Second moment fuzzy-field-theory-like matrix models
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Springer
Received: February 7, 2020 Accepted: May 26, 2020 Published: June 15, 2020
ˇ M´ aria Subjakov´ a and Juraj Tekel Department of Theoretical Physics, Faculty of Mathematics, Physics and Informatics, Comenius University, Mlynsk´ a Dolina, Bratislava, 842 48, Slovakia
E-mail: [email protected], [email protected] Abstract: We solve a multitrace matrix model approximating the real quartic scalar field theory on the fuzzy sphere and obtain its phase diagram. We generalize this method to models with modified kinetic terms and demonstrate its use by investigating models related to the removal of the UV/IR mixing. We show that for the fuzzy sphere a modification of the kinetic part of the action by higher derivative term can change the phase diagram of the theory such that the triple point moves further from the origin. Keywords: Matrix Models, Non-Commutative Geometry ArXiv ePrint: 2002.02317
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)088
JHEP06(2020)088
Second moment fuzzy-field-theory-like matrix models
Contents 1
2 Preliminaries 2.1 Matrix models of fuzzy field theories 2.2 Solutions of the second moment multitrace models 2.3 The fuzzy sphere
3 3 4 5
3 Fuzzy sphere model 3.1 Symmetric one-cut to symmetric two-cut transition 3.2 Asymmetric one-cut to symmetric two-cut transition 3.3 Asymmetric one-cut to symmetric one-cut transition 3.4 Phase diagram and the triple point location
5 6 6 8 9
4 Modified kinetic term models 4.1 General remarks 4.2 UV/IR mixing free theory 4.3 Coupling enhanced fuzzy sphere model 4.3.1 Triple point 4.3.2 Phase diagram 4.4 Higher derivative model 4.4.1 Analysis of the simple model 4.4.2 Analysis of the complete model
10 10 12 13 13 14 16 16 17
5 Conclusions
19
1
Introduction
Spaces with noncommuting coordinates have been a part of theoretical physics for quite some time. As a fundamental concept [1, 2], as an effective description of different phenomena [3, 4], or as various solutions and backgrounds in matrix model descriptions of string theory [5, 6]. Fuzzy spaces are finite mode approximations to compact manifolds [7]. The space is divided into a finite number of cells, not unlike the phase space of quantum mechanics. As such, the field theories on fuzzy spaces have finite number of degrees of freedom and are essentially matrix models. These properties make the fuzzy spaces a very important setting to test the consequences of quantum structure of spacetimes, which is expected to be present in the quantum theory of gravity [8]. Any matrix model begs to be put on a computer. Matrix models describing the fuzzy field theories have been investigated in numerous Monte Carlo studies: for the fuzzy
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1 Introduction
1
See [16] for a review of numerical investigations into noncommutative field theories.
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sphere [9–11], for the fuzzy disc [12], for the fuzzy sphere with a commutative time, i.e. the three dimensional space R × SF2 [13], for the fuzzy torus [14] a
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