Refinement and modularity of immortal dyons

Extending recent results in $$ \mathcal{N} $$ 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 $$ \mathcal{N} $$ N of supersymmetries. We show that this equation allows to recover all known results about modularity (under SL(2, ℤ) duality group) of BPS states in $$ \mathcal{N} $$ 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.

K3 surfaces with involution, equivariant analytic torsion, and automorphic forms on the moduli space IV: The structure of the invariant

Yoshikawa in [Invent. Math. 156 (2004), 53–117] introduces a holomorphic torsion invariant of $K3$ surfaces with involution. In this paper we completely determine its structure as an automorphic function on the moduli space of such $K3$ surfaces. On every component of the moduli space, it is expressed as the product of an explicit Borcherds lift and a classical Siegel modular form. We also introduce its twisted version. We prove its modularity and a certain uniqueness of the modular form corresponding to the twisted holomorphic torsion invariant. This is used to study an equivariant analogue of Borcherds’ conjecture.

Some Einstein nilmanifolds with skew torsion arising in CR geometry

We describe some new examples of nilmanifolds admitting an Einstein with skew torsion invariant Riemannian metric. These are affine CR quadrics, whose CR structure is preserved by the characteristic connection.

Degenerations of Calabi–Yau threefolds and BCOV invariants

In [M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Holomorphic anomalies in topological field theories, Nucl. Phys. B405 (1993) 279–304; M. Bershadsky, S. Cecotti, H. Ooguri and C. Vafa, Kodaira–Spencer theory of gravity and exact results for quantum string amplitudes, Commun. Math. Phys.165 (1994) 311–427], by expressing the physical quantity F1 in two distinct ways, Bershadsky–Cecotti–Ooguri–Vafa discovered a remarkable equivalence between Ray–Singer analytic torsion and elliptic instanton numbers for Calabi–Yau threefolds. After their discovery, in [H. Fang, Z. Lu and K.-I. Yoshikawa, Analytic torsion for Calabi–Yau threefolds, J. Differential Geom.80 (2008) 175–250], a holomorphic torsion invariant for Calabi–Yau threefolds corresponding to F1, called BCOV invariant, was constructed. In this article, we study the asymptotic behavior of BCOV invariants for algebraic one-parameter degenerations of Calabi–Yau threefolds. We prove the rationality of the coefficient of logarithmic divergence and give its geometric expression by using a semi-stable reduction of the given family.

The spiral index of knots

In this paper, we introduce two new invariants that are closely related to Milnor's curvature-torsion invariant. The first, a particularly natural invariant called the spiral index of a knot, captures the number of local maxima in a knot projection that is free of inflection points. This invariant is sandwiched between the bridge and braid index of a knot, and captures more subtle properties. The second invariant, the projective superbridge index, provides a method of counting the greatest number of local maxima that occur in a given projection. In addition to investigating the relationships among these invariants, we use them to classify all those knots for which Milnor's curvature-torsion invariant is 6π.