arXiv subjects

Elias Furrer

Publications and source records attributed to Elias Furrer.

11 recordsLinked to original sources

arXiv2026

Galois Covers of Calabi-Yau Quivers and BPS State Counting

BPS quivers are central to our understanding of BPS states in 4d $\mathcal{N}=2$ supersymmetric field theories and of D-branes at Calabi-Yau threefold singularities. The two subjects are deeply interrelated through geometric engineering in Type II string theory, where a CY$_3$ quiver, also known as a 5d BPS quiver, describes fractional branes at a threefold singularity ${\bf X}$. We study the Galois cover ${\skew{2}\tilde Q}\rightarrow Q$ of any BPS quiver $Q$ by a finite abelian group $\mathbb{G}$, leading to a covering quiver ${\skew{2}\tilde Q}$. The Galois cover is determined by a $\mathbb{G}$-grading of the arrows of the quiver $Q$, which can be understood as an orbifolding procedure. In particular, if $Q$ is a CY$_3$ quiver for ${\bf X}$, then the Galois cover $\skew{2}\tilde Q$ is the CY$_3$ quiver for the orbifold singularity ${\bf X}/\mathbb{G}$. We explore such Galois covering procedures in the language of supersymmetric quiver quantum mechanics, in terms of fixed loci under $\mathbb{G}$ actions on moduli spaces of quiver representations, and in terms of homomorphisms between the Kontsevich-Soibelman algebras of $Q$ and ${\skew{2}\tilde Q}$. Our main result is an explicit covering formula for the BPS invariants of 4d $\mathcal{N}=2$ field theories, wherein the rational BPS invariant $\bar{\Omega}^Q(\gamma)$ of $Q$ is expressed as a sum of BPS invariants of $\skew{2}\tilde Q$. We derive this formula in various special cases, which include the case when $\gamma$ is a primitive charge vector, the case of general charge vectors for quivers without loops, and the case of CY$_3$ quivers for some simple geometries such as the conifold or local del Pezzo surfaces. The general formula is presented as a conjecture that can be verified in many examples.

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arXiv2026

Universal Functions for Topological Correlators

We consider correlation functions of topologically twisted, $\mathcal{N}=2$ supersymmetric Yang-Mills theory with gauge group ${\rm SU}(2)$ and $N_f\leq 3$ massive hypermultiplets in the fundamental representation. For a smooth, compact, oriented four-manifold $X$ with $b_2^+>1$, the correlation functions are expressed in terms of a finite set of universal functions. The mass dependence of these functions encodes intersection numbers of the moduli space of instantons. We determine closed expressions for the universal functions by combining techniques of the Seiberg-Witten geometry, $u$-plane integral and the blowup formula. If $X$ is specialised to a complex algebraic surface $S$, the correlation functions can be identified with generating functions of Segre invariants for moduli spaces of sheaves on $S$. We verify that our results agree with the results by G\"ottsche and Kool for these generating functions.

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arXiv2025

The 3d $A$-model and generalised symmetries, Part I: bosonic Chern-Simons theories

The 3d $A$-model is a two-dimensional approach to the computation of supersymmetric observables of three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories. In principle, it allows us to compute half-BPS partition functions on any compact Seifert three-manifold (as well as of expectation values of half-BPS lines thereon), but previous results focussed on the case where the gauge group $\widetilde G$ is a product of simply-connected and/or unitary gauge groups. We are interested in the more general case of a compact gauge group $G=\widetilde G/\Gamma$, which is obtained from the $\widetilde G$ theory by gauging a discrete one-form symmetry. In this paper, we discuss in detail the case of pure $\mathcal{N}=2$ Chern-Simons theories (without matter) for simple groups $G$. When $G=\widetilde G$ is simply-connected, we demonstrate the exact matching between the supersymmetric approach in terms of Seifert fibering operators and the 3d TQFT approach based on topological surgery in the infrared Chern-Simons theory $\widetilde G_k$, including through the identification of subtle counterterms that relate the two approaches. We then extend this discussion to the case where the Chern-Simons theory $G_k$ can be obtained from $\widetilde G_k$ by the condensation of abelian anyons which are bosonic. Along the way, we revisit the 3d $A$-model formalism by emphasising its 2d TQFT underpinning.

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arXiv2024

Coulomb branch surgery: Holonomy saddles, S-folds and discrete symmetry gaugings

Symmetries of Seiberg-Witten (SW) geometries capture intricate physical aspects of the underlying 4d $\mathcal{N} = 2$ field theories. For rank-one theories, these geometries are rational elliptic surfaces whose automorphism group is a semi-direct product between the Coulomb branch (CB) symmetries and the Mordell-Weil group. We study quotients of the SW geometry by subgroups of its automorphism group, which most naturally become gaugings of discrete 0- and 1-form symmetries. Yet, new interpretations of these surgeries become evident when considering 5d $\mathcal{N}=1$ superconformal field theories. There, certain CB symmetries are related to symmetries of the corresponding $(p,q)$-brane web and, as a result, CB surgeries can give rise to (fractional) S-folds. Another novel interpretation of these quotients is the folding across dimensions: circle compactifications of the 5d $E_{2N_f + 1}$ Seiberg theories lead in the infrared to two copies of locally indistinguishable 4d SU(2) SQCD theories with $N_f$ fundamental flavours. This extends earlier results on holonomy saddles, while also reproducing detailed computations of 5d BPS spectra and predicting new 5d and 6d BPS quivers. Finally, we argue that the semi-direct product structure of the automorphism group of the SW geometry includes mixed 't Hooft anomalies between the 0- and 1-form symmetries, and we also present some new results on non-cyclic CB symmetries.

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arXiv2024

One-form symmetries and the 3d $\mathcal{N}=2$ $A$-model: Topologically twisted indices and CS theories

We study three-dimensional $\mathcal{N}=2$ supersymmetric Chern-Simons-matter gauge theories with a one-form symmetry in the $A$-model formalism on $\Sigma_g\times S^1$. We explicitly compute expectation values of topological line operators that implement the one-form symmetry. This allows us to compute the topologically twisted index on the closed Riemann surface $\Sigma_g$ for any real compact gauge group $G$ as long as the ground states are all bosonic. All computations are carried out in the effective $A$-model on $\Sigma_g$, whose $S^1$ ground states are the so-called Bethe vacua. We discuss how the 3d one-form symmetry acts on the Bethe vacua, and also how its 't Hooft anomaly constrains the vacuum structure. In the special case of the $\text{SU}(N)_K$ $\mathcal{N}=2$ Chern-Simons theory, we obtain results for the $(\text{SU}(N)/\mathbb Z_r)^{\theta}_K$ $\mathcal{N}=2$ Chern-Simons theories, for all non-anomalous $\mathbb Z_r \subseteq \mathbb Z_N$ subgroups of the centre of the gauge group, and with a $\mathbb Z_r$ $\theta$-angle turned on. In the special cases with $N$ even, $\frac{N}{r}$ odd and $\frac{K}{r}$ even, we find a mixed 't Hooft anomaly between gravity and the $\mathbb Z_r^{(1)}$ one-form symmetry of the $\text{SU}(N)_K$ theory, and the infrared 3d TQFT after gauging is spin. In all cases, we count the Bethe states and the higher-genus states in terms of refinements of Jordan's totient function. This counting gives us the twisted indices if and only if the infrared 3d TQFT is bosonic. Our results lead to precise conjectures about integrality of indices, which appear to have a strong number-theoretic flavour. Note: this paper directly builds upon unpublished notes by Brian Willett from 2020.

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arXiv2023

Topological twists of massive SQCD, Part I

We consider topological twists of four-dimensional $\mathcal{N}=2$ supersymmetric QCD with gauge group SU(2) and $N_f\leq 3$ fundamental hypermultiplets. The twists are labelled by a choice of background fluxes for the flavour group, which provides an infinite family of topological partition functions. In this Part I, we demonstrate that in the presence of such fluxes the theories can be formulated for arbitrary gauge bundles on a compact four-manifold. Moreover, we consider arbitrary masses for the hypermultiplets, which introduce new intricacies for the evaluation of the low-energy path integral on the Coulomb branch. We develop techniques for the evaluation of these path integrals. In the forthcoming Part II, we will deal with the explicit evaluation.

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arXiv2023

Topological twists of massive SQCD, Part II

This is the second and final part of ``Topological twists of massive SQCD''. Part I is available at arXiv:2206.08943. In this second part, we evaluate the contribution of the Coulomb branch to topological path integrals for $\mathcal{N}=2$ supersymmetric QCD with $N_f\leq 3$ massive hypermultiplets on compact four-manifolds. Our analysis includes the decoupling of hypermultiplets, the massless limit and the merging of mutually non-local singularities at the Argyres-Douglas points. We give explicit mass expansions for the four-manifolds $\mathbb{P}^2$ and $K3$. For $\mathbb{P}^2$, we find that the correlation functions are polynomial as function of the masses, while infinite series and (potential) singularities occur for $K3$. The mass dependence corresponds mathematically to the integration of the equivariant Chern class of the matter bundle over the moduli space of $Q$-fixed equations. We demonstrate that the physical partition functions agree with mathematical results on Segre numbers of instanton moduli spaces.

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arXiv2022

The $u$-plane integral, mock modularity and enumerative geometry

We revisit the low-energy effective $U(1)$ action of topologically twisted $\mathcal N=2$ SYM theory with gauge group of rank one on a generic oriented smooth 4-manifold $X$ with nontrivial fundamental group. After including a specific new set of $\mathcal Q$-exact operators to the known action, we express the integrand of the path integral of the low-energy $U(1)$ theory as an anti-holomorphic derivative. This allows us to use the theory of mock modular forms and indefinite theta functions for the explicit evaluation of correlation functions of the theory, including but not restricted to those that physically reproduce Donaldson invariants, thus facilitating the computations compared to previously used methods. As an explicit check of our results, we compute the path integral for the product ruled surfaces $X=Σ_g \times \mathbb{CP}^1$ for the reduction on either factor and compare the results with existing literature. In the case of reduction on the Riemann surface $Σ_g$, via an equivalent topological A-model on $\mathbb{CP}^1$, we will be able to express the generating function of genus zero Gromov-Witten invariants of the moduli space of flat rank one connections over $Σ_g$ in terms of an indefinite theta function, whence we would be able to make concrete numerical predictions of these enumerative invariants in terms of modular data, thereby allowing us to derive results in enumerative geometry from number theory.

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arXiv2021

Four flavours, triality and bimodular forms

We consider $\mathcal{N}=2$ supersymmetric $\text{SU}(2)$ gauge theory with $N_f=4$ massive hypermultiplets. The duality group of this theory contains transformations acting on the UV-coupling $τ_{\text{UV}}$ as well as on the running coupling $τ$. We establish that subgroups of the duality group act separately on $τ_{\text{UV}}$ and $τ$, while a larger group acts simultaneously on $τ_{\text{UV}}$ and $τ$. For special choices of the masses, we find that the duality groups can be identified with congruence subgroups of $\text{SL}(2,\mathbb Z)$. We demonstrate that in such cases, the order parameters are instances of bimodular forms with arguments $τ$ and $τ_{\text{UV}}$. Since the UV duality group of the theory contains the triality group of outer automorphisms of the flavour symmetry $\text{SO}(8)$, the duality action gives rise to an orbit of mass configurations. Consequently, the corresponding order parameters combine to vector-valued bimodular forms with $\text{SL}(2,\mathbb Z)$ acting simultaneously on the two couplings.

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arXiv2021

Cutting and gluing with running couplings in $\mathcal{N}=2$ QCD

We consider the order parameter $u=\left<{\rm Tr}ϕ^2\right>$ as function of the running coupling constant $τ\in \mathbb{H}$ of asymptotically free $\mathcal{N}=2$ QCD with gauge group $SU(2)$ and $N_f\leq 3$ massive hypermultiplets. If the domain for $τ$ is restricted to an appropriate fundamental domain $\mathcal{F}_{N_f}$, the function $u$ is one-to-one. We demonstrate that these domains consist of six or less images of an ${\rm SL}(2,\mathbb{Z})$ keyhole fundamental domain, with appropriate identifications of the boundaries. For special choices of the masses, $u$ does not give rise to branch points and cuts, such that $u$ is a modular function for a congruence subgroup $Γ$ of ${\rm SL}(2,\mathbb{Z})$ and the fundamental domain is $Γ\backslash\mathbb{H}$. For generic masses, however, branch points and cuts are present, and subsets of $\mathcal{F}_{N_f}$ are being cut and glued upon varying the mass. We study this mechanism for various phenomena, such as decoupling of hypermultiplets, merging of local singularities, as well as merging of non-local singularities which give rise to superconformal Argyres-Douglas theories.

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arXiv2021

Elliptic Loci of SU(3) Vacua

The space of vacua of many four-dimensional, $\mathcal{N}=2$ supersymmetric gauge theories can famously be identified with a family of complex curves. For gauge group $SU(2)$, this gives a fully explicit description of the low-energy effective theory in terms of an elliptic curve and associated modular fundamental domain. The two-dimensional space of vacua for gauge group $SU(3)$ parametrizes an intricate family of genus two curves. We analyze this family using the so-called Rosenhain form for these curves. We demonstrate that two natural one-dimensional subloci of the space of $SU(3)$ vacua, $\mathcal{E}_u$ and $\mathcal{E}_v$, each parametrize a family of elliptic curves. For these elliptic loci, we describe the order parameters and fundamental domains explicitly. The locus $\mathcal{E}_u$ contains the points where mutually local dyons become massless, and is a fundamental domain for a classical congruence subgroup. Moreover, the locus $\mathcal{E}_v$ contains the superconformal Argyres-Douglas points, and is a fundamental domain for a Fricke group.

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