Holography for (1,0) theories in six dimensions (1404.0711v2)

Abstract: M-theory and string theory predict the existence of many six-dimensional SCFTs. In particular, type IIA brane constructions involving NS5-, D6- and D8-branes conjecturally give rise to a very large class of N=(1,0) CFTs in six dimensions. We point out that these theories sit at the end of RG flows which start from six-dimensional theories which admit an M-theory construction as a M5 stack transverse to $R^4/Z_k \times R$. The flows are triggered by Higgs branch expectation values and correspond to D6's opening up into transverse D8-branes via a Nahm pole. We find a precise correspondence between these CFT's and the AdS$_7$ vacua found in a recent classification in type II theories. Such vacua involve massive IIA regions, and the internal manifold is topologically $S^3$. They are characterized by fluxes for the NS three-form and RR two-form, which can be thought of as the near-horizon version of the NS5's and D6's in the brane picture; the D8's, on the other hand, are still present in the AdS$_7$ solution, in the form of an arbitrary number of concentric shells wrapping round $S^2$'s.

• The paper establishes a connection between six-dimensional (1,0) superconformal field theories (SCFTs) and AdS7 vacua within massive type IIA supergravity through holographic correspondence.
• It proposes that AdS7 solutions are near-horizon limits of D8-D6-NS brane systems and classifies infinite classes of these solutions based on brane configurations and flux integers.
• The findings enrich the understanding and classification of 6d SCFTs, offer methods to predict new theories, and advance insights into string theory and M-theory in higher dimensions.

Holography for $(1,0)$ Theories in Six Dimensions

Summary of the Paper's Findings

This paper addresses the intricate landscape of six-dimensional superconformal field theories (SCFTs) within the framework of M-theory constructions, string theory predictions, and holographic correspondences. The primary focus is on $(1,0)$ theories, which arise from type IIA brane configurations involving NS5-, D6-, and D8-branes. These theories are posited to emerge at the end of RG flows initiated by Higgs branch expectation values. The paper establishes a connection between these SCFTs and AdS$_7$ vacua in massive type IIA supergravity.

Theoretical Framework and Key Results

The paper begins by outlining the significance of six-dimensional CFTs, particularly in the field of theoretical physics, where they provide crucial insights into higher-dimensional field theories. These theories often emerge from M5-branes and are characterized by the emergence of various interesting field theories upon compactification. Particularly, it is noted that $(1,0)$ theories might be interpreted as orbifolds of $(2,0)$ theories, promising a broad class of SCFTs within the spectrum of supergravity solutions.

One of the intriguing points of the paper involves the correspondence between $AdS_7$ vacua and $(1,0)$ CFTs through a detailed examination of brane configurations. The paper proposes that AdS$_7$ solutions are the near-horizon limits of D8-D6-NS systems. This conjecture is supported by analyses of Higgs branch RG flows and Nahm pole structures in seven-dimensional gauge theories.

Through numerical analysis and semi-empirical classifications, the paper identifies infinite classes of AdS$_7$ solutions, categorized by their brane configurations and various flux integers. Specifically, the authors refine earlier classifications and argue for a bijective correspondence between AdS vacua and CFTs such that the brane ordering constraint and anomaly cancellation conditions emerge naturally from the supergravity formalism. They further determine the bounds for flux integers, mapping them to distinct NS5-brane configurations.

Implications and Future Directions

The implications of these results extend to both practical and theoretical domains. Practically, they enrich the roster of known 6d SCFTs by providing a framework to predict previously unclassified theories. This extends the possibilities for formulating new field theories with real-world applications in high-energy physics. Theoretically, the paper fosters the understanding of string theory predictions and M-theory configurations, especially in higher dimensions where predictive constructs like AdS/CFT become pivotal.

The exploration of the degrees of freedom in various configurations reveals that differential equations governing the AdS$_7$ solutions provide significant insight into the number of degrees of freedom associated with $(1,0)$ SCFTs. These estimates offer quantitative measures of quantum states, providing crucial benchmarking for conjectural formulations like orbifolds of $(2,0)$ theories.

The paper suggests future exploration on refining the mathematical tools for classifying AdS vacua and their corresponding SCFTs. Detailed examination of the symmetry-breaking patterns leads to new conjectures and opens up further inquiries into the correlation between brane configurations and field theoretic predictions. Moreover, expanding the analysis to other half-BPS extensions can potentially unlock richer dimensional landscapes.

By solidifying the correspondence between supergravity solutions and field theory constructs, this paper significantly contributes to advancing the understanding of six-dimensional theories. It establishes a valuable foundation for further exploration into realizing their potential within the broader context of string theory and holographic correspondences.