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Geometry of 6D RG Flows

Jonathan J. Heckman, David R. Morrison, Tom Rudelius, Cumrun Vafa

TL;DR

This work develops a geometric framework for renormalization group flows between six-dimensional superconformal field theories realized in F-theory. By mapping tensor-branch deformations to base blowups and Higgs-branch deformations to complex-structure unfoldings, the authors connect UV and IR fixed points and test the flows with explicit anomaly-polynomial matching, including in E-string and small E8 instanton sectors. They demonstrate that the UV–IR differences factor into perfect squares (under unbroken symmetries), providing strong evidence for RG flows and ruling out nontrivial dualities. The results refine the 6D SCFT landscape, clarifying how conformal matter, E-string dynamics, and instanton probes organize into RG trajectories with potential holographic interpretations.

Abstract

We study renormalization group flows between six-dimensional superconformal field theories (SCFTs) using their geometric realizations as singular limits of F-theory compactified on elliptically fibered Calabi-Yau threefolds. There are two general types of flows: One corresponds to giving expectation values to scalars in the tensor multiplets (tensor branch flow) realized as resolving the base of the geometry. The other corresponds to giving expectation values to hypermultiplets (Higgs branch flow) realized as complex structure deformations of the geometry. To corroborate this physical picture we calculate the change in the anomaly polynomial for these theories, finding strong evidence for a flow from a UV fixed point to an IR fixed point. Moreover, we find evidence against non-trivial dualities for 6D SCFTs. In addition we find non-trivial RG flows for theories realizing small E8 instantons on ALE spaces.

Geometry of 6D RG Flows

TL;DR

This work develops a geometric framework for renormalization group flows between six-dimensional superconformal field theories realized in F-theory. By mapping tensor-branch deformations to base blowups and Higgs-branch deformations to complex-structure unfoldings, the authors connect UV and IR fixed points and test the flows with explicit anomaly-polynomial matching, including in E-string and small E8 instanton sectors. They demonstrate that the UV–IR differences factor into perfect squares (under unbroken symmetries), providing strong evidence for RG flows and ruling out nontrivial dualities. The results refine the 6D SCFT landscape, clarifying how conformal matter, E-string dynamics, and instanton probes organize into RG trajectories with potential holographic interpretations.

Abstract

We study renormalization group flows between six-dimensional superconformal field theories (SCFTs) using their geometric realizations as singular limits of F-theory compactified on elliptically fibered Calabi-Yau threefolds. There are two general types of flows: One corresponds to giving expectation values to scalars in the tensor multiplets (tensor branch flow) realized as resolving the base of the geometry. The other corresponds to giving expectation values to hypermultiplets (Higgs branch flow) realized as complex structure deformations of the geometry. To corroborate this physical picture we calculate the change in the anomaly polynomial for these theories, finding strong evidence for a flow from a UV fixed point to an IR fixed point. Moreover, we find evidence against non-trivial dualities for 6D SCFTs. In addition we find non-trivial RG flows for theories realizing small E8 instantons on ALE spaces.

Paper Structure

This paper contains 11 sections, 72 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Geometry of 6D RG Flows
  3. Anomaly Polynomial Analysis
  4. Examples with E-strings
  5. Examples of RG Flows
  6. Flows on the Tensor Branch
  7. Flows on the Higgs Branch
  8. Instantons probing an $E_8$ Wall
  9. Instantons on a $\Gamma_{E_{8}}$ Orbifold
  10. Further Examples of Instanton Flows
  11. Conclusions and Future Directions

Figures (5)

  • Figure 1: M-theory realization of the flow from $E_8 \oplus E_8$ conformal matter to a single tensor multiplet. There is an M5-brane probing the $E_8$ singularity in the UV (left), which lifts off the singularity in the IR (right).
  • Figure 2: Two M5-branes probing an $E_8$ singularity. The theory flows from (a) to (b) to (c).
  • Figure 3: Two M5-branes probing an $E_8$ singularity. A tensor branch flow occurs when the two branes are separated along the singular locus.
  • Figure 4: M-theory realization of the theory of two heterotic small instantons.
  • Figure 5: M-theory realization of the theory of two heterotic small instantons.