arXiv subjects

Albrecht Klemm

Publications and source records attributed to Albrecht Klemm.

At least 37 records · Page 2Linked to original sources

arXiv2021

Elliptic Blowup Equations for 6d SCFTs. IV: Matters

Given the recent geometrical classification of 6d $(1,0)$ SCFTs, a major question is how to compute for this large class their elliptic genera. The latter encode the refined BPS spectrum of the SCFTs, which determines geometric invariants of the associated elliptic non-compact Calabi-Yau threefolds. In this paper we establish for all 6d $(1,0)$ SCFTs in the atomic classification blowup equations that fix these elliptic genera to large extent. The latter fall into two types: the unity- and the vanishing blowup equations. For almost all rank one theories, we find unity blowup equations which determine the elliptic genera completely. We develop several techniques to compute elliptic genera and BPS invariants from the blowup equations, including a recursion formula with respect to the number of strings, a Weyl orbit expansion, a refined BPS expansion and an $ε_1,ε_2$ expansion. For higher-rank theories, we propose a gluing rule to obtain all their blowup equations based on those of rank one theories. For example, we explicitly give the elliptic blowup equations for the three higher-rank non-Higgsable clusters, ADE chain of $-2$ curves and conformal matter theories. We also give the toric construction for many elliptic non-compact Calabi-Yau threefolds which engineer 6d $(1,0)$ SCFTs with various matter representations.

hep-th↗

arXiv2021

Analytic Structure of all Loop Banana Amplitudes

Using the Gelfand-Kapranov-Zelevinsk\uı system for the primitive cohomology of an infinite series of complete intersection Calabi-Yau manifolds, whose dimension is the loop order minus one, we completely clarify the analytic structure of all banana amplitudes with arbitrary masses. In particular, we find that the leading logarithmic structure in the high energy regime, which corresponds to the point of maximal unipotent monodromy, is determined by a novel $\widehat Γ$-class evaluation in the ambient spaces of the mirror, while the imaginary part of the amplitude in this regime is determined by the $\widehat Γ$-class of the mirror Calabi-Yau manifold itself. We provide simple closed all loop formulas for the former as well as for the Frobenius $κ$-constants, which determine the behaviour of the amplitudes, when the momentum square equals the sum of the masses squared, in terms of zeta values. We extend our previous work from three to four loops by providing for the latter case a complete set of (inhomogenous) Picard-Fuchs differential equations for arbitrary masses. This allows to evaluate the amplitude as well as other master integrals with raised powers of the propagators in very short time to very high numerical precision for all values of the physical parameters. Using a recent $p$-adic analysis of the periods we determine the value of the maximal cut equal mass four-loop amplitude at the attractor points in terms of periods of modular weight two and four Hecke eigenforms and the quasiperiods of their meromorphic cousins.

hep-th↗

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.

hep-th↗

arXiv2021

Lost Chapters in CHL Black Holes: Untwisted Quarter-BPS Dyons in the $\mathbb{Z}_2$ Model

Motivated by recent advances in Donaldson-Thomas theory, four-dimensional $\mathcal{N}=4$ string-string duality is examined in a reduced rank theory on a less studied BPS sector. In particular we identify candidate partition functions of "untwisted" quarter-BPS dyons in the heterotic $\mathbb{Z}_2$ CHL model by studying the associated chiral genus two partition function, based on the M-theory lift of string webs argument by Dabholkar and Gaiotto. This yields meromorphic Siegel modular forms for the Iwahori subgroup $B(2) \subset \text{Sp}_4 (\mathbb{Z}) $ which generate BPS indices for dyons with untwisted sector electric charge, in contrast to twisted sector dyons counted by a multiplicative lift of twisted-twining elliptic genera known from Mathieu moonshine. The new partition functions are shown to satisfy the expected constraints coming from wall-crossing and S-duality symmetry as well as the black hole entropy based on the Gauss-Bonnet term in the effective action. In these aspects our analysis confirms and extends work of Banerjee, Sen and Srivastava, which only addressed a subset of the untwisted sector dyons considered here. Our results are also compared with recently conjectured formulae of Bryan and Oberdieck for the partition functions of primitive DT invariants of the CHL orbifold $X=( \text{K3} \times T^2 )/ \mathbb{Z}_2$, as suggested by string duality with type IIA theory on $X$.

hep-th↗

arXiv2020

Towards Refining the Topological Strings on Compact Calabi-Yau 3-folds

We make a proposal for calculating refined Gopakumar-Vafa numbers (GVN) on elliptically fibered Calabi-Yau 3-folds based on refined holomorphic anomaly equations. The key examples are smooth elliptic fibrations over (almost) Fano surfaces. We include a detailed review of existing mathematical methods towards defining and calculating the (unrefined) Gopakumar-Vafa invariants (GVI) and the GVNs on compact Calabi-Yau 3-folds using moduli of stable sheaves, in a language that should be accessible to physicists. In particular, we discuss the dependence of the GVNs on the complex structure moduli and on the choice of an orientation. We calculate the GVNs in many instances and compare the B-model predictions with the geometric calculations. We also derive the modular anomaly equations from the holomorphic anomaly equations by analyzing the quasi-modular properties of the propagators. We speculate about the physical relevance of the mathematical choices that can be made for the orientation.

hep-th↗

arXiv2020

The $l$-loop Banana Amplitude from GKZ Systems and relative Calabi-Yau Periods

We use the GKZ description of periods and certain classes of relative periods on families of Barth-Nieto Calabi-Yau $(l-1)$-folds in order to solve the $l$-loop banana amplitudes with their general mass dependence. As examples we compute the mass dependencies of the banana amplitudes up to the three-loop case and check the results against the known results for special mass values.

hep-th↗

arXiv2019

Elliptic Blowup Equations for 6d SCFTs. III: E-strings, M-strings and Chains

We establish the elliptic blowup equations for E-strings and M-strings and solve elliptic genera and refined BPS invariants from them. Such elliptic blowup equations can be derived from a path integral interpretation. We provide toric hypersurface construction for the Calabi-Yau geometries of M-strings and those of E-strings with up to three mass parameters turned on, as well as an approach to derive the perturbative prepotential directly from the local description of the Calabi-Yau threefolds. We also demonstrate how to systematically obtain blowup equations for all rank one 5d SCFTs from E-string by blow-down operations. Finally, we present blowup equations for E-M and M string chains.

hep-th↗

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.

hep-th↗

arXiv2019

Elliptic Blowup Equations for 6d SCFTs. II: Exceptional Cases

The building blocks of 6d $(1,0)$ SCFTs include certain rank one theories with gauge group $G=SU(3),SO(8),F_4,E_{6,7,8}$. In this paper, we propose a universal recursion formula for the elliptic genera of all such theories. This formula is solved from the elliptic blowup equations introduced in our previous paper. We explicitly compute the elliptic genera and refined BPS invariants, which recover all previous results from topological string theory, modular bootstrap, Hilbert series, 2d quiver gauge theories and 4d $\mathcal{N}=2$ superconformal $H_{G}$ theories. We also observe an intriguing relation between the $k$-string elliptic genus and the Schur indices of rank $k$ $H_{G}$ SCFTs, as a generalization of Lockhart-Zotto's conjecture at the rank one cases. In a subsequent paper, we deal with all other non-Higgsable clusters with matters.

hep-th↗

arXiv2019

Swampland Distance Conjecture for One-Parameter Calabi-Yau Threefolds

We investigate the swampland distance conjecture (SDC) in the complex moduli space of type II compactifications on one-parameter Calabi-Yau threefolds. This class of manifolds contains hundreds of examples and, in particular, a subset of 14 geometries with hypergeometric differential Picard-Fuchs operators. Of the four principal types of singularities that can occur - specified by their limiting mixed Hodge structure - only the K-points and the large radius points (or more generally the M-points) are at infinite distance and therefore of interest to the SDC. We argue that the conjecture is fulfilled at the K- and the M-points, including models with several M-points, using explicit calculations in hypergeometric models which contain typical examples of all these degenerations. Together with previous work on the large radius points, this suggests that the SDC is indeed fulfilled for one-parameter Calabi-Yau spaces.

hep-th↗

arXiv2019

WKB Method and Quantum Periods beyond Genus One

We extend topological string methods in order to perform WKB approximations for quantum mechanical problems with higher order potentials efficiently. This requires techniques for the evaluation of the relevant quantum periods for Riemann surfaces beyond genus one. The basis of these quantum periods is fixed using the leading behaviour of the classical periods. The full expansion of the quantum periods is obtained using a system of Picard-Fuchs like operators for a sequence of integrals of meromorphic forms of the second kind. Discrete automorphisms of simple higher order potentials allow to view the corresponding higher genus curves as covering of a genus one curve. In this case the quantum periods can be alternatively obtained using the holomorphic anomaly solved in the holomorphic limit within the ring of quasi modular forms of a congruent subgroup of SL$(2,\mathbb{Z})$ as we check for a symmetric sextic potential.

hep-th↗

arXiv2018

Topological Strings on Singular Elliptic Calabi-Yau 3-folds and Minimal 6d SCFTs

We apply the modular approach to computing the topological string partition function on non-compact elliptically fibered Calabi-Yau 3-folds with higher Kodaira singularities in the fiber. The approach consists in making an ansatz for the partition function at given base degree, exact in all fiber classes to arbitrary order and to all genus, in terms of a rational function of weak Jacobi forms. Our results yield, at given base degree, the elliptic genus of the corresponding non-critical 6d string, and thus the associated BPS invariants of the 6d theory. The required elliptic indices are determined from the chiral anomaly 4-form of the 2d worldsheet theories, or the 8-form of the corresponding 6d theories, and completely fix the holomorphic anomaly equation constraining the partition function. We introduce subrings of the known rings of Weyl invariant Jacobi forms which are adapted to the additional symmetries of the partition function, making its computation feasible to low base wrapping number. In contradistinction to the case of simpler singularities, generic vanishing conditions on BPS numbers are no longer sufficient to fix the modular ansatz at arbitrary base wrapping degree. We show that to low degree, imposing exact vanishing conditions does suffice, and conjecture this to be the case generally.

hep-th↗

arXiv2018

Refined BPS invariants of 6d SCFTs from anomalies and modularity

F-theory compactifications on appropriate local elliptic Calabi-Yau manifolds engineer six dimensional superconformal field theories and their mass deformations. The partition function $Z_{top}$ of the refined topological string on these geometries captures the particle BPS spectrum of this class of theories compactified on a circle. Organizing $Z_{top}$ in terms of contributions $Z_β$ at base degree $β$ of the elliptic fibration, we find that these, up to a multiplier system, are meromorphic Jacobi forms of weight zero with modular parameter the Kaehler class of the elliptic fiber and elliptic parameters the couplings and mass parameters. The indices with regard to the multiple elliptic parameters are fixed by the refined holomorphic anomaly equations, which we show to be completely determined from knowledge of the chiral anomaly of the corresponding SCFT. We express $Z_β$ as a quotient of weak Jacobi forms, with a universal denominator inspired by its pole structure as suggested by the form of $Z_{top}$ in terms of 5d BPS numbers. The numerator is determined by modularity up to a finite number of coefficients, which we prove to be fixed uniquely by imposing vanishing conditions on 5d BPS numbers as boundary conditions. We demonstrate the feasibility of our approach with many examples, in particular solving the E-string and M-string theories including mass deformations, as well as theories constructed as chains of these. We make contact with previous work by showing that spurious singularities are cancelled when the partition function is written in the form advocated here. Finally, we use the BPS invariants of the E-string thus obtained to test a generalization of the Goettsche-Nakajima-Yoshioka $K$-theoretic blowup equation, as inspired by the Grassi-Hatsuda-Marino conjecture, to generic local Calabi-Yau threefolds.

hep-th↗

arXiv2017

Effective action from M-theory on twisted connected sum $G_2$-manifolds

We study the four-dimensional low energy effective $\mathcal{N}=1$ supergravity theory of the dimensional reduction of M-theory on $G_2$-manifolds, which are constructed by Kovalev's twisted connected sum gluing suitable pairs of asymptotically cylindrical Calabi-Yau threefolds $X_{L/R}$ augmented with a circle $S^1$. In the Kovalev limit the Ricci-flat $G_2$-metrics are approximated by the Ricci-flat metrics on $X_{L/R}$ and we identify the universal modulus - the Kovalevton - that parametrizes this limit. We observe that the low energy effective theory exhibits in this limit gauge theory sectors with extended supersymmetry. We determine the universal (semi-classical) Kähler potential of the effective $\mathcal{N}=1$ supergravity action as a function of the Kovalevton and the volume modulus of the $G_2$-manifold. This Kähler potential fulfills the no-scale inequality such that no anti-de-Sitter vacua are admitted. We describe geometric degenerations in $X_{L/R}$, which lead to non-Abelian gauge symmetries enhancements with various matter content. Studying the resulting gauge theory branches, we argue that they lead to transitions compatible with the gluing construction and provide many new explicit examples of $G_2$-manifolds.

hep-th↗

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.

hep-th↗

arXiv2015

Exact solutions to quantum spectral curves by topological string theory

We generalize the conjectured connection between quantum spectral problems and topological strings to many local almost del Pezzo surfaces with arbitrary mass parameters. The conjecture uses perturbative information of the topological string in the unrefined and the Nekrasov-Shatashvili limit to solve non-perturbatively the quantum spectral problem. We consider the quantum spectral curves for the local almost del Pezzo surfaces of $\mathbb{F}_2$, $\mathbb{F}_1$, the blowup of $\mathbb{P}^2$ in two points and a mass deformation of the $E_8$ del Pezzo corresponding to different deformations of the three-term operators $\mathsf{O}_{1,1}$, $\mathsf{O}_{1,2}$ and $\mathsf{O}_{2,3}$. To check the conjecture, we compare the predictions for the spectrum of these operators with numerical results for the eigenvalues. We also compute the first few fermionic spectral traces from the conjectural spectral determinant, and we compare them to analytic and numerical results in spectral theory. In all these comparisons, we find that the conjecture is fully validated with high numerical precision. For local $\mathbb{F}_2$ we expand the spectral determinant around the orbifold point and find intriguing relations for Jacobi theta functions. We also give an explicit map between the geometries of $\mathbb{F}_0$ and $\mathbb{F}_2$ as well as a systematic way to derive the operators $\mathsf{O}_{m,n}$ from toric geometries.

hep-th↗

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.

hep-th↗

arXiv2015

Topological String on elliptic CY 3-folds and the ring of Jacobi forms

We give evidence that the all genus amplitudes of topological string theory on compact elliptically fibered Calabi-Yau manifolds can be written in terms of meromorphic Jacobi forms whose weight grows linearly and whose index grows quadratically with the base degree. The denominators of these forms have a simple universal form with the property that the poles of the meromorphic form lie only at torsion points. The modular parameter corresponds to the fibre class while the role of the string coupling is played by the elliptic parameter. This leads to very strong all genus results on these geometries, which are checked against results from curve counting. The structure can be viewed as an indication that an N=2 analog of the reciprocal of the Igusa cusp form exists that might govern the topological string theory on these Calabi-Yau manifolds completely.

hep-th↗