Generalized FaddeevVolkov models Rinat Kashaev University of Geneva - - PowerPoint PPT Presentation

Generalized Faddeev–Volkov models

Rinat Kashaev

University of Geneva

RAQIS’16 Geneva, August 22-26, 2016

Rinat Kashaev Generalized Faddeev–Volkov models

Motivation: special functions and integrable lattice models with infinitely many local states

Notable examples The quantum dilogarithm: the Faddeev–Volkov model (Bazhanov–Mangazeev–Sergeev) The elliptic beta function: a master solution of the quantum Yang–Baxter equation (Spiridonov, Bazhanov–Sergeev)

Rinat Kashaev Generalized Faddeev–Volkov models

Review of Faddeev’s quantum dilogarithm

Φb (x) :=

∞

• k=0

1 + e2πbx+πib2(2k+1) 1 + e2πb−1x−πib−2(2k+1) , Im(b2) > 0 Integral representations Φb(x) = exp

• R+i0

e−i2xz 4 sinh(zb) sinh(zb−1)z dz

• (Faddeev)

Φb (x) = exp i 2π

• R

log

• eb2t+2πbx + 1
• dt

et + 1

• (Woronowicz)

⇒ extension to b ∈ C \ iR

Rinat Kashaev Generalized Faddeev–Volkov models

Functional equations:

Φb(z−is) Φb(z+is) = 1 + e4πsz,

s = b±1

2

Poles: z =

• m + 1

2

• ib +
• n + 1

2

• ib−1,

m, n ∈ Z≥0 Zeros = − Poles Unitarity: (1 − |b|) Im b = 0 ⇒ |Φb (x) | = 1 ∀x ∈ R Inversion relation: Φb(x)Φb(−x) = Φb (0)2 eiπx2 Pentagon identity: Φb(p)Φb(q) = Φb(q)Φb(p + q)Φb(p) for self-adjoint Heisenberg operators pq − qp =

1 2πi

(2π = 1)

Rinat Kashaev Generalized Faddeev–Volkov models

The weight function of the Faddeev–Volkov model

WFV (x, y) = Φb (x − y) Φb (x + y)e2πixy Symmetry properties: WFV (x, y) = 1 WFV (x, −y) = WFV (−x, y) The Yang–Baxter relation: WFV (p, x) WFV (q, x + y) WFV (p, y) = WFV (q, y) WFV (p, x + y) WFV (q, x)

Rinat Kashaev Generalized Faddeev–Volkov models

Generalization to arbitrary self-dual LCA groups

Let A be a LCA (locally compact abelian) group with the (Pontryagin) dual group ˆ A = all continuous group homomorphisms from A to the complex circle group T := {z ∈ C : |z| = 1}. A is self-dual if there exists a group isomorphism f : A → ˆ

• A. Assume

that there exists a function ·: A → T (to be called gaussian exponential) such that x = −x, x; y := f (x)(y) = x + y xy Normalized Haar measure dµ(x):

• A2x; y dµ(x) dµ(y) = 1

Examples

1 A = R, x = eπiαx2, x; y = e2πiαxy, dµ(x) =

• |α| dx

2 A = Z/NZ, m = eπi M N m(m+N), m; n = e2πi M N mn,

dµ(m) = 1/ √ N

3 A = T × Z, (z, m) = z±m, (u, m); (v, n) = (unvm)±1,

dµ(e2πit, m) = dt

Rinat Kashaev Generalized Faddeev–Volkov models

The quantum dilogarithm over a self-dual LCA group

Let A be a self-dual LCA group with a gaussian exponential .. The Fourier transformation operator F in L2(A) is defined by the integral kernel x|F|y = x; y. To any function g : A → C, associate operators g(q), g(p) and g(p + q) in L2(A): (g(q)f )(x) = g(x)f (x), ∀f ∈ L2(A), g(p) := Fg(q)F−1, g(p + q) := q−1g(p)q. Definition (Andersen–K) A quantum dilogarithm over (A, .) is a function φ: A → T such that (inversion relation) φ(x)φ(−x) = φ(0)2x, ∀x ∈ A, (pentagon relation) φ(p)φ(q) = φ(q)φ(p + q)φ(p).

Rinat Kashaev Generalized Faddeev–Volkov models

Remark The pentagon relation is equivalent to the integral relation φ(x)φ(y) =

• A3 D(u, x, v, y, w)φ(w)φ(v)φ(u) dµ(u, v, w)

D(u, x, v, y, w) := γ u − x; w − y u − v + w γ :=

• A

z dµ(z)

Rinat Kashaev Generalized Faddeev–Volkov models

Examples

1 A = R, x = eπix2, φ(x) = Φb (x), (1 − |b|) Im b = 0 2 A = T × Z, (z, m) = zm 1

φ(z, m) = zmax(m,0)

2

φ(z, m) = ∞

j=0 1+q1−m+2jz 1+q1−m+2j/z , q ∈] − 1, 1[, related to the

tetrahedron index of Dimofte–Gaiotto–Gukov

3 A = R × Z/kZ, (x, m) = eπix2e−πim(m+k)/k,

φ(x, m) =

k−1

• j=0

Φeiθ (aj(x, m)) aj(x, m) := x √ k + (1 − k) cos θ + je−iθ ik − ieiθ j + m k

• θ ∈ [0, π/2[, {x} := the fractional part of x.

Rinat Kashaev Generalized Faddeev–Volkov models

Generalized Faddeev–Volkov weight function

Let φ(x) be a quantum dilogarithm over a self-dual LCA group (A, .). The associated Faddeev–Volkov weight function is defined by the formula w(x, y) := φ(x − y) φ(x + y)x; y Symmetry properties w(x, y) = 1 w(x, −y) = w(−x, y) Yang–Baxter relation w(p, x) w(q, x + y) w(p, y) = w(q, y) w(p, x + y) w(q, x)

Rinat Kashaev Generalized Faddeev–Volkov models

The standard interpretation as a star-triangle relation

• A

˜ w(u − t, x) w(t, x + y) ˜ w(t − s, y) dµ(t) = w(u, y) ˜ w(u − s, x + y) w(s, x) ˜ w(y, x) :=

• A

y; z w(z, x) dµ(z)

Rinat Kashaev Generalized Faddeev–Volkov models

Weil transformation

Let B ⊂ A be a closed subgroup such that B = B⊥ := {x ∈ A | x; b = 1, ∀b ∈ B} We have a group homomorphism p: A → ˆ B, p(x)(b) = x; b, inducing a natural group isomorphism A/B ≃ ˆ

• B. Define the Weil

transformation of w(x, y): ˇ w(s, t, x) :=

• B

w(s + b, x) t; b dµ(b) The inverse Weil transformation w(s, x) =

• A/B

ˇ w(s, t, x) dµ(t) Quasi B-periodicity properties ˇ w(s + b, t, x) = −b; t ˇ w(s, t, x) , ˇ w(s, t + b, x) = ˇ w(s, t, x) ⇒ ˇ w(s, t, x) is a section of a complex line bundle over (A/B)2.

Rinat Kashaev Generalized Faddeev–Volkov models

Interpretation as IRF-model with abelian gauge symmetry

Let χ: A2 → T be a bi-character such that x; y = χ(x, y)χ(y, x). Then M(x, y, z) := χ(x, y) ˇ w(x, y, z) is quasi B-periodic M(x, y+b, z) = χ(x, b) M(x, y, z), M(x+b, y, z) = ¯ χ(y, b) M(x, y, z) and satisfies a Yang–Baxter relation of IRF-type

• A/B

a b c d e f g α β γ dµ(g) =

• A/B

a b c d e f h α β γ dµ(h) a b c d α β := M(a − c, d − b, α − β)

Rinat Kashaev Generalized Faddeev–Volkov models

Example: the IRF-form of the Faddeev–Volkov model

A = R, x = eπix2, φ(x) = Φb (x) B = Z ⊂ R = A, A/B = R/Z ≃ T ≃ ˆ Z = ˆ B x; y = e2πixy, χ(x, y) = eπixy M(x, y, z)φ(0)2 = eπi(xy−x2−z2) 1 ˇ φ(x − z, u + y − x − 1/2)ˇ φ(−x − z, u) du with the Weil–Gelfand–Zak transformation of the quantum dilogarithm ˇ φ(x, y) =

• k∈Z

φ(x + k)e2πiky

Rinat Kashaev Generalized Faddeev–Volkov models

The special value b = eπi/6

ˇ φ(x, y) = (q; q)∞

• −e2πix; q
• ∞ θq
• −e2π(¯

bx−iy)

(e−2πiy; q)∞

• −e2πi(x+y); q
• ∞ θq
• −e2π¯

bx

• where q := e2πib2 = ie−π

√ 3,

(x; q)∞ :=

∞

• n=0

(1 − xqn), θq(x) :=

• k∈Z

q

k2 2 xk = (q; q)∞ (−√qx; q)∞ (−√q/x; q)∞ Rinat Kashaev Generalized Faddeev–Volkov models