Abstract

Refining previous work by Iso, Kawai and Kitazawa, we discuss bi-local string states as a tool for loop computations in noncommutative field theory and matrix models. Defined in terms of coherent states, they exhibit the stringy features of noncommutative field theory. This leads to a closed form for the 1-loop effective action in position space, capturing the long-range non-local UV/IR mixing for scalar fields. The formalism applies to generic fuzzy spaces. The non-locality is tamed in the maximally supersymmetric IKKT or IIB model, where it gives rise to supergravity. The linearized supergravity interactions are obtained directly in position space at one loop using string states on generic noncommutative branes.

Highlights

  • Noncommutative field theory (NCFT) was conceived as a generalization of field theory to noncommutative or quantized spaces

  • 1 One may consider various limits of noncommutative field theories, which may again become local, see e.g. [14]. This is the subject of the second part of this paper, where the formalism of string states is used to elaborate the 1-loop effective action of the supersymmetric IKKT or IIB model

  • To explain the new technique for evaluating the trace using string states, we focus on the case of an irreducible fuzzy space of brane11 given by the quantization of a symplectic manifold M; stacks of branes will not be discussed here

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Summary

Introduction

Noncommutative field theory (NCFT) was conceived as a generalization of (quantum) field theory to noncommutative or quantized spaces. Attempts to directly use generic (non-supersymmetric) NC field theories as a replacement for ordinary local QFT are doomed, and only the maximally supersymmetric model(s) remain as candidates for a fundamental, “UV-complete” quantum theory This is the subject of the second part of this paper, where the formalism of string states is used to elaborate the 1-loop effective action of the supersymmetric IKKT or IIB model. In this case, the residual non-locality is mild and can be understood as a manifestation of the 10-dimensional supergravity in target space, which leads to a short-range r−8 interaction. This is presented in a separate paper, demonstrating the emergence of 4-dimensional gravity [24]

Coherent states on the fuzzy sphere
String states
One-loop computations using coherent states
Gij μ2 eiki θ ij pj
Higher loops and ’t Hooft approach to NC QFT
The 1-loop effective potential for the IKKT model
Induced interactions and linearized IIB supergravity
Fluctuations on a background
Non-supersymmetric matrix models
Conclusion
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