Conformal blocks in nonrational CFTs with c 1

Transcription

1 Conformal blocks in nonrational CFTs with c 1 Eveliina Peltola Université de Genève Section de Mathématiques < eveliina.peltola@unige.ch > March 15th 2018 Based on various joint works with Steven M. Flores, Alex Karrila and Kalle Kytölä Recent developments in Constructive Field Theory, Columbia University 1

2 PLAN Belavin-Polyakov-Zamolodchikov (BPZ) PDEs in 2D conformal field theory physical solutions : find single-valued ones? basis for solution space: conformal blocks 2

3 PLAN Belavin-Polyakov-Zamolodchikov (BPZ) PDEs in 2D conformal field theory physical solutions : find single-valued ones? basis for solution space: conformal blocks How to solve the PDEs? Coulomb gas formalism action of the quantum group U q (sl 2 ) on the solution space can use representation theory to analyze solution space currently works in the nonrational case 2

4 PLAN Belavin-Polyakov-Zamolodchikov (BPZ) PDEs in 2D conformal field theory physical solutions : find single-valued ones? basis for solution space: conformal blocks How to solve the PDEs? Coulomb gas formalism action of the quantum group U q (sl 2 ) on the solution space can use representation theory to analyze solution space currently works in the nonrational case explicit formulas? integral formulas (Coulomb gas) algebraic formulas when c = 1 and c = 2 2

5 INTRODUCTION: BPZ PARTIAL DIFFERENTIAL EQUATIONS 2

6 CONFORMAL FIELD THEORY (CFT) consider a 2D QFT with fields φ(z) impose conformal symmetry: fields φ(z) carry action of Virasoro algebra Vir Vir: Lie algebra generated by (L n ) n N and central element C [L n, L m ] = (n m)l n+m + C 12 n(n2 1) δ n+m,0, [C, L n ] = 0 central element C acts as a scalar central charge c = 1 2κ (3κ 8)(6 κ) (with κ 0) [ Initiated by Belavin, Polyakov, Zamolodchikov (1984) ] 3

7 PDE OPERATORS FROM VIRASORO GENERATORS at fixed z i, to each generator L m one associates a differential operator L (z i) m that acts on correlations of the fields: φ1 (z 1 ) (L m φ i (z i ) ) φ d (z d ) = L (z i) m φ1 (z 1 ) φ d (z d ) where L (z i) m = j i ( 1 + (1 m) h φ j (z j z i ) m 1 z j (z j z i ) m h φ are conformal weights of the fields φ(z): transformation rule φ(z) = ( f (z) ) hφ φ( f (z)) ) for conformal maps f 4

8 SINGULAR VECTORS What is the Vir -action on a field φ(z)? field φ(z) generates a Vir -module M φ = Vir.φ(z) M φ is isomorphic to a quotient of some Verma module V: M φ V/N 5

9 SINGULAR VECTORS What is the Vir -action on a field φ(z)? field φ(z) generates a Vir -module M φ = Vir.φ(z) M φ is isomorphic to a quotient of some Verma module V: M φ V/N Is this quotient the whole Verma module? 5

10 SINGULAR VECTORS What is the Vir -action on a field φ(z)? field φ(z) generates a Vir -module M φ = Vir.φ(z) M φ is isomorphic to a quotient of some Verma module V: M φ V/N Is this quotient the whole Verma module? in degenerate CFTs, N is non-trivial (containing null fields ) elements generating N are called singular vectors: P(L m : m N).φ(z) N where P is a polynomial in the Virasoro generators 5

11 BPZ PARTIAL DIFFERENTIAL EQUATIONS singular vector P(L m : m N).φ(z) N in the quotient M φ V/N gives rise to a PDE with D (z) = P(L (z) m : m N) where L (z i) m = D (z) φ(z)φ 1 (z 1 ) φ d (z d ) = 0 j i ( 1 + (1 m) h φ j (z j z i ) m 1 z j (z j z i ) m ) 6

12 BPZ PARTIAL DIFFERENTIAL EQUATIONS singular vector P(L m : m N).φ(z) N in the quotient M φ V/N gives rise to a PDE with D (z) = P(L (z) m : m N) where L (z i) m = D (z) φ(z)φ 1 (z 1 ) φ d (z d ) = 0 j i ( 1 + (1 m) h φ j (z j z i ) m 1 z j (z j z i ) m ) Benoit, Saint-Aubin (1988): explicit formulas for singular vectors when conformal weights are of type s(2s + 4 κ) 2κ ( = h1,s+1 or h s+1,1 ) s 0 Recall: c = 1 2κ (3κ 8)(6 κ) 6

13 BENOIT & SAINT-AUBIN PARTIAL DIFFERENTIAL EQUATIONS Benoit, Saint-Aubin (1988): explicit formulas for singular vectors when conformal weights are of type s(2s + 4 κ) ( ) = h1,s+1 or h s+1,1 2κ example: h 1,2 = 6 κ 2κ, field φ 1,2 (or φ 2,1 ): ( 3 L 2 2(2h 1,2 + 1) (L 1) 2) φ 1,2 (z) (with translation invariance) gives rise to the PDE κ 2 d ( 2 z + 2 2h ) 1,2 2 z i z z i (z i z) 2 φ1,2 (z)φ 1 (z 1 ) φ d (z d ) = 0 i=1 7

14 BENOIT & SAINT-AUBIN PARTIAL DIFFERENTIAL EQUATIONS Benoit, Saint-Aubin (1988): explicit formulas for singular vectors when conformal weights are of type s(2s + 4 κ) ( ) = h1,s+1 or h s+1,1 2κ example: h 1,2 = 6 κ 2κ, field φ 1,2 (or φ 2,1 ): ( 3 L 2 2(2h 1,2 + 1) (L 1) 2) φ 1,2 (z) (with translation invariance) gives rise to the PDE κ 2 d ( 2 z + 2 2h ) 1,2 2 z i z z i (z i z) 2 φ1,2 (z)φ 1 (z 1 ) φ d (z d ) = 0 i=1 in general: homogeneous PDE operators of degree s + 1 D (z) s+1 = c n1,...,n j (s, κ) L (z) n 1 L (z) n j n n j =s 7

15 BSA PARTIAL DIFFERENTIAL EQUATIONS We seek solutions F ς (z; z) to the PDE system D (z j) s j +1 F ς(z; z) = 0, D ( z j) s j +1 F ς(z; z) = 0, for all j = 1,..., d for variables z = (z 1, z 2,..., z d ) and c.c. z = ( z 1, z 2,..., z d ) 8

16 BSA PARTIAL DIFFERENTIAL EQUATIONS We seek solutions F ς (z; z) to the PDE system D (z j) s j +1 F ς(z; z) = 0, D ( z j) s j +1 F ς(z; z) = 0, for all j = 1,..., d for variables z = (z 1, z 2,..., z d ) and c.c. z = ( z 1, z 2,..., z d ) That is, we seek correlation functions φs1 (z 1 ; z 1 ) φ sd (z d ; z d ) = F ς (z; z)= F (k) (z 1,..., z d ) F (l) ( z 1,..., z d ) where ς = (s 1,..., s d ) φ s have conformal weights of type θ s = s(2s+4 κ) 2κ (i.e. h 1,s+1 or h s+1,1 in Kac table) k,l 8

17 BSA PARTIAL DIFFERENTIAL EQUATIONS We seek solutions F ς (z; z) to the PDE system D (z j) s j +1 F ς(z; z) = 0, D ( z j) s j +1 F ς(z; z) = 0, for all j = 1,..., d for variables z = (z 1, z 2,..., z d ) and c.c. z That is, we seek correlation functions φs1 (z 1 ; z 1 ) φ sd (z d ; z d ) = F ς (z; z) = F (k) (z 1,..., z d ) F (l) ( z 1,..., z d ) where ς = (s 1,..., s d ) φ s have conformal weights of type θ s = s(2s+4 κ) 2κ (i.e. h 1,s+1 or h s+1,1 in Kac table) k,l 8

18 QUESTION:? SOLUTIONS 8

19 QUESTION:? SINGLE-VALUED SOLUTIONS 8

20 CORRELATION FUNCTIONS Benoit & Saint-Aubin PDE system: D (z j) s j +1 F ς(z; z) = 0, D ( z j) s j +1 F ς(z; z) = 0, for all j = 1,..., d covariance: for all conformal maps f, ( n ( ) ( F ς f (z), f ( z) = f (zi ) f ( z i ) ) ) θ si F ς (z, z) i=1 F ς is defined for { (z 1,..., z d ) C d zi z j for i j } 9

21 CORRELATION FUNCTIONS Benoit & Saint-Aubin PDE system: D (z j) s j +1 F ς(z; z) = 0, D ( z j) s j +1 F ς(z; z) = 0, for all j = 1,..., d covariance: for all conformal maps f, ( n ( ) ( F ς f (z), f ( z) = f (zi ) f ( z i ) ) ) θ si F ς (z, z) i=1 F ς is defined for { (z 1,..., z d ) C d zi z j for i j } Theorem [ Flores, P. (2018+) ] Suppose κ (0, 8) \ Q ( so c = 1 2κ (3κ 8)(6 κ) 1 irrational). There exists a unique single-valued, conformally covariant solution F ς (z, z) to the Benoit & Saint-Aubin PDEs. (Uniqueness holds up to normalization in a natural solution space.) 9

22 MONODROMY INVARIANT SINGLE-VALUED 10 correlation function F ς single-valued in W d = { (z 1,..., z d ) C d zi z j for i j } braid group Br d is the fundamental group of W d invariance: σ j.f ς (z, z) = F ς (z, z) j = 1, 2,..., d 1 where σ j are generators of the braid group Br d

23 Theorem [ Flores, P. (2018+) ] Suppose κ (0, 8) \ Q ( so c = 1 2κ (3κ 8)(6 κ) 1 irrational). There exists a unique collection of smooth functions F ς (z, z) with ς = (s 1,..., s d ) satisfying F (1) = 1 and properties PDE system: D (z j) s j +1 F ς(z; z) = 0, D ( z j) s j +1 F ς(z; z) = 0, j = 1,..., d covariance: for all conformal maps f, ( n ( ) ( F ς f (z), f ( z) = f (zi ) f ( z i ) ) ) θ si F ς (z, z) i=1 monodromy invariance: σ j.f = F for all j = 1,..., d 1 power law growth: there exist C > 0 and p > 0 s.t. F ς C max ( (z j z i ) p, (z j z i ) p) 1 i< j d 11

24 FURTHER PROPERTIES Corollary: Asymptotics / OPE: F ς (z; z) Cs s i,s i+1 (z i+1 z i ) θ s i θ si+1 +θ s c.c. z i, z i+1 ζ where s belongs to a finite index set and C s r,t are structure constants s F (s1,...,s,...,s d )(..., z i 1, ζ, z i+2,... ; c.c. ) 12

25 FURTHER PROPERTIES Corollary: Asymptotics / OPE: F ς (z; z) Cs s i,s i+1 (z i+1 z i ) θ s i θ si+1 +θ s c.c. z i, z i+1 ζ where s belongs to a finite index set and Cr,t s are structure constants: ( B s ) 2 Cr,t s r,t B 0 s,s := s B 0 r,rb 0 t,t [n]! = [1][2] [n 1][n] and [n] = sin(4πn/κ) sin(4π/κ) B s r,t := 1 ( r+t s 2 r+t s 2 )! i=1 F (s1,...,s,...,s d )(..., z i 1, ζ, z i+2,... ; c.c. ) ([s]!) 2 [r + 1][s + 1][t + 1][ r+t s 2 ]! [ r+s+t 2 + 1]![ s+r t 2 ]![ t+s r 2 ]! and Γ ( 1 4 (r i + 1)) Γ ( 1 4 (t i + 1)) Γ ( i) κ κ κ Γ ( ( 2 4 r+s+t i + )) 2 Γ ( ) κ 2 κ Compare with the work of Dotsenko & Fateev in the 1980s: they calculated the 4-point function and structure constants 12

26 13 CONSTRUCTION: CONFORMAL BLOCKS Ref. In preparation; see the introduction section 1D in [arxiv: ] Flores, P. Standard modules, Jones-Wenzl projectors, and the valenced Temperley-Lieb algebra. uniqueness: representation theory + knowledge of the solution space of the PDEs ( I ll give some idea later... )

27 13 CONSTRUCTION: CONFORMAL BLOCKS Ref. In preparation; see the introduction section 1D in [arxiv: ] Flores, P. Standard modules, Jones-Wenzl projectors, and the valenced Temperley-Lieb algebra. uniqueness: representation theory + knowledge of the solution space of the PDEs ( I ll give some idea later... ) construction: we can write F(z; z) = U ϱ (z) U ϱ ( z) U ϱ are conformal blocks, which can be written as contour integrals á la Feigin & Fuchs / Dotsenko & Fateev ϱ

28 CONSTRUCTION: CONFORMAL BLOCKS Ref. In preparation; see the introduction section 1D in [arxiv: ] Flores, P. Standard modules, Jones-Wenzl projectors, and the valenced Temperley-Lieb algebra. uniqueness: representation theory + knowledge of the solution space of the PDEs ( I ll give some idea later... ) construction: we can write F(z; z) = U ϱ (z) U ϱ ( z) ϱ U ϱ are conformal blocks, which can be written as contour integrals á la Feigin & Fuchs / Dotsenko & Fateev Example: when ς = (1, 1,..., 1) U α (z) = (z i z j ) 2/κ (w r z j ) 4/κ (w r w s ) 8/κ dw, r<s i< j Γ α where Γ α are certain integration surfaces See [arxiv: ] Karrila, Kytölä, P. Conformal blocks, q-combinatorics, and quantum group symmetry. r j 13

29 14 Theorem [ Karrila, Kytölä, P. (2017) ] case ς = (1, 1,..., 1) Suppose κ (0, 8) \ Q ( so c = 1 2κ (3κ 8)(6 κ) 1 irrational). There exists a unique collection of smooth functions U α (x) ( with x = (x 1 < < x 2N ) and α planar pair partitions of 2N points ) satisfying U = 1 and properties PDE system: 1 j 2N, κ 2 ( 2 + 6/κ 1 ) 2 x 2 x j i x j x i (x i x j ) U α(x 2 1,..., x 2N) = 0 i j covariance: for all (admissible) Möbius maps µ: H H U ( µ(x 1 ),..., µ(x 2N ) ) = j µ (x j ) κ 6 2κ U α (x 1,..., x 2N ) specific asymptotics (related to fusion) power law growth: there exist C > 0 and p > 0 s.t. U α C max ( (x j x i ) p, (x j x i ) p) i< j

30 Theorem [ Karrila, Kytölä, P. (2017) ] case ς = (1, 1,..., 1) Suppose κ (0, 8) \ Q ( so c = 1 2κ (3κ 8)(6 κ) 1 irrational). There exists a unique collection of smooth functions U α (x) ( with x = (x 1 < < x 2N ) and α planar pair partitions of 2N points ) satisfying U = 1 and properties PDE system covariance asymptotics power law growth Furthermore: U α (x 1,..., x 2N ) form basis for the solution space U α give a formula for the unique single-valued (monodromy invariant) correlation function when ς = (1, 1,..., 1) F (1,1,...,1) (z; z) = α U α (z 1,..., z 2N ) U α ( z 1,..., z 2N ) 15

31 WHAT S NEXT? 1. RATIONAL EXAMPLES 2. IDEAS FOR PROOF 15

32 LOOP-ERASED RANDOM WALK (c = 2) GAUSSIAN FREE FIELD (c = 1) 15

33 16 LEVEL LINE OF GAUSSIAN FREE FIELD: SLE 4 (c = 1) GFF with boundary data ±λ: take boundary values λ on the left, +λ on the right zero level line: SLE κ with κ = 4 [Schramm & Sheffield (2003)] O. Schramm & S. Sheffield

34 CONFORMAL BLOCKS FOR GFF (c = 1) U α (x 1,..., x 2N ) = (x j x i ) 1 2 ϑ α(i, j), where ϑ α (i, j) := 1 i< j 2N +1 if i, j {a 1, a 2,..., a N } or i, j {b 1, b 2,..., b N } 1 otherwise x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 λ +λ +3λ +5λ +3λ +λ +3λ +λ +3λ +λ λ Associate boundary data to walks i.e. planar pairings α = {{a 1, b 1 },..., {a N, b N }} where {a 1,..., b N } = {1,..., 2N}, a 1 < a 2 < < a N, and a j < b j for all j {1,..., N} 17

35 CONFORMAL BLOCKS FOR GFF (c = 1) U α (x 1,..., x 2N ) = (x j x i ) 1 2 ϑ α(i, j), where ϑ α (i, j) := 1 i< j 2N +1 if i, j {a 1, a 2,..., a N } or i, j {b 1, b 2,..., b N } 1 otherwise x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 λ +λ +3λ +5λ +3λ +λ +3λ +λ +3λ +λ λ Associate boundary data to walks i.e. planar pairings α = {{a 1, b 1 },..., {a N, b N }} where {a 1,..., b N } = {1,..., 2N}, a 1 < a 2 < < a N, and a j < b j for all j {1,..., N} 17

36 INTERPRETATION OF U α FOR THE GFF LEVEL LINES Level lines form some connectivity and, for any β γ c βγ U γ (x 1,..., x 2N ) P[connectivity = β] = U α (x 1,..., x 2N ) where the coefficients c βγ are explicit (combinatorial) x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 λ +λ +3λ +5λ +3λ +λ +3λ +λ +3λ +λ λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 λ +λ +3λ +5λ +3λ +λ +3λ +λ +3λ +λ λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 λ +λ +3λ +5λ +3λ +λ +3λ +λ +3λ +λ λ x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 λ +λ +3λ +5λ +3λ +λ +3λ +λ +3λ +λ λ λ +λ +3λ +5λ +3λ +λ +3λ +λ +3λ +λ λ [arxiv: ] P., Wu, Global and local multiple SLEs for κ 4 and connection probabilities for level lines of GFF. 18

37 LOOP-ERASED RANDOM WALK (c = 2) GAUSSIAN FREE FIELD (c = 1) 18

38 CONFORMAL BLOCKS FOR LOOP-ERASED RANDOM WALKS Realize walks as branches in a wired uniform spanning tree: Connectivities encoded to planar pairings α = {{a 1, b 1 },..., {a N, b N }} where {a 1,..., b N } = {1,..., 2N}, a 1 < a 2 < < a N, and a j < b j for all j {1,..., N} The conformal blocks for c = 2 (so κ = 2) are combinatorial determinants á la Fomin, involving Poisson kernels: U α (x 1,..., x 2N ) = det ( (x ai x b j ) 2) N i, j=1 [arxiv: ] Karrila, Kytölä, P., Boundary correlations in planar LERW and UST. 19

39 GENERAL METHOD: HIDDEN QUANTUM GROUP SYMMETRY [arxiv: ] / JEMS, to appear Kytölä, P., Conformally covariant boundary correlation functions with a quantum group. 19

40 20 Theorem [ Flores, P. (2018+) ] Suppose κ (0, 8) \ Q ( so c = 1 2κ (3κ 8)(6 κ) 1 irrational). There exists a unique collection of smooth functions F ς (z, z) with ς = (s 1,..., s d ) satisfying F (1) = 1 and properties PDE system: D (z j) s j +1 F ς(z; z) = 0, D ( z j) s j +1 F ς(z; z) = 0, j = 1,..., d covariance: for all conformal maps f, ( n ( ) ( F ς f (z), f ( z) = f (zi ) f ( z i ) ) ) θ si F ς (z, z) i=1 monodromy invariance: σ j.f = F for all j = 1,..., d 1 power law growth: there exist C > 0 and p > 0 s.t. F ς C max ( (z j z i ) p, (z j z i ) p) 1 i< j d

41 HOW TO FIND SOLUTIONS OF PDES D (z j) s j +1 F = 0 for all j = 1,..., d Write solution in integral form & use Stokes theorem : F(z 1,..., z d ) = f (z 1,..., z d ; w 1,..., w l ) dw 1 dw l Γ 21

42 HOW TO FIND SOLUTIONS OF PDES D (z j) s j +1 F = 0 for all j = 1,..., d Write solution in integral form & use Stokes theorem : F(z 1,..., z d ) = f (z 1,..., z d ; w 1,..., w l ) dw 1 dw l find f s.t. for all j D (z j) s j f = total derivative +1 Γ then D (z j) s j f dw = dη is exact l-form +1 21

43 HOW TO FIND SOLUTIONS OF PDES D (z j) s j +1 F = 0 for all j = 1,..., d Write solution in integral form & use Stokes theorem : F(z 1,..., z d ) = f (z 1,..., z d ; w 1,..., w l ) dw 1 dw l Γ find f s.t. for all j D (z j) s j f = total derivative +1 then D (z j) s j f dw = dη is exact l-form +1 find closed l-surface Γ: Γ = then D (z j) s j +1 F = Γ D(z j) s j +1 f dw = Γ dη = Γ η = η = 0 21

44 HOW TO FIND SOLUTIONS OF PDES D (z j) s j +1 F = 0 for all j = 1,..., d Write solution in integral form & use Stokes theorem : F(z 1,..., z d ) = f (z 1,..., z d ; w 1,..., w l ) dw 1 dw l Task: find f and Γ such that: D (z j) s j f = total derivative +1 Γ = D (z j) s j +1 F = 0 Γ 1. How to find f? [Feigin-Fuchs / Dotsenko-Fateev 84, Coulomb gas ]: f = (z j z i ) 2α iα j (w r z i ) 2α α i (w s w r ) 2α α i< j i,r α i = s i κ and α = 2 κ 21 r<s

45 HOW TO FIND SOLUTIONS OF PDES D (z j) s j +1 F = 0 for all j = 1,..., d Write solution in integral form & use Stokes theorem : F(z 1,..., z d ) = f (z 1,..., z d ; w 1,..., w l ) dw 1 dw l Task: find f and Γ such that: D (z j) s j f = total derivative +1 Γ = D (z j) s j +1 F = 0 Γ 1. How to find f? [Feigin-Fuchs / Dotsenko-Fateev 84, Coulomb gas ] 2. How to choose closed Γ s.t. also get conformal covariance monodromy invariance? 21

46 HOW TO FIND SOLUTIONS OF PDES D (z j) s j +1 F = 0 for all j = 1,..., d Write solution in integral form & use Stokes theorem : F(z 1,..., z d ) = f (z 1,..., z d ; w 1,..., w l ) dw 1 dw l Task: find f and Γ such that: D (z j) s j f = total derivative +1 Γ = D (z j) s j +1 F = 0 Γ 1. How to find f? [Feigin-Fuchs / Dotsenko-Fateev 84, Coulomb gas ] 2. How to find Γ? Idea: Use action of quantum group on integration surfaces Γ! 21

47 SPIN CHAIN COULOMB GAS CORRESPONDENCE Theorem [ Kytölä, P. (2016) & Flores, P. (2018+) ] Suppose κ (0, 8) \ Q ( so c = 1 2κ (3κ 8)(6 κ) 1 irrational). We have an embedding of H ς := { v M ς E.v = 0, K.v = v } onto the solution space F : H ς { solutions Γ f (z; w)dw to BSA PDEs } E, F, K generators of U q (sl 2 ) ( = quantum sl 2 (C) ) M ς = M (s1 ) M (sd ), where M (s) is the irreducible type one U q (sl 2 )-module of dimension s + 1 [arxiv: ] Karrila, Kytölä, P. Pure partition functions of multiple SLEs. CMP,

48 SPIN CHAIN COULOMB GAS CORRESPONDENCE Theorem [ Kytölä, P. (2016) & Flores, P. (2018+) ] Suppose κ (0, 8) \ Q ( so c = 1 2κ (3κ 8)(6 κ) 1 irrational). We have an embedding of H ς := { v M ς E.v = 0, K.v = v } onto the solution space F : H ς { solutions Γ f (z; w)dw to BSA PDEs } E, F, K generators of U q (sl 2 ) ( = quantum sl 2 (C) ) M ς = M (s1 ) M (sd ), where M (s) is the irreducible type one U q (sl 2 )-module of dimension s + 1 when ς = (1, 1,..., 1), then the map F is surjective onto conformally covariant solutions to the 2nd order BPZ PDE system D (z i) 2 F = 0 i with at most power-law growth [arxiv: ] Karrila, Kytölä, P. Pure partition functions of multiple SLEs. CMP,

49 QUANTUM GROUP U q (sl 2 ) = QUANTUM sl 2 (C) generators E, F, K, K 1 deformation parameter q = e 4πi/κ e πiq relations KK 1 = 1 = K 1 K KE = q 2 EK, KF = q 2 FK EF FE = K K 1 q q 1 vectors v M (s1 ) M (sd ) functions F[v](z 1,..., z d ) Theorem [ parts 1-3: Kytölä, P. (JEMS 2018); part 4: folklore ] 1 highest weight vectors ( E.v = 0 ) 2 weight ( K.v = q λ v ) 3 projections onto subrepresentations 4 braiding F monodromy F solutions of PDEs F conformal transformation properties F asymptotics 23

50 HIDDEN QUANTUM GROUP SYMMETRY 24 hidden quantum group: U q (sl 2 ) ( = quantum sl 2 (C) ) vectors v M (s1 ) M (sd ) functions F[v](z 1,..., z d ) basis vectors e l1 e ld F[v](z) = Γ(v) f (z; w)dw F l1 integral functions f dw Γ l1,...,l d x1... ln 1 xn 1 ln xn x0 Γ l1,...,ln

51 HIDDEN QUANTUM GROUP SYMMETRY hidden quantum group: U q (sl 2 ) ( = quantum sl 2 (C) ) vectors v M (s1 ) M (sd ) functions F[v](z 1,..., z d ) basis vectors e l1 e ld F[v](z) = Γ(v) f (z; w)dw F l1 integral functions f dw Γ l1,...,l d x1... ln 1 xn 1 ln xn x0 Theorem [ parts 1-3: Kytölä, P. (JEMS 2018); part 4: folklore ] 1 highest weight vectors ( E.v = 0 ) 2 weight ( K.v = q λ v ) 3 projections onto subrepresentations 4 braiding F monodromy F solutions of PDEs F conformal transformation properties F asymptotics 24

52 IRREDUCIBLE REPRESENTATIONS OF U q (sl 2 ) 25 M (s) = span { e 0,..., e s } e 0 is highest weight vector: E.e 0 = 0 and K.e 0 = q s e 0 e 0 generates M (s) : e k = F k.e 0 associate contours to e k : e k = z k F adds a contour: F. z = z E. E removes a contour: z = [k] q [d k] q z

53 26 REPRESENTATION THEORY OF U q (sl 2 ) using coproduct : U q (sl 2 ) U q (sl 2 ) U q (sl 2 ) can define action on tensor products of representations: U q (sl 2 ) M (s1 ) M (s2 ) z 1 z 2 K.(w 1 w 2 ) := (K)(w 1 w 2 ) = K.w 1 K.w 2 E.(w 1 w 2 ) := (E)(w 1 w 2 ) = 1.w 1 E.w 2 + E.w 1 K.w 2 F.(w 1 w 2 ) := (F)(w 1 w 2 ) = F.w 1 1.w 2 + K 1.w 1 F.w 2

54 CALCULATING MONODROMY BRAIDING 27 vectors v M (s1 ) M (sd ) functions F[v](z 1,..., z d ) basis vectors e l1 e ld F integral functions f dw Γ l1,...,l d consider correlation function F(z 1,..., z d ) = F[v](z) = f (z; w)dw Γ f = (z j z i ) 2α iα j (w r z i ) 2α α i (w s w r ) 2α α i< j i,r monodromy can be calculated by contour deformation method: r<s z 1 z 2 σ e πi(...) z2 z1

55 28 CONCLUSION:! SINGLE-VALUED SOLUTION Theorem [ Flores, P. (2018+) ] Suppose κ (0, 8) \ Q ( so c = 1 2κ (3κ 8)(6 κ) 1 irrational). There exists a unique single-valued, conformally covariant solution F ς (z, z) to the Benoit & Saint-Aubin PDEs.

56 CONCLUSION:! SINGLE-VALUED SOLUTION Theorem [ Flores, P. (2018+) ] Suppose κ (0, 8) \ Q ( so c = 1 2κ (3κ 8)(6 κ) 1 irrational). There exists a unique single-valued, conformally covariant solution F ς (z, z) to the Benoit & Saint-Aubin PDEs. Proposition [ Flores, P (2018+) ] There exists a unique braiding invariant vector v H ς H ς. Idea: observe that { braiding invariant vectors in H ς H ς } Hom Brn (H ς, H ς ) use representation theory to prove that dim Hom Brn (H ς, H ς ) = 1 use the Spin chain Coulomb gas correspondence bijection F : H ς H ς { solution space } to conclude with the theorem 28