Showing papers in "Communications in Number Theory and Physics in 2012"

Anomalies and the euler characteristic of elliptic calabi-yau threefolds

Abstract: We investigate the delicate interplay between the types of singular fibers in elliptic fibrations of Calabi-Yau threefolds (used to formulate F-theory) and the "matter" representation of the associated Lie algebra. The main tool is the analysis and the appropriate interpretation of the anomaly formula for six-dimensional supersymmetric theories. We find that this anomaly formula is geometrically captured by a relation among codimension two cycles on the base of the elliptic fibration, and that this relation holds for elliptic fibrations of any dimension. We introduce a "Tate cycle" which efficiently describes thisrelation- ship, and which is remarkably easy to calculate explicitly from the Weierstrass equation of the fibration. We check the anomaly cancellation formula in a num- ber of situations and show how this formula constrains the geometry (and in particular the Euler characteristic) of the Calabi-Yau threefold.

The Resurgence of Instantons in String Theory

Abstract: Nonperturbative eects in string theory are usually associated to D{branes. In many cases it can be explicitly shown that D{brane instantons control the large{order behavior of string perturbation theory, leading to the well{known (2g)! growth of the genus expansion. This paper presents a detailed treatment of nonperturbative solutions in string theory, and their relation to the large{order behavior of perturbation theory, making use of transseries and resurgent analysis. These are powerful techniques addressing general nonperturbative contributions within non{linear systems, which are developed at length herein as they apply to string theory. The cases of topological strings, the Painlev e I equation describing 2d quantum gravity, and the quartic matrix model, are explicitly addressed. These results generalize to minimal strings and general matrix models. It is shown that, in order to completely understand string theory at a fully nonperturbative level, new sectors are required beyond the standard D{brane sector.

Symmetries of K3 sigma models

Abstract: It is shown that the supersymmetry-preserving automorphisms of any non-linear σ-model on K3 generate a subgroup of the Conway group Co1. This is the stringy generalisation of the classical theorem, due to Mukai and Kondo, showing that the symplectic automorphisms of any K3 manifold form a subgroup of the Mathieu group M23. The Conway group Co1 contains the Mathieu group M24 (and therefore in particular M23) as a subgroup. We confirm the predictions of the Theorem with three explicit CFT realisations of K3: the T 4 /Z2 orbifold at the self-dual point, and the two Gepner models (2) 4 and (1) 6 . In each case we demonstrate that their symmetries do not form a subgroup of M24, but lie inside Co1 as predicted by our

A second-order differential equation for the two-loop sunrise graph with arbitrary masses

Abstract: We derive a second-order differential equation for the two-loop sunrise graph in two dimensions with arbitrary masses. The differential equation is obtained by viewing the Feynman integral as a period of a variation of a mixed Hodge structure, where the variation is with respect to the external momentum squared. The fibre is the complement of an elliptic curve. From the fact that the first cohomology group of this elliptic curve is two-dimensional we obtain a second-order differential equation. This is an improvement compared to the usual way of deriving differential equations: Integration-by-parts identities lead only to a coupled system of four first-order differential equations.

On Rademacher sums, the largest Mathieu group and the holographic modularity of moonshine

Abstract: Recently a conjecture has been proposed which attaches (mock) modular forms to the largest Mathieu group. This may be compared to monstrous moonshine, in which modular functions are attached to elements of the Monster group. One of the most remarkable aspects of monstrous moonshine is the following genus zero property: the modular functions turn out to be the generators for the function fields of their invariance groups. In particular, these invariance groups define genus zero quotients of the upper half plane. It is therefore natural to ask if there is an analogue of this property in the Mathieu case, and at first glance the answer appears to be negative since not all the discrete groups arising there have genus zero. On the other hand, in this article we prove that each (mock) modular form appearing in the Mathieu correspondence coincides with the Rademacher sum constructed from its polar part. This property, inspired by the AdS/CFT correspondence in physics, was shown previously to be equivalent to the genus zero property of monstrous moonshine. Hence we conclude that this “Rademacher summability” property serves as the natural analogue of the genus zero property in the Mathieu case. Our result constitutes further evidence that the Rademacher method provides a powerful framework for understanding the modularity of moonshine, and leads to interesting physical questions regarding the gravitational duals of the relevant conformal field theories.

Quantum geometry of elliptic Calabi-Yau manifolds

Abstract: We study the quantum geometry of the class of Calabi-Yau threefolds, which are elliptic brations over a two-dimensional toric base. A holomorphic anomaly equation for the topological string free energy is proposed, which is iterative in the genus expansion as well as in the curve classes in the base. T -duality on the bre implies that the topological string free energy also captures the BPSinvariants of D4-branes wrapping the elliptic bre and a class in the base. We verify this proposal by explicit computation of the BPS invariants of 3D4-branes on the rational elliptic surface.

A new look at one-loop integrals in string theory

Abstract: We revisit the evaluation of one-loop modular integrals in string theory, employing new methods that, unlike the traditional 'orbit method', keep T-duality manifest throughout. In particular, we apply the Rankin-Selberg-Zagier approach to cases where the integrand function grows at most polynomially in the IR. Furthermore, we introduce new techniques in the case where 'unphysical tachyons' contribute to the one-loop couplings. These methods can be viewed as a modular invariant version of dimensional regularisation. As an example, we treat one-loop BPS-saturated couplings involving the $d$-dimensional Narain lattice and the invariant Klein $j$-function, and relate them to (shifted) constrained Epstein Zeta series of O(d,d;Z). In particular, we recover the well-known results for d=2 in a few easy steps.

BPS invariants of $\CN=4$ gauge theory on Hirzebruch surfaces

Abstract: Generating functions of BPS invariants for $\CN=4$ $U(r)$ gauge theory on aHirzebruch surface with $r\leq 3$ are computed. The BPS invariantsprovide the Betti numbers of moduli spaces of semi-stable sheaves.The generating functions for $r=2$ are expressed in terms of higher level Appellfunctions for a certain polarization of the surface. The level corresponds to the self-intersection of the basecurve of the Hirzebruch surface. The non-holomorphic functions aredetermined, which added to the holomorphic generating functions providefunctions, which transform as a modular form.

Equivalence of three wall-crossing formulæ

Abstract: The wall crossing formula of Kontsevich and Soibelman gives an implicit relation between the BPS indices on two sides of the wall of marginal stability by equating two symplectomorphisms constructed from the indices on two sides of the wall. The wall crossing formulae of Manschot, Pioline and the author give two apparently different explicit expressions for the BPS index on one side of the wall in terms of the BPS indices on the other side. We prove the equivalence of all the three formulae.

Pfaffian Calabi-Yau threefolds and mirror symmetry

Abstract: The aim of this article is to report on recent progress in un- derstanding mirror symmetry for some non-complete intersection Calabi- Yau threefolds. We first construct four new smooth non-complete in- tersection Calabi-Yau threefolds with h 1,1 = 1, whose existence was previously conjectured by C. van Enckevort and D. van Straten in (19). We then compute the period integrals of candidate mirror families of F. Tonoli's degree 13 Calabi-Yau threefold and three of the new Calabi- Yau threefolds. The Picard-Fuchs equations coincide with the expected Calabi-Yau equations listed in (18, 19). Some of the mirror families turn out to have two maximally unipotent monodromy points.

HOMFLY polynomials, stable pairs and motivic Donaldson–Thomas invariants

Abstract: Hilbert scheme topological invariants of plane curve singularities are identified to framed threefold stable pair invariants. As a result, the conjecture of Oblomkov and Shende on HOMFLY polynomials of links of plane curve singularities is given a Calabi-Yau threefold interpretation. The motivic Donaldson-Thomas theory developed by M. Kontsevich and the third author then yields natural motivic invariants for algebraic knots. This construction is motivated by previous work of V. Shende, C. Vafa and the first author on the large $N$-duality derivation of the above conjecture.

Feynman graph integrals and almost modular forms

Abstract: We introduce a type of graph integrals on elliptic curves from the heat kernel. We show that such graph integrals have modular properties under the modular group $SL(2, \Z)$, and prove the polynomial nature of the anti-holomorphic dependence.

Some $tt$* structures and their integral Stokes data

Abstract: In [16] a description was given of all smooth solutions of the two-function tt*-Toda equations in terms of asymptotic data, holomorphic data, and monodromy data. In this supplementary article we focus on the holomorphic data and its interpretation in quantum cohomology, and enumerate those solutions with integral Stokes data. This leads to a characterization of quantum Dmodules for certain complete intersections of Fano type in weighted projective spaces. 1. The tt*-Toda equations The tt* (topological—anti-topological fusion) equations were introduced by S. Cecotti and C. Vafa in their work on deformations of quantum field theories with N=2 supersymmetry (section 8 of [3], and also [4],[5]). This has led to the development of an area known as tt* geometry ([3],[11],[19]), a generalization of special geometry. Solutions of the tt* equations can be interpreted as pluriharmonic maps with values in the noncompact real symmetric space GLnR/On, or as pluriharmonic maps with values in a certain classifying space of variations of polarized (finite or infinite-dimensional) Hodge structure. Frobenius manifolds with real structure, e.g. quantum cohomology algebras, provide a very special class of solutions “of geometric origin” (see [11]). These special solutions lie at the intersection of p.d.e. theory, integrable systems, and (differential, algebraic, and symplectic) geometry. However, very few concrete examples have been worked out in detail, and their study is just beginning. It is relatively straightforward to obtain local solutions, but these special solutions have (or are expected to have) global properties, and these properties are hard to establish. In [17], [16] a family of global solutions was constructed by relatively elementary p.d.e. methods. In this article we shall describe the special solutions in terms of their holomorphic data. This allows us to obtain — in a very restricted situation — an a fortiori characterization of 2000 Mathematics Subject Classification. Primary 81T40; Secondary 53D45, 35J60.

Higher rank stable pairs on k3 surfaces

Abstract: We define and compute higher rank analogs of Pandharipande– Thomas stable pair invariants in primitive classes for K3 surfaces. Higher rank stable pair invariants for Calabi–Yau threefolds have been defined by Sheshmani [26, 27] using moduli of pairs of the form O^n →F for F purely one-dimensional and computed via wall-crossing techniques. These invariants may be thought of as virtually counting embedded curves decorated with a (n − 1)- dimensional linear system. We treat invariants counting pairs O^n → E on a K3 surface for E an arbitrary stable sheaf of a fixed numerical type (“coherent systems” in the language of [16]) whose first Chern class is primitive, and fully compute them geometrically. The ordinary stable pair theory of K3 surfaces is treated by [22]; there they prove the KKV conjecture in primitive classes by showing the resulting partition functions are governed by quasimodular forms. We prove a “higher” KKV conjecture by showing that our higher rank partition functions are modular forms.

Rigid local systems and a question of Wootters

Abstract: Recently we learned from Ron Evans of some fascinating questions raised by Wootters [A-S-S-W]. These questions, which concern exponential sums, arose from his investigations of a particular quantum state with special properties, where the underlying vector space is the space of functions on the finite field Fp := Z/pZ, p a prime which is 3 mod 4. Due to our ignorance of the underlying physics, we concentrate on the exponential sums themselves. In our approach, it costs us nothing to work over an arbitrary finite field Fq of odd characteristic. [Thus Fq is “the” finite field of q elements, q a power of some odd prime p.] We also introduce a parameter a ∈ Fq . In the Wootters setup, where q = p is 3 mod 4, the parameter a is simply a = −1. Ultimately we end up proving identities among exponential sums, but not at all in a straightforward way; we need to invoke the theory of Kloosterman sheaves and their rigidity properties, as well as the fundamental results of [De-Weil II] and [BBD]. It would be interesting to find direct proofs of these identities.

On the Arithmetic of D-brane Superpotentials. Lines and Conics on the Mirror Quintic

Abstract: Irrational invariants from D-brane superpotentials are pursued on the mirror quintic, systematically according to the degree of a representative curve. Lines are completely understood: the contribution from isolated lines vanishes. All other lines can be deformed holomorphically to the van Geemen lines, whose superpotential is determined via the associated inhomogeneous Picard-Fuchs equation. Substantial progress is made for conics: the families found by Mustat¸ya contain conics reducible to isolated lines, hence they have a vanishing superpotential. The search for all conics invariant under a residual Z2 symmetry reduces to an algebraic problem at the limit of our computational capabilities. The main results are of arithmetic flavor: the extension of the moduli space by the algebraic cycle splits in the large complex structure limit into groups each governed by an algebraic number field. The expansion coefficients ofthe superpotential around large volume remain irrational. The integrality of those coefficients is revealed by a new, arithmetic twist of the di-logarithm: the D-logarithm. There are several options for attempting to explain how these invariants could arise from the A-model perspective. A successful spacetime interpretation will require spaces of BPS states to carry number theoretic structures, such as an action of the Galois group.

On multiple higher Mahler measures and Witten zeta values associated with semisimple Lie algebras

Abstract: The Witten zeta-functions associated with semisimple Lie algebras were defined by Zagier, and their special values at even positive integers were first studied by Witten in connection with quantum gauge theory. In this paper, relations between multiple higher Mahler measures for some families of polynomials and special values of Witten zeta-functions at positive integers are showed. Consequently, a geometrical interpretation of the multiple higher Mahler measure as the volume of certain moduli space is given.