Abstract
Abstract We define modular linear differential equations (MLDE) for the level-two congruence subgroups $\Gamma_\theta$, $\Gamma^0(2)$ and $\Gamma_0(2)$ of $\text{SL}_2(\mathbb Z)$. Each subgroup corresponds to one of the spin structures on the torus. The pole structures of the fermionic MLDEs are investigated by exploiting the valence formula for the level-two congruence subgroups. We focus on the first- and second-order holomorphic MLDEs without poles and use them to find a large class of “fermionic rational conformal field theories” (fermionic RCFTs), which have non-negative integer coefficients in the $q$-series expansion of their characters. We study the detailed properties of these fermionic RCFTs, some of which are supersymmetric. This work also provides a starting point for the classification of the fermionic modular tensor category.
Highlights
Introduction and concluding remarksThe classification of all unitary conformal field theories in two dimensions certainly plays a key role in our understanding of the critical phenomena
We extend the modular invariant linear differential equations (MLDEs) method to classify the fermionic rational conformal field theories (CFTs), some of which appear to be supersymmetric
We consider a fermionic rational conformal field theory (RCFT) which has a finite number of characters χiNS, i = 0, 1, ...N −1 with conformal weight hi
Summary
Let us consider the θ valence formula for definiteness. The valence formula for 0(2) and 0(2) could be found . Modular linear differential equation for θ Let us see how to use the above valence formula in the context of fermionic RCFTs. For that purpose, we consider a fermionic RCFT which has a finite number of characters χiNS, i = 0, 1, ...N −1 with conformal weight hi. [8] and in Eq (24), if we define l/2 to be the “number" of zeros inside the fundamental domain, the valence formula (49) gives us the following relation:. For a given fermionic RCFT with known c, hNj S and hRj , the value of the zero determines the structure of the MLDE satisfied by the characters in the NS sector. The above relation provides the information on the b, i.e., the pole structure of the coefficient functions for MLDE of bosonic 3N characters
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