arXiv subjects

Cyril Closset

Publications and source records attributed to Cyril Closset.

At least 37 records · Page 2Linked to original sources

arXiv2019

Three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories and partition functions on Seifert manifolds: A review

We give a pedagogical introduction to the study of supersymmetric partition functions of 3D $\mathcal{N}{=}2$ supersymmetric Chern-Simons-matter theories (with an $R$-symmetry) on half-BPS closed three-manifolds---including $S^3$, $S^2 \times S^1$, and any Seifert three-manifold. Three-dimensional gauge theories can flow to non-trivial fixed points in the infrared. In the presence of 3D $\mathcal{N}{=}2$ supersymmetry, many exact results are known about the strongly-coupled infrared, due in good part to powerful localization techniques. We review some of these techniques and emphasize some more recent developments, which provide a simple and comprehensive formalism for the exact computation of half-BPS observables on closed three-manifolds (partition functions and correlation functions of line operators). Along the way, we also review simple examples of 3D infrared dualities. The computation of supersymmetric partition functions provides exceedingly precise tests of these dualities.

hep-th↗

arXiv2019

't Hooft anomalies and the holomorphy of supersymmetric partition functions

We study the dependence of supersymmetric partition functions on continuous parameters for the flavor symmetry group, $G_F$, for 2d $\mathcal{N} = (0,2)$ and 4d $\mathcal{N}=1$ supersymmetric quantum field theories. In any diffeomorphism-invariant scheme and in the presence of $G_F$ 't Hooft anomalies, the supersymmetric Ward identities imply that the partition function has a non-holomorphic dependence on the flavor parameters. We show this explicitly for the 2d torus partition function, $Z_{T^2}$, and for a large class of 4d partition functions on half-BPS four-manifolds, $Z_{\mathcal{M}_4}$---in particular, for $\mathcal{M}_4=S^3 \times S^1$ and $\mathcal{M}_4=Σ_g \times T^2$. We propose a new expression for $Z_{\mathcal{M}_{d-1} \times S^1}$, which differs from earlier holomorphic results by the introduction of a non-holomorphic `Casimir' pre-factor. The latter is fixed by studying the `high temperature' limit of the partition function. Our proposal agrees with the supersymmetric Ward identities, and with explicit calculations of the absolute value of the partition function using a gauge-invariant zeta-function regularization.

hep-th↗

arXiv2019

Five-dimensional SCFTs and gauge theory phases: an M-theory/type IIA perspective

We revisit the correspondence between Calabi-Yau (CY) threefold isolated singularities $\mathbf{X}$ and five-dimensional superconformal field theories (SCFTs), which arise at low energy in M-theory on the space-time transverse to $\mathbf{X}$. Focussing on the case of toric CY singularities, we analyze the "gauge-theory phases" of the SCFT by exploiting fiberwise M-theory/type IIA duality. In this setup, the low-energy gauge group simply arises on stacks of coincident D6-branes wrapping 2-cycles in some ALE space of type $A_{M-1}$ fibered over a real line, and the map between the Kähler parameters of $\mathbf{X}$ and the Coulomb branch parameters of the field theory (masses and VEVs) can be read off systematically. Different type IIA "reductions" give rise to different gauge theory phases, whose existence depends on the particular (partial) resolutions of the isolated singularity $\mathbf{X}$. We also comment on the case of non-isolated toric singularities. Incidentally, we propose a slightly modified expression for the Coulomb-branch prepotential of 5d $\mathcal{N}=1$ gauge theories.

hep-th↗

arXiv2018

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.

hep-th↗

arXiv2018

Seifert fibering operators in 3d $\mathcal{N}=2$ theories

We study 3d $\mathcal{N}=2$ supersymmetric gauge theories on closed oriented Seifert manifold---circle bundles over an orbifold Riemann surface---, with a gauge group G given by a product of simply-connected and/or unitary Lie groups. Our main result is an exact formula for the supersymmetric partition function on any Seifert manifold, generalizing previous results on lens spaces. We explain how the result for an arbitrary Seifert geometry can be obtained by combining simple building blocks, the "fibering operators." These operators are half-BPS line defects, whose insertion along the $S^1$ fiber has the effect of changing the topology of the Seifert fibration. We also point out that most supersymmetric partition functions on Seifert manifolds admit a discrete refinement, corresponding to the freedom in choosing a three-dimensional spin structure. As a strong consistency check on our result, we show that the Seifert partition functions match exactly across infrared dualities. The duality relations are given by intricate (and seemingly new) mathematical identities, which we tested numerically. Finally, we discuss in detail the supersymmetric partition function on the lens space $L(p,q)_b$ with rational squashing parameter $b^2 \in \mathbb{Q}$, comparing our formalism to previous results, and explaining the relationship between the fibering operators and the three-dimensional holomorphic blocks.

hep-th↗

arXiv2018

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.

hep-th↗

arXiv2017

Supersymmetric gauged matrix models from dimensional reduction on a sphere

It was recently proposed that N=1 supersymmetric gauged matrix models have a duality of order four - that is, a quadrality - reminiscent of infrared dualities of SQCD theories in higher dimensions. In this note, we show that the zero-dimensional quadrality proposal can be infered from the two-dimensional Gadde-Gukov-Putrov triality. We consider two-dimensional N=(0,2) SQCD compactified on a sphere with the half-topological twist. For a convenient choice of R-charge, the zero-mode sector on the sphere gives rise to a simple N=1 gauged matrix model. Triality on the sphere then implies a triality relation for the supersymmetric matrix model, which can be completed to the full quadrality.

hep-th↗

arXiv2017

$\mathcal{N}{=}1$ 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, $Σ_g$. 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 $Σ_g$. 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 $Σ_g \times T^2$. 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.

hep-th↗

arXiv2017

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 $Σ_g$, 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 $Σ_g$ 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.

hep-th↗

arXiv2017

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, $Σ_g$. 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.

hep-th↗

arXiv2016

Comments on twisted indices in 3d supersymmetric gauge theories

We study three-dimensional ${\mathcal N}=2$ supersymmetric gauge theories on ${Σ_g \times S^1}$ with a topological twist along $Σ_g$, 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.

hep-th↗

arXiv2016

Localization of twisted $\mathcal{N}{=}(0,2)$ gauged linear sigma models in two dimensions

We study two-dimensional $\mathcal{N}{=}(0,2)$ supersymmetric gauged linear sigma models (GLSMs) using supersymmetric localization. We consider $\mathcal{N}{=}(0,2)$ theories with an $R$-symmetry, which can always be defined on curved space by a pseudo-topological twist while preserving one of the two supercharges of flat space. For GLSMs which are deformations of $\mathcal{N}{=}(2,2)$ GLSMs and retain a Coulomb branch, we consider the $A/2$-twist and compute the genus-zero correlation functions of certain pseudo-chiral operators, which generalize the simplest twisted chiral ring operators away from the $\mathcal{N}{=}(2,2)$ locus. These correlation functions can be written in terms of a certain residue operation on the Coulomb branch, generalizing the Jeffrey-Kirwan residue prescription relevant for the $\mathcal{N}{=}(2,2)$ locus. For abelian GLSMs, we reproduce existing results with new formulas that render the quantum sheaf cohomology relations and other properties manifest. For non-abelian GLSMs, our methods lead to new results. As an example, we briefly discuss the quantum sheaf cohomology of the Grassmannian manifold.

hep-th↗

arXiv2015

The equivariant A-twist and gauged linear sigma models on the two-sphere

We study two-dimensional $\mathcal{N}=(2,2)$ supersymmetric gauged linear sigma models (GLSM) on the $Ω$-deformed sphere, $S^2_Ω$, which is a one-parameter deformation of the $A$-twisted sphere. We provide an exact formula for the $S^2_Ω$ supersymmetric correlation functions using supersymmetric localization. The contribution of each instanton sector is given in terms of a Jeffrey-Kirwan residue on the Coulomb branch. In the limit of vanishing $Ω$-deformation, the localization formula greatly simplifies the computation of $A$-twisted correlation functions, and leads to new results for non-abelian theories. We discuss a number of examples and comment on the $ε_Ω$-deformation of the quantum cohomology relations. Finally, we present a complementary Higgs branch localization scheme in the special case of abelian gauge groups.

hep-th↗

arXiv2014

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.

hep-th↗

arXiv2014

Comments on N=(2,2) Supersymmetry on Two-Manifolds

We study curved-space rigid supersymmetry for two-dimensional $\mathcal{N}=(2,2)$ supersymmetric fields theories with a vector-like $R$-symmetry by coupling such theories to background supergravity. The associated Killing spinors can be viewed as holomorphic sections of particular complex line bundles over Euclidean space-time, which severely restrict the allowed supersymmetric couplings on compact orientable Riemann surfaces without boundaries. For genus $g>1$, the only consistent non-singular couplings are the ones dictated by the topological $A$-twist. On spaces with $S^2$ topology, there exist additional supersymmetric backgrounds with $m=0$ or $\pm 1$ unit of flux for the $R$-symmetry gauge field. The $m=-1$ case includes the $Ω$-background on the sphere. We also systematically work out the curved-space supersymmetry multiplets and supersymmetric Lagrangians.

hep-th↗

arXiv2014

The $\mathcal{N}=1$ Chiral Multiplet on $T^2\times S^2$ and Supersymmetric Localization

We compute the supersymmetric partition function of an $\mathcal{N}=1$ chiral multiplet coupled to an external Abelian gauge field on complex manifolds with $T^2 \times S^2$ topology. The result is locally holomorphic in the complex structure moduli of $T^2\times S^2$. This computation illustrates in a simple example some recently obtained constraints on the parameter dependence of supersymmetric partition functions. We also devise a simple method to compute the chiral multiplet partition function on any four-manifold $\mathcal{M}_4$ preserving two supercharges of opposite chiralities, via supersymmetric localization. In the case of $\mathcal{M}_4=S^3\times S^1$, we provide a path integral derivation of the previously known result, the elliptic gamma function, which emphasizes its dependence on the $S^3 \times S^1$ complex structure moduli.

hep-th↗

arXiv2013

The Geometry of Supersymmetric Partition Functions

We consider supersymmetric field theories on compact manifolds M and obtain constraints on the parameter dependence of their partition functions Z_M. Our primary focus is the dependence of Z_M on the geometry of M, as well as background gauge fields that couple to continuous flavor symmetries. For N=1 theories with a U(1)_R symmetry in four dimensions, M must be a complex manifold with a Hermitian metric. We find that Z_M is independent of the metric and depends holomorphically on the complex structure moduli. Background gauge fields define holomorphic vector bundles over M and Z_M is a holomorphic function of the corresponding bundle moduli. We also carry out a parallel analysis for three-dimensional N=2 theories with a U(1)_R symmetry, where the necessary geometric structure on M is a transversely holomorphic foliation (THF) with a transversely Hermitian metric. Again, we find that Z_M is independent of the metric and depends holomorphically on the moduli of the THF. We discuss several applications, including manifolds diffeomorphic to S^3 x S^1 or S^2 x S^1, which are related to supersymmetric indices, and manifolds diffeomorphic to S^3 (squashed spheres). In examples where Z_M has been calculated explicitly, our results explain many of its observed properties.

hep-th↗

arXiv2013

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.

hep-th↗