[Paper Review] Supersymmetry and the Modular Double

Ivan Chi Ho Ip, Anton M. Zeitlin|arXiv (Cornell University)|Apr 25, 2013

Advanced Topics in Algebra4 citations

TL;DR

This paper constructs a supersymmetric analogue of the modular double for the quantum superalgebra $\mathcal{U}_q(\mathfrak{osp}(1|2))$ using supersymmetric quantum mechanics, extending the framework of the modular double from $\mathfrak{sl}(2,\mathbb{R})$ to superalgebras. The key result is the explicit construction of an $R$-matrix for this superalgebra using quantum dilogarithm functions, which is essential for describing fusion rules in $\mathcal{N}=1$ SUSY Liouville theory.

ABSTRACT

A counterpart of the modular double for quantum superalgebra $\cU_q(\osp(1|2))$ is constructed by means of supersymmetric quantum mechanics. We also construct the $R$-matrix operator acting in the corresponding representations, which is expressed via quantum dilogarithm.

Motivation & Objective

• To extend the concept of the modular double—previously defined for $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$—to the quantum superalgebra $\mathcal{U}_q(\mathfrak{osp}(1|2))$.
• To establish a representation of the modular double for $\mathcal{U}_q(\mathfrak{osp}(1|2))$ in $L^2(\mathbb{R})$ using supersymmetric quantum mechanics.
• To construct an $R$-matrix operator acting on representations of $\mathcal{U}_q(\mathfrak{osp}(1|2))$, expressed through quantum dilogarithm functions.
• To lay the foundation for a tensor category of representations that may describe fusion rules in $\mathcal{N}=1$ SUSY Liouville field theory.

Proposed method

• Adapts the free field realization of $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$ via Heisenberg algebra generators $U = e^{2\pi b x}$, $V = e^{2\pi b p}$ to the superalgebra $\mathcal{U}_q(\mathfrak{osp}(1|2))$ using supersymmetric quantum mechanics.
• Introduces dual generators $\tilde{U} = U^{1/b^2}$, $\tilde{V} = V^{1/b^2}$ to define a second quantum algebra $\mathcal{U}_{\tilde{q}}(\mathfrak{osp}(1|2))$ with $\tilde{q} = e^{\pi i b^{-2}}$, forming the modular double.
• Constructs the $R$-matrix using the quantum dilogarithm function $\Phi_b$, which satisfies braiding relations and intertwines tensor products of representations.
• Uses the operator $g_{b_*}$ to define the $R$-matrix as a product of exponentials involving $e \otimes K \otimes f$ and $1 \otimes e \otimes f$, with appropriate $\Phi$-transformations.
• Establishes compatibility of the $R$-matrix with the coproduct structure via $\Phi$-automorphisms, ensuring braid group relations.
• Demonstrates that the $R$-matrix is bounded but not unitary, with its inverse given by a related unbounded operator $\tilde{R}$ satisfying $R^* \tilde{R} = 1$.

Experimental results

Research questions

• RQ1 Can the modular double construction for $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$ be generalized to the superalgebra $\mathcal{U}_q(\mathfrak{osp}(1|2))$?
• RQ2 How can the $R$-matrix for $\mathcal{U}_q(\mathfrak{osp}(1|2))$ be explicitly constructed using quantum dilogarithm functions?
• RQ3 Does the representation space of the modular double for $\mathcal{U}_q(\mathfrak{osp}(1|2))$ decompose into even and odd subrepresentations isomorphic to $\mathcal{U}_{q_*\tilde{q}_*}(\mathfrak{sl}(2,\mathbb{R}))$?
• RQ4 Can the tensor category of these representations be constructed, and do its $3j$-symbols match fusion coefficients in $\mathcal{N}=1$ SUSY Liouville theory?

Key findings

• The modular double for $\mathcal{U}_q(\mathfrak{osp}(1|2))$ is constructed as a representation on $L^2(\mathbb{R})$ using supersymmetric quantum mechanics, with generators of $\mathcal{U}_q(\mathfrak{osp}(1|2))$ and $\mathcal{U}_{\tilde{q}}(\mathfrak{osp}(1|2))$ realized via $U$, $V$ and their duals $\tilde{U}$, $\tilde{V}$.
• The $R$-matrix is explicitly constructed as $R = g_{b_*}(i\hat{e} \otimes \mathcal{K} \otimes \hat{f} + i \otimes \hat{e} \otimes \hat{f})$, expressed through the quantum dilogarithm function $\Phi_b$.
• The $R$-matrix satisfies the Yang-Baxter equation and intertwines the coproducts of the superalgebra, with its inverse given by an unbounded operator $\tilde{R}$ such that $R^* \tilde{R} = 1$.
• The representation space $P^s_z$ naturally decomposes into even and odd subrepresentations $P^{(e)}_z$ and $P^{(o)}_z$, analogous to the classical decomposition under $\mathfrak{sl}(2,\mathbb{R})$.
• The $R$-matrix is bounded but not unitary, and its construction relies on $\Phi$-automorphisms to ensure compatibility with the Hopf algebra structure.
• The framework suggests that the tensor category of $P^s_z$ representations may be continuous and isomorphic to that of the modular double of $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$, with potential applications to $\mathcal{N}=1$ SUSY Liouville theory fusion rules.