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Free fermion vertex superalgebras are discussed from the point of view of Urod vertex algebras [T. Arakawa, T. Creutzig and B. Feigin, Urod algebras and translation of [Formula: see text]-algebras, Forum Mathematics Sigma, Vol. 10 (Cambridge University Press, 2022) and M. Bershtein, B. Feigin and A. Litvinov, Coupling of two conformal field theories and Nakajima–Yoshioka blow-up equations, preprint (2013), arXiv:1310.7281]. We present all finite decompositions of the [Formula: see text]-fermion vertex algebra via Virasoro and [Formula: see text] superconformal vertex algebras. We also present decompositions of higher rank fermion algebras using affine [Formula: see text]-algebras, and a conjecture on the existence of the “square root” of the [Formula: see text] fermion algebra. 

Abstract False theta functions form a family of functions with intriguing modular properties and connections to mock modular forms. In this paper, we take the first step towards investigating modular transformations of higher rank false theta functions, following the example of higher depth mock modular forms. In particular, we prove that under quite general conditions, a rank two false theta function is determined in terms of iterated, holomorphic, Eichler-type integrals. This provides a new method for examining their modular properties and we apply it in a variety of situations where rank two false theta functions arise. We first consider generic parafermion characters of vertex algebras of type $$A_2$$ A 2 and $$B_2$$ B 2 . This requires a fairly non-trivial analysis of Fourier coefficients of meromorphic Jacobi forms of negative index, which is of independent interest. Then we discuss modularity of rank two false theta functions coming from superconformal Schur indices. Lastly, we analyze $${\hat{Z}}$$ Z ^ -invariants of Gukov, Pei, Putrov, and Vafa for certain plumbing $$\mathtt{H}$$ H -graphs. Along the way, our method clarifies previous results on depth two quantum modularity.