Attractive strings and five-branes, skew-holomorphic Jacobi forms and moonshine

Abstract

We show that certain BPS counting functions for both fundamental strings and strings arising from fivebranes wrapping divisors in Calabi-Yau threefolds naturally give rise to skew-holomorphic Jacobi forms at rational and attractor points in the moduli space of string compactifications. For M5-branes wrapping divisors these are forms of weight negative one, and in the case of multiple M5-branes skewholomorphic mock Jacobi forms arise. We further find that in simple examples these forms are related to skew-holomorphic (mock) Jacobi forms of weight two that play starring roles in moonshine. We discuss examples involving M5-branes on the complex projective plane, del Pezzo surfaces of degree one, and half-K3 surfaces. For del Pezzo surfaces of degree one and certain half-K3 surfaces we find a corresponding graded (virtual) module for the degree twelve Mathieu group. This suggests a more extensive relationship between Mathieu groups and complex surfaces, and a broader role for M5-branes in the theory of Jacobi forms and moonshine.

TASI Lectures on Moonshine

“…• In [18] it was shown that the modified elliptic genus of the so-called MSW string [19] involves a skew-holomorphic Jacobi form with index specified by the charges of the black hole, provided that the moduli is fixed at their black hole attractor value. See §9 for more discussions on this.…”

“…MSW strings. Another physical setup which delivers examples of modular objects connected to finite groups is M5 branes wrapping divisors of Calabi-Yau three-folds [18]. These give rise to effective strings, the so-called MSW strings, with (0,4) worldsheet supersymmetry [19].…”

“…These give rise to effective strings, the so-called MSW strings, with (0,4) worldsheet supersymmetry [19]. In [18], it was pointed out that the generalised elliptic genera of the MSW string theories are examples of skew-holomorphic Jacobi forms (cf. §2.2), a type of modular object playing a starring role in the family of moonshine generalising the Thompson moonshine, but whose role in physics has thus far not been highlighted.…”

“…§2.2), a type of modular object playing a starring role in the family of moonshine generalising the Thompson moonshine, but whose role in physics has thus far not been highlighted. In particular, the generalised elliptic genera for one or two M5 branes wrapping the surfaces P 2 , del Pezzo 8, and half-K3 were examined in [18]. In the first case, it was known that two M5 branes wrapping the surface P 2 leads to mock skew-holomorphic forms that are closely related to the generating function of the class number mock modular form (2.49).…”

“…(See §2.1 for the definition of E 4 and η. ) In [18] it was shown that f (1) is the graded superdimension of a supermodule for the Mathieu group M 12 , and the corresponding graded characters f (1) g for all g ∈ M 12 are explicitly given. It would be very interesting to have a geometric and string theoretic understanding of this M 12 -supermodule.…”