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Twisted compactifications of 3d \( \mathcal{N} \) = 4 theories and conformal blocks

 Gaiotto Open Access Regular Article - Theoretical Physics First Online: 12 February 2019 Abstract Three-dimensional \( \mathcal{N} \) = 4 supersymmetric quantum field theories admit two topological

Vertex algebras at the corner

Abstract We introduce a class of Vertex Operator Algebras which arise at junctions of supersymmetric interfaces in \( \mathcal{N} \) = 4 Super Yang Mills gauge theory. These vertex algebras satisfy non-trivial duality relations inherited from S-duality of the four-dimensional gauge theory. The gauge theory construction equips the vertex algebras with collections of modules...

Curious aspects of three-dimensional \( \mathcal{N}=1 \) SCFTs

Abstract We study the dynamics of certain 3d \( \mathcal{N}=1 \) time reversal invariant theories. Such theories often have exact moduli spaces of supersymmetric vacua. We propose several dualities and we test these proposals by comparing the deformations and supersymmetric ground states. First, we consider a theory where time reversal symmetry is only emergent in the infrared...

Dual boundary conditions in 3d SCFT’s

Abstract 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.

Surface defect indices and 2d-4d BPS states

Abstract 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...

The Coulomb Branch of 3d \({\mathcal{N}= 4}\) Theories

The Coulomb Branch of 3d N = 4 Theories Mathew Bullimore Tudor Dimofte Davide Gaiotto PS rhiomo lto r NIn sttuitru tl S oir Tn h so rI ntis ti tluPt h yosri As Wv nt rl ooS tuO Ny NPr iLn Yto n CN

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...

Surface defects and chiral algebras

We investigate superconformal surface defects in four-dimensional \( \mathcal{N}=2 \) 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...

Duality walls and defects in 5d \( \mathcal{N}=1 \) 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.

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...

Surface defects and instanton partition functions

Abstract We study the superconformal index of five-dimensional SCFTs and the sphere partition function of four-dimensional gauge theories with eight supercharges in the presence of co-dimension two half-BPS defects. We derive a prescription which is valid for defects which can be given a “vortex construction”, i.e. can be defined by RG flow from vortex configurations in a larger...

Boundaries, mirror symmetry, and symplectic duality in 3d \( \mathcal{N}=4 \) gauge theory

We introduce several families of \( \mathcal{N}=\left(2,\ 2\right) \) UV boundary conditions in 3d \( \mathcal{N}=4 \) gaugetheoriesandstudytheirIRimagesinsigma-modelstotheHiggsandCoulomb 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, respec-tively, whose...

Infrared computations of defect Schur indices

Abstract We conjecture a formula for the Schur index of four-dimensional \( \mathcal{N}=2 \) 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...

\( \mathcal{N}=1 \) theories of class \( {\mathcal{S}}_k \)

AbstractWe construct classes of \( \mathcal{N}=1 \) 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 AN − 1 \( \mathcal{N}=2 \) theories of class \( \mathcal{S} \) and simple...

Holography for (1,0) theories in six dimensions

Abstract M-theory and string theory predict the existence of many six-dimensional SCFTs. In particular, type IIA brane constructions involving NS5-, D6- and D8-branes conjecturally give rise to a very large class of \( \mathcal{N}=\left(1,0\right) \) CFTs in six dimensions. We point out that these theories sit at the end of RG flows which start from six-dimensional theories which...

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...

tt * geometry in 3 and 4 dimensions

We consider the vacuum geometry of supersymmetric theories with 4 supercharges, on a flat toroidal geometry. The 2 dimensional vacuum geometry is known to be captured by the tt * geometry. In the case of 3 dimensions, the parameter space is (T 2 × \( \mathbb{R} \)) N and the vacuum geometry turns out to be a solution to a generalization of monopole equations in 3N dimensions...

Bootstrapping the 3d Ising twist defect

Recent numerical results point to the existence of a conformally invariant twist defect in the critical 3d Ising model. In this note we show that this fact is supported by both epsilon expansion and conformal bootstrap calculations. We find that our results are in good agreement with the numerical data. We also make new predictions for operator dimensions and OPE coefficients...

On 6d N = (2, 0) theory compactified on a Riemann surface with finite area

Prog. Theor. Exp. Phys. On 6d (2, 0) theory compactified N = on a Riemann surface with finite area Davide Gaiotto 2 Gregory W. Moore 1 Yuji Tachikawa 0 Subject Index 0 IPMU, University of Tokyo