6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories

Transcription

1 6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories G. S. Vartanov in collaboration with J. Teschner DESY, Hamburg arxiv: String Math 2012, 18 July 2012

2 Introduction fusion kernel/b-6j symbol; relation to 3d hyperbolic geometry; application to SuSy theories;

3 Fusion kernel I V α4 (z 4, z 4 )V α3 (z 3, z 3 )V α2 (z 2, z 2 )V α1 (z 1, z 1 ) = = dα s C(α 4,α 3,α s )C(Q α s,α 2,α 1 )F α (s) s (A Z)F α (s) s (A Z) Q/2+iR = dα t C(α 4,α t,α 1 )C(Q α t,α 3,α 2 )F α (t) t (A Z)F α (t) t (A Z) Q/2+iR where A = (α 1,α 2,α 3,α 4 ), Z = (z 1, z 2, z 3, z 4 ), and DOZZ 3-point function is C(α 1,α 2,α 3 ) = (πµγ(b 2 )b 2 2b2 ) 1 b (Q α 1 α 2 α 3 ) Υ 0 Υ(2α 1 )Υ(2α 2 )Υ(2α 3 ) Υ(α 1 +α 2 +α 3 Q)Υ(α 1 +α 3 α 2 )Υ(α 1 +α 2 α 3 )Υ(α 2 +α 3 α 1 ), Υ(x) = Γ b (x)γ b (Q x) and Q = b + b 1.

4 Fusion kernel II α 2 α 3 α 2 α 3 α s = [ dα t F α3 ] α αsα t α 2 4 α 1 α t α 1 α 4 α 1 α 4

5 Fusion kernel III F (s) α s (A Z) = Q/2+iR where F -kernel is Ponsot, Teschner 99; Teschner 01 where F αsα t [ α3 α 4 α 2 dµ(α t ) F αsα t [ α3 α 4 α 2 α 1 ] F (t) α t (A Z), Teschner 01 ] N(α s,α 2,α 1 )N(α 4,α 3,α s ) α 1 = N(α t,α 3,α 2 )N(α 4,α t,α 1 ) { α1 α 3 α 2 α 4 αs α t }PT, Γ N(α 3,α 2,α 1 ) = b (2Q 2α 3 )Γ b (2α 2 )Γ b (2α 1 ) Γ b (2Q α 1 α 2 α 3 )Γ b (Q α 1 α 2 +α 3 ) 1 Γ b (α 1 +α 3 α 2 )Γ b (α 2 +α 3 α 1 ).

6 Special functions I The function Γ b (x) is closely related to the double Barnes function logγ b (x) = 0 ( dt t e xt e Qt/2 (1 e bt )(1 e t/b ) 2x)2 (Q 8e t Q 2x ) t. Important properties of Γ b (x) are functional equation Γ b (x + b) = 2πb bx 1 2Γ 1 (bx)γ b (x). analyticity Γ b (x) is meromorphic, poles x = nb mb 1, n, m Z 0.

7 Special functions II Double Sine functions satisfies self duality functional equation S b (x) = Γ b(x) Γ b (Q x), S b (x) = S b 1(x), S b (x + b ±1 ) = 2 sin(πb ±1 x) S b (x), reflection property S b (x) S b (Q x) = 1.

8 Fusion kernel IV { α1 α 2 α 3 ᾱ 4 αs C α t } PT = S b(α 2 +α s α 1 )S b (α t +α 1 α 4 ) S b (α 2 +α t α 3 )S b (α s +α 3 α 4 ) du S b ( α 2 ±(α 1 Q/2)+u)S b ( α 4 ±(α 3 Q/2)+u) S b (α 2 +α 4 ±(α t Q/2) u)s b (Q ±(α s Q/2) u). Here, S b (α±u) := S b (α+u)s b (α u). A contour C approaches Q + ir near infinity, and passes the real axis in (Q/2, Q).

9 Volume of non-ideal tetrahedron I A non-ideal tetrahedron defined by 6 dihedral angles η i, i = 1,...,6 η 1 η 2 η3 η 4 η 5 η 6

10 Volume of non-ideal tetrahedron II The volume Murakami Yano 05 Vol(A) = 1 2 Im[ U(u +, A)+ (A) ] = 1 2 Im[ U(u, A)+ (A) ], where A k = e iη k and u ± are the two roots du(u, A) du = 2πi u. Here U(u, A) = Li 2 (u)+li 2 (A st13 u)+ Li 2 (A st24 u)+li 2 (A 1234 u) Li 2 ( A 12s u) Li 2 ( A s34 u) Li 2 ( A 4t1 u) Li 2 ( A 32t u), where A ijk := A i A j A k, A ijkl := A i A j A k A l.

11 New integral representation A new integral representation Teschner, GV 12 { α1 α 2 α s α 3 α 4 where α t }b = (α s,α 2,α 1 ) (α 4,α 3,α s ) (α t,α 3,α 2 ) (α 4,α t,α 1 ) du S b (u α 12s )S b (u α s34 )S b (u α 23t )S b (u α 1t4 ) C S b (α 1234 u)s b (α st13 u)s b (α st24 u)s b (2Q u), α ijk = α i +α j +α k, α ijkl = α i +α j +α k +α l. (α 3,α 2,α 1 ) ( = S b (α 1 +α 2 +α s Q) S b (α 1 +α 2 α s )S b (α 1 +α s α 2 )S b (α 2 +α s α 1 ) a contour C which approaches 2Q + ir near infinity, and passes the real axis in the interval (3Q/2, 2Q). ) 1 2.

12 Semiclassical limit Let us reparameterize variables e 2πibα k+πi A k, k {1, 2, 3, 4, s, t}. Introducing v := 2πb(u Q/2) we get an integral of the form dv I = D(α) J(a, b; v) 2πb C Q/2 whose integrand J(a, b; v) has quasi-classical asymptotics ( ) 1 ( J(a, b; v) = exp 2πb 2 U(eiv, A) 1+O(b )) 2, since S b (x) for b 0 is given as ( ν ) S b 2πb ( = e i 2πb 2( 1 4 ν2 π 2 ν+ 1 6 π2 ) exp 1 ) ( 2πib 2 Li 2(e iν ) 1+O(b )) 2.

13 b-6j symbol, identities b-6j symbol satisfies Q/2+iR + dµ(δ 1 ) { α 1 α 3 α 2 β 2 β 1 δ 1 }b = { β 1 α 4 α 3 α 5 β 2 γ 1 }b { α1 γ 1 α 2 α 5 β 1 Q/2+iR + dµ(α s ) { α 1 α 3 α 2 α 4 α s α t } b { α1 { δ 1 β 2 α2 α 3 δ 1 α 4 α 5 γ 2 }b α 4 γ 2 γ 1 }b γ 2 }b, { α1 α 2 α s α 3 α 4 α t } b = (M(α t)) 1 δ(α t α t), which together with the semiclassical limit b 0 suggests Turaev-Viro type state-sum models.

14 3d gauge theories on duality walls I Recently a nice generalization of AGT duality was proposed Drukker, Gaiotto, Gomis 10: one may consider two four-dimensional theories from class S (4d N = 2 SYM theories) on the upperand lower semispheres of S 4, respectively, coupled to a three-dimensional theory on the defect S 3 separating the two semi-spheres. dα s dα t (G α (s) s (A Z)) [ ] G αsα t... G (t) α t (A Z ). (Q/2+iR) 2

15 PF for 3d N = 2 SuSy theories PF on S 3 Kapustin, Willett, Yaakov 10, Jafferis 10; Hama, Hosomichi, Lee 10 and PF on S 3 b Hama, Hosomichi, Lee 11 Z(f) = i i rank G j=1 du j J(u)Z vec (u) I ZΦ chir I (f, u). Here f k are mass parameters of matter while u j -variables Weyl weights for the Cartan subalgebra of the gauge group G. For Chern-Simons theories one has J(u) = e πik rank G j=1 uj 2, where k is the level of CS-term, and for SYM theories one has J(u) = e 2πiλ rank G j=1 u j, where λ is the Fayet-Illiopoulos term. Z vec SU(2) (u) = 1 S b (±2u).

16 3d gauge theories on duality walls, N = 2 I Explicitly the above idea was checked only for 4d N = 2 SYM theory where on the domain wall the so-called T[SU(2)] theory lives. Explicit check of DGG idea Hosomichi, Lee, Park 10. 3d 3d relation Terashima, Yamazaki 11; Dimofte, Gukov 11; Dimofte, Gaiotto, Gukov 11. Here, G α (s) s (A Z), G α (t) t (A Z) are one-point CBs on a torus and G αsα t is the so-called S-kernel transformation Teschner 03 i S = 23/2 S b (Q/4 µ+m/2±z) S b (m) i S b (3Q/4 µ m/2±z) e4πiξz dz

17 3d gauge theories on duality walls, N = 2 II There is an alternative interpretation in terms of 3d N = 2 CS theory with SU(2) gauge group and 4 quarks Spiridonov, GV 11; Teschner, GV 12 i i S b (Q/4 µ+m/2±z) S b (3Q/4 µ m/2±z) e4πiξz dz = 1 ( Q 2 e2πi(ξ2 4 + m 2 )2 +µ 2 ) S b (Q/2 m±2ξ) i i S b ( Q 4 + m 2 ±µ±ξ ± y) e 2πiy 2 dy. S b (±2y)

18 3d gauge theories on duality walls, N f = 4 I Now we consider 4d N = 2 SYM with SU(2) gauge group and N f = 4 hypermultiplets. dα s dα t (G α (s) s (A Z)) [ G α3 ] α 2 αsα t α 4 α 1 G (t) α t (A Z ). (Q/2+iR) 2 G (s) α s (A Z), G (t) α t (A Z) are four-point CBs on a sphere and G αsα t is the F -kernel.

19 3d gauge theories on duality walls, N f = 4 II The defined F -kernel/b-6j symbol is equal to Teschner, GV 12 ( Q αt α 1 α 4 3Q α A 1 I 2 +α t α 1 α 4 Q+α s 2 α 1 α 4 +α t s 2 α 3 Q α 1 +α 4 +α t Q α 2 +α 1 +α 4 +α t Q+α 2 2 α 1 α 4 +α t 2 2 +α 3 where ( µ1 µ I 2 µ 3 µ 4 µ 5 µ 6 ) = 1 2 i i 6 i=1 S b(µ i ±α) dα. S b (±2α) which is PF for 3d N = 2 SYM theory with SU(2) gauge group and 6 quarks. ),

20 Conclusion we found a new integral representation for b-6j symbol which 1. defines a natural normalization from Liouville and Teichmüller theory (quantization of the Fenchel-Nielsen coordinates Teschner s talk); 2. in a semiclassical limit reproduces the volume of a non-ideal tetrahedron; 3d N = 2 non-abelian theory arising on a domain wall of 4d N = 2 SYM theory with SU(2) gauge group and N f = 4 was identified; we construct 3d 3d starting from non-ideal tetrahedrons.