[Paper Review] Duality in N=4 Liouville Theory and Moonshine Phenomena

Tohru Eguchi, Yuji Sugawara|arXiv (Cornell University)|Mar 9, 2016

Black Holes and Theoretical Physics17 references4 citations

TL;DR

This paper investigates duality in N=4 Liouville theory at two special values of the linear dilaton coupling constant Q, corresponding to central charges c=6 and c=6(N−1), and demonstrates that these dual theories exhibit Mathieu and umbral moonshine phenomena. By analyzing their elliptic genera and shadows under A-type modular invariants, the authors derive algebraic identities and confirm numerical matches with mock modular forms, establishing a dual description of moonshine structures in terms of dual CFTs with comparable effective degrees of freedom.

ABSTRACT

We consider the ${\cal N}=4$ Liouville theory by varying the linear dilaton coupling constant $\cal{Q}$. It is known that at two different values of coupling constant ${\cal Q}=\sqrt{2\over N},-(N-1)\sqrt{2\over N}$ system exhibits two different small ${\cal N}=4$ superconformal symmetries with central charge $c=6$ and $c=6(N-1)$, respectively. In the context of string theory these two theories are considered to describe Coulumb and Higgs branch of the theory and expected to be dual to each other. We study the Mathieu and umbral moonshine phenomena in these two theories and discuss their dual description. We mainly consider the case of $A_N$ type modular invariants.

Motivation & Objective

• To investigate the duality between two N=4 Liouville theories with central charges c=6 and c=6(N−1) at special values of the linear dilaton coupling Q.
• To explore the realization of Mathieu and umbral moonshine phenomena in these dual conformal field theories.
• To derive algebraic identities and inequalities for the elliptic genera of the two dual theories under A-type modular invariants.
• To confirm the correspondence between the elliptic genera and mock modular forms by matching shadows and numerical coefficients.
• To relate the effective central charge and second helicity supertrace to the structure of the moonshine functions via number-theoretic formulas.

Proposed method

• The authors analyze the free field realization of the large N=4 superconformal algebra and its deformation via the linear dilaton coupling Q.
• They identify two special values of Q, Q=√(2/N) and Q=−(N−1)√(2/N), which yield small N=4 SCAs with central charges c=6 and c=6(N−1), respectively.
• The method involves constructing the elliptic genera h^{(N)}(τ) and shadows H^{(N)}(τ) for the two dual CFTs using modular properties and free field realizations.
• They derive a key identity H^{(N)}(τ) = 1/12 χ²^{(N,1)}(τ), linking the shadow function to the second helicity supertrace from string-theoretic constructions.
• The authors use number-theoretic identities involving Eisenstein series E₂(τ) and sums over lattice points to verify the match between H^{(N)}(τ) and the supertrace formula.
• They numerically confirm the match between the derived series expansions of h^{(N)}(τ) and known moonshine mock modular forms for N=2,3,4.

Experimental results

Research questions

• RQ1 How do the elliptic genera of the two dual N=4 Liouville CFTs with c=6 and c=6(N−1) relate to each other under duality?
• RQ2 Can the Mathieu and umbral moonshine phenomena be consistently realized in both the Coulomb and Higgs branch descriptions of N=4 Liouville theory?
• RQ3 What is the precise relation between the shadow functions H^{(N)}(τ) and the second helicity supertrace χ²^{(k,d)}(τ) in string-theoretic models?
• RQ4 Do the elliptic genera of the dual theories satisfy algebraic identities that reflect the duality structure and modular invariance?
• RQ5 How do the A-type modular invariants constrain the form of the moonshine functions and their coefficients?

Key findings

• The elliptic genus h^{(2)}(τ) for the c=6 theory matches the known Mathieu moonshine function, with coefficients -1, 45, 231, 770, 2277, 5796, ..., confirming the duality prediction.
• For the c=6(N−1) theory with N=3 (A₂¹²), the identity -12H^{(3)}(τ) = h₁^{(3)}(τ)S₁,₃(τ) + h₂^{(3)}(τ)S₂,₃(τ) holds numerically, yielding coefficients -2, 72, 216, 216, 336, ...
• For N=4 (A₃⁸), the identity -8H^{(4)}(τ) = h₁^{(4)}(τ)S₁,₄(τ) + h₂^{(4)}(τ)S₂,₄(τ) + h₃^{(4)}(τ)S₃,₄(τ) is numerically confirmed with coefficients -2, 64, 192, 256, 384, ...
• The shadow function H^{(N)}(τ) is proven to equal 1/12 χ²^{(N,1)}(τ), establishing a direct link between the moonshine function and the second helicity supertrace in string theory.
• The derivation confirms that H^{(N)}(τ) = (N/12)E₂(τ) − 1/12 + 2∑∑(2Nm+n)q^{Nm²+mn} − 2N∑mq^{Nm²}, matching the supertrace formula exactly.
• The results support the duality between the Coulomb and Higgs branches of N=4 Liouville theory, with moonshine structures preserved across the duality despite differing central charges.