Anomaly polynomials, 6d (1,0) SCFTs and 4d N=1 SCFTs

Gabi Zafrir, IPMU

C. Vafa, S. S. Razamat and G.Z: arXiv 1610.09178

I. Bah, A. Hanany, K. Maruyoshi, S. S. Razamat, Y. Tachikawa, G.Z: arXiv 1702.04740

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Motivation

• Compactification of 6d SCFTs on Riemann surfaces can be used to better understand dynamics of 4d SCFTs.

• Has been successfully carried out for the case of the 6d SCFT living on 𝑁 M5-branes, for 4d N=2 and N=1 SCFTs (class 𝑆).

• Recently extended also to the 6d SCFT living on 𝑁 M5-branes probing a 𝐶2/𝑍𝑘 singularity, and 4d N=1 SCFTs (class 𝑆𝑘).

• In this talk we shall use the 6d construction of class 𝑆𝑘 theories to derive various properties of the 4d theories in some simple cases. These can then be compared against the 4d results.

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Outline

1. Introduction

2. 6d expectations

3. 4d results and conclusions

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6d reduction for class 𝑆 theories

Class 𝑆𝑘 theories

Class 𝑆 theories

• Provide a systematic construction of 4d N=2 SCFTs by compactification on a Riemann surface of the 6d SCFT living on 𝑁 M5-branes.

• Different realizations of the same Riemann surface imply duality relations for the 4d SCFT.

• For example: take the Riemann surface to be a torus. Leads to 4d maximally supersymmetric Yang-Mills. The complex structure moduli of the torus becomes the 4d complexified coupling constant.

• Modular invariance of the torus then leads to the 𝑆𝐿(2, 𝑍) symmetry of 4d maximally supersymmetric Yang-Mills theory.

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6d construction

• The curvature of the Riemann surface breaks SUSY.

• In order to preserve SUSY we must perform a twist 𝑆𝑂(2)𝑆 →𝑆𝑂(2)𝑆 − 𝑈(1)𝑅 for 𝑈(1)𝑅 a subgroup of 𝑈(1)𝑅 × 𝑆𝑈(2)𝐽 ⊂ 𝑆𝑂(4) ⊂ 𝑆𝑂(5)𝑅.

• The twist makes some of the spinors covariantly constant and preserves N=1 SUSY in 4d.

• This breaks 𝑆𝑂(5)𝑅 to 𝑈(1)𝑅 × 𝑆𝑈(2)𝐽.

• We can also give flux for the 𝑈(1)𝐽 Cartan of the 𝑆𝑈(2)𝐽.

• This is necessary to get N=2 SUSY, but for N=1 we have a choice for the total flux.

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6d construction

• The resulting theories are labeled by the Riemann surface and flux value.

• Gaiotto case: flux chosen to preserve N=2. Gives the N=2 class 𝑆theories [Gaiotto, 2009].

• Sicillian case: flux is zero. Have an extra 𝑆𝑈(2)𝐽 global symmetry [Benini, Tachikawa, Wecht, 2010].

• BBBW case: generic case. Have an extra 𝑈(1)𝐽 global symmetry [Bah, Beem, Bobev, Wecht, 2012].

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Conformal manifold

• The conformal manifold is given by the complex structure moduli of the Riemann surface.

• In addition one can turn on flat connections, holonomies, for 𝑆𝑈(2)𝐽 or 𝑈(1)𝐽symmetry. These preserve only N=1 SUSY.

• For a genus 𝑔 > 1 Riemann surface with no punctures, we can turn on such holonomies on each of the 2𝑔 cycles 𝐴𝑖 , 𝐵𝑗. These are subject to one relation and the action of global flavor rotation.

• For Sicilian case:

• For BBBW case:

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[Benini, Tachikawa, Wecht, 2010][Bah, Beem, Bobev, Wecht, 2012]

Class 𝑆𝑘 theories

• 4d SCFTs can be constructed by the compactification of the 6d SCFT living on 𝑁 M5-branes probing a 𝐶2/𝑍𝑘 singularity. This in general produces N=1 theories.

• For generic 𝑁, 𝑘 the 6d theory has an 𝑆𝑈(𝑘) × 𝑆𝑈(𝑘) × 𝑈(1) global symmetry.

• A class of field theories arising from this construction was conjectured in [Gaiotto, Razamat, 2015].

• When Riemann surface is a sphere or torus: theories may have Lagrangiandescriptions. These are generally N=1 quivers of 𝑆𝑈(𝑁) gauge groups.

• For genus > 1 Riemann surfaces, the theories are generally non-Lagrangian.

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• Consider the 6d SCFT living on 𝑁 M5-branes probing a 𝐶2/𝑍𝑘singularity compactified on a Riemann surface, and derive expectations for the resulting 4d theory.

• Concentrate on specific cases:

Plan

Riemann surfaces with no punctures.

The 𝑁 = 𝑘 = 2 case and genus > 1 Riemann surface.

Generic 𝑁, 𝑘 and torus.

6d reduction: 𝑁 = 𝑘 = 2 case

• Take the 6d SCFT on 2 M5-branes probing a 𝐶2/𝑍2 singularity and compactify it to 4d on a genus 𝑔 > 1 Riemann surface.

• To preserve SUSY must twist: 𝑆𝑂(2)𝑆 → 𝑆𝑂(2)𝑆 − 𝑈 1 𝑅 , for 𝑈 1 𝑅⊂𝑆𝑈 2 𝑅.

• The 𝑆𝑈(𝑘) × 𝑆𝑈(𝑘) × 𝑈(1) global symmetry is thought to enhance to 𝑆𝑂(7) for the 𝑁 = 𝑘 = 2 case [Ohmori, Shimizu, Tachikawa, Yonekura, 2014].

• Can also have non-zero flux in an abelian subgroup of 𝑆𝑂(7).

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6d reduction: fluxes

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• Decompose the flavor symmetry:

• Define a flux vector ℱ = β, γ, 𝑡 .

• The flux breaks 𝑆𝑂(7) to a subgroup 𝐺𝑚𝑎𝑥 with abelian part 𝐿.

• The possible values for 𝐺𝑚𝑎𝑥 are:

6d reduction: conformal manifold

• Complex structure moduli.

• Can also turn on flat connections for global symmetries. Conformal manifold:

• 𝐺𝑚𝑎𝑥 is the global symmetry preserved by the flux. In 4d should be the maximal global symmetry on the conformal manifold.

• 𝐿 is the abelian part of 𝐺𝑚𝑎𝑥. In 4d should be the global symmetry on a generic point on the conformal manifold.

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6d reduction: anomalies

• Can calculate the 4d t’ Hooft anomalies by integrating the 6d anomaly polynomial eight form.

• The 6d theory anomaly polynomial was evaluated in [Ohmori, Shimizu, Tachikawa, Yonekura, 2014].

• Example for 𝑆𝑂(7) compactification:

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6d reduction: torus and general 𝑁, 𝑘

• Compactification: no twist. Fluxes in 𝑆𝑈(𝑘) × 𝑆𝑈(𝑘) × 𝑈(1) breaks SUSY to N=1.

• Global symmetry: given by the commutant of the flux in the 𝑆𝑈(𝑘) ×𝑆𝑈(𝑘) × 𝑈(1) global symmetry of the 6d theory. Enhanced to 𝑆𝑈(2𝑘) for 𝑁 = 2, 𝑘 > 2, and 𝑆𝑈(2)3 for 𝑘 = 2,𝑁 > 2.

• Conformal manifold: the homotopy group relation for the torus forces holonomies to be abelian. Conformal manifold dimension then is 2𝑘. Appears to be too naïve for the torus.

• Anomalies: can use the 6d anomaly polynomial to compute the 4d anomalies.

6d reduction: global properties of flavor symmetries

• Fluxes must be quantize. The exact quantization depends on the global properties of the group, for instance: 𝑆𝑝𝑖𝑛(7) verses 𝑆𝑂(7).

• In the general case: 𝑆𝑈(𝑘) × 𝑆𝑈(𝑘) verses (𝑆𝑈 𝑘 × 𝑆𝑈 𝑘 )/𝑍𝑘.

• In the latter case: additional choices in compactification. Turning on a non-trivial Stiefel-Whitney class.

• Can materialize through fractional flux.

6d reduction: global properties

• The Stiefel-Whitney class can also materialize through holonomies:

• These in general break the global symmetry.

• Analysis of global symmetries and anomalies can be generalized also to this case.

• Understanding the possible 4d theories may teach us about properties of the 6d SCFT.

Matching to 4d

• The 6d expectations are then to be matched against the 4d results.

• For the 𝑁 = 𝑘 = 2 case we construct examples with every 𝐺𝑚𝑎𝑥. The global symmetry, conformal manifold and anomalies for these theories are all consistent with the 6d results.

• For the case of the torus we find that the 4d theories have Lagrangiandescription when the 𝑈(1) flux is zero. In those cases we match the 4d anomaly polynomial to the 6d one for any value of 𝑆𝑈(𝑘) × 𝑆𝑈(𝑘)flux. However 4d theories generically contain singlet fields.

• We find 4d theories corresponding to fractional fluxes. These fluxes are only consistent if the 6d global symmetry is (𝑆𝑈 𝑘 × 𝑆𝑈 𝑘 )/𝑍𝑘.

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Conclusions

• We studied the compactification of the 6d SCFT living on 𝑁 M5-branes probing a 𝐶2/𝑍𝑘 singularity on a Riemann surface with no punctures for selected cases.

• The possible 4d theories are labeled by the Riemann surface and fluxes for global symmetries.

• For each choice we can deduce, using the 6d construction, expectations for the global symmetry, conformal manifold and anomalies of the 4d theory.

• Choices may be sensitive to global properties of the global symmetry.

• Can be applied for general 6d SCFTs.

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Thank you

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