arXiv subjects

Hans Jockers

Publications and source records attributed to Hans Jockers.

At least 19 recordsLinked to original sources

arXiv2026

Quantum Elliptic Cohomology From Four-Dimensional Minimal Supersymmetric Gauge Theories

In this paper, we study non-perturbative vortex partition functions of four-dimensional N=1 supersymmetric gauge theories on the space-time geometry $T^2 \times D^2$, and we propose that these partition functions offer a notion of quantum elliptic cohomology. For particular U(1)-gauge theories we calculate these partition functions explicitly by applying equivariant localization methods to Handsaw quiver varieties that realize for this particular class of U(1)-gauge theories the moduli spaces of the non-perturbative vortex sectors. The determined vortex partition functions are annihilated by difference operators, which are interpreted as Ward identities among N=(0,2) BPS surface defects. Compared to lower dimensional gauge theories with four supercharges, anomalies play an essential role for a consistent formulation of the four-dimensional partition functions. In our proposal towards a mathematical formulation of the vortex partition functions in terms of equivariant elliptic cohomology, the gauge theory anomalies relate to geometric properties of the Thom sheaves corresponding to the relevant quasimap moduli spaces. Motivated by the explicit computations we reflect on the existence of a `virtual structure sheaf' on the moduli space of quasimaps for a general mathematical theory of quantum elliptic cohomology.

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arXiv2025

Hodge Structures of Complex Multiplication Type from Rational Conformal Field Theories

Under certain assumptions, we show that unitary rational $\mathcal{N}=(2,2)$ conformal field theories together with a certain generating set of Cardy boundary states in the associated boundary conformal field theories give rise to rational Hodge structures of complex multiplication type. We argue that these rational Hodge structures for such rational conformal field theories arising from infrared fixed points of $\mathcal{N}=(2,2)$ non-linear sigma models with Calabi-Yau target spaces coincide with the rational Hodge structures of the middle-dimensional cohomology of the target space geometry. This gives non-trivial evidence of the general expectation in the literature that rational $\mathcal{N}=(2,2)$ supersymmetric conformal field theories associated to Calabi-Yau target spaces yield middle dimensional cohomological rational Hodge structures with complex multiplication. We exemplify our general results with the $\mathcal{N}=2$ A-type minimal model series - which do not have a geometric origin as a non-linear sigma model - and with two explicit $\mathcal{N}=(2,2)$ Gepner models that correspond to particular non-linear sigma models with specific Calabi-Yau threefold target spaces.

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arXiv2024

Non-Perturbative Corrections to 3d BPS Indices and Topological Strings

For a 3d gauged linear sigma model parametrized by a Kahler manifold X, the 3d BPS index defines a q-series that can be analytically continued in the Kahler modulus by standard methods. It is argued that an SL(2,Z)-transform of the Birkhoff connection matrix captures non-perturbative corrections to the 3d GLSM. As an application, a 3d lift of the standard 2d GLSM for the resolved conifold is shown to provide a world-volume dual for the non-perturbative topological string on the resolved conifold. The perturbative 3d BPS index computes the Gopakumar-Vafa partition function, while the analytic continuation matches existing proposals for a non-perturbative completion of the topological string.

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arXiv2024

Minimally Extended Current Algebras of Toroidal Conformal Field Theories

It is well-known that families of two-dimensional toroidal conformal field theories possess a dense subset of rational toroidal conformal field theories, which makes such families an interesting testing ground about rationality of conformal field theories in families in general. Rational toroidal conformal field theories possess an extended chiral and anti-chiral algebra known as W-algebras. Their partition functions decompose into a finite sum of products of holomorphic and anti-holomorphic characters of these W-algebras. Instead of considering these characters, we decompose the partition functions into products of characters of minimal extensions of $\widehat{\mathfrak{u}}(1)$ current algebras, which already appear for rational conformal field theories with target space $S^1$. We present an explicit construction that determines such decompositions. While these decompositions are not unique, they are universal in the sense that any rational toroidal conformal field theory with a target space torus of arbitrary dimension admits such decompositions. We illustrate these decompositions with a few representative examples of rational toroidal conformal field theories with two- and three-dimensional target space tori.

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arXiv2024

On the Geometry of N=2 Minkowski Vacua of Gauged N=2 Supergravity Theories in Four Dimensions

Gauging isometries of four-dimensional N=2 supergravity theories yields an N=2 supersymmetric theory with a scalar potential. In this note, we study the well-known constraints for four-dimensional N=2 Minkowski vacua of such theories. We propose that classically a projective special K\"ahler submanifold of the projective K\"ahler target space of the ungauged theory describes the moduli space of the complex scalar fields of massless vector multiplets for N=2 Minkowski vacua configurations, which then receives quantum corrections from integrating out massive fields. Subloci of projective special K\"ahler manifolds appear as supersymmetric flux vacua in the context of type IIB Calabi-Yau threefold compactifications with background fluxes as well. While these flux vacua equations arise from the critical locus of an N=1 superpotential, we show that these equations can also be obtained from the N=2 supersymmetric Minkowski vacuum equations of gauged N=2 supergravity theories upon gauging suitable isometries in the semi-classical universal hypermultiplet sector of type IIB string Calabi-Yau threefold compactifications. Thus, we give an intrinsic N=2 supersymmetric interpretation to the flux vacua equations.

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arXiv2024

A Calabi-Yau-to-Curve Correspondence for Feynman Integrals

It has long been known that the maximal cut of the equal-mass four-loop banana integral is a period of a family of Calabi-Yau threefolds that depends on the kinematic variable $z=m^2/p^2$. We show that it can also be interpreted as a period of a family of genus-two curves. We do this by introducing a general Calabi-Yau-to-curve correspondence, which in this case locally relates the original period of the family of Calabi-Yau threefolds to a period of a family of genus-two curves that varies holomorphically with the kinematic variable $z$. In addition to working out the concrete details of this correspondence for the equal-mass four-loop banana integral, we outline when we expect a correspondence of this type to hold.

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arXiv2024

Ensemble Averages of $\mathbb{Z}_2$ Orbifold Classes of Narain CFTs

In this work we study families of $\mathbb{Z}_2$ orbifolds of toroidal conformal field theories based on both factorizable and non-factorizable target space tori. For these classes of theories, we analyze their moduli spaces, and compute their partition functions. Building on previous work, we express the calculated partition functions in terms of suitable Siegel-Narain theta functions that allow us to determine their ensemble averages. We express the derived averaged partition functions of the studied families of conformal field theories in a manifest modular invariant finite sum of products of real analytic Eisenstein series. We speculate on a tentative holographic three-dimensional dual bulk interpretations for the considered $\mathbb{Z}_2$ orbifold classes of ensembles of conformal field theories.

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arXiv2023

Modular Calabi-Yau Fourfolds and Connections to M-Theory Fluxes

In this work, we study the local zeta functions of Calabi-Yau fourfolds. This is done by developing arithmetic deformation techniques to compute the factor of the zeta function that is attributed to the horizontal four-form cohomology. This, in turn, is sensitive to the complex structure of the fourfold. Focusing mainly on examples of fourfolds with a single complex structure parameter, it is demonstrated that the proposed arithmetic techniques are both applicable and consistent. We present a Calabi-Yau fourfold for which a factor of the horizontal four-form cohomology further splits into two pieces of Hodge type $(4,0)+(2,2)+(0,4)$ and $(3,1)+(1,3)$. The latter factor corresponds to a weight-3 modular form, which allows expressing the value of the periods in terms of critical values of the L-function of this modular form, in accordance with Deligne's conjecture. The arithmetic considerations are related to M-theory Calabi-Yau fourfold compactifications with background four-form fluxes. We classify such background fluxes according to their Hodge type. For those fluxes associated to modular forms, we express their couplings in the low-energy effective action in terms of L-function values.

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arXiv2021

Deformations of JT Gravity via Topological Gravity and Applications

We study the duality between JT gravity and the double-scaled matrix model including their respective deformations. For these deformed theories we relate the thermal partition function to the generating function of topological gravity correlators that are determined as solutions to the KdV hierarchy. We specialise to those deformations of JT gravity coupled to a gas of defects, which conforms with known results in the literature. We express the (asymptotic) thermal partition functions in a low temperature limit, in which non-perturbative corrections are suppressed and the thermal partition function becomes exact. In this limit we demonstrate that there is a Hawking-Page phase transition between connected and disconnected surfaces for this instance of JT gravity with a transition temperature affected by the presence of defects. Furthermore, the calculated spectral form factors show the qualitative behaviour expected for a Hawking-Page phase transition. The considered deformations cause the ramp to be shifted along the real time axis. Finally, we comment on recent results related to conical Weil-Petersson volumes and the analytic continuation to two-dimensional de Sitter space.

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arXiv2021

BPS Indices, Modularity and Perturbations in Quantum K-theory

We study a perturbation family of N=2 3d gauge theories and its relation to quantum K-theory. A 3d version of the Intriligator-Vafa formula is given for the quantum K-theory ring of Grassmannians. The 3d BPS half-index of the gauge theory is connected to the theory of bilateral hypergeometric q-series, and to modular q-characters of a class of conformal field theories in a certain massless limit. Turning on 3d Wilson lines at torsion points leads to mock modular behavior. Perturbed correlators in the IR regime are computed by determining the UV-IR map in the presence of deformations.

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arXiv2019

Wilson loop algebras and quantum K-theory for Grassmannians

We study the algebra of Wilson line operators in three-dimensional N=2 supersymmetric U(M) gauge theories with a Higgs phase related to a complex Grassmannian Gr(M,N), and its connection to K-theoretic Gromov-Witten invariants for Gr(M,N). For different Chern-Simons levels, the Wilson loop algebra realizes either the quantum cohomology of Gr(M,N), isomorphic to the Verlinde algebra for U(M), or the quantum K-theoretic ring of Schubert structure sheaves studied by mathematicians, or closely related algebras.

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arXiv2019

Quantum K-Theory of Calabi-Yau Manifolds

The disk partition function of certain 3d N=2 supersymmetric gauge theories computes a quantum K-theoretic ring for Kahler manifolds X. We study the 3d gauge theory/quantum K-theory correspondence for global and local Calabi-Yau manifolds with several Kahler moduli. We propose a multi-cover formula that relates the 3d BPS world-volume degeneracies computed by quantum K-theory to Gopakumar-Vafa invariants.

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arXiv2018

A 3d Gauge Theory/Quantum K-Theory Correspondence

The 2d gauged linear sigma model (GLSM) gives a UV model for quantum cohomology on a Kahler manifold X, which is reproduced in the IR limit. We propose and explore a 3d lift of this correspondence, where the UV model is the N=2 supersymmetric 3d gauge theory and the IR limit is given by Givental's permutation equivariant quantum K-theory on X. This gives a one-parameter deformation of the 2d GLSM/quantum cohomology correspondence and recovers it in a small radius limit. We study some novelties of the 3d case regarding integral BPS invariants, chiral rings, deformation spaces and mirror symmetry.

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arXiv2018

The Geometry of Gauged Linear Sigma Model Correlation Functions

Applying advances in exact computations of supersymmetric gauge theories, we study the structure of correlation functions in two-dimensional N=(2,2) Abelian and non-Abelian gauge theories. We determine universal relations among correlation functions, which yield differential equations governing the dependence of the gauge theory ground state on the Fayet-Iliopoulos parameters of the gauge theory. For gauge theories with a non-trivial infrared N=(2,2) superconformal fixed point, these differential equations become the Picard-Fuchs operators governing the moduli-dependent vacuum ground state in a Hilbert space interpretation. For gauge theories with geometric target spaces, a quadratic expression in the Givental I-function generates the analyzed correlators. This gives a geometric interpretation for the correlators, their relations, and the differential equations. For classes of Calabi-Yau target spaces, such as threefolds with up to two Kahler moduli and fourfolds with a single Kahler modulus, we give general and universally applicable expressions for Picard-Fuchs operators in terms of correlators. We illustrate our results with representative examples of two-dimensional N=(2,2) gauge theories.

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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.

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arXiv2017

SU(N) transitions in M-theory on Calabi-Yau fourfolds and background fluxes

We study M-theory on a Calabi-Yau fourfold with a smooth surface $S$ of $A_{N-1}$ singularities. The resulting three-dimensional theory has a $\mathcal{N}=2$ $SU(N)$ gauge theory sector, which we obtain from a twisted dimensional reduction of a seven-dimensional $\mathcal{N}=1$ $SU(N)$ gauge theory on the surface $S$. A variant of the Vafa-Witten equations governs the moduli space of the gauge theory, which, for a trivial $SU(N)$ principal bundle over $S$, admits a Coulomb and a Higgs branch. In M-theory these two gauge theory branches arise from a resolution and a deformation to smooth Calabi-Yau fourfolds, respectively. We find that the deformed Calabi-Yau fourfold associated to the Higgs branch requires for consistency a non-trivial four-form background flux in M-theory. The flat directions of the flux-induced superpotential are in agreement with the gauge theory prediction for the moduli space of the Higgs branch. We illustrate our findings with explicit examples that realize the Coulomb and Higgs phase transition in Calabi-Yau fourfolds embedded in weighted projective spaces. We generalize and enlarge this class of examples to Calabi-Yau fourfolds embedded in toric varieties with an $A_{N-1}$ singularity in codimension two.

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arXiv2017

Dual Pairs of Gauged Linear Sigma Models and Derived Equivalences of Calabi-Yau threefolds

In this work we study the phase structure of skew symplectic sigma models, which are a certain class of two-dimensional N = (2,2) non-Abelian gauged linear sigma models. At low energies some of them flow to non-linear sigma models with Calabi-Yau target spaces, which emerge from non-Abelian strong coupling dynamics. The observed phase structure results in a non-trivial duality proposal among skew symplectic sigma models and connects non-complete intersection Calabi-Yau threefolds, that are non-birational among another, in a common quantum Kahler moduli space. As a consequence we find non-trivial identifications of spectra of topological B-branes, which from a modern algebraic geometry perspective imply derived equivalences among Calabi-Yau varieties. To further support our proposals, we calculate the two sphere partition function of skew symplectic sigma models to determine geometric invariants, which confirm the anticipated Calabi-Yau threefold phases. We show that the two sphere partition functions of a pair of dual skew symplectic sigma models agree in a non-trivial fashion. To carry out these calculations, we develop a systematic approach to study higher-dimensional Mellin-Barnes type integrals. In particular, these techniques admit the evaluation of two sphere partition functions for gauged linear sigma models with higher rank gauge groups, but are applicable in other contexts as well.

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arXiv2016

Quantum periods of Calabi-Yau fourfolds

In this work we study the quantum periods together with their Picard-Fuchs differential equations of Calabi-Yau fourfolds. In contrast to Calabi-Yau threefolds, we argue that the large volume points of Calabi-Yau fourfolds generically are regular singular points of the Picard-Fuchs operators of non-maximally unipotent monodromy. We demonstrate this property in explicit examples of Calabi-Yau fourfolds with a single Kahler modulus. For these examples we construct integral quantum periods and study their global properties in the quantum Kahler moduli space with the help of numerical analytic continuation techniques. Furthermore, we determine their genus zero Gromov-Witten invariants, their Klemm-Pandharipande meeting invariants, and their genus one BPS invariants. In our computations we emphasize the features attributed to the non-maximally unipotent monodromy property. For instance, it implies the existence of integral quantum periods that at large volume are purely worldsheet instanton generated. To verify our results, we also present intersection theory techniques to enumerate lines with a marked point on complete intersection Calabi-Yau fourfolds in Grassmannian varieties.

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