[Paper Review] More on the Matter of 6D SCFTs

Jonathan J. Heckman|arXiv (Cornell University)|Jul 31, 2014

Black Holes and Theoretical Physics12 citations

TL;DR

This paper computes the scaling dimension of the brane recombination operator in 6D (1,0) superconformal field theories (SCFTs) arising from M5-branes probing ADE singularities using F-theory compactifications. By analyzing the holomorphic three-form and Kähler geometry of the resolved Calabi-Yau threefold in a 5D Kaluza-Klein regulated theory, the authors derive the scaling dimension of the operator parameterizing M5-brane motion off the singularity, finding it to be greater than six in all interacting cases, thus confirming it as an irrelevant deformation in the 6D SCFT.

ABSTRACT

M5-branes probing an ADE singularity lead to 6D SCFTs with (1,0) supersymmetry. On the tensor branch, the M5-branes specify domain walls of a 7D Super Yang-Mills theory with gauge group G of ADE-type, thus providing conformal matter for a broad class of generalized quiver theories. Additionally, these theories have G x G flavor symmetry, and a corresponding Higgs branch. In this note we use the F-theory realization of these theories to calculate the scaling dimension of the operator parameterizing seven-brane recombination, i.e. motion of the stack of M5-branes off of the orbifold singularity. In all cases with an interacting fixed point, we find that this operator has scaling dimension at least six, and defines a marginal irrelevant deformation.

Motivation & Objective

• To determine the scaling dimension of the operator governing M5-brane recombination off an ADE singularity in 6D (1,0) SCFTs.
• To extend the use of F-theory geometry and holomorphic three-form scaling to extract operator dimensions in higher-dimensional superconformal field theories.
• To clarify the nature of the Higgs branch operator associated with moving M5-branes away from the orbifold fixed locus.
• To establish that the recombination operator is an irrelevant deformation in all interacting 6D SCFTs arising from M5-branes on ADE singularities.

Proposed method

• Use F-theory compactification on non-compact singular elliptically fibered Calabi-Yau threefolds to realize the 6D SCFTs.
• Analyze the holomorphic three-form Ω = dx/y ∧ du ∧ dv and its scaling behavior under Kähler moduli deformations.
• Relate the scaling of Ω to the tension of tensionless strings from D3-branes wrapping collapsing P1 curves, using J ∧ J ∧ J ∼ Ω ∧ Ω.
• Perform a 5D Kaluza-Klein reduction to compute the scaling dimension in a regulated 5D theory.
• Apply the relation ∆6D = (4/3)∆5D,KK to lift the 5D scaling dimensions to 6D.
• Use homogeneity arguments on the F-theory geometry to determine the scaling of the recombination parameter r, including in the quotiented T(G,N) theories.

Experimental results

Research questions

• RQ1 What is the scaling dimension of the operator that parameterizes the motion of M5-branes off an ADE singularity in 6D SCFTs?
• RQ2 How does the F-theory geometry encode the scaling dimension of the Higgs branch operator for GL × GR flavor symmetry breaking?
• RQ3 Why does the A-type case (Ak, Ak) not allow for a fixed absolute scaling of the holomorphic three-form?
• RQ4 Is the brane recombination operator an irrelevant deformation in all interacting 6D SCFTs?
• RQ5 How does the scaling dimension of the recombination operator change under the quotient (u,v) → (ζu, ζ⁻¹v) in T(G,N) theories?

Key findings

• The scaling dimension of the brane recombination operator in the 6D SCFT is found to be 24N for (E8,E8), 16N for (E7,E7), 12N for (E6,E6), 8N for (Dp,Dp), and 4N for (Ak,Ak), where N ≥ 1 (N ≥ 2 for Ak).
• In all cases with an interacting fixed point, the scaling dimension of the recombination operator exceeds six, confirming it as an irrelevant deformation in the 6D SCFT.
• The A-type case (Ak,Ak) does not support a physical tensionless string or a fixed scaling of the holomorphic three-form, rendering the method inapplicable for absolute scaling.
• The 5D Kaluza-Klein regulated theory provides a consistent framework to compute scaling dimensions, which are then uplifted to 6D using ∆6D = (4/3)∆5D,KK.
• The recombination operator in T(G,N) theories scales as dim r(N) = N × dim r(N=1), reflecting the N-fold quotient of the base coordinates.
• The analysis confirms that the Higgs branch operator for GL × GR flavor symmetry breaking is irrelevant in all interacting 6D SCFTs arising from M5-branes on ADE singularities.