34 research outputs found

    4d/2d -> 3d/1d: A song of protected operator algebras

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    Superconformal field theories (SCFT) are known to possess solvable yet nontrivial sectors in their full operator algebras. Two prime examples are the chiral algebra sector on a two dimensional plane in four dimensional N=2 SCFTs, and the topological quantum mechanics (TQM) sector on a line in three dimensional N=4 SCFTs. Under Weyl transformation, they respectively map to operator algebras on a great torus in S^1Γ—S^3 and a great circle in S^3, and are naturally related by reduction along the S^1 factor, which amounts to taking the Cardy (high-temperature) limit of the four dimensional theory on S1Γ—S3. We elaborate on this relation by explicit examples that involve both Lagrangian and non-Lagrangian theories in four dimensions, where the chiral algebra sector is generally described by a certain W-algebra, while the three dimensional descendant SCFT always has a (mirror) Lagrangian description. By taking into account a subtle R-symmetry mixing, we provide explicit dictionaries between selected operator product expansion (OPE) data in the four and three dimensional SCFTs, which we verify in the examples using recent localization results in four and three dimensions. Our methods thus provide nontrivial support for various chiral algebra proposals in the literature. Along the way, we also identify three dimensional mirrors for Argyres-Douglas theories of type (A1,D2n+1) reduced on S^1, and find more evidence for earlier proposals in the case of (A_1,A_(2nβˆ’2)), which both realize certain superconformal boundary conditions for the four dimensional N=4 super-Yang-Mills. This is a companion paper to arXiv:1911.05741

    Chiral Algebra, Localization, Modularity, Surface defects, And All That

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    We study the 2D vertex operator algebra (VOA) construction in 4D N=2\mathcal{N}=2 superconformal field theories (SCFT) on S3Γ—S1S^3 \times S^1, focusing both on old puzzles as well as new observations. The VOA lives on a two-torus T2βŠ‚S3Γ—S1\mathbb{T}^2\subset S^3\times S^1, it is 12Z\frac12\mathbb{Z}-graded, and this torus is equipped with the natural choice of spin structure (1,0) for the Z+12\mathbb{Z} +\frac12-graded operators, corresponding to the NS sector vacuum character. By analyzing the possible refinements of the Schur index that preserve the VOA, we find that it admits discrete deformations, which allow access to the remaining spin structures (1,1), (0,1) and (0,0), of which the latter two involve the inclusion of a particular surface defect. For Lagrangian theories, we perform the detailed analysis: we describe the natural supersymmetric background, perform localization, and derive the gauged symplectic boson action on a torus in any spin structure. In the absence of flavor fugacities, the 2D and 4D path integrals precisely match, including the Casimir factors. We further analyze the 2D theory: we identify its integration cycle, the two-point functions, and interpret flavor holonomies as screening charges in the VOA. Next, we make some observations about modularity; the TT-transformation acts on our four partition functions and lifts to a large diffeomorphism on S3Γ—S1S^3\times S^1. More interestingly, we generalize the four partition functions on the torus to an infinite family labeled both by the spin structure and the integration cycle inside the complexified maximal torus of the gauge group. Members of this family transform into one another under the full modular group, and we confirm the recent observation that the SS-transform of the Schur index in Lagrangian theories exhibits logarithmic behavior. Finally, we comment on how locally our background reproduces the Ξ©\Omega-background.Comment: 100 pages, 0 tables and figure

    Gluing II: Boundary Localization and Gluing Formulas

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    We describe applications of the gluing formalism discussed in the companion paper. When a dd-dimensional local theory QFTd\text{QFT}_d is supersymmetric, and if we can find a supersymmetric polarization for QFTd\text{QFT}_d quantized on a (dβˆ’1)(d-1)-manifold WW, gluing along WW is described by a non-local QFTdβˆ’1\text{QFT}_{d-1} that has an induced supersymmetry. Applying supersymmetric localization to QFTdβˆ’1\text{QFT}_{d-1}, which we refer to as the boundary localization, allows in some cases to represent gluing by finite-dimensional integrals over appropriate spaces of supersymmetric boundary conditions. We follow this strategy to derive a number of `gluing formulas' in various dimensions, some of which are new and some of which have been previously conjectured. First we show how gluing in supersymmetric quantum mechanics can reduce to a sum over a finite set of boundary conditions. Then we derive two gluing formulas for 3D N=4\mathcal{N}=4 theories on spheres: one providing the Coulomb branch representation of gluing, and another providing the Higgs branch representation. This allows to study various properties of their (2,2)(2,2)-preserving boundary conditions in relation to Mirror Symmetry. After that we derive a gluing formula in 4D N=2\mathcal{N}=2 theories on spheres, both squashed and round. First we apply it to predict the hemisphere partition function, then we apply it to the study of boundary conditions and domain walls in these theories. Finally, we mention how to glue half-indices of 4D N=2\mathcal{N}=2 theories.Comment: 68 pages, 4 figures; v2: references adde

    Remarks on Berry Connection in QFT, Anomalies, and Applications

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    Berry connection has been recently generalized to higher-dimensional QFT, where it can be thought of as a topological term in the effective action for background couplings. Via the inflow, this term corresponds to the boundary anomaly in the space of couplings, another notion recently introduced in the literature. In this note we address the question of whether the old-fashioned Berry connection (for time-dependent couplings) still makes sense in a QFT on Ξ£(d)Γ—R\Sigma^{(d)}\times \mathbb{R}, where Ξ£(d)\Sigma^{(d)} is a dd-dimensional compact space and R\mathbb{R} is time. Compactness of Ξ£(d)\Sigma^{(d)} relieves us of the IR divergences, so we only have to address the UV issues. We describe a number of cases when the Berry connection is well defined (which includes the ttβˆ—tt^* equations), and when it is not. We also mention a relation to the boundary anomalies and boundary states on the Euclidean Ξ£(d)Γ—Rβ‰₯0\Sigma^{(d)} \times \mathbb{R}_{\geq 0}. We then work out the examples of a free 3D Dirac fermion and a 3D N=2\mathcal{N}=2 chiral multiplet. Finally, we consider 3D theories on T2Γ—R\mathbb{T}^2\times \mathbb{R}, where the space T2\mathbb{T}^2 is a two-torus, and apply our machinery to clarify some aspects of the relation between 3D SUSY vacua and elliptic cohomology. We also comment on the generalization to higher genus.Comment: 52 pages plus references; v2: references added; v3: minor improvement

    From VOAs to short star products in SCFT

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    We build a bridge between two algebraic structures in SCFT: a VOA in the Schur sector of 4d N = 2 theories and an associative algebra in the Higgs sector of 3d N = 4. The natural setting is a 4d N = 2 SCFT placed on S^3 Γ—S^1: by sending the radius of S^1 to zero, we recover the 3d N = 4 theory, and the corresponding VOA on the torus degenerates to the associative algebra on the circle. We prove that: 1) the Higgs branch operators remain in the cohomology; 2) all the Schur operators of the non-Higgs type are lifted by line operators wrapped on the S^1; 3) no new cohomology classes are added. We show that the algebra in 3d is given by the quotient AH = Zhus(V )/N, where Zhus(V ) is the non-commutative Zhu algebra of the VOA V (for s ∈ Aut(V )), and N is a certain ideal. This ideal is the null space of the (s-twisted) trace map Ts : Zhus(V ) β†’ C determined by the torus 1-point function in the high temperature (or small complex structure) limit. It therefore equips AH with a non-degenerate (twisted) trace, leading to a short star-product according to the recent results of Etingof and Stryker. The map Ts is easy to determine for unitary VOAs, but has a much subtler structure for non-unitary and non-C2-cofinite VOAs of our interest. We comment on relation to the Beem-Rastelli conjecture on the Higgs branch and the associated variety. A companion paper will explore further details, examples, and some applications of these ideas

    Gluing I: Integrals and Symmetries

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    We review some aspects of the cutting and gluing law in local quantum field theory. In particular, we emphasize the description of gluing by a path integral over a space of polarized boundary conditions, which are given by leaves of some Lagrangian foliation in the phase space. We think of this path integral as a non-local (dβˆ’1)(d-1)-dimensional gluing theory associated to the parent local dd-dimensional theory. We describe various properties of this procedure and spell out conditions under which symmetries of the parent theory lead to symmetries of the gluing theory. The purpose of this paper is to set up a playground for the companion paper where these techniques are applied to obtain new results in supersymmetric theories
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