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Higher depth quantum modular forms, multiple Eichler integrals, and $\frak{sl}_3$ false theta functions.
TL;DR: In this paper, the authors introduced and studied higher depth quantum modular forms of positive integral weight, and proved that the false theta of the vertex algebra W^0(p)_{A_2} can be expressed as the sum of two depth two quantum modular functions of positive weight.
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Abstract: We introduce and study higher depth quantum modular forms. We construct two families of examples coming from rank two false theta functions, whose "companions" in the lower half-plane can be also realized both as double Eichler integrals and as non-holomorphic theta series having values of "double error" functions as coefficients. In particular, we prove that the false theta of $\frak{sl}_3$, appearing in the character of the vertex algebra $W^0(p)_{A_2}$, can be written as the sum of two depth two quantum modular forms of positive integral weight.
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Citations
S-duality and refined BPS indices
TL;DR: In this article, the modularity of generalized Donaldson-Thomas invariants counting D4-D2-D0 brane bound states in type IIA strings on a Calabi-Yau threefold manifold was studied.
27
Higher depth quantum modular forms and plumbed $3$-manifolds
TL;DR: In this paper, it was shown that for any positive definite unimodular plumbing matrix, a depth two quantum modular form on the manifold is a depth-two quantum form on a plumbed $3-manifold.
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On parafermion vertex algebras of $\frak{sl}(2)_{-3/2}$ and $\frak{sl}(3)_{-3/2}$
TL;DR: In this article, the irreducibility of parafermion vertex algebras was proved for the Zamolodchikov algebra and the logarithmic vertex algebra.
8
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Rank two false theta functions and Jacobi forms of negative definite matrix index
TL;DR: It is shown that these functions appear as Fourier coefficients of a meromorphic Jacobi form of negative definite matrix index of type $A_2$ as well as Hypergeometric $q$-series identities.
6
References
On Modular Forms of Half Integral Weight
TL;DR: In this article, the connection of modular forms with zeta functions was clarified, and a more affirmative aspect of the subject was revealed, which might have given a rather negative and somewhat misleading impression that one would not be able to do much except in some special cases.
1K
Vassiliev invariants and a strange identity related to the Dedekind eta-function
TL;DR: In this article, it was shown that the coefficients of the Taylor expansion at q = 1 are equal to the numbers ξD of regular linearized chord diagrams as defined by Stoimenow and hence give an upper bound for the number of linearly independent Vassiliev invariants of degree D. The same values and derivatives of all orders at all roots of unity are obtained as the limiting value of the function − 1 2 ∑ n∈ Z (−1) n |6n+1|q (3n 2 +n)/2, the "der
311
Quantum modular forms
Don Zagier
- 01 Jan 2010
TL;DR: In this article, the authors introduce the notion of quantum modular forms, which are objects which live at the boundary of the space X, are defined only asymptotically, rather than exactly, and have a transformation behavior of a quite different type with respect to some modular group.
230
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Logarithmic CFTs connected with simple Lie algebras
Boris Feigin,I. Yu. Tipunin +1 more
TL;DR: For any root system corresponding to a semisimple simply-laced Lie algebra, a logarithmic CFT is constructed as discussed by the authors, where characters of irreducible representations are calculated in terms of theta functions.
83
Torus knot and minimal model
TL;DR: In this article, the authors reveal an intimate connection between the quantum knot invariant for torus knots and the character of the minimal model M (s,t), where s and t are relatively prime integers.
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