Lens Hyperbolic Gamma Solution

• Lens hyperbolic gamma solution is the lens-space extension of hyperbolic gamma functions that generate sum–integral identities used in supersymmetric partition functions and quantum integrability.
• It intertwines continuous and discrete structures via Z₍r₎-valued labels, yielding finite-difference operators and star–triangle relations essential to lattice models.
• The construction underpins gauge theoretic dualities on squashed lens spaces and leads to the novel lens hyperbolic modular double, enriching the study of exactly solvable models.

Searching arXiv for papers on the lens hyperbolic gamma function and related integrability/algebraic structures. The lens hyperbolic gamma solution is the lens-space extension of hyperbolic hypergeometric constructions that appear simultaneously in supersymmetric partition functions, exactly solvable lattice models, and modular-double representation theory. In this setting, “lens” refers not to an optical device but to the squashed lens space $S_b^3/\mathbb Z_r$ and, algebraically, to the presence of a continuous variable together with a discrete $\mathbb Z_r$-valued label. The central objects are the hyperbolic gamma function $\gamma^{(2)}$, its lens-space refinement $\gamma_h(u,y)$, and the sum–integral identities built from them. These identities furnish solutions of the star-triangle and star-star relations, while an intertwining integral operator built from the same kernel realizes a lens hyperbolic hypergeometric solution of the Yang–Baxter equation (Bülbül et al., 9 Nov 2025, Mullahasanoglu et al., 2021).

1. Basic special-function structure

At the foundational level, the construction uses the standard hyperbolic gamma function

$$\gamma^{(2)}(u;\omega_1,\omega_2) = e^{B_{2,2}(u;\omega_1,\omega_2)} \frac{(e^{2\pi i u/\omega_1}\widetilde q;\widetilde q)_\infty}{(e^{2\pi i u/\omega_2};q)_\infty}, \qquad (t;q)_\infty=\prod_{k=0}^{\infty}(1-tq^k),$$

with

$$q=e^{2\pi i\omega_1/\omega_2}, \qquad \widetilde q=e^{-2\pi i\omega_2/\omega_1},$$

and

$$B_{2,2}(u;\omega_1,\omega_2) = \frac{1}{\omega_1\omega_2} \left( \left(u-\frac{\omega_1+\omega_2}{2}\right)^2 -\frac{\omega_1^2+\omega_2^2}{12} \right).$$

A parallel presentation used in the lens-partition-function literature writes

$$\gamma^{(2)}(z;\omega_{1},\omega_{2}) = e^{\frac{\pi i}{2}B_{2,2}(z;\omega_{1},\omega_{2})} \frac{(e^{2\pi i z/\omega_{2}};\tilde q)_\infty}{(e^{2\pi i z/\omega_{1}};q)_\infty},$$

with the reflection identity

$$\gamma^{(2)}(\omega_1+\omega_2-z;\omega_1,\omega_2)\gamma^{(2)}(z;\omega_1,\omega_2)=1.$$

These equivalent formulations encode the same hyperbolic special-function input (Bülbül et al., 9 Nov 2025, Mullahasanoglu et al., 2021).

The lens refinement is the lens hyperbolic gamma function

$$\gamma_h(u,y) = \gamma^{(2)}(u+\omega_1 y;\omega_1 r,\omega_1+\omega_2)\, \gamma^{(2)}(u+\omega_2(r-y);\omega_2 r,\omega_1+\omega_2),$$

where the discrete variable $\mathbb Z_r$0 records the $\mathbb Z_r$1 sector. Its first-order shift relations are

$\mathbb Z_r$2

and

$\mathbb Z_r$3

These relations are structurally decisive because they turn integral kernels into finite-difference operators. A plausible implication is that the lens construction is not merely a discrete decoration of the hyperbolic theory; it reorganizes the analytic structure so that continuous and discrete shifts are intertwined from the outset (Bülbül et al., 9 Nov 2025).

2. Gauge-theoretic origin on squashed lens spaces

A major origin of the lens hyperbolic gamma solution is the partition function of $\mathbb Z_r$4, $\mathbb Z_r$5 gauge theories on the squashed lens space

$\mathbb Z_r$6

The squashed three-sphere is described by

$\mathbb Z_r$7

and the lens quotient is

$\mathbb Z_r$8

The resulting partition function takes the form

$\mathbb Z_r$9

with measure

$\gamma^{(2)}$0

and holonomy

$\gamma^{(2)}$1

The lens character is visible in the simultaneous presence of the integral over Cartan variables and the sum over discrete holonomies (Mullahasanoglu et al., 2021).

The one-loop determinants are written in terms of the improved double sine function $\gamma^{(2)}$2,

$\gamma^{(2)}$3

with

$\gamma^{(2)}$4

The vector and chiral multiplet contributions are

$\gamma^{(2)}$5

and

$\gamma^{(2)}$6

where $\gamma^{(2)}$7. Equality of such lens partition functions under Seiberg-like dualities yields hyperbolic hypergeometric integral identities, and through the gauge/YBE correspondence those identities become integrability relations for lattice models (Mullahasanoglu et al., 2021).

3. Hyperbolic hypergeometric identities

The phrase “solution” most naturally refers to a class of sum–integral identities whose kernels are built from the lens hyperbolic gamma function or its $\gamma^{(2)}$8-based equivalent. A central example is the identity associated with the $\gamma^{(2)}$9 / $\gamma_h(u,y)$0 dual pair, with balancing conditions

$\gamma_h(u,y)$1

Its left-hand side is a discrete sum over $\gamma_h(u,y)$2 and a continuous integral over $\gamma_h(u,y)$3, weighted by products of $\gamma_h(u,y)$4 functions with parameters $\gamma_h(u,y)$5 and $\gamma_h(u,y)$6, while the right-hand side factorizes into a product over pairs $\gamma_h(u,y)$7. In the gauge-theory interpretation, this is the equality of lens partition functions for Theory A, an $\gamma_h(u,y)$8 gauge theory with six chiral multiplets and flavor symmetry $\gamma_h(u,y)$9, and Theory B, a gauge-singlet theory with 15 chiral multiplets in the antisymmetric tensor of $$\gamma^{(2)}(u;\omega_1,\omega_2) = e^{B_{2,2}(u;\omega_1,\omega_2)} \frac{(e^{2\pi i u/\omega_1}\widetilde q;\widetilde q)_\infty}{(e^{2\pi i u/\omega_2};q)_\infty}, \qquad (t;q)_\infty=\prod_{k=0}^{\infty}(1-tq^k),$$0 (Mullahasanoglu et al., 2021).

A second, more directly named result is the new star-star identity with eight chiral parameters. It is written as a pair of sum–integral expressions related by explicit products of $$\gamma^{(2)}(u;\omega_1,\omega_2) = e^{B_{2,2}(u;\omega_1,\omega_2)} \frac{(e^{2\pi i u/\omega_1}\widetilde q;\widetilde q)_\infty}{(e^{2\pi i u/\omega_2};q)_\infty}, \qquad (t;q)_\infty=\prod_{k=0}^{\infty}(1-tq^k),$$1-factors and shifted parameters

$$\gamma^{(2)}(u;\omega_1,\omega_2) = e^{B_{2,2}(u;\omega_1,\omega_2)} \frac{(e^{2\pi i u/\omega_1}\widetilde q;\widetilde q)_\infty}{(e^{2\pi i u/\omega_2};q)_\infty}, \qquad (t;q)_\infty=\prod_{k=0}^{\infty}(1-tq^k),$$2

$$\gamma^{(2)}(u;\omega_1,\omega_2) = e^{B_{2,2}(u;\omega_1,\omega_2)} \frac{(e^{2\pi i u/\omega_1}\widetilde q;\widetilde q)_\infty}{(e^{2\pi i u/\omega_2};q)_\infty}, \qquad (t;q)_\infty=\prod_{k=0}^{\infty}(1-tq^k),$$3

where

$$\gamma^{(2)}(u;\omega_1,\omega_2) = e^{B_{2,2}(u;\omega_1,\omega_2)} \frac{(e^{2\pi i u/\omega_1}\widetilde q;\widetilde q)_\infty}{(e^{2\pi i u/\omega_2};q)_\infty}, \qquad (t;q)_\infty=\prod_{k=0}^{\infty}(1-tq^k),$$4

and the balancing conditions are

$$\gamma^{(2)}(u;\omega_1,\omega_2) = e^{B_{2,2}(u;\omega_1,\omega_2)} \frac{(e^{2\pi i u/\omega_1}\widetilde q;\widetilde q)_\infty}{(e^{2\pi i u/\omega_2};q)_\infty}, \qquad (t;q)_\infty=\prod_{k=0}^{\infty}(1-tq^k),$$5

For $$\gamma^{(2)}(u;\omega_1,\omega_2) = e^{B_{2,2}(u;\omega_1,\omega_2)} \frac{(e^{2\pi i u/\omega_1}\widetilde q;\widetilde q)_\infty}{(e^{2\pi i u/\omega_2};q)_\infty}, \qquad (t;q)_\infty=\prod_{k=0}^{\infty}(1-tq^k),$$6, this reduces to the hyperbolic identity of Sarkissian et al. The appearance of an eight-parameter star-star relation is important because it extends the catalog of known hyperbolic hypergeometric solutions beyond the more familiar star-triangle class (Mullahasanoglu et al., 2021).

In the later modular-double formulation, the same analytic content is recast as a hyperbolic beta-type integral identity

$$\gamma^{(2)}(u;\omega_1,\omega_2) = e^{B_{2,2}(u;\omega_1,\omega_2)} \frac{(e^{2\pi i u/\omega_1}\widetilde q;\widetilde q)_\infty}{(e^{2\pi i u/\omega_2};q)_\infty}, \qquad (t;q)_\infty=\prod_{k=0}^{\infty}(1-tq^k),$$7

$$\gamma^{(2)}(u;\omega_1,\omega_2) = e^{B_{2,2}(u;\omega_1,\omega_2)} \frac{(e^{2\pi i u/\omega_1}\widetilde q;\widetilde q)_\infty}{(e^{2\pi i u/\omega_2};q)_\infty}, \qquad (t;q)_\infty=\prod_{k=0}^{\infty}(1-tq^k),$$8

with

$$\gamma^{(2)}(u;\omega_1,\omega_2) = e^{B_{2,2}(u;\omega_1,\omega_2)} \frac{(e^{2\pi i u/\omega_1}\widetilde q;\widetilde q)_\infty}{(e^{2\pi i u/\omega_2};q)_\infty}, \qquad (t;q)_\infty=\prod_{k=0}^{\infty}(1-tq^k),$$9

This is described as the “lens hyperbolic hypergeometric” identity and functions as the analytic core of the Yang–Baxter solution (Bülbül et al., 9 Nov 2025).

4. Statistical-mechanical interpretation

Through the gauge/YBE correspondence, the same identities define Boltzmann weights for exactly solvable lattice models. The spins are

$$q=e^{2\pi i\omega_1/\omega_2}, \qquad \widetilde q=e^{-2\pi i\omega_2/\omega_1},$$0

so each spin carries both a continuous and a discrete component. The horizontal edge weight is

$$q=e^{2\pi i\omega_1/\omega_2}, \qquad \widetilde q=e^{-2\pi i\omega_2/\omega_1},$$1

and the vertical edge weight is

$$q=e^{2\pi i\omega_1/\omega_2}, \qquad \widetilde q=e^{-2\pi i\omega_2/\omega_1},$$2

with

$$q=e^{2\pi i\omega_1/\omega_2}, \qquad \widetilde q=e^{-2\pi i\omega_2/\omega_1},$$3

The self-interaction is

$$q=e^{2\pi i\omega_1/\omega_2}, \qquad \widetilde q=e^{-2\pi i\omega_2/\omega_1},$$4

This makes the lens hyperbolic gamma function the direct special-function input of the lattice model rather than a posteriori normalization factor (Bülbül et al., 9 Nov 2025).

Integrability is encoded in the star-triangle relation

$$q=e^{2\pi i\omega_1/\omega_2}, \qquad \widetilde q=e^{-2\pi i\omega_2/\omega_1},$$5

where the right-hand side is the product over the three pairs $$q=e^{2\pi i\omega_1/\omega_2}, \qquad \widetilde q=e^{-2\pi i\omega_2/\omega_1},$$6 and $$q=e^{2\pi i\omega_1/\omega_2}, \qquad \widetilde q=e^{-2\pi i\omega_2/\omega_1},$$7 is a spin-independent normalization. In the gauge-theory presentation, satisfaction of the star-triangle relation implies that transfer matrices commute, while the star-star relation implies an IRF-type Yang–Baxter equation. The discrete prefactor

$$q=e^{2\pi i\omega_1/\omega_2}, \qquad \widetilde q=e^{-2\pi i\omega_2/\omega_1},$$8

except at $$q=e^{2\pi i\omega_1/\omega_2}, \qquad \widetilde q=e^{-2\pi i\omega_2/\omega_1},$$9 and $$B_{2,2}(u;\omega_1,\omega_2) = \frac{1}{\omega_1\omega_2} \left( \left(u-\frac{\omega_1+\omega_2}{2}\right)^2 -\frac{\omega_1^2+\omega_2^2}{12} \right).$$0, where $$B_{2,2}(u;\omega_1,\omega_2) = \frac{1}{\omega_1\omega_2} \left( \left(u-\frac{\omega_1+\omega_2}{2}\right)^2 -\frac{\omega_1^2+\omega_2^2}{12} \right).$$1, is a specifically lens-space feature of these constructions (Mullahasanoglu et al., 2021).

A common misconception is to interpret the adjective “lens” geometrically in the optical sense. In this context, the term instead refers to the quotient structure $$B_{2,2}(u;\omega_1,\omega_2) = \frac{1}{\omega_1\omega_2} \left( \left(u-\frac{\omega_1+\omega_2}{2}\right)^2 -\frac{\omega_1^2+\omega_2^2}{12} \right).$$2 and to the associated discrete holonomy sectors. The “solution” is therefore a hyperbolic hypergeometric or Yang–Baxter solution indexed by lens-space data, not a solution describing physical refraction.

5. The lens hyperbolic modular double

The algebraic culmination of the subject is the construction of the lens hyperbolic modular double, introduced as a new algebraic structure whose intertwining operator produces a lens hyperbolic hypergeometric solution of the Yang–Baxter equation. It is a lens-space generalization of the hyperbolic modular-double framework, and in the limit $$B_{2,2}(u;\omega_1,\omega_2) = \frac{1}{\omega_1\omega_2} \left( \left(u-\frac{\omega_1+\omega_2}{2}\right)^2 -\frac{\omega_1^2+\omega_2^2}{12} \right).$$3 it collapses to the standard hyperbolic modular double (Bülbül et al., 9 Nov 2025).

Its key operator is the intertwiner $$B_{2,2}(u;\omega_1,\omega_2) = \frac{1}{\omega_1\omega_2} \left( \left(u-\frac{\omega_1+\omega_2}{2}\right)^2 -\frac{\omega_1^2+\omega_2^2}{12} \right).$$4, acting on functions of a continuous variable and a discrete $$B_{2,2}(u;\omega_1,\omega_2) = \frac{1}{\omega_1\omega_2} \left( \left(u-\frac{\omega_1+\omega_2}{2}\right)^2 -\frac{\omega_1^2+\omega_2^2}{12} \right).$$5-label. The kernel is

$$B_{2,2}(u;\omega_1,\omega_2) = \frac{1}{\omega_1\omega_2} \left( \left(u-\frac{\omega_1+\omega_2}{2}\right)^2 -\frac{\omega_1^2+\omega_2^2}{12} \right).$$6

The kernel is symmetric under the lens reflection

$$B_{2,2}(u;\omega_1,\omega_2) = \frac{1}{\omega_1\omega_2} \left( \left(u-\frac{\omega_1+\omega_2}{2}\right)^2 -\frac{\omega_1^2+\omega_2^2}{12} \right).$$7

which yields, as $$B_{2,2}(u;\omega_1,\omega_2) = \frac{1}{\omega_1\omega_2} \left( \left(u-\frac{\omega_1+\omega_2}{2}\right)^2 -\frac{\omega_1^2+\omega_2^2}{12} \right).$$8,

$$B_{2,2}(u;\omega_1,\omega_2) = \frac{1}{\omega_1\omega_2} \left( \left(u-\frac{\omega_1+\omega_2}{2}\right)^2 -\frac{\omega_1^2+\omega_2^2}{12} \right).$$9

so that $$\gamma^{(2)}(z;\omega_{1},\omega_{2}) = e^{\frac{\pi i}{2}B_{2,2}(z;\omega_{1},\omega_{2})} \frac{(e^{2\pi i z/\omega_{2}};\tilde q)_\infty}{(e^{2\pi i z/\omega_{1}};q)_\infty},$$0 after the appropriate residue analysis. This limiting property shows that the intertwiner interpolates between identity-type behavior and a nontrivial transformation controlled by the lens hyperbolic gamma kernel (Bülbül et al., 9 Nov 2025).

The operator satisfies modular-double type finite-difference relations: $$\gamma^{(2)}(z;\omega_{1},\omega_{2}) = e^{\frac{\pi i}{2}B_{2,2}(z;\omega_{1},\omega_{2})} \frac{(e^{2\pi i z/\omega_{2}};\tilde q)_\infty}{(e^{2\pi i z/\omega_{1}};q)_\infty},$$1

$$\gamma^{(2)}(z;\omega_{1},\omega_{2}) = e^{\frac{\pi i}{2}B_{2,2}(z;\omega_{1},\omega_{2})} \frac{(e^{2\pi i z/\omega_{2}};\tilde q)_\infty}{(e^{2\pi i z/\omega_{1}};q)_\infty},$$2

Iterating them exposes the two dual shift directions associated with $$\gamma^{(2)}(z;\omega_{1},\omega_{2}) = e^{\frac{\pi i}{2}B_{2,2}(z;\omega_{1},\omega_{2})} \frac{(e^{2\pi i z/\omega_{2}};\tilde q)_\infty}{(e^{2\pi i z/\omega_{1}};q)_\infty},$$3 and $$\gamma^{(2)}(z;\omega_{1},\omega_{2}) = e^{\frac{\pi i}{2}B_{2,2}(z;\omega_{1},\omega_{2})} \frac{(e^{2\pi i z/\omega_{2}};\tilde q)_\infty}{(e^{2\pi i z/\omega_{1}};q)_\infty},$$4. The algebra itself has generators $$\gamma^{(2)}(z;\omega_{1},\omega_{2}) = e^{\frac{\pi i}{2}B_{2,2}(z;\omega_{1},\omega_{2})} \frac{(e^{2\pi i z/\omega_{2}};\tilde q)_\infty}{(e^{2\pi i z/\omega_{1}};q)_\infty},$$5 obeying trigonometric commutation relations with

$$\gamma^{(2)}(z;\omega_{1},\omega_{2}) = e^{\frac{\pi i}{2}B_{2,2}(z;\omega_{1},\omega_{2})} \frac{(e^{2\pi i z/\omega_{2}};\tilde q)_\infty}{(e^{2\pi i z/\omega_{1}};q)_\infty},$$6

and a dual copy $$\gamma^{(2)}(z;\omega_{1},\omega_{2}) = e^{\frac{\pi i}{2}B_{2,2}(z;\omega_{1},\omega_{2})} \frac{(e^{2\pi i z/\omega_{2}};\tilde q)_\infty}{(e^{2\pi i z/\omega_{1}};q)_\infty},$$7 obtained by exchanging

$$\gamma^{(2)}(z;\omega_{1},\omega_{2}) = e^{\frac{\pi i}{2}B_{2,2}(z;\omega_{1},\omega_{2})} \frac{(e^{2\pi i z/\omega_{2}};\tilde q)_\infty}{(e^{2\pi i z/\omega_{1}};q)_\infty},$$8

The intertwiner simultaneously satisfies

$$\gamma^{(2)}(z;\omega_{1},\omega_{2}) = e^{\frac{\pi i}{2}B_{2,2}(z;\omega_{1},\omega_{2})} \frac{(e^{2\pi i z/\omega_{2}};\tilde q)_\infty}{(e^{2\pi i z/\omega_{1}};q)_\infty},$$9

for both untilded and tilded generators. This simultaneous intertwining property is precisely what justifies the term “modular double” in the lens setting (Bülbül et al., 9 Nov 2025).

6. Limits, reductions, and broader significance

Several reductions organize the subject. First, when

$$\gamma^{(2)}(\omega_1+\omega_2-z;\omega_1,\omega_2)\gamma^{(2)}(z;\omega_1,\omega_2)=1.$$0

the discrete lens structure disappears. On the gauge-theory side, lens-space partition-function identities reduce to the ordinary squashed-sphere identities on $$\gamma^{(2)}(\omega_1+\omega_2-z;\omega_1,\omega_2)\gamma^{(2)}(z;\omega_1,\omega_2)=1.$$1. On the algebraic side, the lens hyperbolic modular double reduces to the standard hyperbolic modular double, and the lens hyperbolic gamma function reduces to its usual hyperbolic form (Bülbül et al., 9 Nov 2025, Mullahasanoglu et al., 2021).

Second, in the fundamental representation

$$\gamma^{(2)}(\omega_1+\omega_2-z;\omega_1,\omega_2)\gamma^{(2)}(z;\omega_1,\omega_2)=1.$$2

the lens hyperbolic gamma factors collapse to trigonometric functions, and the reduced operator becomes a $$\gamma^{(2)}(\omega_1+\omega_2-z;\omega_1,\omega_2)\gamma^{(2)}(z;\omega_1,\omega_2)=1.$$3 Lax matrix with entries built from

$$\gamma^{(2)}(\omega_1+\omega_2-z;\omega_1,\omega_2)\gamma^{(2)}(z;\omega_1,\omega_2)=1.$$4

and shift operators. This gives a concrete bridge from the hypergeometric integral kernel to a Lax representation of the Yang–Baxter structure (Bülbül et al., 9 Nov 2025).

Third, gauge symmetry breaking and asymptotic decoupling produce further lens hyperbolic identities. The partition-function analysis uses the asymptotics of $$\gamma^{(2)}(\omega_1+\omega_2-z;\omega_1,\omega_2)\gamma^{(2)}(z;\omega_1,\omega_2)=1.$$5 to implement reductions such as

$$\gamma^{(2)}(\omega_1+\omega_2-z;\omega_1,\omega_2)\gamma^{(2)}(z;\omega_1,\omega_2)=1.$$6

together with flavor-symmetry reductions including

$$\gamma^{(2)}(\omega_1+\omega_2-z;\omega_1,\omega_2)\gamma^{(2)}(z;\omega_1,\omega_2)=1.$$7

These generate additional integral identities, including $$\gamma^{(2)}(\omega_1+\omega_2-z;\omega_1,\omega_2)\gamma^{(2)}(z;\omega_1,\omega_2)=1.$$8-gauge versions of the lens-space duality and broken star-star relations. A plausible implication is that the lens hyperbolic gamma solution should be regarded as a hierarchy of related identities rather than a single formula (Mullahasanoglu et al., 2021).

Finally, the subject occupies a junction of several research programs. It is the hyperbolic degeneration of lens elliptic identities; it encodes equality of supersymmetric partition functions on $$\gamma^{(2)}(\omega_1+\omega_2-z;\omega_1,\omega_2)\gamma^{(2)}(z;\omega_1,\omega_2)=1.$$9; it provides exactly solvable Boltzmann weights for continuous-discrete spin systems; and it admits an operator-theoretic formulation through the lens hyperbolic modular double. In that precise sense, the lens hyperbolic gamma solution is both a special-function construction and a unifying mechanism linking gauge theory, hypergeometric analysis, and quantum integrability (Bülbül et al., 9 Nov 2025, Mullahasanoglu et al., 2021).