arXiv subjects

Davide Gaiotto

Publications and source records attributed to Davide Gaiotto.

At least 73 records · Page 4Linked to original sources

arXiv2018

Vortices and Vermas

In three-dimensional gauge theories, monopole operators create and destroy vortices. We explore this idea in the context of 3d N=4 gauge theories in the presence of an Omega-background. In this case, monopole operators generate a non-commutative algebra that quantizes the Coulomb-branch chiral ring. The monopole operators act naturally on a Hilbert space, which is realized concretely as the equivariant cohomology of a moduli space of vortices. The action furnishes the space with the structure of a Verma module for the Coulomb-branch algebra. This leads to a new mathematical definition of the Coulomb-branch algebra itself, related to that of Braverman-Finkelberg-Nakajima. By introducing additional boundary conditions, we find a construction of vortex partition functions of 2d N=(2,2) theories as overlaps of coherent states (Whittaker vectors) for Coulomb-branch algebras, generalizing work of Braverman-Feigin-Finkelberg-Rybnikov on a finite version of the AGT correspondence. In the case of 3d linear quiver gauge theories, we use brane constructions to exhibit vortex moduli spaces as handsaw quiver varieties, and realize monopole operators as interfaces between handsaw-quiver quantum mechanics, generalizing work of Nakajima.

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arXiv2018

Dual Boundary Conditions in 3d SCFT's

We propose matching pairs of half-BPS boundary conditions related by IR dualities of 3d $\mathcal{N}=2$ gauge theories. From these matching pairs we construct duality interfaces. We test our proposals by anomaly matching and the computation of supersymmetric indices. Examples include basic abelian dualities, level-rank dualities, and Aharony dualities.

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arXiv2017

Time-Reversal Breaking in QCD$_4$, Walls, and Dualities in 2+1 Dimensions

We study $SU(N)$ Quantum Chromodynamics (QCD) in 3+1 dimensions with $N_f$ degenerate fundamental quarks with mass $m$ and a $θ$-parameter. For generic $m$ and $θ$ the theory has a single gapped vacuum. However, as $θ$ is varied through $θ=π$ for large $m$ there is a first order transition. For $N_f=1$ the first order transition line ends at a point with a massless $η'$ particle (for all $N$) and for $N_f>1$ the first order transition ends at $m=0$, where, depending on the value of $N_f$, the IR theory has free Nambu-Goldstone bosons, an interacting conformal field theory, or a free gauge theory. Even when the $4d$ bulk is smooth, domain walls and interfaces can have interesting phase transitions separating different $3d$ phases. These turn out to be the phases of the recently studied $3d$ Chern-Simons matter theories, thus relating the dynamics of QCD$_4$ and QCD$_3$, and, in particular, making contact with the recently discussed dualities in 2+1 dimensions. For example, when the massless $4d$ theory has an $SU(N_f)$ sigma model, the domain wall theory at low (nonzero) mass supports a $3d$ massless $CP^{N_f-1}$ nonlinear $σ$-model with a Wess-Zumino term, in agreement with the conjectured dynamics in 2+1 dimensions.

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arXiv2017

Theta, Time Reversal, and Temperature

$SU(N)$ gauge theory is time reversal invariant at $θ=0$ and $θ=π$. We show that at $θ=π$ there is a discrete 't Hooft anomaly involving time reversal and the center symmetry. This anomaly leads to constraints on the vacua of the theory. It follows that at $θ=π$ the vacuum cannot be a trivial non-degenerate gapped state. (By contrast, the vacuum at $θ=0$ is gapped, non-degenerate, and trivial.) Due to the anomaly, the theory admits nontrivial domain walls supporting lower-dimensional theories. Depending on the nature of the vacuum at $θ=π$, several phase diagrams are possible. Assuming area law for space-like loops, one arrives at an inequality involving the temperatures at which CP and the center symmetry are restored. We also analyze alternative scenarios for $SU(2)$ gauge theory. The underlying symmetry at $θ=π$ is the dihedral group of 8 elements. If deconfined loops are allowed, one can have two $O(2)$-symmetric fixed points. It may also be that the four-dimensional theory around $θ=π$ is gapless, e.g. a Coulomb phase could match the underlying anomalies.

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arXiv2017

Vertex Algebras for S-duality

We define new deformable families of vertex operator algebras $\mathfrak{A}[\mathfrak{g}, Ψ, σ]$ associated to a large set of S-duality operations in four-dimensional supersymmetric gauge theory. They are defined as algebras of protected operators for two-dimensional supersymmetric junctions which interpolate between a Dirichlet boundary condition and its S-duality image. The $\mathfrak{A}[\mathfrak{g}, Ψ, σ]$ VOAs are equipped with two $\mathfrak{g}$ affine vertex subalgebras whose levels are related by the S-duality operation. They compose accordingly under a natural convolution operation and can be used to define an action of the S-duality operations on a certain space of VOAs equipped with a $\mathfrak{g}$ affine vertex subalgebra. We give a self-contained definition of the S-duality action on that space of VOAs. The space of conformal blocks (in the derived sense, i.e. chiral homology) for $\mathfrak{A}[\mathfrak{g}, Ψ, σ]$ is expected to play an important role in a broad generalization of the quantum Geometric Langlands program. Namely, we expect the S-duality action on VOAs to extend to an action on the corresponding spaces of conformal blocks. This action should coincide with and generalize the usual quantum Geometric Langlands correspondence. The strategy we use to define the $\mathfrak{A}[\mathfrak{g}, Ψ, σ]$ VOAs is of broader applicability and leads to many new results and conjectures about deformable families of VOAs.

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arXiv2017

Infrared Computations of Defect Schur Indices

We conjecture a formula for the Schur index of N=2 four-dimensional theories in the presence of boundary conditions and/or line defects, in terms of the low-energy effective Seiberg-Witten description of the system together with massive BPS excitations. We test our proposal in a variety of examples for SU(2) gauge theories, either conformal or asymptotically free. We use the conjecture to compute these defect-enriched Schur indices for theories which lack a Lagrangian description, such as Argyres-Douglas theories. We demonstrate in various examples that line defect indices can be expressed as sums of characters of the associated two-dimensional chiral algebra and that for Argyres-Douglas theories the line defect OPE reduces in the index to the Verlinde algebra.

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arXiv2017

Surface Defects and Chiral Algebras

We investigate superconformal surface defects in four-dimensional N=2 superconformal theories. Each such defect gives rise to a module of the associated chiral algebra and the surface defect Schur index is the character of this module. Various natural chiral algebra operations such as Drinfeld-Sokolov reduction and spectral flow can be interpreted as constructions involving four-dimensional surface defects. We compute the index of these defects in the free hypermultiplet theory and Argyres-Douglas theories, using both infrared techniques involving BPS states, as well as renormalization group flows onto Higgs branches. In each case we find perfect agreement with the predicted characters.

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arXiv2017

Surface Defect Indices and 2d-4d BPS States

We conjecture a formula for the Schur index of four-dimensional $\mathcal{N}=2$ theories coupled to $(2,2)$ surface defects in terms of the $2d$-$4d$ BPS spectrum in the Coulomb phase of the theory. The key ingredient in our conjecture is a refined $2d$-$4d$ wall-crossing invariant, which we also formulate. Our result intertwines recent conjectures expressing the four-dimensional Schur index in terms of infrared BPS particles, with the Cecotti-Vafa formula for limits of the elliptic genus in terms of two-dimensional BPS solitons. We extend our discussion to framed $2d$-$4d$ BPS states, and use this to demonstrate a general relationship between surface defect indices and line defect indices. We illustrate our results in the example of $SU(2)$ super Yang-Mills coupled to the $\mathbb{CP}^1$ sigma model defect.

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arXiv2017

Supersymmetric SYK models

We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev model. These are quantum mechanical models involving $N$ Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with random coefficients. The Hamiltonian is the square of the supercharge. The ${\cal N}=1$ model with a single supercharge has unbroken supersymmetry at large $N$, but non-perturbatively spontaneously broken supersymmetry in the exact theory. We analyze the model by looking at the large $N$ equation, and also by performing numerical computations for small values of $N$. We also compute the large $N$ spectrum of "singlet" operators, where we find a structure qualitatively similar to the ordinary SYK model. We also discuss an ${\cal N}=2$ version. In this case, the model preserves supersymmetry in the exact theory and we can compute a suitably weighted Witten index to count the number of ground states, which agrees with the large $N$ computation of the entropy. In both cases, we discuss the supersymmetric generalizations of the Schwarzian action which give the dominant effects at low energies.

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arXiv2016

S-duality of boundary conditions and the Geometric Langlands program

Maximally supersymmetric gauge theory in four dimensions admits local boundary conditions which preserve half of the bulk supersymmetries. The S-duality of the bulk gauge theory can be extended in a natural fashion to act on such half-BPS boundary conditions. The purpose of this note is to explain the role these boundary conditions can play in the Geometric Langlands program. In particular, we describe how to obtain pairs of Geometric Langland dual objects from S-dual pairs of half-BPS boundary conditions.

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arXiv2016

Boundaries, Mirror Symmetry, and Symplectic Duality in 3d $\mathcal{N}=4$ Gauge Theory

We introduce several families of $\mathcal{N}=(2,2)$ UV boundary conditions in 3d $\mathcal N=4$ gauge theories and study their IR images in sigma-models to the Higgs and Coulomb branches. In the presence of Omega deformations, a UV boundary condition defines a pair of modules for quantized algebras of chiral Higgs- and Coulomb-branch operators, respectively, whose structure we derive. In the case of abelian theories, we use the formalism of hyperplane arrangements to make our constructions very explicit, and construct a half-BPS interface that implements the action of 3d mirror symmetry on gauge theories and boundary conditions. Finally, by studying two-dimensional compactifications of 3d $\mathcal{N}=4$ gauge theories and their boundary conditions, we propose a physical origin for symplectic duality - an equivalence of categories of modules associated to families of Higgs and Coulomb branches that has recently appeared in the mathematics literature, and generalizes classic results on Koszul duality in geometric representation theory. We make several predictions about the structure of symplectic duality, and identify Koszul duality as a special case of wall crossing.

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arXiv2016

Topological conformal defects with tensor networks

The critical 2d classical Ising model on the square lattice has two topological conformal defects: the $\mathbb{Z}_2$ symmetry defect $D_ε$ and the Kramers-Wannier duality defect $D_σ$. These two defects implement antiperiodic boundary conditions and a more exotic form of twisted boundary conditions, respectively. On the torus, the partition function $Z_{D}$ of the critical Ising model in the presence of a topological conformal defect $D$ is expressed in terms of the scaling dimensions $Δ_α$ and conformal spins $s_α$ of a distinct set of primary fields (and their descendants, or conformal towers) of the Ising CFT. This characteristic conformal data $\{Δ_α, s_α\}_{D}$ can be extracted from the eigenvalue spectrum of a transfer matrix $M_{D}$ for the partition function $Z_D$. In this paper we investigate the use of tensor network techniques to both represent and coarse-grain the partition functions $Z_{D_ε}$ and $Z_{D_σ}$ of the critical Ising model with either a symmetry defect $D_ε$ or a duality defect $D_σ$. We also explain how to coarse-grain the corresponding transfer matrices $M_{D_ε}$ and $M_{D_σ}$, from which we can extract accurate numerical estimates of $\{Δ_α, s_α\}_{D_ε}$ and $\{Δ_α, s_α\}_{D_σ}$. Two key new ingredients of our approach are (i) coarse-graining of the defect $D$, which applies to any (i.e. not just topological) conformal defect and yields a set of associated scaling dimensions $Δ_α$, and (ii) construction and coarse-graining of a generalized translation operator using a local unitary transformation that moves the defect, which only exist for topological conformal defects and yields the corresponding conformal spins $s_α$.

cond-mat.str-el↗

arXiv2016

State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter

It is possible to describe fermionic phases of matter and spin-topological field theories in 2+1d in terms of bosonic "shadow" theories, which are obtained from the original theory by "gauging fermionic parity". The fermionic/spin theories are recovered from their shadow by a process of fermionic anyon condensation: gauging a one-form symmetry generated by quasi-particles with fermionic statistics. We apply the formalism to theories which admit gapped boundary conditions. We obtain Turaev-Viro-like and Levin-Wen-like constructions of fermionic phases of matter. We describe the group structure of fermionic SPT phases protected by the product of fermion parity and internal symmetry G. The quaternion group makes a surprise appearance.

cond-mat.str-el↗

arXiv2016

An Introduction To The Web-Based Formalism

This paper summarizes our rather lengthy paper, "Algebra of the Infrared: String Field Theoretic Structures in Massive ${\cal N}=(2,2)$ Field Theory In Two Dimensions," and is meant to be an informal, yet detailed, introduction and summary of that larger work.

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arXiv2015

Spin TQFTs and fermionic phases of matter

We study lattice constructions of gapped fermionic phases of matter. We show that the construction of fermionic Symmetry Protected Topological orders by Gu and Wen has a hidden dependence on a discrete spin structure on the Euclidean space-time. The spin structure is needed to resolve ambiguities which are otherwise present. An identical ambiguity is shown to arise in the fermionic analog of the string-net construction of 2D topological orders. We argue that the need for a spin structure is a general feature of lattice models with local fermionic degrees of freedom and is a lattice analog of the spin-statistics relation.

cond-mat.str-el↗

arXiv2015

N=1 theories of class S_k

We construct classes of ${\cal N}=1$ superconformal theories elements of which are labeled by punctured Riemann surfaces. Degenerations of the surfaces correspond, in some cases, to weak coupling limits. Different classes are labeled by two integers (N,k). The k=1 case coincides with A_{N-1} ${\cal N}=2$ theories of class S and simple examples of theories with k>1 are Z_k orbifolds of some of the A_{N-1} class S theories. For the space of ${\cal N}=1$ theories to be complete in an appropriate sense we find it necessary to conjecture existence of new ${\cal N}=1$ strongly coupled SCFTs. These SCFTs when coupled to additional matter can be related by dualities to gauge theories. We discuss in detail the A_1 case with k=2 using the supersymmetric index as our analysis tool. The index of theories in classes with k>1 can be constructed using eigenfunctions of elliptic quantum mechanical models generalizing the Ruijsenaars-Schneider integrable model. When the elliptic curve of the model degenerates these eigenfunctions become polynomials with coefficients being algebraic expressions in fugacities, generalizing the Macdonald polynomials with rational coefficients appearing when k=1.

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arXiv2015

Algebra of the Infrared: String Field Theoretic Structures in Massive ${\cal N}=(2,2)$ Field Theory In Two Dimensions

We introduce a "web-based formalism" for describing the category of half-supersymmetric boundary conditions in $1+1$ dimensional massive field theories with ${\cal N}=(2,2)$ supersymmetry and unbroken $U(1)_R$ symmetry. We show that the category can be completely constructed from data available in the far infrared, namely, the vacua, the central charges of soliton sectors, and the spaces of soliton states on $\mathbb{R}$, together with certain "interaction and boundary emission amplitudes". These amplitudes are shown to satisfy a system of algebraic constraints related to the theory of $A_\infty$ and $L_\infty$ algebras. The web-based formalism also gives a method of finding the BPS states for the theory on a half-line and on an interval. We investigate half-supersymmetric interfaces between theories and show that they have, in a certain sense, an associative "operator product." We derive a categorification of wall-crossing formulae. The example of Landau-Ginzburg theories is described in depth drawing on ideas from Morse theory, and its interpretation in terms of supersymmetric quantum mechanics. In this context we show that the web-based category is equivalent to a version of the Fukaya-Seidel $A_\infty$-category associated to a holomorphic Lefschetz fibration, and we describe unusual local operators that appear in massive Landau-Ginzburg theories. We indicate potential applications to the theory of surface defects in theories of class S and to the gauge-theoretic approach to knot homology.

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arXiv2015

Duality walls and defects in 5d N=1 theories

We propose an explicit description of duality walls which encode at low energy the global symmetry enhancement expected in the UV completion of certain five-dimensional gauge theories. The proposal is supported by explicit localization computations and implies that the instanton partition function of these theories satisfies novel and unexpected integral equations.

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