A bstract We calculate the generating functions of BPS indices using their modular properties in Type II and M-theory compactifications on compact genus one fibered CY 3-folds with singular fibers additional rational sections or just N -sections, order to study string dualities four five dimensions as well rigid limits which gravity decouples. The are Jacobi-forms Γ 1 ( ) complexified fiber volume parameter. coupling λ , ϵ ± parameters limit, masses charged hypermultiplets non-Abelian gauge...

Abstract In this paper we propose a definition of torsion refined Gopakumar–Vafa (GV) invariants for Calabi–Yau threefolds with terminal nodal singularities that do not admit Kähler crepant resolutions. Physically, the refinement takes into account charge five-dimensional BPS states under discrete gauge symmetry in M-theory. We mathematical terms geometry all non-Kähler resolutions taken together. The are encoded A-model topological string partition functions associated to non-commutative...

We give further evidence that genus-one fibers with multi-sections are mirror dual to Mordell-Weil torsion. In the physics of F-theory compactifications this implies a relation between models non-simply connected gauge group and those discrete symmetries. provide combinatorial explanation phenomenon for toric hypersurfaces. particular leads criterion deduce torsion directly from polytope. For all 3134 complete intersection curves in three-dimensional ambient spaces we confirm conjecture by...

A bstract We present a novel technique to obtain base independent expressions for the matter loci of fibrations complete intersection Calabi-Yau onefolds in toric ambient spaces. These can be used systematically construct elliptically and genus one fibered d -folds that lead desired gauge groups spectra F-theory. The technique, which we refer as GV-spectroscopy, is based on calculation fiber Gopakumar-Vafa invariants using Batyrev-Borisov construction mirror pairs application so-called...

A bstract We study global anomalies of discrete gauge symmetries in six-dimensional supergravities and their realizations F-theory. explicitly construct a Green-Schwarz mechanism that depends on the choice coupling constant certain quadratic refinement differential cohomology. By geometrically engineering theories with G = ℤ 3 symmetry no tensor multiplets, we observe particular is singled out This implies new Swampland constraints charge spectra 6d supergravities. On other hand, geometry...

This work considers aspects of almost holomorphic and meromorphic Siegel modular forms from the perspective physics mathematics.The first part is concerned with (refined) topological string theory direct integration anomaly equations.Here, a central object to compute higher genus amplitudes, which serve as generating functions various enumerative invariants, provided by so-called propagator.We derive universal expression for propagator geometries that have mirror curves two given derivative...

A bstract In this note we describe a method to calculate the action of particular Fourier-Mukai transformation on basis brane charges elliptically fibered Calabi-Yau threefolds with and without section. The kernel is ideal sheaf relative diagonal for fibrations that admit section essentially Poincaré sheaf. We find in case it induces an modular group 2-branes.

We extend the dictionary between BPS spectrum of Heterotic strings and one F-/M-theory compactifications on $K3$ fibered Calabi-Yau 3-folds to cases with higher rank non-Abelian gauge groups in particular dual pairs CHL orbifolds a compatible genus fibration. show how obtain new supersymmetric index purely from geometry by reconstructing Noether-Lefschetz generators, which are vector-valued modular forms. There is an isomorphism latter objects lattice Jacobi forms, relates them elliptic...

A bstract We show that the stringy Kähler moduli space of a generic genus one curve degree N , for ≤ 5, is Γ 1 ( ) modular X ). This implies correspondence between cusps curves and certain large volume limits in spaces fibered Calabi-Yau manifolds with -sections. Using Higgs transitions M-theory F-theory as well properties topological string partition function, we identify these elements Tate-Shafarevich group fibration. Singular appear form non-commutative resolutions torsional B-field at...

We discuss the period geometry and topological string amplitudes on elliptically fibered Calabi-Yau fourfolds in toric ambient spaces. In particular, we describe a general procedure to fix integral periods. Using some elementary facts from homological mirror symmetry then obtain Bridgelands involution its monodromy action basis for non-singular fourfolds. The full group contains subgroup that acts as PSL(2,Z) Kähler modulus of fiber analyze consequences this modularity genus zero one well...

Elliptic and genus one fibered Calabi-Yau spaces play a prominent role in string theory mathematics. In this article we discuss class of threefolds with 5-sections from various perspectives. algebraic geometry, such Calabi-Yaus can be constructed as complete intersections Grassmannian fibrations Pfaffian varieties. These constructions naturally fit into the framework homological projective duality lead to dual pairs Calabi-Yaus. From physics perspective, these realised low-energy...

In this paper we study compactifications of the $$ \mathcal{N} = 2 heterotic E8× E8 string on (K3 × T2)/ℤ3 with various gauge backgrounds and calculate topological couplings in effective supergravity action that arise from one-loop amplitudes. We then identify candidates for dual type IIA Calabi-Yau threefolds compare results corresponding find geometries are K3 fibrations also genus one fibered three- sections. Moreover, show intersection form polarization lattice fibration has to be three...

Motivated in part by the modular properties of enumerative invariants K3-fibered Calabi-Yau threefolds, we introduce a family 39 mirror pairs $(X,Y)$ with $h_{1,1}(X)=h_{2,1}(Y)=2$, labelled certain integer quadruples $(m,i,j,s)$ $m\leq 11$. On A-model side, $X$ arises as complete intersection projective bundle over Fano fourfold $V_m^{[i,j]}$, and admits Tyurin degeneration into pair degree $m$ threefolds $F_m^{[i]}\cup F_m^{[j]}$ intersecting on an anticanonical K3 divisor $2m$. B-model...

By exploiting new mathematical relations between Pandharipande-Thomas (PT) invariants, closely related to Gopakumar-Vafa (GV) and rank 0 Donaldson-Thomas (DT) invariants counting D4-D2-D0 BPS bound states, we rigorously compute the first few terms in generating series of Abelian indices for compact one-parameter Calabi-Yau threefolds hypergeometric type. In all cases where GV can be computed sufficiently high genus, find striking confirmation that is modular, predict infinite indices....

We study topological strings on non-commutative resolutions of singular Calabi-Yau threefolds that are double covers $\mathbb{P}^3$, ramified over determinantal octic surfaces. Using conifold transitions to complete intersections in toric ambient spaces, we prove any small resolution has 2-torsional exceptional curves and is necessarily non-K\"ahler. The same imply M-theory develops a $\mathbb{Z}_2$ gauge symmetry the space. then construct gauged linear sigma models with hybrid phases flow...

This work considers aspects of almost holomorphic and meromorphic Siegel modular forms from the perspective physics mathematics. The first part is concerned with (refined) topological string theory direct integration anomaly equations. Here, a central object to compute higher genus amplitudes, which serve as generating functions various enumerative invariants, provided by so-called propagator. We derive universal expression for propagator geometries that have mirror curves two given...

We discuss the period geometry and topological string amplitudes on elliptically fibered Calabi-Yau fourfolds in toric ambient spaces. In particular, we describe a general procedure to fix integral periods. Using some elementary facts from homological mirror symmetry then obtain Bridgelands involution its monodromy action basis for non-singular fourfolds. The full group contains subgroup that acts as PSL(2,Z) K\"ahler modulus of fiber analyze consequences this modularity genus zero one well...