We prove resurgence properties for the Borel transform of a formal power series associated to elements in the Habiro ring that come from radial limits of partial theta series via strange identities. As an application, we prove a conjecture in quantum topology due to Costin and Garoufalidis for two families of torus knots.

In 1992 Wirthmüller showed that for any irreducible root system not of type E8 the ring of weak Jacobi forms invariant under Weyl group is a polynomial algebra. However, it has recently been proved that for E8 the ring is not a polynomial algebra. Weyl invariant E8 Jacobi forms have many applications in string theory and it is an open problem to describe such forms. The scaled refined free energies of E-strings with certain η-function factors are conjectured to be Weyl invariant E8 quasi holomorphic Jacobi forms. It is further observed that the scaled refined free energies up to some powers of E4 can be written as polynomials in nine Sakai's E8 Jacobi forms and Eisenstein series E2, E4, E6. Motivated by the physical conjectures, we prove that for any Weyl invariant E8 Jacobi form φt of index t the function E /6] φt can be expressed uniquely as a polynomial in E4, E6 and Sakai's forms, where [x] is the integer part of x. This means that a Weyl invariant E8 Jacobi form is completely determined by a solution of some linear equations. By solving the linear systems, we determine the generators of the free module of Weyl invariant E8 weak (resp. holomorphic) Jacobi forms of given index t when t ≤ 13 (resp. t ≤ 11). is a polynomial algebra generated by r + 1 Jacobi forms over the ring M * (SL 2 (Z)) = C[E 4 , E 6 ] of SL 2 (Z)-modular forms. For example, the algebra of W (A 1 )-invariant weak Jacobi forms, that is, the ring of even-weight weak Jacobi forms introduced by Eichler and Zagier, is freely generated by two forms of index 1 and weight 0 and −2. Recently, the second named author proved in [22] that the ring of W (E 8 )-invariant weak Jacobi forms is not a polynomial algebra. In other words, there exist some algebraic relations among generators. It is still unknown if this ring is finitely generated. In the study on the Seiberg-Witten curve of E-string theory [6], Sakai [18] constructed nine W (E 8 )-invariant holomorphic Jacobi forms denoted A

We provide Fourier expansions of vector-valued eigenfunctions of the hyperbolic Laplacian that are twist-periodic in a horocycle direction. The twist may be given by any endomorphism of a finite-dimensional vector space; no assumptions on invertibility or unitarity are made. Examples of such eigenfunctions include vector-valued twisted automorphic forms of Fuchsian groups. We further provide a detailed description of the Fourier coefficients and explicitly identify each of their constituents, which intimately depend on the eigenvalues of the twisting endomorphism and the size of its Jordan blocks. In addition, we determine the growth properties of the Fourier coefficients.

We analyze the coefficients of partition functions of Vafa-Witten theory for the complex projective plane CP 2 . We experimentally study the growth of the coefficients for gauge group SU (2) and SU (3), which are examples of mock modular forms of depth 1 and 2 respectively. We also introduce the notion of "mock cusp form", and study an example of weight 3 related to the SU (3) partition function. Numerical experiments on the first 200 coefficients suggest that the coefficients of a mock modular form of weight k grow as the coefficients of a modular form of weight k, that is to say as n k−1 . On the other hand the coefficients of the mock cusp form appear to grow as n 3/2 , which exceeds the growth of classical cusp forms of weight 3. We provide bounds using saddle point analysis, which however largely exceed the experimental observation.

We present basic constructions and properties in arithmetic Chern-Simons theory with finite gauge group along the line of topological quantum field theory. For a finite set S of finite primes of a number field k, we construct arithmetic analogues of the Chern-Simons 1-cocycle, the prequantization bundle for a surface and the Chern-Simons functional for a 3-manifold. We then construct arithmetic analogues for k and S of the quantum Hilbert space (space of conformal blocks) and the Dijkgraaf-Witten partition function in (2+1)-dimensional Chern-Simons TQFT. We show some basic and functorial properties of those arithmetic analogues. Finally we show decomposition and gluing formulas for arithmetic Chern-Simons invariants and arithmetic Dijkgraaf-Witten partition functions. Contents.

The quantization of the mirror curve to a toric Calabi-Yau threefold gives rise to quantum-mechanical operators, whose fermionic spectral traces produce factorially divergent power series in the Planck constant. These asymptotic expansions can be promoted to resurgent trans-series. They show infinite towers of periodic singularities in their Borel plane and infinitely many rational Stokes constants, which are encoded in generating functions expressed in closed form in terms of q-series. We provide an exact solution to the resurgent structure of the first fermionic spectral trace of the local P 2 geometry in the semiclassical limit of the spectral theory, corresponding to the strongly-coupled regime of topological string theory on the same background in the conjectural TS/ST correspondence. Our approach straightforwardly applies to the dual weakly-coupled limit of the topological string. We present and prove closed formulae for the Stokes constants as explicit arithmetic functions and for the perturbative coefficients as special values of known L-functions, while the duality between the two scaling regimes of strong and weak string coupling constant appears in number-theoretic form. A preliminary numerical investigation of the local F 0 geometry unveils a more complicated resurgent structure with logarithmic subleading asymptotics. Finally, we obtain a new analytic prediction on the asymptotic behavior of the fermionic spectral traces in an appropriate WKB double-scaling regime, which is captured by the refined topological string in the Nekrasov-Shatashvili limit.

We investigate $W(E_8)$-invariant Jacobi forms which are the Jacobi forms invariant under the action of the Weyl group of the root system $E_8$. This type of Jacobi forms has applications in mathematics and physics, but very little has been known about its structure. In this paper we show that the bigraded ring of weak $W(E_8)$-invariant Jacobi forms is not a polynomial algebra over $C$ and prove that every $W(E_8)$-invariant Jacobi form can be expressed uniquely as a polynomial in nine algebraically independent holomorphic Jacobi forms introduced by Sakai with coefficients which are meromorphic $SL_2(Z)$ modular forms. The latter result implies that the graded ring of weak $W(E_8)$-invariant Jacobi forms of fixed index is a free module over the ring of $SL_2(Z)$ modular forms and the number of generators can be calculated by a generating series. We also determine and construct all generators of small index. These results extend Wirthm\{u}ller's theorem proved in 1992 to the last open case.