• Direction d'équipe

• Physique/Physique des Hautes Energies - Théorie
  

The generating functions of degeneracies of D4-D2-D0 black holes in Type II string compactifications on Calabi-Yau threefolds are examples of (higher depth) mock modular forms. I'll explain how S-duality can be used to derive an explicit form for their modular completions, which becomes particularly simple in the presence of a refinement. This result turns out to have many applications going beyond the original context. In particular, I'll show that it can be used i) to reproduce and generalize in an easy way the known results on modular properties of the generating functions of BPS dyons in N=4 string compactifications; ii) to find Vafa-Witten invariants of arbitrary(!) rank on CP^2, Hirzebruch and del Pezzo surfaces; iii) to obtain holomorphic anomaly equations for BPS partition functions; iv) to reveal a non-commutative structure induced by the refinement on the moduli space of compactified theory.

Extending recent results in ${\cal N}=2$ string compactifications, we propose that the holomorphic anomaly equation satisfied by the modular completions of the generating functions of refined BPS indices has a universal structure independent of the number ${\cal N}$ of supersymmetries. We show that this equation allows to recover all known results about modularity (under $SL(2,\mathbb{Z})$ duality group) of BPS states in ${\cal N}=4$ string theory. In particular, we reproduce the holomorphic anomaly characterizing the mock modular behavior of quarter-BPS dyons and generalize it to the case of non-trivial torsion invariant.

We derive explicit expressions for the generating functions of refined Vafa-Witten invariants $\Omega(\gamma,y)$ of $\mathbb{P}^2$ of arbitrary rank $N$ and for their non-holomorphic modular completions. In the course of derivation we also provide: i) a generalization of the recently found generating functions of $\Omega(\gamma,y)$ and their completions for Hirzebruch and del Pezzo surfaces in the canonical chamber of the moduli space to a generic chamber; ii) a version of the blow-up formula expressed directly in terms of these generating functions and its reformulation in a manifestly modular form.

Recently, a universal formula for a non-holomorphic modular completion of the generating functions of refined BPS indices in various theories with $N=2$ supersymmetry has been suggested. It expresses the completion through the holomorphic generating functions of lower ranks. Here we show that for $U(N)$ Vafa-Witten theory on Hirzebruch and del Pezzo surfaces this formula can be used to extract the holomorphic functions themselves, thereby providing the Betti numbers of instanton moduli spaces on such surfaces. As a result, we derive a closed formula for the generating functions and their completions for all $N$. Besides, our construction reveals in a simple way instances of fiber-base duality, which can be used to derive new non-trivial identities for generalized Appell functions. It also suggests the existence of new invariants, whose meaning however remains obscure.

I'll present recent results on modular properties of generating functions of degeneracies of BPS black holes appearing in string theory compactifications on Calabi-Yau threefolds. These generating functions turn out to be examples of (higher depth) mock modular forms and therefore have modular anomalies. I'll present the explicit form for their modular completions which becomes particularly simple in the presence of a refinement. In this case the construction suggests that the refinement induces a quantization of the corresponding moduli space. I'll also discuss implications of these results for Vafa-Witten theory and holomorphic anomaly equations for BPS partition functions.