[1] K. Wilson and J. B. Kogut, “The Renormalization group and the ϵ expansion,” Phys. Rept. 12 (1974) 75–199.
[2] A. M. Polyakov, “Conformal symmetry of critical fluctuations,” JETP Lett. 12 (1970) 381–383.
[3] A. Zamolodchikov, “Conformal Symmetry and Multicritical Points in Two-Dimensional Quantum Field Theory. (In Russian),” Sov. J. Nucl. Phys. 44 (1986) 529–533.
[4] V. Ginzburg and L. Landau, “On the Theory of superconductivity,” Zh. Eksp. Teor. Fiz. 20 (1950) 1064–1082.
[5] B. Feigin and D. Fuks, “Invariant skew symmetric differential operators on the line and verma modules over the Virasoro algebra,” Funct. Anal. Appl. 16 (1982) 114–126.
[6] B. Feigin and D. Fuks, “Verma modules over the Virasoro algebra,” Funct. Anal. Appl. 17 (1983) 241–241.
[7] V. Dotsenko and V. Fateev, “Conformal Algebra and Multipoint Correlation Functions in Two-Dimensional Statistical Models,” Nucl. Phys. B 240 (1984) 312.
[8] G. Felder, “BRST Approach to Minimal Models,” Nucl. Phys. B 317 (1989) 215. [Erratum: Nucl.Phys.B 324, 548 (1989)].
[9] J.-L. Gervais and B. Sakita, “Field Theory Interpretation of Supergauges in Dual Models,” Nucl. Phys. B 34 (1971) 632–639.
[10] Y. Golfand and E. Likhtman, “Extension of the Algebra of Poincare Group Generators and Violation of P Invariance,” JETP Lett. 13 (1971) 323–326.
[11] D. Volkov and V. Akulov, “Possible universal neutrino interaction,” JETP Lett. 16 (1972) 438–440.
[12] T. Yoneya, “Quantum gravity and the zero slope limit of the generalized Virasoro model,” Lett. Nuovo Cim. 8 (1973) 951–955
[13] T. Yoneya, “Connection of Dual Models to Electrodynamics and Gravidynamics,” Prog. Theor. Phys. 51 (1974) 1907–1920.
[14] T. Yoneya, “Geometry, Gravity and Dual Strings,” Prog. Theor. Phys. 56 (1976) 1310.
[15] J. Scherk, “An Introduction to the Theory of Dual Models and Strings,” Rev. Mod. Phys. 47 (1975) 123–164.
[16] J. H. Schwarz, “Superstring Theory,” Phys. Rept. 89 (1982) 223–322.
[17] P. Ramond, “Dual Theory for Free Fermions,” Phys. Rev. D 3 (1971) 2415–2418.
[18] A. Neveu and J. Schwarz, “Quark Model of Dual Pions,” Phys. Rev. D 4 (1971) 1109–1111.
[19] A. Neveu and J. Schwarz, “Factorizable dual model of pions,” Nucl. Phys. B 31 (1971) 86–112.
[20] P. Goddard, C. Rebbi, and C. B. Thorn, “Lorentz covariance and the physical states in dual resonance models,” Nuovo Cim. A 12 (1972) 425–441.
[21] J. Schwarz, “Physical States and Pomeron Poles in the Dual Pion Model,” Nucl. Phys. B 46 (1972) 61–74.
[22] R. Brower and K. Friedman, “Spectrum Generating Algebra and No Ghost Theorem for the Neveu-schwarz Model,” Phys. Rev. D 7 (1973) 535–539.
[23] J. H. Schwarz, “Dual resonance theory,” Phys. Rept. 8 (1973) 269–335.
[24] E. Calabi, “On K¨ahler manifolds with vanishing canonical class,” in Algebraic Geometry and Topology: A Symposium in Honor of S. Lefschetz, pp. 78–89. Princeton University Press, 1957.
[25] S.-T. Yau, “Calabi’s Conjecture and some new results in algebraic geometry,” Proc. Nat. Acad. Sci. 74 (1977) 1798–1799.
[26] D. Gepner, “Space-Time Supersymmetry in Compactified String Theory and Superconformal Models,” Nucl. Phys. B 296 (1988) 757.
[27] D. Gepner, “Exactly Solvable String Compactifications on Manifolds of SU(N) Holonomy,” Phys. Lett. B 199 (1987) 380–388.
[28] P. Di Vecchia, J. Petersen, and H. Zheng, “N=2 Extended Superconformal Theories in Two-Dimensions,” Phys. Lett. B 162 (1985) 327–332.
[29] P. Di Vecchia, J. Petersen, and M. Yu, “On the Unitary Representations of N=2 Superconformal Theory,” Phys. Lett. B 172 (1986) 211–215.
[30] P. Di Vecchia, J. Petersen, M. Yu, and H. Zheng, “Explicit Construction of Unitary Representations of the N=2 Superconformal Algebra,” Phys. Lett. B 174 (1986) 280–284.
[31] W. Boucher, D. Friedan, and A. Kent, “Determinant Formulae and Unitarity for the N=2 Superconformal Algebras in Two-Dimensions or Exact Results on String Compactification,” Phys. Lett. B 172 (1986) 316.
[32] D. Gepner, “On the Spectrum of 2D Conformal Field Theories,” Nucl. Phys. B 287 (1987) 111–130.
[33] A. Cappelli, C. Itzykson, and J. Zuber, “Modular Invariant Partition Functions in Two-Dimensions,” Nucl. Phys. B 280 (1987) 445–465.
[34] A. Cappelli, “Modular Invariant Partition Functions of Superconformal Theories,” Phys. Lett. B 185 (1987) 82–88.
[35] D. Gepner and Z.-a. Qiu, “Modular Invariant Partition Functions for Parafermionic Field Theories,” Nucl. Phys. B 285 (1987) 423.
[36] A. Cappelli, C. Itzykson, and J. Zuber, “The ADE Classification of Minimal and A1(1) Conformal Invariant Theories,” Commun. Math. Phys. 113 (1987) 1.
[37] A. Kato, “Classification of Modular Invariant Partition Functions in Two-dimensions,” Mod. Phys. Lett. A 2 (1987) 585.
[38] J. Wess and B. Zumino, “Supergauge Transformations in Four-Dimensions,” Nucl. Phys. B 70 (1974) 39–50.
[39] E. J. Martinec, “Algebraic Geometry and Effective Lagrangians,” Phys. Lett. B 217 (1989) 431–437.
[40] S. Cecotti, “Geometry of N=2 Landau-Ginzburg families,” Nucl. Phys. B 355 (1991) 755–775.
[41] B. R. Greene, C. Vafa, and N. Warner, “Calabi-Yau Manifolds and Renormalization Group Flows,” Nucl. Phys. B 324 (1989) 371
[42] E. Witten, “Phases of N=2 theories in two-dimensions,” AMS/IP Stud. Adv. Math. 1 (1996) 143–211, arXiv:hep-th/9301042.
[43] D. Kastor, E. Martinec, and S. Shenker, “RG Flow in N=1 Discrete Series,” Nucl. Phys. B 316 (1989) 590–608.
[44] C. Vafa and N. P. Warner, “Catastrophes and the Classification of Conformal Theories,” Phys. Lett. B 218 (1989) 51–58.
[45] W. Lerche, C. Vafa, and N. P. Warner, “Chiral Rings in N=2 Superconformal Theories,” Nucl. Phys. B 324 (1989) 427–474.
[46] P. S. Howe and P. C. West, “N = 2 Superconformal Models, Landau-ginzburg Hamiltonians and the ϵ Expansion,” Phys. Lett. B 223 (1989) 377–385.
[47] S. Cecotti, L. Girardello, and A. Pasquinucci, “Nonperturbative Aspects and Exact Results for the N = 2 Landau-ginzburg Models,” Nucl. Phys. B 328 (1989) 701–722.
[48] P. S. Howe and P. C. West, “Chiral Correlators in Landau-ginzburg Theories and N = 2 Superconformal Models,” Phys. Lett. B 227 (1989) 397–405.
[49] S. Cecotti, L. Girardello, and A. Pasquinucci, “Singularity Theory and N = 2 Supersymmetry,” Int. J. Mod. Phys. A 6 (1991) 2427–2496.
[50] S. Cecotti, “N=2 Landau-Ginzburg versus Calabi-Yau sigma models: Nonperturbative aspects,” Int. J. Mod. Phys. A 6 (1991) 1749–1814.
[51] P. S. Howe and P. C. West, “Fixed points in multifield Landau-Ginsburg models,” Phys. Lett. B 244 (1990) 270–274.
[52] E. Witten, “On the Landau-Ginzburg description of N=2 minimal models,” Int. J. Mod. Phys. A 9 (1994) 4783–4800, arXiv:hep-th/9304026.
[53] J. Polchinski, String theory. Vol. 2: Superstring theory and beyond. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1998.
[54] K. Hori, S. Katz, A. Klemm, R. Pandharipande, R. Thomas, C. Vafa, R. Vakil, and E. Zaslow, Mirror symmetry. Clay mathematics monographs. AMS, Providence, USA, 2003.
[55] K. G. Wilson, “Confinement of Quarks,” Phys. Rev. D 10 (1974) 2445–2459.
[56] G. Curci and G. Veneziano, “Supersymmetry and the Lattice: A Reconciliation?,” Nucl. Phys. B 292 (1987) 555–572.
[57] D. B. Kaplan, “Dynamical Generation of Supersymmetry,” Phys. Lett. B 136 (1984) 162–164.
[58] DESY-Munster-Roma, F. Farchioni, C. Gebert, R. Kirchner, I. Montvay, A. Feo, G. Munster, T. Galla, and A. Vladikas, “The Supersymmetric Ward identities on the lattice,” Eur. Phys. J. C 23 (2002) 719–734, arXiv:hep-lat/0111008.
[59] S. Ali, H. Gerber, I. Montvay, G. Mu¨nster, S. Piemonte, P. Scior, and G. Bergner, “Analysis of Ward identities in supersymmetric Yang–Mills theory,” Eur. Phys. J. C 78 no. 5, (2018) 404, arXiv:1802.07067 [hep-lat].
[60] D. Kadoh, “Recent progress in lattice supersymmetry: from lattice gauge theory to black holes,” PoS LATTICE2015 (2016) 017, arXiv:1607.01170 [hep-lat].
[61] H. Kawai and Y. Kikukawa, “A Lattice study of N=2 Landau-Ginzburg model using a Nicolai map,” Phys. Rev. D 83 (2011) 074502, arXiv:1005.4671 [hep-lat].
[62] Y. Kikukawa and Y. Nakayama, “Nicolai mapping versus exact chiral symmetry on the lattice,” Phys. Rev. D 66 (2002) 094508, arXiv:hep-lat/0207013.
[63] M. Beccaria, G. Curci, and E. D’Ambrosio, “Simulation of supersymmetric models with a local Nicolai map,” Phys. Rev. D 58 (1998) 065009, arXiv:hep-lat/9804010.
[64] S. Catterall and S. Karamov, “Exact lattice supersymmetry: The Two-dimensional N=2 Wess-Zumino model,” Phys. Rev. D 65 (2002) 094501, arXiv:hep-lat/0108024.
[65] J. Giedt, “R-symmetry in the Q-exact (2,2) 2-D lattice Wess-Zumino model,” Nucl. Phys. B 726 (2005) 210–232, arXiv:hep-lat/0507016.
[66] G. Bergner, T. Kaestner, S. Uhlmann, and A. Wipf, “Low-dimensional Supersymmetric Lattice Models,” Annals Phys. 323 (2008) 946–988, arXiv:0705.2212 [hep-lat].
[67] T. Kastner, G. Bergner, S. Uhlmann, A. Wipf, and C. Wozar, “Two-Dimensional Wess-Zumino Models at Intermediate Couplings,” Phys. Rev. D 78 (2008) 095001, arXiv:0807.1905 [hep-lat].
[68] N. Sakai and M. Sakamoto, “Lattice Supersymmetry and the Nicolai Mapping,” Nucl. Phys. B 229 (1983) 173–188.
[69] H. Nicolai, “On a New Characterization of Scalar Supersymmetric Theories,” Phys. Lett. B 89 (1980) 341.
[70] H. Nicolai, “Supersymmetry and Functional Integration Measures,” Nucl. Phys. B 176 (1980) 419–428.
[71] G. Parisi and N. Sourlas, “Supersymmetric Field Theories and Stochastic Differential Equations,” Nucl. Phys. B 206 (1982) 321–332.
[72] S. Cecotti and L. Girardello, “STOCHASTIC AND PARASTOCHASTIC ASPECTS OF SUPERSYMMETRIC FUNCTIONAL MEASURES: A NEW NONPERTURBATIVE APPROACH TO SUPERSYMMETRY,” Annals Phys. 145 (1983) 81–99.
[73] J. Giedt and E. Poppitz, “Lattice supersymmetry, superfields and renormalization,” JHEP 09 (2004) 029, arXiv:hep-th/0407135.
[74] D. Kadoh and H. Suzuki, “Supersymmetry restoration in lattice formulations of 2D N = (2, 2) WZ model based on the Nicolai map,” Phys. Lett. B 696 (2011) 163–166, arXiv:1011.0788 [hep-lat].
[75] S. Kamata and H. Suzuki, “Numerical simulation of the N = (2, 2) Landau-Ginzburg model,” Nucl. Phys. B 854 (2012) 552–574, arXiv:1107.1367 [hep-lat].
[76] D. Kadoh and H. Suzuki, “Supersymmetric nonperturbative formulation of the WZ model in lower dimensions,” Phys. Lett. B 684 (2010) 167–172, arXiv:0909.3686 [hep-th].
[77] J. Bartels and J. Bronzan, “SUPERSYMMETRY ON A LATTICE,” Phys. Rev. D 28 (1983) 818.
[78] S. Drell, M. Weinstein, and S. Yankielowicz, “Variational Approach to Strong Coupling Field Theory. 1. Phi**4 Theory,” Phys. Rev. D 14 (1976) 487.
[79] S. Drell, M. Weinstein, and S. Yankielowicz, “Strong Coupling Field Theories. 2. Fermions and Gauge Fields on a Lattice,” Phys. Rev. D 14 (1976) 1627.
[80] O. Morikawa and H. Suzuki, “Numerical study of the N = 2 Landau–Ginzburg model,” PTEP 2018 no. 8, (2018) 083B05, arXiv:1805.10735 [hep-lat].
[81] O. Morikawa, “Numerical study of the N = 2 Landau–Ginzburg model with two superfields,” JHEP 12 (2018) 045, arXiv:1810.02519 [hep-lat].
[82] A. Zamolodchikov, “Irreversibility of the Flux of the Renormalization Group in a 2D Field Theory,” JETP Lett. 43 (1986) 730–732.
[83] A. Cappelli and J. I. Latorre, “Perturbation Theory of Higher Spin Conserved Currents Off Criticality,” Nucl. Phys. B 340 (1990) 659–691.
[84] O. Morikawa, “Continuum limit in numerical simulations of the N = 2 Landau–Ginzburg model,” PTEP 2019 no. 10, (2019) 103B03, arXiv:1906.00653 [hep-lat].
[85] J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string. Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1998.
[86] H. Weyl, “Gravitation and electricity,” Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1918 (1918) 465.
[87] H. Weyl, “A New Extension of Relativity Theory,” Annalen Phys. 59 (1919) 101–133.
[88] D. Capper and M. Duff, “Trace anomalies in dimensional regularization,” Nuovo Cim. A 23 (1974) 173–183.
[89] S. Deser, M. Duff, and C. Isham, “Nonlocal Conformal Anomalies,” Nucl. Phys. B 111 (1976) 45–55.
[90] M. Duff, “Twenty years of the Weyl anomaly,” Class. Quant. Grav. 11 (1994) 1387–1404, arXiv:hep-th/9308075.
[91] J. C. Ward, “An Identity in Quantum Electrodynamics,” Phys. Rev. 78 (1950) 182.
[92] Y. Takahashi, “On the generalized Ward identity,” Nuovo Cim. 6 (1957) 371.
[93] A. Belavin, A. M. Polyakov, and A. Zamolodchikov, “Infinite Conformal Symmetry in Two-Dimensional Quantum Field Theory,” Nucl. Phys. B 241 (1984) 333–380.
[94] M. Virasoro, “Subsidiary conditions and ghosts in dual resonance models,” Phys. Rev. D 1 (1970) 2933–2936.
[95] D. Verma, “Structure of certain induced representations of complex semisimple Lie algebras,” Bull. Amer. Math. Soc. 70 (1968) 160–166.
[96] I. Bernstein, I. Gelfand, and S. Gelfand, “Structure of representations generated by vectors of highest weight,” Funkts. Anal. Prilozh. 5 (1971) 1–9.
[97] V. Kac, “Contravariant Form for Infinite Dimensional Lie Algebras and Superalgebras,” in 7th International Group Theory Colloquium: The Integrative Conference on Group Theory and Mathematical Physics, pp. 441–445. 1, 1978.
[98] G. Andrews, R. Baxter, and P. Forrester, “Eight vertex SOS model and generalized Rogers-Ramanujan type identities,” J. Statist. Phys. 35 (1984) 193–266.
[99] D. A. Huse, “Exact exponents for infinitely many new multicritical points,” Phys. Rev. B 30 (1984) 3908–3915.
[100] E. Witten, “Constraints on Supersymmetry Breaking,” Nucl. Phys. B 202 (1982) 253.
[101] S. Cecotti and L. Girardello, “Functional Measure, Topology and Dynamical Supersymmetry Breaking,” Phys. Lett. B 110 (1982) 39.
[102] F. Ravanini and S.-K. Yang, “Modular Invariance in N = 2 Superconformal Field Theories,” Phys. Lett. B 195 (1987) 202–208.
[103] J. Polchinski, “Scale and Conformal Invariance in Quantum Field Theory,” Nucl. Phys. B 303 (1988) 226–236.
[104] Y. Nakayama, “Scale invariance vs conformal invariance,” Phys. Rept. 569 (2015) 1–93, arXiv:1302.0884 [hep-th].
[105] A. A. Tseytlin, “Conditions of Weyl Invariance of Two-dimensional σ Model From Equations of Stationarity of ’Central Charge’ Action,” Phys. Lett. B 194 (1987) 63.
[106] K. Osterwalder and R. Schrader, “AXIOMS FOR EUCLIDEAN GREEN’S FUNCTIONS,” Commun. Math. Phys. 31 (1973) 83–112.
[107] K. Osterwalder and R. Schrader, “Axioms for Euclidean Green’s Functions. 2.,” Commun. Math. Phys. 42 (1975) 281.
[108] M. T. Grisaru, W. Siegel, and M. Rocek, “Improved Methods for Supergraphs,” Nucl. Phys. B 159 (1979) 429.
[109] N. Seiberg, “Naturalness versus supersymmetric nonrenormalization theorems,” Phys. Lett. B 318 (1993) 469–475, arXiv:hep-ph/9309335.
[110] N. Seiberg, “The Power of holomorphy: Exact results in 4-D SUSY field theories,” in Particles, Strings, and Cosmology (PASCOS 94), pp. 0357–369. 5, 1994. arXiv:hep-th/9408013.
[111] P. Fayet and J. Iliopoulos, “Spontaneously Broken Supergauge Symmetries and Goldstone Spinors,” Phys. Lett. B 51 (1974) 461–464.
[112] P. Dondi and H. Nicolai, “Lattice Supersymmetry,” Nuovo Cim. A 41 (1977) 1.
[113] L. H. Karsten and J. Smit, “The Vacuum Polarization With {SLAC} Lattice Fermions,” Phys. Lett. B 85 (1979) 100–102.
[114] M. Kato, M. Sakamoto, and H. So, “Taming the Leibniz Rule on the Lattice,” JHEP 05 (2008) 057, arXiv:0803.3121 [hep-lat].
[115] G. Bergner, “Complete supersymmetry on the lattice and a No-Go theorem,” JHEP 01 (2010) 024, arXiv:0909.4791 [hep-lat].
[116] M. Lu¨scher, “Trivializing maps, the Wilson flow and the HMC algorithm,” Commun. Math. Phys. 293 (2010) 899–919, arXiv:0907.5491 [hep-lat].
[117] “Eigen.” http://eigen.tuxfamily.org/.
[118] “The julia programming language.” http://julialang.org/.
[119] V. B. S. J. Bezanson, S. Karpinski and A. Edelman, “Julia: A fast dynamic language for technical computing,” arXiv:1209.5145 [cs.PL].
[120] S. K. J. Bezanson, A. Edelman and V. B. Shah, “Julia: A fresh approach to numerical computing,” arXiv:1411.1607 [cs.MS].
[121] M. Lu¨scher, P. Weisz, and U. Wolff, “A Numerical method to compute the running coupling in asymptotically free theories,” Nucl. Phys. B 359 (1991) 221–243.
[122] M. Lu¨scher, “Properties and uses of the Wilson flow in lattice QCD,” JHEP 08 (2010) 071, arXiv:1006.4518 [hep-lat]. [Erratum: JHEP 03, 092 (2014)].
[123] H. Suzuki, “Energy–momentum tensor from the Yang–Mills gradient flow,” PTEP 2013 (2013) 083B03, arXiv:1304.0533 [hep-lat]. [Erratum: PTEP 2015, 079201 (2015)].
[124] H. Makino and H. Suzuki, “Lattice energy–momentum tensor from the Yang–Mills gradient flow—inclusion of fermion fields,” PTEP 2014 (2014) 063B02, arXiv:1403.4772 [hep-lat]. [Erratum: PTEP 2015, 079202 (2015)].
[125] H. Makino, F. Sugino, and H. Suzuki, “Large-N limit of the gradient flow in the 2D O(N ) nonlinear sigma model,” PTEP 2015 no. 4, (2015) 043B07, arXiv:1412.8218 [hep-lat].
[126] K. Hieda, A. Kasai, H. Makino, and H. Suzuki, “4D N = 1 SYM supercurrent in terms of the gradient flow,” PTEP 2017 no. 6, (2017) 063B03, arXiv:1703.04802 [hep-lat].
[127] A. Kasai, O. Morikawa, and H. Suzuki, “Gradient flow representation of the four-dimensional N = 2 super Yang–Mills supercurrent,” PTEP 2018 no. 11, (2018) 113B02, arXiv:1808.07300 [hep-lat].
[128] M. Lu¨scher, “Future applications of the Yang-Mills gradient flow in lattice QCD,” PoS LATTICE2013 (2014) 016, arXiv:1308.5598 [hep-lat].
[129] A. Kagimura, A. Tomiya, and R. Yamamura, “Effective lattice action for the configurations smeared by the Wilson flow,” arXiv:1508.04986 [hep-lat].
[130] R. Yamamura, “The Yang—Mills gradient flow and lattice effective action,” PTEP 2016 no. 7, (2016) 073B10, arXiv:1510.08208 [hep-lat].
[131] S. Aoki, J. Balog, T. Onogi, and P. Weisz, “Flow equation for the large N scalar model and induced geometries,” PTEP 2016 no. 8, (2016) 083B04, arXiv:1605.02413 [hep-th].
[132] H. Makino, O. Morikawa, and H. Suzuki, “Gradient flow and the Wilsonian renormalization group flow,” PTEP 2018 no. 5, (2018) 053B02, arXiv:1802.07897 [hep-th].
[133] Y. Abe and M. Fukuma, “Gradient flow and the renormalization group,” PTEP 2018 no. 8, (2018) 083B02, arXiv:1805.12094 [hep-th].
[134] A. Carosso, A. Hasenfratz, and E. T. Neil, “Nonperturbative Renormalization of Operators in Near-Conformal Systems Using Gradient Flows,” Phys. Rev. Lett. 121 no. 20, (2018) 201601, arXiv:1806.01385 [hep-lat].
[135] H. Sonoda and H. Suzuki, “Derivation of a gradient flow from the exact renormalization group,” PTEP 2019 no. 3, (2019) 033B05, arXiv:1901.05169 [hep-th].
[136] A. Carosso, “Stochastic Renormalization Group and Gradient Flow,” JHEP 01 (2020) 172, arXiv:1904.13057 [hep-th].
[137] A. Carosso, A. Hasenfratz, and E. T. Neil, “Stochastic Renormalization Group and Gradient Flow in Scalar Field Theory,” PoS LATTICE2019 (2019) 287, arXiv:1912.01766 [hep-lat].
[138] M. Matsumoto, G. Tanaka, and A. Tsuchiya, “Renormalization group and diffusion equation,” arXiv:2011.14687 [hep-th].
論文の公開元へ
