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Brian Willett

Publications and source records attributed to Brian Willett.

At least 19 recordsLinked to original sources
arXiv2020

Higher-Form Symmetries in 5d

We study higher-form symmetries in 5d quantum field theories, whose charged operators include extended operators such as Wilson line and 't Hooft operators. We outline criteria for the existence of higher-form symmetries both from a field theory point of view as well as from the geometric realization in M-theory on non-compact Calabi-Yau threefolds. A geometric criterion for determining the higher-form symmetry from the intersection data of the Calabi-Yau is provided, and we test it in a multitude of examples, including toric geometries. We further check that the higher-form symmetry is consistent with dualities and is invariant under flop transitions, which relate theories with the same UV-fixed point. We explore extensions to higher-form symmetries in other compactifications of M-theory, such as $G_2$-holonomy manifolds, which give rise to 4d $\mathcal{N}=1$ theories.

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arXiv2020

On the reduction of 4d N=1 theories on S^2

We discuss reductions of general N=1 four dimensional gauge theories on S^2. The effective two dimensional theory one obtains depends on the details of the coupling of the theory to background fields, which can be translated to a choice of R-symmetry. We argue that, for special choices of R-symmetry, the resulting two dimensional theory has a natural interpretation as an N=(0,2) gauge theory. As an application of our general observations, we discuss reductions of N=1 and N=2 dualities and argue that they imply certain two dimensional dualities.

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arXiv2019

Star shaped quivers with flux

We study the compactification of the 6d ${\cal N}=(2,0)$ SCFT on the product of a Riemann surface with flux and a circle. On the one hand, this can be understood by first reducing on the Riemann surface, giving rise to 4d ${\cal N}=1$ and ${\cal N}=2$ class ${\cal S}$ theories, which we then reduce on $S^1$ to get 3d ${\cal N}=2$ and ${\cal N}=4$ class ${\cal S}$ theories. On the other hand, we may first compactify on $S^1$ to get the 5d ${\cal N}=2$ Yang-Mills theory. By studying its reduction on a Riemann surface, we obtain a mirror dual description of 3d class ${\cal S}$ theories, generalizing the star-shaped quiver theories of Benini, Tachikawa, and Xie. We comment on some global properties of the gauge group in these reductions, and test the dualities by computing various supersymmetric partition functions.

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arXiv2019

Higher-Form Symmetries, Bethe Vacua, and the 3d-3d Correspondence

By incorporating higher-form symmetries, we propose a refined definition of the theories obtained by compactification of the 6d $(2,0)$ theory on a three-manifold $M_3$. This generalization is applicable to both the 3d $\mathcal{N}=2$ and $\mathcal{N}=1$ supersymmetric reductions. An observable that is sensitive to the higher-form symmetries is the Witten index, which can be computed by counting solutions to a set of Bethe equations that are determined by $M_3$. This is carried out in detail for $M_3$ a Seifert manifold, where we compute a refined version of the Witten index. In the context of the 3d-3d correspondence, we complement this analysis in the dual topological theory, and determine the refined counting of flat connections on $M_3$, which matches the Witten index computation that takes the higher-form symmetries into account.

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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.

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arXiv2018

From 3d duality to 2d duality

In this paper we discuss $3d$ ${\cal N}=2$ supersymmetric gauge theories and their IR dualities when they are compactified on a circle of radius $r$, and when we take the $2d$ limit in which $r\to 0$. The $2d$ limit depends on how the mass parameters are scaled as $r\to 0$, and often vacua become infinitely distant in the $2d$ limit, leading to a direct sum of different $2d$ theories. For generic mass parameters, when we take the same limit on both sides of a duality, we obtain $2d$ dualities (between gauge theories and/or Landau-Ginzburg theories) that pass all the usual tests. However, when there are non-compact branches the discussion is subtle because the metric on the moduli space, which is not controlled by supersymmetry, plays an important role in the low-energy dynamics after compactification. Generally speaking, for IR dualities of gauge theories, we conjecture that dualities involving non-compact Higgs branches survive. On the other hand when there is a non-compact Coulomb branch on at least one side of the duality, the duality fails already when the $3d$ theories are compactified on a circle. Using the valid reductions we reproduce many known $2d$ IR dualities, giving further evidence for their validity, and we also find new $2d$ dualities.

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arXiv2018

Partition functions on 3d circle bundles and their gravity duals

The partition function of a three-dimensional $\mathcal{N} =2$ theory on the manifold $\mathcal{M}_{g,p}$, an $S^1$ bundle of degree $p$ over a closed Riemann surface $Σ_g$, was recently computed via supersymmetric localization. In this paper, we compute these partition functions at large $N$ in a class of quiver gauge theories with holographic M-theory duals. We provide the supergravity bulk dual having as conformal boundary such three-dimensional circle bundles. These configurations are solutions to $\mathcal{N}=2$ minimal gauged supergravity and pertain to the class of Taub-NUT-AdS and Taub-Bolt-AdS preserving $1/4$ of the supersymmetries. We discuss the conditions for the uplift of these solutions to M-theory, and compute the on-shell action via holographic renormalization. We show that the uplift condition and on-shell action for the Bolt solutions are correctly reproduced by the large $N$ limit of the partition function of the dual superconformal field theory. In particular, the $Σ_g \times S^1 \cong \mathcal{M}_{g,0}$ partition function, which was recently shown to match the entropy of $AdS_4$ black holes, and the $S^3 \cong \mathcal{M}_{0,1}$ free energy, occur as special cases of our formalism, and we comment on relations between them.

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arXiv2018

5d Partition Functions with A Twist

We derive the partition function of 5d ${\cal N}=1$ gauge theories on the manifold $S^3_b \times Σ_{\frak g}$ with a partial topological twist along the Riemann surface, $Σ_{\frak g}$. This setup is a higher dimensional uplift of the two-dimensional A-twist, and the result can be expressed as a sum over solutions of Bethe-Ansatz-type equations, with the computation receiving nontrivial non-perturbative contributions. We study this partition function in the large $N$ limit, where it is related to holographic RG flows between asymptotically locally AdS$_6$ and AdS$_4$ spacetimes, reproducing known holographic relations between the corresponding free energies on $S^{5}$ and $S^{3}$ and predicting new ones. We also consider cases where the 5d theory admits a UV completion as a 6d SCFT, such as the maximally supersymmetric ${\cal N}=2$ Yang-Mills theory, in which case the partition function computes the 4d index of general class ${\cal S}$ theories, which we verify in certain simplifying limits. Finally, we comment on the generalization to ${\cal M}_3 \times Σ_{\frak g}$ with more general three-manifolds ${\cal M}_3$ and focus in particular on ${\cal M}_3=Σ_{\frak g'}\times S^{1}$, in which case the partition function relates to the entropy of black holes in AdS$_6$.

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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.

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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.

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arXiv2016

The long flow to freedom

Two-dimensional field theories do not have a moduli space of vacua. Instead, it is common that their low-energy behavior is a sigma model with a target space. When this target space is compact its renormalization group flow is standard. When it is non-compact the continuous spectrum of operators can change the qualitative behavior. Here we discuss two-dimensional gauge theories with N=(2,2) supersymmetry. We focus on two specific theories, for which we argue that they flow to free chiral multiplets at low energies: the U(1) gauge theory with one flavor (two chiral superfields with charges plus and minus one) and a non-zero Fayet-Iliopoulos term, and pure SU(N) gauge theories. We argue that the renormalization group flow of these theories has an interesting order of limits issue. Holding the position on the target space fixed, the space flattens out under the renormalization group. On the other hand, if we first go to infinity on the target space and then perform the renormalization group, we always have a non-trivial space, e.g. a cone with a deficit angle. We explain how to interpret low-energy dualities between theories with non-compact target spaces. We expect a similar qualitative behavior also for other non-compact sigma models, even when they do not flow to free theories.

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arXiv2016

Localization on three-dimensional manifolds

In this review article we describe the localization of three dimensional N=2 supersymmetric theories on compact manifolds, including the squashed sphere, S^3_b, the lens space, S^3_b/Z_p, and S^2 x S^1. We describe how to write supersymmetric actions on these spaces, and then compute the partition functions and other supersymmetric observables by employing the localization argument. We briefly survey some applications of these computations.

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arXiv2015

A "Lagrangian" for a non-Lagrangian theory

We suggest an N=1 Lagrangian flowing in the infra-red to the N=2 rank one superconformal field theory with E_6 flavor symmetry. We utilize this description to compute several supersymmetric partition functions.

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arXiv2015

Generalized Global Symmetries

A $q$-form global symmetry is a global symmetry for which the charged operators are of space-time dimension $q$; e.g. Wilson lines, surface defects, etc., and the charged excitations have $q$ spatial dimensions; e.g. strings, membranes, etc. Many of the properties of ordinary global symmetries ($q$=0) apply here. They lead to Ward identities and hence to selection rules on amplitudes. Such global symmetries can be coupled to classical background fields and they can be gauged by summing over these classical fields. These generalized global symmetries can be spontaneously broken (either completely or to a subgroup). They can also have 't Hooft anomalies, which prevent us from gauging them, but lead to 't Hooft anomaly matching conditions. Such anomalies can also lead to anomaly inflow on various defects and exotic Symmetry Protected Topological phases. Our analysis of these symmetries gives a new unified perspective of many known phenomena and uncovers new results.

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arXiv2014

Down the rabbit hole with theories of class S

We review some of the properties of 3d N=4 theories obtained by dimensionally reducing theories of class S. We study 3d partition functions, and certain limits thereof, for such theories, and the properties implied for these by 3d mirror symmetry.

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arXiv2013

Global Properties of Supersymmetric Theories and the Lens Space

We compute the supersymmetric partition function on L(r,1)xS^1, the lens space index, for 4d gauge theories related by supersymmetric dualities and involving non simply-connected groups. This computation is sensitive to the global properties of the underlying gauge group and to discrete theta angle parameters and thus distinguishes versions of dualities differing by such. We explicitly discuss N=1 so(N_c) Seiberg dualities and N=4 su(N_c) S-dualities.

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arXiv2013

3d dualities from 4d dualities for orthogonal groups

We extend recent work on the relation of 4d and 3d IR dualities of supersymmetric gauge theories with four supercharges to the case of orthogonal gauge groups. The distinction between different SO(N) gauge theories in 4d plays an important role in this relation. We show that the 4d duality leads to a 3d duality between an SO(N_c) gauge theory with N_f flavors and an SO(N_f-N_c+2) theory with N_f flavors and extra singlets, and we derive its generalization in the presence of Chern-Simons terms. There are two different O(N) theories in 3d, which we denote by O(N)_\pm, and we also show that the O(N_c)_- gauge theory is dual to a Spin(N_f-N_c+2) theory, and derive from 4d the known duality between O(N_c)_+ and O(N_f-N_c+2)_+. We verify the consistency of these 3d dualities by various methods, including index computations.

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