Publications: DR Masanori Hanada

( 2025 ) . Operator algebra, quantum entanglement, and emergent geometry from matrix degrees of freedom . Journal of High Energy Physics vol. 2025 , ( 1 ) Article 19 ,

Hanada M, Ohata H, Shimada H, Watanabe H ( 2024 ) . Partial deconfinement in QCD at N = 3 and N = ∞ . Proceedings of Science . vol. 453 ,

Bergner G, Gautam V, Hanada M, Holden J ( 2024 ) . Wilson loops and random matrices . Journal of High Energy Physics vol. 2024 , ( 7 ) Article 203 ,

Hanada M, Jevicki A, Liu X, Rinaldi E, Tezuka M ( 2024 ) . A model of randomly-coupled Pauli spins . Journal of High Energy Physics vol. 2024 , ( 5 ) Article 280 ,

Bergner G, Hanada M, Rinaldi E, Schäfer A ( 2024 ) . Toward QCD on quantum computer: orbifold lattice approach . Journal of High Energy Physics vol. 2024 , ( 5 ) Article 234 ,

Hanada M, Ohata H, Shimada H, Watanabe H ( 2024 ) . A New Perspective on Thermal Transition in QCD . Progress of Theoretical and Experimental Physics vol. 2024 , ( 4 ) Article 041B02 ,

Bergner G, Gautam V, Hanada M ( 2024 ) . Color confinement and random matrices. A random walk down group manifold toward Casimir scaling . Journal of High Energy Physics vol. 2024 , ( 3 ) Article 13 ,

Hanada M, Watanabe H ( 2024 ) . On Thermal Transition in QCD . Progress of Theoretical and Experimental Physics vol. 2024 , ( 4 ) Article 043B02 ,

Hanada M, Liu J, Rinaldi E, Tezuka M ( 2023 ) . Estimating truncation effects of quantum bosonic systems using sampling algorithms . Machine Learning: Science and Technology vol. 4 , ( 4 ) 045021 - 045021 .

Bauer CW, Davoudi Z, Balantekin AB, Bhattacharya T, Carena M, de Jong WA, Draper P, El-Khadra A et al. ( 2023 ) . Quantum Simulation for High-Energy Physics . PRX Quantum vol. 4 , ( 2 ) Article 027001 ,

Gautam V, Hanada M, Holden J, Rinaldi E ( 2023 ) . Linear confinement in the partially-deconfined phase . Journal of High Energy Physics vol. 2023 , ( 3 ) Article 195 ,

Pateloudis S, Bergner G, Hanada M, Rinaldi E, Schäfer A, Vranas P, Watanabe H, Bodendorfer N ( 2023 ) . Precision test of gauge/gravity duality in D0-brane matrix model at low temperature . Journal of High Energy Physics vol. 2023 , ( 3 ) Article 71 ,

Tezuka M, Oktay O, Rinaldi E, Hanada M, Nori F ( 2023 ) . Binary-coupling sparse Sachdev-Ye-Kitaev model: An improved model of quantum chaos and holography . Physical Review B vol. 107 , ( 8 ) Article L081103 ,

Gautam V, Hanada M, Jevicki A, Peng C ( 2023 ) . Matrix entanglement . Journal of High Energy Physics vol. 2023 , ( 1 ) Article 3 ,

Hanada M ( 2022 ) . Large-N limit as a second quantization . Proceedings of Corfu Summer Institute 2021 \School and Workshops on Elementary Particle Physics and Gravity\ — PoS(CORFU2021) . Conference: Corfu Summer Institute 2021 \School and Workshops on Elementary Particle Physics and Gravity\ from: 29/08/2021260 - 260 .

Hanada M, Watanabe H ( 2022 ) . Partial deconfinement: a brief overview . The European Physical Journal Special Topics vol. 232 , ( 3 ) 333 - 337 .

Pateloudis S, Bergner G, Bodendorfer N, Hanada M, Rinaldi E, Schäfer A ( 2022 ) . Nonperturbative test of the Maldacena-Milekhin conjecture for the BMN matrix model . Journal of High Energy Physics vol. 2022 , ( 8 ) Article 178 ,

Bergner G, Bodendorfer N, Hanada M, Pateloudis S, Rinaldi E, Schäfer A, Vranas P, Watanabe H ( 2022 ) . Confinement/deconfinement transition in the D0-brane matrix model — A signature of M-theory? . Journal of High Energy Physics vol. 2022 , ( 5 ) Article 96 ,

Hanada M, Holden J, Knaggs M, O’Bannon A ( 2022 ) . Global symmetries and partial confinement . Journal of High Energy Physics vol. 2022 , ( 3 ) Article 118 ,

Rinaldi E, Han X, Hassan M, Feng Y, Nori F, McGuigan M, Hanada M ( 2022 ) . Matrix-Model Simulations Using Quantum Computing, Deep Learning, and Lattice Monte Carlo . PRX Quantum vol. 3 , ( 1 ) Article 010324 ,

Buser AJ, Gharibyan H, Hanada M, Honda M, Liu J ( 2021 ) . Quantum simulation of gauge theory via orbifold lattice . Journal of High Energy Physics vol. 2021 , ( 9 ) Article 34 ,

Hanada M, Shimada H, Wintergerst N ( 2021 ) . Color confinement and Bose-Einstein condensation . Journal of High Energy Physics vol. 2021 , ( 8 ) Article 39 ,

Gharibyan H, Hanada M, Honda M, Liu J ( 2021 ) . Toward simulating superstring/M-theory on a quantum computer . Journal of High Energy Physics vol. 2021 , ( 7 ) Article 140 ,

Hanada M ( 2021 ) . Bulk geometry in gauge/gravity duality and color degrees of freedom . Physical Review D vol. 103 , ( 10 ) Article 106007 ,

Alet F, Hanada M, Jevicki A, Peng C ( 2021 ) . Entanglement and confinement in coupled quantum systems . Journal of High Energy Physics vol. 2021 , ( 2 ) Article 34 ,

Watanabe H, Bergner G, Bodendorfer N, Funai SS, Hanada M, Rinaldi E, Schäfer A, Vranas P ( 2021 ) . Partial deconfinement at strong coupling on the lattice . Journal of High Energy Physics vol. 2021 , ( 2 ) Article 4 ,

Hanada M, Robinson B ( 2020 ) . Partial-symmetry-breaking phase transitions . Physical Review D vol. 102 , ( 9 ) Article 096013 ,

Gharibyan H, Hanada M, Swingle B, Tezuka M ( 2020 ) . Characterization of quantum chaos by two-point correlation functions . Physical Review E vol. 102 , ( 2 ) Article 022213 ,

Bergner G, Bodendorfer N, Hanada M, Rinaldi E, Schäfer A, Vranas P ( 2020 ) . Thermal phase transition in Yang-Mills matrix model . Journal of High Energy Physics vol. 2020 , ( 1 ) Article 53 ,

Hanada M, Jevicki A, Peng C, Wintergerst N ( 2019 ) . Anatomy of deconfinement . Journal of High Energy Physics vol. 2019 , ( 12 ) Article 167 ,

Gharibyan H, Hanada M, Swingle B, Tezuka M ( 2019 ) . Quantum Lyapunov spectrum . Journal of High Energy Physics vol. 2019 , ( 4 ) Article 82 ,

Hanada M, Ishiki G, Watanabe H ( 2019 ) . Partial Deconfinement . Journal of High Energy Physics vol. 2019 , ( 3 ) Article 145 ,

Buividovich PV, Hanada M, Schäfer A ( 2019 ) . Quantum chaos, thermalization, and entanglement generation in real-time simulations of the Banks-Fischler-Shenker-Susskind matrix model . Physical Review D vol. 99 , ( 4 ) Article 046011 ,

Hanada M, Romatschke P ( 2019 ) . Real time quantum gravity dynamics from classical statistical Yang-Mills simulations . Journal of High Energy Physics vol. 2019 , ( 1 ) Article 201 ,

Gharibyan H, Hanada M, Shenker SH, Tezuka M ( 2018 ) . Onset of random matrix behavior in scrambling systems . Journal of High Energy Physics vol. 2018 , ( 7 ) Article 124 ,

Berkowitz E, Hanada M, Rinaldi E, Vranas P ( 2018 ) . Gauged and ungauged: a nonperturbative test . Journal of High Energy Physics vol. 2018 , ( 6 ) Article 124 ,

Hanada M, Kadoh D, Matsuura S, Sugino F ( 2018 ) . <mml:math xmlns:mml=\http://www.w3.org/1998/Math/MathML\ altimg=\si1.gif\ overflow=\scroll\><mml:mi mathvariant=\script\>O</mml:mi><mml:mo stretchy=\false\>(</mml:mo><mml:mi>a</mml:mi><mml:mo stretchy=\false\>)</mml:mo></mml:math> improvement of 2D <mml:math xmlns:mml=\http://www.w3.org/1998/Math/MathML\ altimg=\si2.gif\ overflow=\scroll\><mml:mi mathvariant=\script\>N</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\false\>(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\false\>)</mml:mo></mml:math> lattice SYM theory . Nuclear Physics B vol. 929 , 266 - 297 .

Hanada M, Shimada H, Tezuka M ( 2018 ) . Universality in chaos: Lyapunov spectrum and random matrix theory . Physical Review E vol. 97 , ( 2 ) Article 022224 ,

Rinaldi E, Berkowitz E, Hanada M, Maltz J, Vranas P ( 2018 ) . Toward holographic reconstruction of bulk geometry from lattice simulations . Journal of High Energy Physics vol. 2018 , ( 2 ) Article 42 ,

Hanada M, Romatschke P ( 2017 ) . Lattice simulations of 10d Yang-Mills toroidally compactified to 1d, 2d, and 4d . Physical Review D vol. 96 , ( 9 ) Article 094502 ,

Danshita I, Hanada M, Tezuka M ( 2017 ) . Creating and probing the Sachdev–Ye–Kitaev model with ultracold gases: Towards experimental studies of quantum gravity . Progress of Theoretical and Experimental Physics vol. 2017 , ( 8 )

Cotler JS, Gur-Ari G, Hanada M, Polchinski J, Saad P, Shenker SH, Stanford D, Streicher A et al. ( 2017 ) . Black holes and random matrices . Journal of High Energy Physics vol. 2017 , ( 5 ) Article 118 ,

Hanada M, Maltz J ( 2017 ) . A proposal of the gauge theory description of the small Schwarzschild black hole in AdS5 × S5 . Journal of High Energy Physics vol. 2017 , ( 2 ) Article 12 ,

Berkowitz E, Hanada M, Maltz J ( 2016 ) . Chaos in matrix models and black hole evaporation . Physical Review D vol. 94 , ( 12 ) Article 126009 ,

Berkowitz E, Hanada M, Maltz J ( 2016 ) . A microscopic description of black hole evaporation via holography . International Journal of Modern Physics D vol. 25 , ( 12 ) 1644002 - 1644002 .

Berkowitz E, Rinaldi E, Hanada M, Ishiki G, Shimasaki S, Vranas P ( 2016 ) . Precision lattice test of the gauge/gravity duality at large \n<mml:math xmlns:mml=\http://www.w3.org/1998/Math/MathML\ display=\inline\><mml:mi>N</mml:mi></mml:math> . Physical Review D vol. 94 , ( 9 ) Article 094501 ,

Hanada M, Hyakutake Y, Ishiki G, Nishimura J ( 2016 ) . Numerical tests of the gauge/gravity duality conjecture for D0-branes at finite temperature and finite \n<mml:math xmlns:mml=\http://www.w3.org/1998/Math/MathML\ display=\inline\><mml:mi>N</mml:mi></mml:math> . Physical Review D vol. 94 , ( 8 ) Article 086010 ,

Hanada M ( 2016 ) . What lattice theorists can do for superstring/M-theory . International Journal of Modern Physics A vol. 31 , ( 22 ) 1643006 - 1643006 .

Gur-Ari G, Hanada M, Shenker SH ( 2016 ) . Chaos in classical D0-brane mechanics . Journal of High Energy Physics vol. 2016 , ( 2 ) Article 91 ,

Hanada M, Matsuo Y, Morita T ( 2015 ) . Instanton dynamics in finite temperature QCD via holography . Nuclear Physics B vol. 899 , 631 - 650 .

Aoki S, Hanada M, Iizuka N ( 2015 ) . Quantum black hole formation in the BFSS matrix model . Journal of High Energy Physics vol. 2015 , ( 7 ) Article 29 ,

Aoki S, Hanada M, Nakamura A ( 2015 ) . Taming the pion condensation in QCD at finite baryon density: a numerical test in a random matrix model . Journal of High Energy Physics vol. 2015 , ( 5 ) Article 71 ,

Hanada M, Shimada H ( 2015 ) . On the continuity of the commutative limit of the 4d <mml:math xmlns:mml=\http://www.w3.org/1998/Math/MathML\ altimg=\si1.gif\ overflow=\scroll\><mml:mi mathvariant=\script\>N</mml:mi><mml:mo>=</mml:mo><mml:mn>4</mml:mn></mml:math> non-commutative super Yang–Mills theory . Nuclear Physics B vol. 892 , 449 - 474 .

Hanada M, Maltz J, Susskind L ( 2014 ) . Deconfinement transition as black hole formation by the condensation of QCD strings . Physical Review D vol. 90 , ( 10 ) Article 105019 ,

Hanada M, Hyakutake Y, Ishiki G, Nishimura J ( 2014 ) . Holographic description of a quantum black hole on a computer . Science vol. 344 , ( 6186 ) 882 - 885 .

Azeyanagi T, Hanada M, Honda M, Matsuo Y, Shiba S ( 2014 ) . A new look at instantons and large-N limit . Journal of High Energy Physics vol. 2014 , ( 5 ) Article 8 ,

Hanada M, Lee J-W, Yamada N ( 2013 ) . Large-<mml:math xmlns:mml=\http://www.w3.org/1998/Math/MathML\ display=\inline\><mml:msub><mml:mi>N</mml:mi><mml:mi>c</mml:mi></mml:msub></mml:math>gauge theory and chiral random matrix theory . Physical Review D vol. 88 , ( 2 ) Article 025046 ,

( 2013 ) . String inspired solution to the sign problem and overlapping problem . Journal of Physics: Conference Series vol. 432 , 012008 - 012008 .

Azeyanagi T, Fujita M, Hanada M ( 2013 ) . From the Planar Limit to M Theory . Physical Review Letters vol. 110 , ( 12 ) Article 121601 ,

Hanada M, Matsuo Y, Yamamoto N ( 2012 ) . Sign problem and phase quenching in finite-density QCD: Models, holography, and lattice . Physical Review D vol. 86 , ( 7 ) Article 074510 ,

Hanada M, Hoyos C, Karch A, Yaffe LG ( 2012 ) . Holographic realization of large-N c orbifold equivalence with non-zero chemical potential . Journal of High Energy Physics vol. 2012 , ( 8 ) Article 81 ,

Fujita M, Hanada M, Hoyos C ( 2012 ) . New large-<mml:math xmlns:mml=\http://www.w3.org/1998/Math/MathML\ display=\inline\><mml:mi>N</mml:mi></mml:math>limit and the planar equivalence outside the planar limit . Physical Review D vol. 86 , ( 2 ) Article 026007 ,

Hanada M, Honda M, Honma Y, Nishimura J, Shiba S, Yoshida Y ( 2012 ) . Numerical studies of the ABJM theory for arbitrary N at arbitrary coupling constant . Journal of High Energy Physics vol. 2012 , ( 5 ) Article 121 ,

Hanada M, Matsuura S, Sugino F ( 2012 ) . Non-perturbative construction of 2D and 4D supersymmetric Yang–Mills theories with 8 supercharges . Nuclear Physics B vol. 857 , ( 3 ) 335 - 361 .

Hanada M, Yamamoto N ( 2012 ) . Universality of Phases in QCD and QCD-like Theories . Journal of High Energy Physics vol. 2012 , ( 2 ) Article 138 ,

Hanada M, Hoyos C, Shimada H ( 2012 ) . On a new type of orbifold equivalence and M-theoretic <mml:math xmlns:mml=\http://www.w3.org/1998/Math/MathML\ altimg=\si1.gif\ overflow=\scroll\><mml:msub><mml:mi mathvariant=\italic\>AdS</mml:mi><mml:mn>4</mml:mn></mml:msub><mml:mo stretchy=\false\>/</mml:mo><mml:msub><mml:mi mathvariant=\italic\>CFT</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:math> duality . Physics Letters B vol. 707 , ( 3-4 ) 394 - 397 .

Hanada M, Hoyos C, Karch A ( 2012 ) . Generating new dualities through the orbifold equivalence: a demonstration in ABJM and four-dimensional quivers . Journal of High Energy Physics vol. 2012 , ( 1 ) Article 68 ,

Hanada M, Nishimura J, Sekino Y, Yoneya T ( 2011 ) . Direct test of the gauge-gravity correspondence for Matrix theory correlation functions . Journal of High Energy Physics vol. 2011 , ( 12 ) Article 20 ,

Hanada M, Matsuura S, Sugino F ( 2011 ) . Two-Dimensional Lattice for Four-Dimensional = 4 Supersymmetric Yang-Mills . Progress of Theoretical Physics vol. 126 , ( 4 ) 597 - 611 .

Cherman A, Hanada M, Robles-Llana D ( 2011 ) . Orbifold Equivalence and the Sign Problem at Finite Baryon Density . Physical Review Letters vol. 106 , ( 9 ) Article 091603 ,

Hanada M, Matsuura S, Nishimura J, Robles-Llana D ( 2011 ) . Nonperturbative studies of supersymmetric matrix quantum mechanics with 4 and 8 supercharges at finite temperature . Journal of High Energy Physics vol. 2011 , ( 2 ) Article 60 ,

Hanada M, Kanamori I ( 2011 ) . Absence of sign problem in two-dimensional $ \\mathcal{N} = \\left( {2,2} \\right) $ super Yang-Mills on lattice . Journal of High Energy Physics vol. 2011 , ( 1 ) Article 58 ,

Azeyanagi T, Hanada M, Ünsal M, Yacoby R ( 2010 ) . Large-<mml:math xmlns:mml=\http://www.w3.org/1998/Math/MathML\ display=\inline\><mml:mi>N</mml:mi></mml:math>reduction in QCD-like theories with massive adjoint fermions . Physical Review D vol. 82 , ( 12 ) Article 125013 ,

Hanada M ( 2010 ) . A proposal of a fine tuning free formulation of 4d $ \\mathcal{N} = 4 $ super Yang-Mills . Journal of High Energy Physics vol. 2010 , ( 11 ) Article 112 ,

Hanada M, Nishimura J, Sekino Y, Yoneya T ( 2010 ) . Monte Carlo Studies of Matrix Theory Correlation Functions . Physical Review Letters vol. 104 , ( 15 ) Article 151601 ,

Hanada M, Mannelli L, Matsuo Y ( 2009 ) . Four-dimensional<mml:math xmlns:mml=\http://www.w3.org/1998/Math/MathML\ display=\inline\><mml:mi mathvariant=\script\>N</mml:mi><mml:mo>=</mml:mo><mml:mn>1</mml:mn></mml:math>super Yang-Mills theory from a matrix model . Physical Review D vol. 80 , ( 12 ) Article 125001 ,

Hanada M, Mannelli L, Matsuo Y ( 2009 ) . Large-<i>N</i>reduced models of supersymmetric quiver, Chern-Simons gauge theories and ABJM . Journal of High Energy Physics vol. 2009 , ( 11 ) 087 - 087 .

Hanada M, Kanamori I ( 2009 ) . Lattice study of two-dimensional<mml:math xmlns:mml=\http://www.w3.org/1998/Math/MathML\ display=\inline\><mml:mi mathvariant=\script\>N</mml:mi><mml:mo>=</mml:mo><mml:mo stretchy=\false\>(</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo stretchy=\false\>)</mml:mo></mml:math>super Yang-Mills theory at large<mml:math xmlns:mml=\http://www.w3.org/1998/Math/MathML\ display=\inline\><mml:mi>N</mml:mi></mml:math> . Physical Review D vol. 80 , ( 6 ) Article 065014 ,

Azeyanagi T, Hanada M, Kawai H, Matsuo Y ( 2009 ) . Worldsheet analysis of gauge/gravity dualities . Nuclear Physics B vol. 816 , ( 1-2 ) 278 - 292 .

Hanada M, Hyakutake Y, Nishimura J, Takeuchi S ( 2009 ) . Higher Derivative Corrections to Black Hole Thermodynamics from Supersymmetric Matrix Quantum Mechanics . Physical Review Letters vol. 102 , ( 19 ) Article 191602 ,

Hanada M, Miwa A, Nishimura J, Takeuchi S ( 2009 ) . Schwarzschild Radius from Monte Carlo Calculation of the Wilson Loop in Supersymmetric Matrix Quantum Mechanics . Physical Review Letters vol. 102 , ( 18 ) Article 181602 ,

Azeyanagi T, Hanada M, Hirata T, Shimada H ( 2009 ) . On the shape of a D-brane bound state and its topology change . Journal of High Energy Physics vol. 2009 , ( 03 ) 121 - 121 .

Berenstein DE, Hanada M, Hartnoll SA ( 2009 ) . Multi-matrix models and emergent geometry . Journal of High Energy Physics vol. 2009 , ( 02 ) 010 - 010 .

Azeyanagi T, Hanada M, Hirata T ( 2008 ) . On matrix model formulations of noncommutative Yang-Mills theories . Physical Review D vol. 78 , ( 10 ) Article 105017 ,

Ishikawa T ( 2008 ) . Phase structure of twisted Eguchi-Kawai model . Proceedings of The XXV International Symposium on Lattice Field Theory — PoS(LATTICE 2007) . Conference: The XXV International Symposium on Lattice Field Theory from: 30/07/2007054 - 054 .

Nishimura J ( 2008 ) . Putting M theory on a computer . Proceedings of The XXV International Symposium on Lattice Field Theory — PoS(LATTICE 2007) . Conference: The XXV International Symposium on Lattice Field Theory from: 30/07/2007059 - 059 .

Anagnostopoulos KN, Hanada M, Nishimura J, Takeuchi S ( 2008 ) . Monte Carlo Studies of Supersymmetric Matrix Quantum Mechanics with Sixteen Supercharges at Finite Temperature . Physical Review Letters vol. 100 , ( 2 ) Article 021601 ,

Azeyanagi T, Hirata T, Hanada M, Ishikawa T ( 2008 ) . Phase structure of twisted Eguchi-Kawai model . Journal of High Energy Physics vol. 2008 , ( 01 ) 025 - 025 .

Hanada M, Nishimura J, Takeuchi S ( 2007 ) . Nonlattice Simulation for Supersymmetric Gauge Theories in One Dimension . Physical Review Letters vol. 99 , ( 16 ) Article 161602 ,

Hanada M, Nishioka T ( 2007 ) . Cascade of Gregory-Laflamme transitions and U(1) breakdown in super Yang-Mills . Journal of High Energy Physics vol. 2007 , ( 09 ) 012 - 012 .

Furuta K, Hanada M, Kawai H, Kimura Y ( 2007 ) . Field equations of massless fields in the new interpretation of the matrix model . NUCL PHYS B vol. 767 , ( 1-2 ) 82 - 99 .

Hanada M, Kubo F ( 2007 ) . String Tension and String Susceptibility in Two-Dimensional Generalized Weingarten Model . Progress of Theoretical Physics vol. 117 , ( 2 ) 379 - 393 .

Hanada M, Kanai T, Kawai H, Kubo F ( 2006 ) . Phase Structure of the Large-N Reduced Gauge Theory and the Generalized Weingarten Model . Progress of Theoretical Physics vol. 115 , ( 6 ) 1167 - 1177 .

Hanada M ( 2006 ) . Regularization of the Covariant Derivative on Curved Space by Finite Matrices . Progress of Theoretical Physics vol. 115 , ( 6 ) 1189 - 1209 .

Hanada M, Kawai H, Kimura Y ( 2006 ) . Curved superspaces and local supersymmetry in supermatrix model . PROG THEOR PHYS vol. 115 , ( 5 ) 1003 - 1025 .

Hanada A, Kawai H, Kimura Y ( 2005 ) . Describing curved spaces by matrices . PROG THEOR PHYS vol. 114 , ( 6 ) 1295 - 1316 .

Bal S, Hanada M, Kawai H, Kubo F ( 2005 ) . Fuzzy torus in matrix model . Nuclear Physics B vol. 727 , ( 1-2 ) 196 - 217 .

Hanada M, Hayakawa M, Ishibashi N, Kawai H, Kuroki T, Matsuo Y, Tada T ( 2004 ) . Loops versus Matrices . Progress of Theoretical Physics vol. 112 , ( 1 ) 131 - 181 .