statistical ensemble of slits η i=1 is obtained. Then, we would like to ask the probability law for the resulting{ curves:} Problem 2. Determine the probability law for the slits η(i) N . { }i=1 Note that the anchor points of these curves on R are also random variables: (i) (i) η0 = η0 (ω, x), i = 1,...,N, while this seems to prevent us from capturing the ensemble of slits. Thus we also set the following problem as a sub-problem of Problem 2: Problem 3. Find a γ-QS-(N + 1)-MBPs of the (X,α)-standard type such that the anchor points η(i) N are deterministic. { 0 }i=1 In the case of N = 1, this problem reduces to the one addressed by Sheffield < [She16]. The space γ,N+1 is fibered over Aut(H) ConfN+1(∂H) by forgetting the S \ < distributions (see Appendix A). Owing to the fact that the space Aut(H) Conf2 (∂H)= [0, ] consists of a single element, Problem 3 becomes trivial in this\ case. We {could∞ say} that Sheffield [She16] addressed Problems 1 and 2 for a γ-QS-2-MBP of CONFORMAL WELDING PROBLEM 11
2 the (X,α)-standard type with α1 = γ and that he found a one-parameter family of solutions to Problem 1; the reverse flow of an SLE gives the required conformal equivalences. Consequently, the resulting curve obeys the probability law of the one for the SLE of parameter κ = γ2 as the solution to Problem 2. Result. To present a result, let us recall the Dyson model. In the usual conven- tion, the Dyson model of parameter β > 0 is the system of SDEs on XDβ = { t (XDβ(1),...,XDβ(N)) RN : t 0 [Dys62, Kat15]: t t ∈ ≥ } N Dβ(i) (i) β 1 (3.6) dXt = dBt + dt, t 0, i =1,...,N. 2 Dβ(i) Dβ(j) ≥ j=1 Xt Xt Xj=i − 6 It is known that when β 1, the Dyson model with an arbitrary finite number ≥ of particles N Z 2 has a strong and pathwise unique non-colliding solution for general initial conditions∈ ≥ [RS93,CL97,GM13,GM14]. The non-colliding condition β 1 for the Dyson model corresponds to κ (0, 8] through the relation β =8/κ. See≥ Section 6.2 for more detail on the relations∈ among parameters. We obtain a solution to Problems 1 to 3 for γ-QSs-(N + 1)-MBPs as follows:
Theorem 3.10. Let γ = √κ (0, 2) and N Z 1. Suppose that Xt = ≥ (1) (N) ∈ ∈ D8/κ { (Xt ,...,Xt ): t 0 is a time change of the Dyson model Xκt : t 0 starting at a deterministic≥ } initial state X = x Conf< (R).{ Then, at≥ each} 0 ∈ N XT ,α time T (0, ), the conformal welding problem for H,HH , (XT , ) with ∈ ∞ ∞ γ 2 2 h i (α1,...,αN )=( γ ,..., γ ) is solved as follows: 1. The solution of the Loewner equation (2.2) driven by the time change of the D8/κ Dyson model Xκt : 0 t T gives a solution to Problem 1. In other 1 { η ≤ ≤ } words, g− : H H is the desired conformal equivalence. T → T 2. The probability law for resulting slits η(i) N is the one for the multiple { }i=1 SLEκ. This gives a solution to Problem 2. 3. Problem 3 is answered positively with η0 = x a.s. This theorem will be proved in Sect. 4. 3.2. Flow line problem. Another topic in random geometry that stems from GFF is the imaginary geometry [MS16a, MS16b, MS16c, MS17], which sees the √ 1H/χ flow line of the vector field e − , where H is a C0∞(D)′-valued random field and χ> 0 is a parameter. Let us temporarily suppose that h were a smooth function √ 1h/χ on D (C. Then e − is a smooth vector field and its flow line η : [0, ) D starting at x D is defined as the solution of the ordinary differential equation∞ → 0 ∈ dη(t) √ 1h(η(t))/χ = e − , t 0, η(0) = x D. dt ≥ 0 ∈ For another simply connected domain D˜ and a conformal equivalence ψ : D˜ D, 1 → we can consider the pull-backη ˜ = ψ− η of the flow line η by ψ. Thenη ˜ satisfies ◦ 12 MAKOTOKATORIANDSHINJIKOSHIDA
the following differential equation:
dη˜(t) 1 √ 1(h ψ χ arg ψ′)(˜η(t))/χ = e − ◦ − , t 0. dt ψ (˜η(t)) ≥ | ′ | t When we adopt a time changeη ˜f (t)=˜η(f(t)) with f(t)= 0 ψ′(˜η(s)) ds, we see that | | R dη˜f (t) √ 1(h ψ χ arg ψ′)(˜η (t))/χ = e − ◦ − f , t 0. dt ≥ Since a time reparametrization does not change the whole curve, we can say that the domains D with a smooth function h and D˜ with h ψ χ arg ψ′ are equivalent √ 1h/χ √ 1(h◦ψ −χ arg ψ′)/χ as long as flow lines of vector fields e − and e − ◦ − are concerned. Interestingly, this equivalence relation also makes sense even when we work with a C0∞(D)′-valued random field H instead of a smooth function [MS16a, MS16b, MS16c, MS17]. Definition 3.11. A χ-imaginary surface (χ-IS) is an equivalence class of pairs (D,H) of simply connected domains D (C and C0∞(D)′-valued random fields H under the equivalence relation 1 (D,H) (D,˜ H˜ ) := (ψ− (D),H ψ χ arg ψ′), ∼χ ◦ − where D˜ (C is another simply connected domain and ψ : D˜ D is a conformal equivalence. → It has been shown in [MS16a, She16] that for a χ-IS (H,HDir 2 arg( )) with H − √κ · χ = 2 √κ , the flow line starting at the origin can be identified with the curve √κ − 2 for the SLEκ. Note that the arg function is a harmonic function on H with the boundary value changes at the origin and infinity by π. In this sense, above imaginary surface is said to have boundary condition changing points (BCCPs) at the origin and infinity. We settle the flow line problem for a χ-IS with more BCCPs than two as follows: < Let x =(x1,...,xN ) ConfN (R) and β1,...,βN be real numbers. The C0∞(H)′- valued random field ∈ N x H ,β,I = HDir β arg( x ) H H − i · − i i=1 X on H defines a χ-IS with BCCPs (x1,...,xN , ), whose boundary value has dis- ∞ N continuity at xi by πβi, i =1,...,N and at by π i=1 βi. Here I stands for imaginary. ∞ − P Problem 4. Determine the probability law for multiple flow lines of a χ-IS with x,β,I N + 1 BCCPs (χ-IS-(N + 1)-BCCPs) HH starting at boundary points x < ∈ ConfN (R). This problem is solved as follows: CONFORMAL WELDING PROBLEM 13
−1∗ η H η H (H, g T H|Hη ) ( T , H |H ) ( , H ) T (2) T η(1) η η(N) gT
(1) (2) (N) X X XN N XT XT XT 1 2 X1 X2 X
Figure 4.1. Cutting operation by multiple SLE.
Theorem 3.12. Let κ (0, 4] and N Z 1. The flow line problem for the ∈ x ∈ ≥ χ-IS-(N + 1)-BCCPs (H,H ,β,I) with χ = 2 √κ is solved for any boundary H √κ − 2 points x = (x ,...,x ) Conf< (R) if β = (β ,...,β ) is given by β = 2 , 1 N ∈ N 1 N i √κ i = 1,...,N. The probability law of the flow lines is given by the multiple SLEκ D driven by the time change of the Dyson model X 8/κ : t 0 . { κt ≥ } This theorem will be proved in Sect. 5. In the case when κ = 4, a related problem was addressed in [PW19], where the connection probability of the multiple level lines for the GFF was investigated.
4. Proof of Theorem 3.10 4.1. Reduction of Theorem 3.10. To solve Problems 1 to 3, we define the cutting operation on ^Rot (H)-valued random fields associated with a multiple Sγ,N+1 SLE. Let (H, X) be an ^Rot (H)-valued random field (see the right picture in Sγ,N+1 Fig. 4.1). Suppose that we have an N-tuple of interacting stochastic processes (1) (N) N (i) Xt = (Xt ,...,Xt ) R : t 0 with the initial conditions X0 = Xi, {i = 1,...,N, which is conditionally∈ ≥ independent} of (H, X). We assume that it (i) N determines random slits η i=1 through the correspondence in Theorem 2.1. Then these slits are anchored{ } at the initial marked boundary points X, i.e., (i) limt 0 η (t) = Xi, i =1,...,N, a.s. The configuration space for Xt : t 0 is → < { ≥ } identified with ConfN (R) [0, ). We denote the probability law on the space < × ∞ SLE ConfN (R) [0, ) which governs Xt : t 0 by P , since it also governs the multiple SLE× given∞ in the form (2.2).{ Then≥ at} each time T [0, ), PSLE induces < ∈ ∞ a probability measure on ConfN (R) by PSLE(dx ,...,dx )= PSLE(X(1) dx ,...,X(N) dx ). T 1 N T ∈ 1 T ∈ N We fix a time T [0, ) and restrict the random distribution H on the domain η ∈ ∞ HT ( H to obtain a new γ-QS-(N + 1)-MBPs (see the middle picture in Fig. 4.1)
η (1) (N) η (4.1) HT , (η (T ),...,η (T ), ),H H . ∞ | T γ h i 14 MAKOTOKATORIANDSHINJIKOSHIDA
It is manifest from this construction that we have γ (i) γ (i) ν (η (0, t]L)= ν (η (0, t]R), t [0, T ], i =1,...,N, a.s. H Hη H Hη | T | T ∈ We define an ^Rot (H)-valued random field (see the left picture in Fig. 4.1) Sγ,N+1 (Xt:0 t T ) 1 (1) (N) (4.2) A ≤ ≤ (H, X) := g− ∗H Hη , (X ,...,X ) , T | T T T
1 η η 1 1 where we wrote gT− ∗H H = H H gT− + Q log gT− ′ . Then, uniformizing the η | T | T ◦ | | domain HT to H by the conformal mapping gT , we can find that the γ-QS-(N +1)- (Xt:0 t T ) MBPs (4.1) coincides with πγ,N∞ +1(A ≤ ≤ (H, X)). We call this assignment
(Xt:0 t T ) ^Rot ^Rot A ≤ ≤ of the (H)-valued random field (4.2) to a given (H)-valued Sγ,N+1 Sγ,N+1 random field (H, X) the cutting operation associated with the multiple SLE driven by Xt :0 t T . { ≤ ≤ } (Xt:0 t T ) Notice that for a quantum surface πγ,N∞ +1(A ≤ ≤ (H, X)) obtained by the cutting operation associated with the multiple SLE, the versions of Problems 1 to 3 are easily solved. Indeed, the mapping gT gives the desired conformal equivalence to solve Problem 1. Problem 3 can also be answered: Since the Loewner theory ensures that the slits η(i) N are anchored at X, i.e., η := (η(1),...,η(N))= X, { }i=1 0 0 0 a.s., the anchor points η0 are deterministic if and only if the initial configuration (i) N X is deterministic. In this case, the probability law of the resulting slits η i=1 is given by the law of the multiple SLE, answering Problem 2. { } Therefore, if a γ-QS-(N + 1)-MBPs obtained by the composition of the cut- ting operation and the surjection πγ,N∞ +1 is of the (X,α)-standard type for some < ConfN (R)-valued random variable X, the conformal welding problem for it is solved in the above arguments. The following propositionf gives such examples.
X f Proposition 4.1. Let (H ,α, X) be an ^Rot (H)-valued random field introduced H Sγ,N+1 in (3.4) with (α ,...,α )=( 2 ,..., 2 ) and assume that X =(X(1),...,X(N)): 1 N γ γ { t t t t 0 solves the set of SDEs (2.3) with κ = γ2 and ≥ } N 4 (4.3) F (i)(x)= , i =1,...,N, x x j=1 i − j Xj=i 6 starting at X. Then, for an arbitrary T [0, ), we have ∈ ∞ X (law) X (Xt:0 t T ) ,α T ,α (4.4) A ≤ ≤ (HH , X) = HH , XT ^Rot (Xt:0 t T ) as (H)-valued random fields. In particular, π∞ (A ≤ ≤ (H, X)) is a Sγ,N+1 γ,N+1 γ-QS-(N + 1)-MBPs of the (XT ,α)-standard type. Note that Theorem 3.10 is concluded from of Proposition 4.1 owing to the argu- ments followed by Proposition 4.1 above. Indeed the stochastic process Xt : t 0 solving (2.3) with (4.3) is just the desired time change of the Dyson{ model.≥ To } CONFORMAL WELDING PROBLEM 15 make the argument more precise, one only has to notice that a multiple SLE driven by such a Dyson model is absolutely continuous with respect to multiple of independent SLEs [Gra07], and therefore, the argument in [She16] can be applied. In the subsequent subsections, we prove Proposition 4.1. It suffices to prove XT ,α 1 X,α that H and g− ∗H Hη obey the same probability law. The central idea H T H | T is to interpolate these two C∞(H)′-valued random fields by a single stochastic ∇ (Xt:0 t T ) process and show its stationarity. The cutting operation A ≤ ≤ indeed defines a candidate of such an interpolating stochastic process, but it turns out that the reverse flow behaves much better. Before proceeding, let us introduce the reverse flow of a multiple Loewner chain needed in our proof.
(1) (N) 4.2. Reverse flow of a multiple Loewner chain. Let Xt =(Xt ,...,Xt ): t 0 be a set of continuous functions of t that drives the{ Loewner equation (2.2). ≥ } We assume that given Xt : t 0 , the Lowener equation (2.2) has a unique { ≥ } (i) N solution and determines an N-tuple of slits η i=1. We fix a time T (0, ) (i) (i) { } ∈ ∞ and set YT ;t := XT t, t [0, T ], i = 1,...,N. The reverse flow of the Loewner chain is defined as the− solution∈ of d N 2 (4.5) f T (z)= , t [0, T ], f T (z)= z H. dt t − T (i) ∈ 0 ∈ i=1 ft (z) Y X − T ;t T 1 1 η Lemma 4.2. The identity fT = gT− holds, where gT− : H HT is the inverse map of the uniformizing map g : Hη H that solves the Loewner→ equation (2.2). T T → T T Proof. Set ut (z) := fT t(z), z H, t [0, T ]. Then it satisfies − ∈ ∈ d N 2 uT (z)= , t [0, T ], uT (z)= f T (z), z H. dt t T (i) ∈ 0 T ∈ i=1 ut (z) Xt X − Since we have assumed that the multiple Loewner equation (2.2) has a unique T T solution, this implies that ut (z)= gt(fT (z)). Indeed, both sides satisfy the same differential equation with the same initial condition. In particular, at time t = T , T T T T 1 u (z)= f (z)= z = g (f (z)), z H, implying that f = g− . T 0 T T ∈ T T The reverse flow of the multiple SLE can also be formulated in connection to CFT as described in Appendix B.
4.3. Interpolation of random fields. We assume that the set of driving pro- (1) (N) cesses Xt = (Xt ,...,Xt ): t 0 is given by the system of SDEs (2.3). { ≥ } (i) (i) For a fixed T (0, ), we set the time reversed process YT ;t = XT t, t [0, T ], ∈ ∞ T − ∈ i = 1,...,N, and let ft ( ): t [0, T ] be the reverse flow defined in Eq. (4.5) (1){ · (N)∈ } driven by YT ;t =(YT :t ,...,YT ;t ): t [0, T ] . From Lemma 4.2, we can conclude 1 { T ∈ } that g− ∗H Hη = f ∗H Hη a.s. T | T T | T 16 MAKOTOKATORIANDSHINJIKOSHIDA
Let us define a stochastic process
N Y 2 n (z)= u T ;t,(2/γ,...,2/γ)(z)= log z Y (i) , t [0, T ], z H. t H γ | − T ;t| ∈ ∈ i=1 X
Fr (i) N We also consider HH which is independent of Bt : t 0 i=1. Then we see that, Fr { ≥ } at each time t [0, T ], [H,HH + nt, (YT ;t, )]γ is a γ-QS-(N + 1)-MBPs of the ∈ ∞2 2 (YT ;t,α)-standard type with (α1,...,αN )=( γ ,..., γ ). We set
T T h (z) := n (f (z)) + Q log f ′(z) , z H, t [0, T ] t t t | t | ∈ ∈ 2 γ with Q = γ + 2 and set
p = h + HFr f T , t [0, T ]. t t H ◦ t ∈
Then we can see that the stochastic process pt, t [0, T ] interpolates two C∞(H)′- X X T ,α ∈ T ,α ∇ valued random fields so that p0 = H and pT = f ∗H Hη . H T H | T
4.4. Stationarity of the stochastic process. We claim that p0 and pT obey the same probability law. The proof relies on the following key lemmas:
Lemma 4.3. The stochastic process ht(z), z H, t [0, T ] is a local martingale with increment ∈ ∈
N 2 (i) (4.6) dht(z)= Re − dB , z H, t [0, T ], T (i) t ∈ ∈ i=1 ft (z) YT ;t X − if κ = γ2 and the functions F (i)(x) N are chosen as (4.3). { }i=1
Proof. Note that ht(z) is the real part of
N 2 T (i) T h∗(z)= log(f (z) Y )+ Q log f ′(z), z H, t [0, T ]. t γ t − T ;t t ∈ ∈ i=1 X 2 We will show that ht∗(z), z H, t [0, T ] is a local martingale if κ = γ and the functions F (i)(x) N are chosen∈ as∈ (4.3). Owing to the time reversibility of the { }i=1 Brownian motions, the set of time reversed driving processes YT ;t : 0 t T solves the following system of SDEs: { ≤ ≤ }
(4.7) dY (i) = √κdB(i) F (i)(Y )dt, t [0, T ], i =1,...,N. T ;t t − T ;t ∈ CONFORMAL WELDING PROBLEM 17
By Eqs. (4.5) and (4.7), Itˆo’s formula gives
N T (i) 1 2dt (i) (i) d log(f (z) Y )= − √κdB + F (YT ;t)dt t − T ;t T (i) T (j) − t ft (z) Y j=1 ft (z) Y ! − T ;t X − T ;t 1 κdt , − 2 (f T (z) Y (i))2 t − T ;t N T 2dt d log ft ′(z)= . T (i) 2 i=1 (ft (z) Y ) X − T ;t They are assembled to give the increment of h∗(z), z H, t ∈ 2 N N (i) N (i) 4 1 2dBt 2 F (YT ;t)dt dh∗(z)= − dt + t √κ T (i) − T (i) √κ T (i) i=1 ft (z) Y ! i=1 ft (z) Y i=1 ft (z) Y X − T ;t X − T ;t X − T ;t N √κdt N dt +2Q , t [0, T ], − T (i) 2 T (i) 2 ∈ i=1 (ft (z) Y ) i=1 (ft (z) Y ) X − T ;t X − T ;t where we have used the relation κ = γ2. It can be verified that
N 1 1 1 N 1 = . T (i) (i) (j) 2 T (i) T (j) i,j=1 ft (z) YT ;t YT ;t YT ;t i,j=1 (ft (z) YT ;t)(ft (z) YT ;t ) Xi=j − − Xi=j − − 6 6
Using this, we see that the increment of h∗(z), z H becomes t ∈
N N 2 1 (i) 4 dh∗(z)= F (YT ;t) dt t √κ T (i) − (i) (j) i=1 ft (z) YT ;t j=1 YT ;t YT ;t X − Xj=i − 6 N (i) 2dB t , t [0, T ] − T (i) ∈ i=1 ft (z) YT ;t X − and conclude that the stochastic process ht∗(z) is a local martingale if the functions (i) N F (x) i=1 are chosen as (4.3). Moreover, under such a choice of the functions {F (i)(x)}N , Eq. (4.6) is obtained. { }i=1 Thus, at each z H, the stochastic process ht(z): t [0, T ] can be regarded as a Brownian motion∈ after an appropriate time{ change.∈ In the} following, we as- (i) N sume that the functions F (x) i=1 are as (4.3). By Eq. (4.6), the cross variation between h (z) and h (w),{ z,w }H is given by t t ∈ N 2 2 d h(z), h(w) = Re Re dt, z,w H. h it T (i) T (i) ∈ i=1 ft (z) Y ! ft (w) Y ! X − T ;t − T ;t 18 MAKOTOKATORIANDSHINJIKOSHIDA
Fr Fr T T Lemma 4.4. Define GHη (z,w) := GH (ft (z), ft (w)), t [0, T ], z,w H. Then t ∈ ∈ Fr d h(z), h(w) = dGHη (z,w), t [0, T ], z,w H. h it − t ∈ ∈ Proof. This can be verified by direct computation. By definition, we have Fr T T T T GHη (z,w)= log ft (z) ft (w) log ft (z) ft (w) . t − | − | − | − | Thus its increment is computed as T T T T Fr dft (z) dft (w) dft (z) dft (w) dG η (z,w)= Re − Re − Ht T T T T − ft (z) ft (w) − f (z) f (w) − t − t N 2dt = Re − T (i) T (i) i=1 (ft (z) Y )(ft (w) Y ) X − T ;t − T ;t N 2dt + Re − T (i) T (i) i=1 (ft (z) Y )(ft (w) Y ) X − T ;t − T ;t N 2 2 = Re Re dt − T (i) T (i) i=1 ft (z) Y ! ft (w) Y ! X − T ;t − T ;t which is the same as d h(z), h(w) , t [0, T ], z,w H. − h it ∈ ∈ For a test function ρ C∞(H), we have ∈ ∇ d (h, ρ), (h, ρ) = dEFr(ρ), h it − t where Fr Fr E (ρ)= ρ(z)G η (z,w)ρ(w)d(µ µ)(z,w) t Ht H H ⊗ Z × is non-increasing in the time variable t [0, T ]. This implies that (ht, ρ), t [0, T ], ∈ Fr ∈ is a Brownian motion such that we can regard Et (ρ) as a time variable. Thus − Fr Fr (hT , ρ) is normally distributed with mean (h0, ρ) and variance ET (ρ) ( E0 (ρ)). Fr T − − − The random variable (HH fT , ρ) is also normally distributed with mean zero Fr ◦ and variance ET (ρ) by the conformal invariance of the GFF. Since the random Fr T variable (HH fT , ρ) is conditionally independent of (hT , ρ), their sum (pT , ρ) is ◦ Fr a normal random variable with mean (h0, ρ) and variance E0 (ρ) coinciding with Fr (law) (h0 +HH , ρ)=(p0, ρ) in probability law. This implies pT = p0 as C∞(H)′-valued random fields. The proof of Proposition 4.1 is complete. ∇
5. Proof of Theorem 3.12 This section is devoted to a proof of another main result Theorem 3.12 in (1) (N) the present paper. Again, we let Xt = (Xt ,...,Xt ): t 0 be a set of driving processes satisfying (2.3) associated{ with parameter κ >≥0 and} functions (i) N (1) (N) < F (x) i=1 with initial conditions (X0 ,...,X0 )=(x1,...,xN ) ConfN (R). {We also} assume that the Loewner equation (2.2) driven by X ; t ∈ 0 has the { t ≥ } CONFORMAL WELDING PROBLEM 19
(i) N unique solution gt, t 0 and determines N non-intersecting slits η i=1 in H anchored on R. ≥ { }
5.1. Key statement. At each z H, consider a stochastic process ∈ 2 N nI(z)= − arg(z X(i)), t 0. t √κ − t ≥ i=1 X I At each time t [0, ), the random field nt is harmonic on H and its boundary ∈ ∞ (i) Dir value changes at N points Xt R, i = 1,...,N. Let HH be independent of X ∈ B(i) : t 0 N . Put H t,β,I = nI + HDir, where (β ,...,β )=( 2 ,..., 2 ) are { t ≥ }i=1 H t H 1 N √κ √κ fixed here and in the sequel. We also define I I h (z) := n (g (z)) χ arg g′(z) t t t − t with χ R and ∈ I Xt,β,I I Dir p := H g χ arg g′ = h + H g . t H ◦ t − t t H ◦ t η Xt,β,I Xt,β,I Note that (Ht ,HH gt χ arg gt′) χ (H,HH ). Due to the initial condition ◦ − I ∼ x,β,I < g0(z) = z H, we can see that p0 = HH , where x =(x1,...,xN ) ConfN (R) are the initial∈ values of the driving processes. ∈
Proposition 5.1. Let κ (0, 4], N Z 1 and χ R. Suppose that Xt = (1) (N) ∈ ∈ ≥ ∈ { (Xt ,...,Xt ): t 0 is the solution of the system of SDEs (2.3) starting at ≥ }< 2 √κ (i) N x =(x1,...,xN ) ConfN (R). If χ = √κ 2 and the functions F (x) i=1 are ∈ − { } I given by (4.3), then at each time T [0, ), two C0∞(H)′-valued random fields p0 I ∈ ∞ and pT obey the same probability law. 5.2. Proof of Proposition 5.1. The proof is very similar to that of Proposition 4.1. Thus we omit the computational details. The following lemmas play key roles:
I Lemma 5.2. At each z H, the stochastic process ht (z), t 0 is a local martin- gale with increment ∈ ≥ N 2 dhI(z)= Im dB(i), t 0, t (i) t ≥ i=1 gt(z) Xt X − if the functions F (i)(x) N are chosen as (4.3). { }i=1 I I Thus, at each t [0, ), the cross variation between ht (z) and ht (w) is given by ∈ ∞ N 2 2 d hI(z), hI(w) = Im Im dt, z,w H, h it (i) (i) ∈ i=1 gt(z) Xt ! gt(w) Xt ! X − − which turns out to be expressed using the Green function. 20 MAKOTOKATORIANDSHINJIKOSHIDA
Dir Dir η Lemma 5.3. Let GHη (z,w) := GH (gt(z),gt(w)), t 0, z,w Ht . Then we t ≥ ∈ have I I Dir η d h (z), h (w) = dGHη (z,w), t 0, z,w Ht . h it − t ≥ ∈ I For ρ C∞(H), the quadratic variation of (h , ρ) becomes ∈ 0 t d (hI, ρ), (hI, ρ) = dEDir(ρ), h i − t where Dir E (ρ)= ρ(z)G η (z,w)ρ(w)d(µ µ)(z,w) t Ht Hη Hη ⊗ t × t Z η is the Dirichlet energy of ρ in the domain Ht , t 0. Here ρ is supposed to be η ≥ η a function on Ht by restriction. Since the process Ht , t 0 is non-increasing, Dir ≥ I regarding Et (ρ) as a new time variable, the stochastic process (ht , ρ) is a Brownian motion.− This implies that at any fixed time T [0, ), the random I I ∈ Dir ∞ Dir variable (hT , ρ) is normal with mean (h0, ρ) and variance ET (ρ) ( E0 (ρ)). Dir − − − The random variable (HH gT , ρ) is also a mean-zero normal variable with vari- Dir I ◦ Dir ance ET (ρ). Since (hT , ρ) and (HH gT , ρ) are conditionally independent, their I ◦ I Dir sum (pT , ρ) is normally distributed with mean (h0, ρ) and variance E0 (ρ), thus I it coincides with (p0, ρ) in probability law. 5.3. Arguments. We here present a geometric interpretation of Proposition 5.1 and see that it indeed proves Theorem 3.12. That is, Proposition 5.1 provides two distinct samplings of C0∞(H)′-valued random fields which obey the same probabil- I x,β,I ity law. The first one is to directly sample the random field p0 = HH (see the upper-left picture in Fig. 5.1), and the other one is first to sample multiple SLE (i) N I I Dir paths η i=1 up to time T (0, ), sample the random field pT = hT +HH gT { } η ∈ ∞ ◦ on the domain HT , and then extend it to H (see the upper-right picture of Fig. 5.1). Coincidence in probability law between these two samplings roughly means that there is a one-to-one correspondence among instances of two samplings with the I same weights. In particular, with each instance h of p0, one can associate an (i) N N-tuple of multiple SLE paths η i=1. We describe this correspondence more concretely in the sequel. { } Notice that nI, t 0, is the unique harmonic function with boundary conditions t ≥ 2π nI(x)= (N i), if x (X(i),X(i+1)), i =0, 1,...,N. t −√κ − ∈ t t (0) (N+1) Here we follow the convention that Xt = and Xt =+ . In particular, (i) −∞ ∞ Xt,β,I it has discontinuity at Xt by 2π/√κ along R, i = 1,...,N. Note that HH , I t 0 is regarded as the GFF with the same boundary condition as nt , t 0. ≥Let us investigate the behavior of pI , T [0, ) near the boundary≥ of Hη . T ∈ ∞ T Near a point x (xi, xi+1) on the real axis, we have limz x+0√ 1 arg gT′ (z) = 0. I ∈ I→ − Thus pT , T [0, ) has the same boundary value as p0 on the real axis. We next suppose∈ that∞x lies on the i-th strand of the N-tuple of SLE paths, i.e., x η(i)(0, T ). Although the strand η(i)(0, T ) is not a smooth curve, we temporarily ∈ CONFORMAL WELDING PROBLEM 21
x,β,I I (H, H H ) H , p (2) ( T T ) η(1) η η(i) η(N) (law)
λi+1 + χθ − χπ
λi + χθ
λ λ1 λ2 λ3 λN N+1 λ1 λ2 λ3 λN λN+1 x1 x2 xN x1 x2 xi xN gT
XT ,β,I (H, H H )
λ1 λ2 λ3 λN λN+1 (1) (2) (N) XT XT XT Figure 5.1. Geometric interpretation of Proposition 5.1: The boundary values on the real axis are defined by λi = 2π(N i 1)/√κ, i =1,...,N + 1. − − − Hη T η(i)
x θ(x)
xi
Figure 5.2. The angle θ(x)
let θ(x) be the angle of the tangent line of η(i)(0, T ) at x (see Fig. 5.2). Then we have limz x+ arg gT′ (z) = π θ(x) where z approaches x from the right, while → − 22 MAKOTOKATORIANDSHINJIKOSHIDA
limz x arg gT′ (z)= θ(x) where z approaches x from the left. Thus → − − I 2π lim pT (z)= (N i)+ χθ(x) χπ, z x+ → −√κ − − I 2π lim pT (z)= (N i +1)+ χθ(x) z x → − −√κ − (see the upper- and lower-right pictures of Fig. 5.1). Although these two val- ues themselves are not well-defined because θ(x) is not, their difference does not depend on θ(x) yielding
I I √κπ lim pT (z) lim pT (z)= , z x+ z x → − → − 2 where we have used the relation χ = 2 √κ . Thus we will see that pI has discon- √κ − 2 T tinuity by √κπ/2 from the left side to the right side of a strand. Conversely, from I x,β,I Proposition 5.1, one can find in an instance h of p0 = HH strands evolving from xi, i = 1,...,N so that the value of h has discontinuity by √κπ/2 from the left side to the right side of a strand. Following the argument in [SS13,MS16a,She16], x I √ 1H ,β, /χ these strands can be regarded as the flow lines of the vector field e − H starting from xi, i =1,...,N. Moreover, the law of these strands agrees with the (i) N one of the slits η i=1 determined by the multiple SLEκ. Therefore, Theorem 3.12 is proved. { }
6. Concluding remarks As conclusion, we make some discussions and remarks on related topics and future directions.
6.1. Pathwise uniqueness of coupling. In the present paper, we found cou- plings between GFFs and multiple SLEs in suitable senses. For both of the con- formal welding problem and the flow line problem, we heuristically argued in Subsects. 3.1 and 5.3 that multiple SLEs are uniquely determined by the GFFs. In [Dub09, MS16a], the authors rigorously proved this kind of pathwise unique- ness for a single curve with assistance of the dual SLE. It is an interesting and important future direction to consider how to generalize their arguments to our case, where we treat multiple curves at once.
6.2. Dynamics of the Dyson model. The conformal welding problem intro- duced in Section 3.1 requires precise description of the statistical behavior of slits (i) N η i=1 in H, which are non-intersecting. Our strategy to solve this problem is { } (i) N to identify η i=1 in H with “N-tuple of SLE curves.” Based on Theorem 2.1 from [RS17]{ for} the deterministic multiple Loewner equation (2.2), we have re- (i) N duced the problem for η i=1 in H to the problem to find a stochastic process (1) (N) { N} Xt = (Xt ,...,Xt ) R , t 0, of N particles on R. Then we have assumed that X , t 0 satisfies∈ a system≥ of SDEs in a general form (2.3). There, the drift t ≥ CONFORMAL WELDING PROBLEM 23
t (i) terms 0 F (Xs)ds, i =1,...,N, t 0, which determine the interaction among N particles, are arbitrary. ≥ In theR formulation of Problem 1, we have assumed that the slits η(i) N are { }i=1 non-intersecting, which implies that the stochastic process Xt, t 0 must be non- colliding. The main result in the present paper given by Theorem≥ 3.10 states that the solution can be given by a proper time change of the Dyson model, which may be the most studied process in non-colliding particle systems in probability theory and random matrix theory (see, for instance [For10,AGZ10,Kat15]). As shown in (3.6), in the Dyson model, the repulsive force acts between any pair of particles, whose strength is proportional to the inverse of distances between the particles and the proportionality coefficient is given by β/2, β (0, ). We note that the γ-QS- (N +1)-MBPs of the (X,α)- standard type has the∈ set∞ of parameters γ (0, 2) and ∈ αi, i =1,...,N, and the multiple SLE does one parameter κ (0, ). Theorem 3.10 determines the relations among them as ∈ ∞ 2 8 κ = γ2, α = (i =1,...,N), β = . i γ κ 2 The equality κ = γ is the same as that given in [She16] and αi = 2/γ, i = 1,...,N are a simple N-variable extension of his result α1 =2/γ for the original conformal welding problem with two marked boundary points. The equality β = 8/κ is found in the literatures [Car03a,Car03b,Car04,BBK05], but its derivations heavily depended on CFT and the so-called group theoretical formulation of SLE [BB03,BB04,Kos18] (see Appendix B). Our derivation given in the proof of Lemma 4.3 is purely probability theoretical and simple. Since γ (0, 2) for the LQG, we have β > 2. Therefore, the resulting time change of the∈ Dyson model indeed is non-colliding. Note that the non-colliding condition β 1 for the Dyson model [RS93, CL97, GM13,GM14] corresponds to κ (0, 8] for the≥ multiple SLE, while when κ> 4 the multiple SLE curves will collide∈ with each other and become self-intersecting in H. In the region κ (4, 8], the correspondence between slits and driving processes will be explained as∈ follows. If two slits collide with each other, this event is classified into two cases. (I) Two tips of slits collide with each other (see Fig. 6.1). (II) A tip collides with an already existing slit (see Fig. 6.2). Since each of the driving processes is the image of a tip of a slit under the uniformization map, two of the driving processes collide with each other when the event (I) occurs. On the other hand, when the event (II) occurs, the driving processes are non-colliding even though the corresponding SLE slits are colliding. Following this argument, we could expect that the Dyson model will fall into three classes. When β 2, the particles are non-colliding and the corresponding SLE slits are non-intersecting.≥ When β [1, 2), the particles are non-colliding, but the event (II) almost surely occurs. When∈ β (0, 1), the particles collide, and correspondingly, the event (I) almost surely occurs.∈ Though it is known [RS93, CL97, GM13, GM14] that the colliding/non-colliding transition occurs at β = 1, the possible phenomenon that the characteristics of the Dyson model changes at β = 2 has not been well 24 MAKOTOKATORIANDSHINJIKOSHIDA
(1) (2) η0 η0
Figure 6.1. (I) Tips collide
(1) (2) η0 η0
Figure 6.2. (II) A tip collides with an already existing slit
studied so far. It would be an interesting future direction to find a property that distinguishes the Dyson model of β [1, 2) and that of β 2. We will also have to make discussions analogous to that∈ in [RS05] to settle this≥ classification.
6.3. Other driving processes. As interacting particle systems related with ran- dom matrix theory, a variety of non-colliding particle systems have been studied (see, for instance, [KT04]). We hope that we can address the conformal welding problems in other situations and, in solving them, interesting relations between non-colliding particle systems and multiple SLEs will be discovered. We will de- pict examples of such other situations for which the conformal welding problem is solvable and produces an another type of driving processes. CONFORMAL WELDING PROBLEM 25
6.3.1. Inhomogeneous systems. The setting (4.5) and (4.7) for the conformal weld- ing problem can be generalized as follows: d N 2λ (6.1) f T (z)= i , t [0, T ], f T (z)= z H, dt t − T (i) ∈ 0 ∈ i=1 ft (z) YT ;t X − (6.2) dY (i) = √κ dB(i) F (i)(Y )dt, t [0, T ], i =1,...,N, T ;t i t − T ;t ∈ N where λi > 0, κi > 0, i =1,...,N with i=1 λi = N [Sch13,RS17,dMHS18]. Let N T (i) P T h∗(z)= α log(f (z) Y )+ Q log f ′(z), z H, t [0, T ], t i t − T ;t t ∈ ∈ i=1 X 2 γ where αi, i = 1,...,N are indeterminate real numbers and Q = γ + 2 . By the similar calculation to that given in the proof of Lemma 4.3, we can show that N N αiκi (i) 2C(αi, κi,λi,γ) dht∗(z)= dBt + dt − T (i) T (i) 2 i=1 ft (z) YT ;t i=1 (ft (z) YT ;t) X − X − N N 1 (i) αiλj + αjλi + αiF (YT ;t) 2 dt, t [0, T ], T (i) − (i) (j) ∈ i=1 ft (z) YT ;t j=1 YT ;t YT ;t X − Xj=i − 6 with κ (6.3) C(α , κ ,λ ,γ)= λ + i α + Qλ , i =1,...,N. i i i − i 4 i i Hence if
(6.4) C(αi, κi,λi,γ)=0, i =1,...,N and 2 N α λ + α λ (6.5) F (i)(x)= i j j i α x x i j=1 i − j Xj=i 6 N λ 2λ N α =2 j + i j , i =1,...,N, x x α x x j=1 i − j i j=1 i − j Xj=i Xj=i 6 6 then ht∗(z), z H, t [0, T ] becomes a local martingale. When λi = 1, κi = κ, i =1,...,N, (6.4)∈ with∈ (6.3) gives 4Q 2Q αi = = , i =1,...,N. 4+ κ 2 √κ √κ( √κ + 2 ) That is, the weights to the marked boundary points are homogeneous. When we 2 2 further assume the relation κ = γ , then αi = γ as we have seen in one of our main theorems (Theorem 3.10). 26 MAKOTOKATORIANDSHINJIKOSHIDA
Inhomogeneous setting of the conformal welding problem with α = α , i = j i 6 j 6 in general (as well as the inhomogeneous flow line problem with βi = βj, i = j in general) will be studied in which inhomogeneous multiple SLE (6.1)6 drive6 n by inhomogeneous interacting particles on R (6.2) shall be analyzed under the conditions (6.4) with (6.3), and (6.5) to solve the problems. 6.3.2. Multiple quadrant SLE and the Wishart process. In this paper, we have for- X,α mulated the conformal welding problem for a γ-QS-(N+1)-MBPs [H,HH , (X, )]γ of the (X,α)-standard type. In solving this, we have adopted the form of multiple∞ Loewner equation (2.2) and assumed that the set of driving processes is determined (i) N by the system of SDEs (2.3) with drift functions F (x) i=1 motivated by preced- ing works [BBK05,RS17]. As a result, we have found{ that} the parameters κ = γ2 2 and αi = γ are determined and the functions are chosen as (4.3) to obtain a one- XT ,α parameter family [H,HH , (XT , )]γ : T (0, ) of γ-QSs-(N + 1)-MBPs, for each of which the{ conformal welding∞ problem∈ is∞ solvable.} Notice that there is room for changing the model of uniformization maps. As a generalized multiple Loewner equation for N slits, we consider the following form d (6.6) g (z)=Ψ(g (z), X ), t 0, g (z)= z, dt t t t ≥ 0
where Ψ(z, x) is a suitable functions of z and x = (x1,...,xN ), and Xt = (1) (N) { (Xt ,...,Xt ): t 0 is a set of driving processes. We do not specify the domain of definition for≥ the} function Ψ(z, x) since it will depend on models. When we take N 2 Ψ(z, x)= , z H, x RN , z x ∈ ∈ i=1 i X − the associated Loewner equation (6.6) reduces to (2.2). Let us see the case for another choice of Ψ(z, x). Let O := z C Rez > 0, Imz > 0 be an orthant in C. We adopt { ∈ | } N 2 2 4δ Ψ(z, x)=Ψ (z, x) := + + , z O, x (R )N . O z x z + x z ∈ ∈ >0 i=1 i i X − Here δ R is a parameter and R>0 = x R x > 0 is the set of positive real numbers.∈ The associated Loewner equation{ ∈ (6.6)| becomes} d N 2 2 4δ (6.7) g (z)= + + , t 0, dt t (i) (i) g (z) ≥ i=1 gt(z) Xt gt(z)+ Xt ! t X − g (z)= z O. 0 ∈ (1) (N) We again assume that the set of driving processes Xt = (Xt ,...,Xt ) (R )N : t 0 solves the sytem of SDEs { ∈ >0 ≥ } dX(i) = √κdB(i) + F (i)(X )dt, t 0, i =1,...,N, t t t ≥ CONFORMAL WELDING PROBLEM 27
(i) N where κ> 0 is a parameter, Bt : t 0 i=1 are mutually independent standard (i) { N ≥ } Brownian motions and F (x) i=1 are suitable functions of x = (x1,...,xN ) so N { } that Xt lies in (R>0) for all t 0. The equation (6.7) is the multiple version of the quadrant Loewner equation considered≥ in [Tak14]. We assume that, if the initial (1) (2) (N) value of Xt satisfies 0
Wβ,ν (i) (i) β(ν + 1) 1 1 dXt =dBt + − 2 Wβ,ν (i) " Xt β N 1 1 + + dt, 2 Wβ,ν (i) Wβ,ν (j) Wβ,ν (i) Wβ,ν (j) j=1 Xt Xt Xt + Xt !# Xj=i − 6 t 0, i =1,...,N. ≥ Using the function ΨO(z, x) as a model of uniformization maps, we can obtain solutions to the conformal welding problem.
(1) (N) Theorem 6.1. Let γ (0, 2) and N Z 1. Suppose that Xt =(Xt ,...,Xt ): ≥ ∈ ∈ W8/κ,δ{ t 0 is a time change of the Wishart process Xκt : t 0 starting at a deterministic≥ } initial state X = x Conf< (R ).{ Then, at each≥ time} T (0, ), 0 ∈ N >0 ∈ ∞ XT ,α the conformal welding problem for O,HO , (XT , ) with (α1,...,αN ) = ∞ γ 2 2 h i ( γ ,..., γ ) is solved as follows: 28 MAKOTOKATORIANDSHINJIKOSHIDA
1. The solution of the Loewner equation (6.7) driven by the time change of the W8/κ,δ Wishart process Xκt : 0 t T gives a solution to Problem 1. In 1 { η ≤ ≤ } other words, g− : O O is the desired conformal equivalence. T → T 2. The probability law for resulting slits η(i) N is the one for the quadrant { }i=1 multiple SLEκ. This gives a solution to Problem 2. 3. Problem 3 is answered positively with η0 = x a.s. From the above observation, we could say that interacting particle systems such as the Dyson model and the Wishart process are associated with models Ψ(z, x) of uniformization maps. Along this line, we could expect a new classification of interacting particle systems from a perspective of the Loewner theroy and coupling with GFF. A detail of this subject including the associated flow line problem will be pub- lished elsewhere [KK20].
6.4. Other variants of SLE. As we have noted above, there is room for variants of multiple SLEs in the form of (6.6). Indeed, the radial and dipole multiple SLEs are special cases of (6.6). It was shown in [SW05] that, in the case of N = 1, the chordal, radial and dipole SLEs are transformed one another by conformal mappings in the framework of the SLEκ;ρ, while the force points are allowed to be interior of the domain. For example, the radial SLEκ is transformed to the chordal SLEκ;κ 6 with an interior force point by a M¨obius transformation. − In [MS16a, She16], the coupling with an SLEκ;ρ and an GFF was formulated, and our result in the present paper is also expected to be extended to the case concerning a multiple SLEκ;ρ. Then, it would be interesting to study how the transformation of these SLEs can be compatible to the coordinate transformation of quantum surfaces in Eq.(3.1) under the connection between the SLE and the LQG.
6.5. The limit N . It would be interesting to consider the conformal weld- ing problem and the→∞ flow line problem in the case with infinitely many boundary points. In the present paper, the multiple SLE driven by an N-particle Dyson model arose as the solution to the conformal welding problem for a γ-QS-(N +1)- MBPs of the (X,α)-standard type. If the method in the present paper is appli- cable at the limit N , it can be expected that the multiple SLE driven by an infinite dimensional Dyson→∞ model [KT10,Osa12,Osa13,Tsa16,OT16,KO18,OT20] would appear in such systems. Although the multiple SLE driven by infinitely many driving processes is not well-posed so far, we hope that it is captured when the coupling with GFF is considered. Another limit of N is the hydrodynamic limit of the multiple SLE [dMS16, HK18]. In this case,→ ∞ the ensemble of slits gets deterministic as N . Accordingly, a quantum surface must be subject to a boundary condition.→ An∞ interesting question is, then, how multiple slits condition the GFF on the domain and finally impose a boundary condition. CONFORMAL WELDING PROBLEM 29
6.6. Discrete models converging to the present systems. It has been re- ported that random planar maps converge to an SLE-decorated LQG in several topology (see [GMS19, HS19, GMS20] and references therein). While a chordal SLE describes the scaling limit of a single interface in various critical lattice mod- els, a multiple SLE describes scaling limits of collections of interfaces in critical lattice models with alternating boundary conditions (see [BPW18] and references therein). In the present paper we introduced new kinds of continuous systems, the γ-quantum surface with N +1 marked boundary points (γ-QS-(N + 1)-MBPs) and the χ-imaginary surface with N +1 boundary condition changing points (χ- IS-(N + 1)-BCCPs) for N Z 1. Both have been related with the multiple SLE driven by a Dyson model in∈ solving≥ the conformal welding problem and the flow line problem. Discrete counterparts of these random systems and corresponding problems will be studied.
Appendix A. Construction of and Sγ Sγ,N+1 In this appendix, we construct the spaces γ and γ,N+1 as orbifolds and study their structures. S S
A.1. Without marked boundary points. Let us begin with γ , γ (0, 2). Consider the following lift of : S ∈ Sγ univ := C∞(D)′, S ∇ D(C: 1-conn.[ where D (C runs over all simply connected domains. We consider the canonical univ surjection ։ γ . Notice that each component C∞(D)′ carries a right action S S ∇ aD,γ : C∞(D)′ Aut(D) C∞(D)′ of Aut(D) depending on the parameter γ defined by∇ × → ∇
aD,γ(h, ψ) := h ψ + Q log ψ′ , h C∞(D), ψ Aut(D), ◦ | | ∈ ∇ ∈ 2 γ where we set Q = γ + 2 . The chain rule ensures that aD,γ defines a right action of the group. We write the quotient space C∞(D)′/aD,γ as C∞(D)γ′ red (we read “γ-red” as “γ-reduced”), and set ∇ ∇ −
γ := C∞(D)γ′ red. S ∇ − D(C: 1-conn.[ f Let us introduce a groupoid = ( 0, 1) whose objects are simply connected proper subdomains in C: =G D (CG :G simply connected and the set of mor- G0 { } phisms of which is given by 1(D1,D2) = Aut(D2) Iso(D1,D2)/Aut(D1), D1,D2 . It is obvious that eachG set (D ,D ) consists\ of a single element, which we∈ G0 G1 1 2 denote by the symbol (D D ). Then the anti-action of on is given as 1 → 2 G Sγ follows. We consider a mapping c : which is defined for pairs γ G1 × Sγ → Sγ ((D1 D2), h C∞(D2)γ′ red), D1,D2 0 as f → ∈ ∇ − ∈ G f f cγ ((D1 D2), h) := h ψ + Q log ψ′ C∞(D1)γ′ red, h C∞(D2)γ′ red, → ◦ | | ∈ ∇ − ∈ ∇ − 30 MAKOTOKATORIANDSHINJIKOSHIDA
where Q = 2 + γ and ψ : D D is a conformal equivalence. It can be verified γ 2 1 → 2 that the above definition does not depend on the choice of a conformal equivalence
ψ. Then the quotient γ /cγ is just the collection of γ-pre-quantum surfaces γ . Consequently, the canonicalS quotient map univ is the composition S S → Sγ f / S a /c univ D D,γ γ . S −−−−−→ Sγ −−→Sγ By uniformizing any domain D to the upper half plane, the collection γ of all γ-pre-quantum surfaces is identified withf the space of γ-reduced distributionsS
C∞(H)γ′ red on H. ∇ − A.2. With marked boundary points. Let us move on to , γ (0, 2), Sγ,N+1 ∈ N Z 0. We consider the following space ∈ ≥ univ < N+1 = C∞(D)′ ConfN+1(∂D), S ∇ × D(C: 1 [conn. − where D (C runs over all simply connected domains. As we have seen, for each D ( C, the component C∞(D)′ has a right action aγ,D of Aut(D) depending on ∇ < the parameter γ. The same group also acts on ConfN+1(∂D) from the left. We < write the diagonal action of Aut(D) on C∞(D)′ ConfN+1(∂D) as aD,γ,N+1 and set ∇ × ^ < γ,N+1 := C∞(D)′ aD,γ,N+1 ConfN+1(∂D). S ∇ × D(C: 1 [conn. −
It can be verified that the groupoid again acts on ^γ,N+1. Then the space γ,N+1 is constructed as G S S / S aD,γ.N+1 / univ D ^ G . SN+1 −−−−−−−−→ Sγ,N+1 −→Sγ,N+1 Let us write each component of ^ as Sγ,N+1 ^ < γ,N+1(D) := C∞(D)′ aD,γ,N+1 ConfN+1(∂D). S ∇ × Because the action of the groupoid is simply transitive, the space is G Sγ,N+1 noncanonically isomorphic to ^ (D) for every D. Sγ,N+1 For simplicity, let us identify γ,N+1 with ^γ,N+1(H) and consider the following commutative diagram: S S < / / < C∞(H)′ ConfN+1(∂H) ConfN+1(∂H) ∇ × ^ (H) π / / Aut(H) Conf< (∂H). Sγ,N+1 \ N+1 < 1. If N 2, the space Aut(H) ConfN+1(∂H) = consists of a single ele- ≤ \ {∗} 1 ment. Thus the space is isomorphic to the fiber π− ( ). Sγ,N+1 ∗ CONFORMAL WELDING PROBLEM 31
< (a) If N = 0, the point Conf1 (∂H) is fixed by the subgroup Aff(H)= z az + b a > 0,∞ b ∈R of affine transformations in Aut(H). Thus, the{ 7→ fiber over| [ ] Aut(∈ }H) Conf<(∂H) becomes ∞ ∈ \ 1 1 π− [ ] C∞(H)′/aH,γ(Aff(H)). ∞ ≃ ∇ < (b) If N = 1, the point (0, ) Conf2 (∂H) is fixed by the subgroup Scl(H) = z az a > 0∞of∈ scale transformations in Aut(H). Thus, the fiber over{ 7→ [0, |] Aut(} H) Conf<(∂H) becomes ∞ ∈ \ 2 1 π− [0, ] C∞(H)′/aH,γ(Scl(H)). ∞ ≃ ∇ < (c) If N = 2, the action of Aut(H) on Conf3 (∂H) is simply transitive. Thus, the fiber over [0, 1, ] Aut(H) Conf<(∂H) becomes ∞ ∈ \ 3 1 π− [0, 1, ] C∞(H)′. ∞ ≃ ∇ < 2. If N 3, the space Aut(H) ConfN+1(∂H) is (N 2)-dimensional over R and each≥ fiber becomes \ − 1 < π− [x1,...,xN+1] C∞(H)′, (x1,...,xN+1) ConfN+1(∂H). ≃ ∇ ∈ We show an alternative construction of ^Rot (H) used to define a γ-QS-(N +1)- Sγ,N+1 MBPs of the (X,α)-standard type in Sect. 3. Let Rot(H) Aut(H) be the subgroup consisting of rotations of H. We consider the following⊂ object:
^Rot < γ,N+1(H) := C∞(H)′ aH,γ,N+1(Rot(H)) ConfN+1(∂H), S ∇ × < which is a fiber bundle over Rot(H) ConfN+1(∂H). By sending the (N + 1)-st point to , we have \ ∞ Rot(H) Conf< (∂H) Conf< (R), \ N+1 ≃ N < and the fiber over x = (x1,...,xN ) ConfN (R) is isomorphic to C∞(H)′. Since ∈ ∇ Conf< (R) is contractible, the fiber bundle ^Rot (H) Conf< (R) is trivial: N Sγ,N+1 → N ^Rot < γ,N+1(H) C∞(H)′ ConfN (R), S ≃ ∇ × reducing to Eq. (3.2). In this construction, it becomes clear that the surjection
πγ,N∞ +1 defined in (3.5) is just the quotient map.
Appendix B. Driving processes of a multiple SLE from an auxiliary function A time change of the Dyson model also appeared in [BBK05] as a particular example of a set of driving processes. In their work, in connection to CFT, a set (1) (N) of driving processes Xt =(Xt ,...,Xt ), t 0, N Z 1, was derived from an ≥ ∈ ≥ auxiliary function Z(x1,...,xN ) annihilated by operators κ 1 h = ∂2 +2 ∂ + κ , i =1,...,N, i xi xj 2 D 2 xi xj (xi xj) j;j=i X6 − − 32 MAKOTOKATORIANDSHINJIKOSHIDA
κ 6 (1) (N) where hκ = 2−κ so that Xt = (Xt ,...,Xt ): t 0 satisfies the system of SDEs { ≥ } 2dt dX(i) = √κdB(i) + κ(∂ log Z)(X )dt + , t 0, i =1,...,N. t t xi t (i) (j) ≥ j;j=i Xt Xt X6 − Our set of driving processes (2.3) associated with functions (4.3) can be obtained by taking the following auxiliary function: 2 Z(x ,...,x )= (x x ) κ . 1 N i − j i
Z(x1, , xN )= h ΨhR (x1) ΨhR (xN ) 0 , ··· h ∞| κ ··· κ | i CONFORMAL WELDING PROBLEM 33
R the central charge for which is cκ . It follows from the existence of a singular κ 2 R R vector (2L 2 + 2 L 1) cκ , hκ that the correlation function Z(x1, , xN ) solves the following− system− of| differentiali equations: ··· (B.1) RZ =0, i =1,...,N, Di where κ 1 hR R = ∂2 2 ∂ κ , i =1, ,N. i xi xj 2 D 2 − xj xi − (xj xi) ··· j;j=i X6 − − Example B.1. The following function gives a solution to the system of differential equations (B.1): 2 (B.2) Z(x , , x )= (x x )− κ . 1 ··· N i − j i Definition B.4. Let Z(x1, , xN ) be a solution to the system of differential equations (B.1) and Y : t ··· 0 be the associated driving processes. The re- { t ≥ } verse flow of an multiple SLEκ associated with these data is a stochastic process R gt ( ) t 0 solving the following multiple version of the reverse flow: { · } ≥ d N 2 gR(z)= , t 0, gR(z)= z. dt t − R (i) ≥ 0 i=1 gt (z) Yt X − The reverse flow defined above is connected to CFT in the following sense: Theorem B.5. 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