Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Anomalies, Renormalization Group Flows, and the a-Theorem in Six-Dimensional (1,0) Theories

Clay Cordova, Thomas T. Dumitrescu, Kenneth Intriligator

TL;DR

The paper proves a universal linear relation between the six-dimensional $a$-type Weyl anomaly and the ’t Hooft anomalies for $SU(2)_R$ and gravity in $(1,0)$ SCFTs, and establishes an $a$-theorem for tensor-branch RG flows via a dilaton effective action and Green–Schwarz anomaly matching.It derives a universal formula $a = \tfrac{16}{7}(\alpha - \beta + \gamma) + \tfrac{6}{7}\delta$, with the coefficient fixed by a free hypermultiplet, and shows this relation holds across rank-one and higher-rank tensor branches, including consistency with known $(2,0)$ theories.The authors apply the framework to exact computations of $a$-anomalies for $N$ small $E_8$ instantons and $N$ M5-branes on orbifolds, verify monotonicity for both tensor and certain Higgs branches, and discuss extensions to tensor branches with vector multiplets where IR theories are scale-invariant but not conformally invariant.

Abstract

We establish a linear relation between the $a$-type Weyl anomaly and the 't Hooft anomaly coefficients for the $R$-symmetry and gravitational anomalies in six-dimensional $(1,0)$ superconformal field theories. For RG flows onto the tensor branch, where conformal symmetry is spontaneously broken, supersymmetry relates the anomaly mismatch $Δa$ to the square of a four-derivative interaction for the dilaton. This establishes the $a$-theorem for all such flows. The four-derivative dilaton interaction is in turn related to the Green-Schwarz-like terms that are needed to match the 't Hooft anomalies on the tensor branch, thus fixing their relation to $Δa$. We use our formula to obtain exact expressions for the $a$-anomaly of $N$ small $E_8$ instantons, as well as $N$ M5-branes probing an orbifold singularity, and verify the $a$-theorem for RG flows onto their Higgs branches. We also discuss aspects of supersymmetric RG flows that terminate in scale but not conformally invariant theories with massless gauge fields.

Anomalies, Renormalization Group Flows, and the a-Theorem in Six-Dimensional (1,0) Theories

TL;DR

The paper proves a universal linear relation between the six-dimensional $a$-type Weyl anomaly and the ’t Hooft anomalies for $SU(2)_R$ and gravity in $(1,0)$ SCFTs, and establishes an $a$-theorem for tensor-branch RG flows via a dilaton effective action and Green–Schwarz anomaly matching.It derives a universal formula $a = \tfrac{16}{7}(\alpha - \beta + \gamma) + \tfrac{6}{7}\delta$, with the coefficient fixed by a free hypermultiplet, and shows this relation holds across rank-one and higher-rank tensor branches, including consistency with known $(2,0)$ theories.The authors apply the framework to exact computations of $a$-anomalies for $N$ small $E_8$ instantons and $N$ M5-branes on orbifolds, verify monotonicity for both tensor and certain Higgs branches, and discuss extensions to tensor branches with vector multiplets where IR theories are scale-invariant but not conformally invariant.

Abstract

We establish a linear relation between the -type Weyl anomaly and the 't Hooft anomaly coefficients for the -symmetry and gravitational anomalies in six-dimensional superconformal field theories. For RG flows onto the tensor branch, where conformal symmetry is spontaneously broken, supersymmetry relates the anomaly mismatch to the square of a four-derivative interaction for the dilaton. This establishes the -theorem for all such flows. The four-derivative dilaton interaction is in turn related to the Green-Schwarz-like terms that are needed to match the 't Hooft anomalies on the tensor branch, thus fixing their relation to . We use our formula to obtain exact expressions for the -anomaly of small instantons, as well as M5-branes probing an orbifold singularity, and verify the -theorem for RG flows onto their Higgs branches. We also discuss aspects of supersymmetric RG flows that terminate in scale but not conformally invariant theories with massless gauge fields.

Paper Structure

This paper contains 17 sections, 76 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Anomaly Matching in Six Dimensions
  3. Anomaly Polynomials
  4. Matching 't Hooft Anomalies on the Moduli Space
  5. The Dilaton Effective Lagrangian and the $a$-Anomaly
  6. The $a$-Theorem for Tensor Branch Flows
  7. Relating the $a$-Anomaly to 't Hooft Anomalies
  8. Rank One Tensor Branches
  9. Tensor Branches of Higher Rank
  10. A Universal Formula for the $a$-Anomaly
  11. Example: The Theory of $N$ Small $E_8$ Instantons
  12. Tensor Branches with Vector Multiplets
  13. Scale and Conformal Invariance on the Tensor Branch
  14. The $a$-Theorem for Tensor Branch Flows with Vector Multiplets
  15. Unitarity Constraints on Tensor Branch Effective Actions
  16. ...and 2 more sections

Figures (3)

  • Figure 1: Factorization of a six-point dilaton amplitude proportional to $\Delta a$ through a pair of four-point amplitudes proportional to $b$. This explains the quadratic relation $\Delta a \sim b^2$.
  • Figure 2: M-Theory description of the $(1,0)$ SCFT ${\mathcal{E}}_N$ of $N$ small $E_8$ instantons. In (a) there are $N$ M5-branes (represented by dots) embedded in the Hořava-Witten wall. In (b) a flow onto the tensor branch is initiated by pulling a single M5-brane off the wall. In (c) a Higgs branch flow is initiated by dissolving the branes inside the wall.
  • Figure 3: The orbifold theory and its RG flows. In (a) there are $N$ NS5-branes (represented by dots) embedded in a stack of $k$ D6-branes. In (b) the flow onto the tensor branch is obtained by separating the NS5-branes inside the D6-branes. In (c) the flow onto a mixed branch is obtained by moving the NS5-branes off of the D6-branes.