Comments on twisted indices in 3d supersymmetric gauge theories

We study three-dimensional ${\mathcal N}=2$ supersymmetric gauge theories on ${\Sigma_g \times S^1}$ with a topological twist along $\Sigma_g$, a genus-$g$ Riemann surface. The twisted supersymmetric index at genus $g$ and the correlation functions of half-BPS loop operators on $S^1$ can be computed exactly by supersymmetric localization. For $g=1$, this gives a simple UV computation of the 3d Witten index. Twisted indices provide us with a clean derivation of the quantum algebra of supersymmetric Wilson loops, for any Yang-Mills-Chern-Simons-matter theory, in terms of the associated Bethe equations for the theory on ${\mathbb R}^2 \times S^1$. This also provides a powerful and simple tool to study 3d ${\mathcal N}=2$ Seiberg dualities. Finally, we study A- and B-twisted indices for ${\mathcal N}=4$ supersymmetric gauge theories, which turns out to be very useful for quantitative studies of three-dimensional mirror symmetry. We also briefly comment on a relation between the $S^2 \times S^1$ twisted indices and the Hilbert series of ${\mathcal N}=4$ moduli spaces.Comment: 66 pages plus appendix; v2: corrected typos and added reference

Supersymmetric partition functions and the three-dimensional A-twist

We study three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories on $\mathcal{M}_{g,p}$, an oriented circle bundle of degree $p$ over a closed Riemann surface, $\Sigma_g$. We compute the $\mathcal{M}_{g,p}$ supersymmetric partition function and correlation functions of supersymmetric loop operators. This uncovers interesting relations between observables on manifolds of different topologies. In particular, the familiar supersymmetric partition function on the round $S^3$ can be understood as the expectation value of a so-called "fibering operator" on $S^2 \times S^1$ with a topological twist. More generally, we show that the 3d $\mathcal{N}=2$ supersymmetric partition functions (and supersymmetric Wilson loop correlation functions) on $\mathcal{M}_{g,p}$ are fully determined by the two-dimensional A-twisted topological field theory obtained by compactifying the 3d theory on a circle. We give two complementary derivations of the result. We also discuss applications to F-maximization and to three-dimensional supersymmetric dualities.Comment: 84 pages plus appendix, 8 figures; v2: added reference

B-branes and supersymmetric quivers in 2d

We study 2d $\mathcal{N}=(0,2)$ supersymmetric quiver gauge theories that describe the low-energy dynamics of D1-branes at Calabi-Yau fourfold (CY$_4$) singularities. On general grounds, the holomorphic sector of these theories---matter content and (classical) superpotential interactions---should be fully captured by the topological $B$-model on the CY$_4$. By studying a number of examples, we confirm this expectation and flesh out the dictionary between B-brane category and supersymmetric quiver: the matter content of the supersymmetric quiver is encoded in morphisms between B-branes (that is, Ext groups of coherent sheaves), while the superpotential interactions are encoded in the $A_\infty$ algebra satisfied by the morphisms. This provides us with a derivation of the supersymmetric quiver directly from the CY$_4$ geometry. We also suggest a relation between triality of $\mathcal{N}=(0,2)$ gauge theories and certain mutations of exceptional collections of sheaves. 0d $\mathcal{N}=1$ supersymmetric quivers, corresponding to D-instantons probing CY$_5$ singularities, can be discussed similarly.Comment: 63 pages plus appendix, 21 figures; v2: corrected typos and added reference. JHEP version; v3: added referenc

N=1\mathcal{N}{=}1 supersymmetric indices and the four-dimensional A-model

We compute the supersymmetric partition function of $\mathcal{N}{=}1$ supersymmetric gauge theories with an $R$-symmetry on $\mathcal{M}_4 \cong \mathcal{M}_{g,p}\times S^1$, a principal elliptic fiber bundle of degree $p$ over a genus-$g$ Riemann surface, $\Sigma_g$. Equivalently, we compute the generalized supersymmetric index $I_{\mathcal{M}_{g,p}}$, with the supersymmetric three-manifold ${\mathcal{M}_{g,p}}$ as the spatial slice. The ordinary $\mathcal{N}{=}1$ supersymmetric index on the round three-sphere is recovered as a special case. We approach this computation from the point of view of a topological $A$-model for the abelianized gauge fields on the base $\Sigma_g$. This $A$-model---or $A$-twisted two-dimensional $\mathcal{N}{=}(2,2)$ gauge theory---encodes all the information about the generalized indices, which are viewed as expectations values of some canonically-defined surface defects wrapped on $T^2$ inside $\Sigma_g \times T^2$. Being defined by compactification on the torus, the $A$-model also enjoys natural modular properties, governed by the four-dimensional 't Hooft anomalies. As an application of our results, we provide new tests of Seiberg duality. We also present a new evaluation formula for the three-sphere index as a sum over two-dimensional vacua.Comment: 91 pages including appendices; v2: corrected typos and added references, JHEP versio

A-twisted correlators and Hori dualities

The Hori-Tong and Hori dualities are infrared dualities between two-dimensional gauge theories with $\mathcal{N}{=}(2,2)$ supersymmetry, which are reminiscent of four-dimensional Seiberg dualities. We provide additional evidence for those dualities with $U(N_c)$, $USp(2N_c)$, $SO(N)$ and $O(N)$ gauge groups, by matching correlation functions of Coulomb branch operators on a Riemann surface $\Sigma_g$, in the presence of the topological $A$-twist. The $O(N)$ theories studied, denoted by $O_+ (N)$ and $O_- (N)$, can be understood as $\mathbb{Z}_2$ orbifolds of an $SO(N)$ theory. The correlators of these theories on $\Sigma_g$ with $g > 0$ are obtained by computing correlators with $\mathbb{Z}_2$-twisted boundary conditions and summing them up with weights determined by the orbifold projection.Comment: 45 pages plus appendix; v2: updated bibliography and acknowledgement

Graded quivers and B-branes at Calabi-Yau singularities

A graded quiver with superpotential is a quiver whose arrows are assigned degrees $c\in \{0, 1, \cdots, m\}$, for some integer $m \geq 0$, with relations generated by a superpotential of degree $m-1$. Ordinary quivers ($m=1)$ often describe the open string sector of D-brane systems; in particular, they capture the physics of D3-branes at local Calabi-Yau (CY) 3-fold singularities in type IIB string theory, in the guise of 4d $\mathcal{N}=1$ supersymmetric quiver gauge theories. It was pointed out recently that graded quivers with $m=2$ and $m=3$ similarly describe systems of D-branes at CY 4-fold and 5-fold singularities, as 2d $\mathcal{N}=(0,2)$ and 0d $\mathcal{N}=1$ gauge theories, respectively. In this work, we further explore the correspondence between $m$-graded quivers with superpotential, $Q_{(m)}$, and CY $(m+2)$-fold singularities, ${\mathbf X}_{m+2}$. For any $m$, the open string sector of the topological B-model on ${\mathbf X}_{m+2}$ can be described in terms of a graded quiver. We illustrate this correspondence explicitly with a few infinite families of toric singularities indexed by $m \in \mathbb{N}$, for which we derive "toric" graded quivers associated to the geometry, using several complementary perspectives. Many interesting aspects of supersymmetric quiver gauge theories can be formally extended to any $m$; for instance, for one family of singularities, dubbed $C(Y^{1,0}(\mathbb{P}^m))$, that generalizes the conifold singularity to $m>1$, we point out the existence of a formal "duality cascade" for the corresponding graded quivers.Comment: 82 pages, 20 figure

Brainworm (Elaphostrongylus rangiferi) abundance in wild reindeer (Rangifer tarandus tarandus) in relation to gastropod densities

Elaphostrongylus rangiferi is a nematode parasite in reindeer (Rangifer tarandus) which can cause considerable neurological damage and could affect the survival chances of the last European wild tundra reindeer. The parasite has terrestrial gastropods as intermediate hosts. Previous research has shown that the development of E. rangiferi inside gastropods is highly temperature-dependent, with faster development at higher temperatures. Additionally, the prevalence and abundance of E. rangiferi has previously been reported to be lower in reindeer grazing at high altitudes, but whether this difference in infection rate is connected to gastropod densities is unknown. Here I showed that overall prevalence and abundance of E. rangiferi was significantly higher for reindeer that are in summer pastures with a high predicted gastropod density. These areas were mainly forested areas at low altitudes. The prevalence and abundance of E. rangiferi changed over time, with maximal output in faecal samples during early spring. Overall prevalence and abundance were considered to be relatively low compared to other studies on both wild and semi-domesticated reindeer. Previous studies suggested that the higher prevalence of E. rangiferi in reindeer grazing in low altitudes was mainly connected to higher temperatures. My results provide a new dimension into understanding risk areas for E. rangiferi transmission. My study showed that the parasite was common in the wild reindeer population of Rondane, a population from which there was little prior information. In light of climate change, prevalence and density of this parasite in reindeer is expected to increase. This makes E. rangiferi a parasite of increasing concern. My findings, in combination with previous research, could be used by both reindeer herders and conservation managers for management and mitigation strategies of reindeer to prevent future outbreaks of E. rangiferi

Toric geometry and local Calabi-Yau varieties: An introduction to toric geometry (for physicists)

These lecture notes are an introduction to toric geometry. Particular focus is put on the description of toric local Calabi-Yau varieties, such as needed in applications to the AdS/CFT correspondence in string theory. The point of view taken in these lectures is mostly algebro-geometric but no prior knowledge of algebraic geometry is assumed. After introducing the necessary mathematical definitions, we discuss the construction of toric varieties as holomorphic quotients. We discuss the resolution and deformation of toric Calabi-Yau singularities. We also explain the gauged linear sigma-model (GLSM) Kahler quotient construction.Comment: Based on lectures given at the Modave Summer School in Mathematical Physics 2008. 35 pages. v2: Added reference

From Rigid Supersymmetry to Twisted Holomorphic Theories

We study N=1 field theories with a U(1)_R symmetry on compact four-manifolds M. Supersymmetry requires M to be a complex manifold. The supersymmetric theory on M can be described in terms of conventional fields coupled to background supergravity, or in terms of twisted fields adapted to the complex geometry of M. Many properties of the theory that are difficult to see in one formulation are simpler in the other one. We use the twisted description to study the dependence of the partition function Z_M on the geometry of M, as well as coupling constants and background gauge fields, recovering and extending previous results. We also indicate how to generalize our analysis to three-dimensional N=2 theories with a U(1)_R symmetry. In this case supersymmetry requires M to carry a transversely holomorphic foliation, which endows it with a near-perfect analogue of complex geometry. Finally, we present new explicit formulas for the dependence of Z_M on the choice of U(1)_R symmetry in four and three dimensions, and illustrate them for complex manifolds diffeomorphic to S^3 x S^1, as well as general squashed three-spheres.Comment: 55 pages; minor change

Supersymmetric Field Theories on Three-Manifolds

We construct supersymmetric field theories on Riemannian three-manifolds M, focusing on N=2 theories with a U(1)_R symmetry. Our approach is based on the rigid limit of new minimal supergravity in three dimensions, which couples to the flat-space supermultiplet containing the R-current and the energy-momentum tensor. The field theory on M possesses a single supercharge, if and only if M admits an almost contact metric structure that satisfies a certain integrability condition. This may lead to global restrictions on M, even though we can always construct one supercharge on any given patch. We also analyze the conditions for the presence of additional supercharges. In particular, two supercharges of opposite R-charge exist on every Seifert manifold. We present general supersymmetric Lagrangians on M and discuss their flat-space limit, which can be analyzed using the R-current supermultiplet. As an application, we show how the flat-space two-point function of the energy-momentum tensor in N=2 superconformal theories can be calculated using localization on a squashed sphere.Comment: 53 pages; minor change