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Interpret the negative-dimension auxiliary AdS for p=6

Clarify the physical meaning of the auxiliary AdS space of negative dimension that arises for p = 6 (η = −9) in the auxiliary AdS_{p+2+η} uplift of D6-brane backgrounds, and ascertain a consistent interpretation of the formal dimensional reduction from 10 dimensions to 1 dimension and its implications for the strong-coupling limit of seven-dimensional maximally supersymmetric Yang–Mills theory.

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Background

In the auxiliary AdS construction, the near-horizon Dp-brane backgrounds can be viewed as arising from a higher-dimensional AdS_{p+2+η}×S{8−p} geometry with fractional η. For p=6, one finds η = −9, implying an AdS space of negative dimension after a formal dimensional reduction to 1 dimension.

The authors state that the meaning of this negative-dimensional AdS and its relation to the unusual strong-coupling behavior of 7d SYM is unclear, highlighting a conceptual gap in understanding the p=6 case.

References

For $p=6$ we have $\eta = -9$ and therefore we find a formal dimensional reduction from 10d to 1d and an AdS space of negative dimension. It is unclear how to think of this exotic structure which may be related to the unusual features of the strong-coupling limit of the 7d SYM theory mentioned in Footnote~\ref{ft:D6branes}.

— Correlation functions for non-conformal D$p$-brane holography (2503.18770 - Bobev et al., 24 Mar 2025) in Subsection 2.2 (Auxiliary AdS), footnote