arXiv subjects

Thorsten Schimannek

Publications and source records attributed to Thorsten Schimannek.

18 recordsLinked to original sources

arXiv2026

Revisiting the Quantum Geometry of Torus-fibered Calabi-Yau Threefolds

About ten years ago, Katz, Klemm and Huang conjectured that topological string amplitudes on compact, elliptically fibered Calabi-Yau threefolds at fixed base degree could be expressed in terms of meromorphic Jacobi forms for $SL(2,\mathbb{Z})$, giving access to Gromov-Witten invariants at arbitrary genus. This was later generalized to torus-fibered CY threefolds with $N$-sections, where topological string amplitudes are conjecturally governed by meromorphic Jacobi forms under the congruence subgroup $Γ_1(N)$. In this work, we show that these modularity properties follow from (and are equivalent to) the wave-function property of the topological string partition function $Z_{\rm top}$ under a relative conifold monodromy, implementing a particular Fourier-Mukai transformation on the derived category of coherent sheaves. In particular, we introduce a variant of $Z_{\rm top}$ which is both holomorphic and modular covariant. Under the same relative conifold monodromy, the generating series of genus 0 Gopakumar-Vafa invariants at fixed base degree is mapped to the generating series of rank 0 Donaldson-Thomas indices counting D4-D2-D0-brane bound states wrapped on the torus fiber. We show that the quasimodularity of the generating series of GV invariants matches the expected mock-modular behavior of the generating series of D4-D2-D0 indices, despite having different multi-cover contributions. We analyze and tabulate a large number of CY threefolds fibered over del Pezzo surfaces, with an $N$-section for $N\leq 5$, including several new examples beyond the realm of toric geometry.

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arXiv2025

The twisted geometry of 6d F-theory vacua with discrete gauge symmetries

We study the fate of discrete gauge groups and discrete charges of gravitational theories under twisted circle compactification. We then apply our results to six-dimensional F-theory vacua with discrete gauge symmetries and relate them to the geometry of the genus one fibered Calabi-Yau threefolds that underlie the dual M-theory compactifications. This leads us to introduce a class of geometries, which we call almost generic elliptic/genus one fibered Calabi-Yau threefolds, and to make detailed conjectures about their properties. A second twisted circle compactification relates these M-theory vacua to Type IIA compactifications with flat but topologically non-trivial B-fields along the internal geometry. The A-model topological string partition function on such configurations is intimately tied to the twisted-twined elliptic genera of the six-dimensional non-critical strings of the associated F-theory vacuum. The modular properties of the twisted-twined elliptic genera imply new twisted derived equivalences. We thus recover and significantly extend earlier results from both the physical and the mathematical literature. An important outcome of our study is that if the discrete gauge symmetry is not cyclic, then no smooth genus one fibration exists that represents the associated axio-dilaton profile.

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arXiv2025

In search of almost generic Calabi-Yau 3-folds

We call a projective Calabi-Yau (CY) 3-fold almost generic if it has only isolated nodes as singularities and the homology classes of all of the exceptional curves in an analytic small resolution are non-trivial but torsion. Such a Calabi-Yau supports a topologically non-trivial flat B-field and the corresponding A-model topological string partition function encodes a torsion refinement of the Gopakumar-Vafa invariants of the smooth deformation. Our goal in this paper is to find new examples of almost generic CY 3-folds, using both conifold transitions as well as the integral structure of the periods of the mirrors. In this way we explicitly construct two quintic CY 3-folds with $\mathbb{Z}_2$-torsion, two octics with $\mathbb{Z}_3$-torsion and deduce the existence of a complete intersection $X_{(6,6)}\subset\mathbb{P}^5_{1,1,2,2,3,3}$ with $\mathbb{Z}_5$-torsion. Via mirror symmetry, the examples give new geometric interpretations to several AESZ Calabi-Yau operators. The mirror periods of the almost generic $X_{(6,6)}$ with non-trivial B-field topology are annihilated by an irrational Picard-Fuchs operator. We describe how the usual integral structure of the periods has to be modified and in all of the cases we calculate the monodromies around the singular points to verify integrality. Additional points of maximally unipotent monodromy in the moduli spaces lead us to find several more examples of smooth or almost generic CY 3-folds and to conjecture new twisted derived equivalences. We integrate the holomorphic anomaly equations and extract the torsion refined Gopakumar-Vafa invariants up to varying genus. For our construction of the almost generic octic CY 3-folds, we also give a short introduction to the subject of hypermatrices and hyperdeterminants.

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arXiv2024

Enumerative geometry and modularity in two-modulus K3-fibered Calabi-Yau threefolds

Motivated in part by the modular properties of enumerative invariants of K3-fibered Calabi-Yau threefolds, we introduce a family of 39 Calabi-Yau mirror pairs $(X,Y)$ with $h_{1,1}(X)=h_{2,1}(Y)=2$, labelled by certain integer quadruples $(m,i,j,s)$ with $m\leq 11$. On the A-model side, $X$ arises as a complete intersection in a projective bundle over a Fano fourfold $V_m^{[i,j]}$, and admits a Tyurin degeneration into a pair of degree $m$ Fano threefolds $F_m^{[i]}\cup F_m^{[j]}$ intersecting on an anticanonical K3 divisor of degree $2m$. On the B-model side, $Y$ is fibered by $M_{m}$-polarized K3-surfaces of Picard rank 19, and determined by a branched covering of $\mathbb{P}^1$, consistent with the Doran-Harder-Thompson mirror conjecture. When $s=0$, $Y$ itself acquires a Tyurin degeneration, and correspondingly $X$ acquires a fibration by degree $2m$ K3 surfaces, such that the two Kähler moduli control the size of the K3-fiber and base $\mathbb{P}^1$. While the mirror pairs with $m\leq 4$ can be realized as complete intersections in products of projective spaces or as hypersurfaces in toric varieties, the examples with $m\geq 5$ are intrinsically non-toric. We obtain uniform formulae for the genus 0 and 1 topological free energies near the Tyurin degeneration (mirror to the large base limit), exhibiting modular properties under the Fricke-extended congruence group $Γ_0(m)^+$. We use these results to compute the vertical Gopakumar-Vafa and Noether-Lefschetz invariants and check that their generating functions satisfy the expected modular properties. We also compute generating series of Gopakumar-Vafa invariants with fixed non-zero base degree and exhibit their modular properties.

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arXiv2024

Quantum geometry, stability and modularity

By exploiting new mathematical relations between Pandharipande-Thomas (PT) invariants, closely related to Gopakumar-Vafa (GV) invariants, and rank 0 Donaldson-Thomas (DT) invariants counting D4-D2-D0 BPS bound states, we rigorously compute the first few terms in the generating series of Abelian D4-D2-D0 indices for compact one-parameter Calabi-Yau threefolds of hypergeometric type. In all cases where GV invariants can be computed to sufficiently high genus, we find striking confirmation that the generating series is modular, and predict infinite series of Abelian D4-D2-D0 indices. Conversely, we use these results to provide new constraints for the direct integration method, which allows to compute GV invariants (and therefore the topological string partition function) to higher genus than hitherto possible. The triangle of relations between GV/PT/DT invariants is powered by a new explicit formula relating PT and rank 0 DT invariants, which is proven in an Appendix by the second named author. As a corollary, we obtain rigorous Castelnuovo-type bounds for PT and GV invariants for CY threefolds with Picard rank one.

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arXiv2023

Topological Strings on Non-Commutative Resolutions

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 the charge of five-dimensional BPS states under a discrete gauge symmetry in M-theory. We propose a mathematical definition of the invariants in terms of the geometry of all non-Kähler crepant resolutions taken together. The invariants are encoded in the A-model topological string partition functions associated to non-commutative (nc) resolutions of the Calabi-Yau. Our main example will be a singular degeneration of the generic Calabi-Yau double cover of $\mathbb{P}^3$ and leads to an enumerative interpretation of the topological string partition function of a hybrid Landau-Ginzburg model. Our results generalize a recent physical proposal made in the context of torus fibered Calabi-Yau manifolds by one of the authors and clarify the associated enumerative geometry.

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arXiv2023

New non-commutative resolutions of determinantal Calabi-Yau threefolds from hybrid GLSM

We study topological strings on non-commutative resolutions of singular Calabi-Yau threefolds that are double covers of $\mathbb{P}^3$, ramified over determinantal octic surfaces. Using conifold transitions to complete intersections in toric ambient spaces, we prove that any small resolution has 2-torsional exceptional curves and is necessarily non-Kähler. The same transitions imply that M-theory develops a $\mathbb{Z}_2$ gauge symmetry on the singular space. We then construct gauged linear sigma models with hybrid phases that flow to the worldsheet theories of strings propagating on the determinantal double solids in the presence of a flat but topologically non-trivial B-field. Localizing the sphere partition function allows us to calculate the fundamental periods of the mirror Calabi-Yau manifolds, then we check agreement with the periods of the Borisov-Li mirrors. We find that the corresponding variations of Hodge structure either correspond to one of the 14 hypergeometric cases or to a double cover thereof. We then use mirror symmetry and integrate the holomorphic anomaly equations to calculate $\mathbb{Z}_2$-refined Gopakumar-Vafa invariants for several examples.

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arXiv2022

The discrete Green-Schwarz mechanism in 6D F-Theory and Elliptic Genera of Non-Critical Strings

We study global anomalies of discrete gauge symmetries in six-dimensional supergravities and their realizations in F-theory. We explicitly construct a discrete Green-Schwarz mechanism that depends on the choice of a coupling constant and on a certain quadratic refinement in differential cohomology. By geometrically engineering theories with $G=\mathbb{Z}_3$ gauge symmetry and no tensor multiplets, we observe that a particular choice of the quadratic refinement is singled out in F-theory. This implies new Swampland constraints on the discrete charge spectra of 6d supergravities. On the other hand, the discrete Green-Schwarz coupling depends on the geometry of the Calabi-Yau. We use anomaly inflow to relate this to a 't Hooft anomaly of the induced global symmetry in the worldsheet theories of non-critical strings. Using topological symmetry lines, we further relate this anomaly to the modular properties of twisted-twined elliptic genera. We then argue that the latter are encoded in the A-model topological string partition functions on different torus fibrations that are equipped with a flat torsional B-field. This allows us to derive a geometric expression for the global discrete anomaly in terms of the height-pairing of a multi-section on a genus one fibered Calabi-Yau.

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arXiv2021

Modular curves, the Tate-Shafarevich group and Gopakumar-Vafa invariants with discrete charges

We show that the stringy Kähler moduli space of a generic genus one curve of degree $N$, for $N\le 5$, is the $Γ_1(N)$ modular curve $X_1(N)$. This implies a correspondence between the cusps of the modular curves and certain large volume limits in the stringy Kähler moduli spaces of genus one fibered Calabi-Yau manifolds with $N$-sections. Using Higgs transitions in M-theory and F-theory as well as modular properties of the topological string partition function, we identify these large volume limits with elements of the Tate-Shafarevich group of the genus one fibration. Singular elements appear in the form of non-commutative resolutions with a torsional B-field at the singularity. The topological string amplitudes that arise at the various large volume limits are related by modular transformations. In particular, we find that the topological string partition function of a smooth genus one fibered Calabi-Yau threefold is transformed into that of a non-commutative resolution of the Jacobian by a Fricke involution. In the case of Calabi-Yau threefolds, we propose an expansion of the partition functions of a singular fibration and its non-commutative resolutions in terms of Gopakumar-Vafa invariants that are associated to BPS states with discrete charges. For genus one fibrations with 5-sections, this provides an enumerative interpretation for the partition functions that arise at certain irrational points of maximally unipotent monodromy.

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arXiv2021

On genus one fibered Calabi-Yau threefolds with 5-sections

Elliptic and genus one fibered Calabi-Yau spaces play a prominent role in string theory and mathematics. In this article we discuss a class of genus one fibered Calabi-Yau threefolds with 5-sections from various perspectives. In algebraic geometry, such Calabi-Yaus can be constructed as complete intersections in Grassmannian fibrations and as Pfaffian varieties. These constructions naturally fit into the framework of homological projective duality and lead to dual pairs of Calabi-Yaus. From a physics perspective, these spaces can be realised as low-energy configurations ("phases") of gauged linear sigma models (GLSMs) with non-Abelian gauge groups, where the dual geometries arise as phases of the same GLSM. Using the modular bootstrap approach of topological string theory, one can compute all-genus Gopakumar-Vafa invariants of these Calabi-Yaus. We observe that homological projective duality acts as an element of $Γ_0(5)$ on the topological string partition function and the partition functions of dual geometries transform into each other. Moreover, we study the geometries from an M-/F-theory perspective. We compute the F-theory spectrum and show how the genus one-fibered Calabi-Yaus are connected to certain Calabi-Yaus in toric varieties via a series of Higgs transitions. Based on the F-theory physics, we conjecture that dual geometries are elements of the same Tate-Shafarevich group. Our analysis also leads to a classification of 5-section geometries, as well as the construction of F-theory models with charge 5 hypermultiplets.

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arXiv2021

State counting on fibered CY-3 folds and the non-Abelian Weak Gravity Conjecture

We extend the dictionary between the BPS spectrum of Heterotic strings and the one of F-/M-theory compactifications on $K3$ fibered Calabi-Yau 3-folds to cases with higher rank non-Abelian gauge groups and in particular to dual pairs between Heterotic CHL orbifolds and compactifications on Calabi-Yau 3-folds with a compatible genus one fibration. We show how to obtain the new supersymmetric index purely from the Calabi-Yau geometry by reconstructing the Noether-Lefschetz generators, which are vector-valued modular forms. There is an isomorphism between the latter objects and vector-valued lattice Jacobi forms, which relates them to the elliptic genera and twisted-twined elliptic genera of six- and five-dimensional Heterotic strings. The meromorphic Jacobi forms generate the dimensions of the refined cohomology of the Hilbert schemes of symmetric products of the fiber and allow us to refine the BPS indices in the fiber and therefore to obtain, conjecturally, actual state counts. Using the properties of the vector-valued lattice Jacobi forms we also provide a mathematical proof of the non-Abelian weak gravity conjecture for F-/M-theory compactified on this general class of fibered Calabi-Yau 3-folds.

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arXiv2019

GV-Spectroscopy for F-theory on genus-one fibrations

We present a novel technique to obtain base independent expressions for the matter loci of fibrations of complete intersection Calabi-Yau onefolds in toric ambient spaces. These can be used to systematically construct elliptically and genus one fibered Calabi-Yau $d$-folds that lead to desired gauge groups and spectra in F-theory. The technique, which we refer to as GV-spectroscopy, is based on the calculation of fiber Gopakumar-Vafa invariants using the Batyrev-Borisov construction of mirror pairs and application of the so-called Frobenius method to the data of a parametrized auxiliary polytope. In particular for fibers that generically lead to multiple sections, only multi-sections or that are complete intersections in higher codimension, our technique is vastly more efficient than classical approaches. As an application we study two Higgs chains of six-dimensional supergravities that are engineered by fibrations of codimension two complete intersection fibers. Both chains end on a vacuum with $G=\mathbb{Z}_4$ that is engineered by fibrations of bi-quadrics in $\mathbb{P}^3$. We use the detailed knowledge of the structure of the reducible fibers that we obtain from GV-spectroscopy to comment on the corresponding Tate-Shafarevich group. We also show that for all fibers the six-dimensional supergravity anomalies including the discrete anomalies generically cancel.

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arXiv2019

Heterotic strings on $(K3\times T^2)/\mathbb{Z}_3$ and their dual Calabi-Yau threefolds

In this paper we study compactifications of the ${\cal N}=2$ heterotic $E_8\times E_8$ string on $(K3\times T^2)/\mathbb{Z}_3$ with various gauge backgrounds and calculate the topological couplings in the effective supergravity action that arise from one-loop amplitudes. We then identify candidates for dual type IIA compactifications on Calabi-Yau threefolds and compare the heterotic results with the corresponding topological string amplitudes. We find that the dual Calabi-Yau geometries are $K3$ fibrations that are also genus one fibered with three-sections. Moreover, we show that the intersection form on the polarization lattice of the $K3$ fibration has to be three times the intersection form on the Narain lattice $Γ^{1,1}$.

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arXiv2019

Topological strings on genus one fibered Calabi-Yau 3-folds and string dualities

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 and additional rational sections or just $N$-sections, in order to study string dualities in four and five dimensions as well as rigid limits in which gravity decouples. The generating functions are Jacobi-forms of $Γ_1(N)$ with the complexified fiber volume as modular parameter. The string coupling $λ$, or the $ε_\pm$ parameters in the rigid limit, as well as the masses of charged hypermultiplets and non-Abelian gauge bosons are elliptic parameters. To understand this structure, we show that specific auto-equivalences act on the category of topological B-branes on these geometries and generate an action of $Γ_1(N)$ on the stringy Kähler moduli space. We argue that these actions can always be expressed in terms of the generic Seidel-Thomas twist with respect to the 6-brane together with shifts of the B-field and are thus monodromies. This implies the elliptic transformation law that is satisfied by the generating functions. We use Higgs transitions in F-theory to extend the ansatz for the modular bootstrap to genus one fibrations with $N$-sections and boundary conditions fix the all genus generating functions for small base degrees completely. This allows us to study in depth a wide range of new, non-perturbative theories, which are Type II theory duals to the CHL $\mathbb{Z}_N$ orbifolds of the heterotic string on $K3\times T_2$. In particular, we compare the BPS degeneracies in the large base limit to the perturbative heterotic one-loop amplitude with $R_+^2 F_+^{2g-2}$ insertions for many new Type II geometries. In the rigid limit we can refine the ansatz and obtain the elliptic genus of superconformal theories in 5d.

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arXiv2019

Modularity from Monodromy

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

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arXiv2017

Modular Amplitudes and Flux-Superpotentials on elliptic Calabi-Yau fourfolds

We discuss the period geometry and the 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 we then obtain Bridgelands involution and its monodromy action on the integral basis for non-singular elliptically fibered fourfolds. The full monodromy group contains a subgroup that acts as PSL(2,Z) on the Kähler modulus of the fiber and we analyze the consequences of this modularity for the genus zero and genus one amplitudes as well as the associated geometric invariants. We find holomorphic anomaly equations for the amplitudes, reflecting precisely the failure of exact PSL(2,Z) invariance that relates them to quasi-modular forms. Finally we use the integral basis of periods to study the horizontal flux superpotential and the leading order Kähler potential for the moduli fields in F-theory compactifications globally on the complex structure moduli space. For a particular example we verify attractor behaviour at the generic conifold given an aligned choice of flux which we expect to be universal. Furthermore we analyze the superpotential at the orbifold points but find no stable vacua.

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arXiv2016

Mordell-Weil Torsion in the Mirror of Multi-Sections

We give further evidence that genus-one fibers with multi-sections are mirror dual to fibers with Mordell-Weil torsion. In the physics of F-theory compactifications this implies a relation between models with a non-simply connected gauge group and those with discrete symmetries. We provide a combinatorial explanation of this phenomenon for toric hypersurfaces. In particular this leads to a criterion to deduce Mordell-Weil torsion directly from the polytope. For all 3134 complete intersection genus-one curves in three-dimensional toric ambient spaces we confirm the conjecture by explicit calculation. We comment on several new features of these models: The Weierstrass forms of many models can be identified by relabeling the coefficient sections. This reduces the number of models to 1024 inequivalent ones. We give an example of a fiber which contains only non-toric sections one of which becomes toric when the fiber is realized in a different ambient space. Similarly a singularity in codimension one can have a toric resolution in one representation while it is non-toric in another. Finally we give a list of 24 inequivalent genus-one fibers that simultaneously exhibit multi-sections and Mordell-Weil torsion in the Jacobian. We discuss a self-mirror example from this list in detail.

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arXiv2015

Direct Integration for Mirror Curves of Genus Two and an Almost Meromorphic Siegel Modular Form

This work considers aspects of almost holomorphic and meromorphic Siegel modular forms from the perspective of physics and mathematics. The first part is concerned with (refined) topological string theory and the direct integration of the holomorphic anomaly equations. Here, a central object to compute higher genus amplitudes, which serve as the generating functions of various enumerative invariants, is provided by the so-called propagator. We derive a universal expression for the propagator for geometries that have mirror curves of genus two which is given by the derivative of the logarithm of Igusa's cusp form of weight 10. In addition, we illustrate our findings by solving the refined topological string on the resolutions of the three toric orbifolds of order three, five and six. In the second part, we give explicit expressions for lowering and raising operators on Siegel modular forms, and define almost holomorphic Siegel modular forms based on them. Extending the theory of Fourier-Jacobi expansions to almost holomorphic Siegel modular forms and building up on recent work by Pitale, Saha, and Schmidt, we can show that there is no analogue of the almost holomorphic elliptic second Eisenstein series. In the case of genus 2, we provide an almost meromorphic substitute for it. This, in particular, leads us to a generalization of Ramanujan's differential equation for the second Eisenstein series. The two parts are intertwined by the observation that the meromorphic analogue of the almost holomorphic second Eisenstein series coincides with the physical propagator. In addition, the generalized Ramanujan identities match precisely the physical consistency conditions that need to be imposed on the propagator.

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