Fermionic rational conformal field theories and modular linear differential equations

Abstract

We define Modular Linear Differential Equations (MLDE) for the level-two congruence subgroups Γθ, Γ0(2) and Γ0(2) of SL2(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’, 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.

Fermionization and boundary states in 1+1 dimensions

“…[17][18][19] for recent works. 3 See also [20] for a recent study of fermionic RCFTs with small number of irreducible representations. 4 This theory is also known as the Arf theory, since its partition function on a closed two-dimensional surface with spin structure is given by its Arf invariant.…”

Report on scipost_202106_00046v2

“…Here we point out that the supersymmetric RCFTs are not always obtained by orbifold or fermionization with help of an non-anomalous Z 2 symmetry. A prominent example is the so-called non-BPS solution of the fermionic second-order modular differential equation at c = 39 2 found in [34]. Two independent solutions in both NS and R-sectors at c = 39 2 were shown to be expressed in terms of the (E 6 ) 4 characters.…”

Report on scipost_202106_00046v1

“…The above two characters (3.12) has appeared in [34] as the solutions of the secondorder modular linear differential equation for Γ θ . Similarly, one can obtain the partition functions in other sectors as follows,…”

“…Therefore, the NS-sector partition function (4.2) can be identified with the singlecharacter solution of the fermionic modular differential equation [34].…”

“…from which one can identify the partition functions (4.7) as the solution of the first order fermionic modular differential equation [34].…”