Signs of the time: Melonic theories over diverse number systems

Abstract

Melonic field theories are defined over the $p$-adic numbers with the help of a sign character. Our construction works over the reals as well as the $p$-adics, and it includes the fermionic and bosonic Klebanov-Tarnopolsky models as special cases; depending on the sign character, the symmetry group of the field theory can be either orthogonal or symplectic. Analysis of the Schwinger-Dyson equation for the two-point function in the leading melonic limit shows that power law scaling behavior in the infrared arises for fermionic theories when the sign character is non-trivial, and for bosonic theories when the sign character is trivial. In certain cases, the Schwinger-Dyson equation can be solved exactly using a quartic polynomial equation, and the solution interpolates between the ultraviolet scaling controlled by the spectral parameter and the universal infrared scaling. As a by-product of our analysis, we see that melonic field theories defined over the real numbers can be modified by replacing the time derivative by a bilocal kinetic term with a continuously variable spectral parameter. The infrared scaling of the resulting two-point function is universal, independent of the spectral parameter of the ultraviolet theory.

40 pages, 2 figures. v2: Small changes, references added

References in corpus (4)

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• The F-theorem in the melonic limit
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• Recursion Relations in $p$-adic Mellin Space
• Non-Archimedean Quantum Mechanics via Quantum Groups
• Building Archimedean Space
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• Probing holography in $p$-adic CFT
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• Pseudodifferential Operators on ${\mathbf{Q}}_p$ and $L$-Series
• Toward Holography on Biregular Trees
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