Evidence for C-theorems in 6D SCFTs

TL;DR

The paper investigates RG flow monotonicity in 6D SCFTs by constructing C-functions as linear combinations of anomaly-polynomial coefficients. Leveraging the 6D SCFT classification from F-theory, it maps the monotonic region in m-space via exhaustive analysis of Higgs- and tensor-branch flows, including the use of bogus theories to facilitate decomposition. It finds large families of weak C-functions exist, with analytic and numeric bounds identifying the monotonic region; in particular, the a-type Euler density combination a6D lies inside this region, supporting monotonicity for known flows. The work also highlights methodological scaffolding, such as node-by-node Higgs-flow decomposition, and points to future directions linking these C-functions to conventional anomaly data and expanding the RG-flow landscape.

Abstract

Using the recently established classification of 6D SCFTs we present evidence for the existence of families of weak C-functions, that is, quantities which decrease in a flow from the UV to the IR. Introducing a background R-symmetry field strength R and a non-trivial tangent bundle T on the 6D spacetime, we consider C-functions given by the linear combinations C = m1 alpha + m2 beta + m3 gamma, where alpha, beta and gamma are the anomaly polynomial coefficients for the formal characteristic classes c2(R)^2, c2(R)p1(T) and p1(T)^2. By performing a detailed sweep over many theories, we determine the shape of the unbounded monotonic region in "m-space" compatible with both Higgs branch flows and tensor branch flows. We also verify that --as expected-- the Euler density conformal anomaly falls in the admissible region.

Figures (9)

• Figure 1: The bounds on the monotonic region $(m_1,m_2,m_3,m_4=0)$ from Higgs branch flows. The only meaningful bound is $m2 / m_1 < 26/11$, and it comes from the simplest possible Higgs branch flow, $1 \overset{RG}\rightarrow \hbox{Free Hypers.}$
• Figure 2: The minimum value of $-\Delta\alpha/\Delta\beta$ for each classical family, up to 25 tensor nodes. Since each value of this quantity provides a bound $m_2 / m_1 < -\Delta\alpha/\Delta\beta$, only the smallest value is meaningful, while the rest of the bounds are redundant. Clearly, the bounds weaken as the number of tensor branch nodes increases, giving us strong reason to believe that the only meaningful bound comes from (\ref{['Higgsbound']}).
• Figure 3: The bounds on the monotonic region $(m_1,m_2,m_3,m_4=0)$ from tensor branch flows of $\underset{[N_f = n - 8]}{\overset{\mathfrak{so}(n)}4}$ theories (blue, negative slope lines) and $\underset{[N_f = 8n + 2]}{\overset{\mathfrak{sp}(n)}1}$ theories (red, positive slope lines). For large $|m_2 / m_1|$, the shaded region $m_1 m_3 > m_2^2$ is well approximated by the bounds from these flows.
• Figure 4: The bounds on the monotonic region $(m_1,m_2,m_3,m_4=0)$ (shaded) from tensor branch flows. The only meaningful restrictions come from the simple flows shown and give the bounds in (\ref{['monotonicbounds']}). The vector $\vec{m_a}$ for the $a$-type Weyl anomaly (red) fits comfortably in the monotonic region.
• Figure 5: The minimum value of $|\Delta\alpha/\Delta\beta|$ for each classical family, up to 25 tensor nodes. The values increase monotonically with the number of tensor nodes for each classical family, indicating that the bounds are getting weaker as the number of tensor nodes increases.