×

zbMATH — the first resource for mathematics

A dimer ABC. (English) Zbl 1345.82006
The paper provides a review of results obtained for models of statistical physics based on the distribution of two different species of particles (hence the name of “dimer systems”) on sites of two-dimensional square lattices, so that two neighboring particles belong to different species. The system may be integrable. If the boundary conditions imposed on the system are periodic, the lattice is actually defined on a toroidal surface, and the analysis becomes tantamount to solving certain problems from the graph theory. The main issues comprised by the review are symmetry of the configurations, representation theory and resolution of singularities, and a combination of both of these topics.

MSC:
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
82B26 Phase transitions (general) in equilibrium statistical mechanics
97K30 Graph theory (educational aspects)
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Abouzaid
[2] Aspinwall · Zbl 1050.14032
[3] DOI: 10.1098/rspa.1958.0181 · Zbl 0135.21301 · doi:10.1098/rspa.1958.0181
[4] Baur
[5] DOI: 10.1016/j.jpaa.2007.03.009 · Zbl 1132.16017 · doi:10.1016/j.jpaa.2007.03.009
[6] DOI: 10.1017/S0017089512000080 · Zbl 1244.14042 · doi:10.1017/S0017089512000080
[7] DOI: 10.1016/j.jalgebra.2012.03.040 · Zbl 1263.14006 · doi:10.1016/j.jalgebra.2012.03.040
[8] DOI: 10.1007/s00209-012-1006-z · Zbl 1278.16017 · doi:10.1007/s00209-012-1006-z
[9] Bocklandt, ’Noncommutative mirror symmetry for punctured surfaces, Trans. Amer. Math. Soc. 368 pp 429– (2016) · Zbl 1383.16016 · doi:10.1090/tran/6375
[10] DOI: 10.1023/A:1002470302976 · Zbl 0994.18007 · doi:10.1023/A:1002470302976
[11] DOI: 10.1112/S0024609398004998 · Zbl 0937.18012 · doi:10.1112/S0024609398004998
[12] DOI: 10.1007/s002220100185 · Zbl 1085.14017 · doi:10.1007/s002220100185
[13] DOI: 10.4007/annals.2007.166.317 · Zbl 1137.18008 · doi:10.4007/annals.2007.166.317
[14] DOI: 10.1090/S0894-0347-01-00368-X · Zbl 0966.14028 · doi:10.1090/S0894-0347-01-00368-X
[15] Broomhead N. , Dimer models and Calabi–Yau algebras, vol. 211 (American Mathematical Society, Providence, RI, 2012). · Zbl 1237.14002
[16] Butti, R-charges from toric diagrams and the equivalence of a-maximization and z-minimization, J. High Energy Phys. 2005 pp 019– (2005) · doi:10.1088/1126-6708/2005/11/019
[17] DOI: 10.1006/jcta.1997.2799 · Zbl 0897.05063 · doi:10.1006/jcta.1997.2799
[18] DOI: 10.1215/S0012-7094-04-12422-4 · Zbl 1082.14009 · doi:10.1215/S0012-7094-04-12422-4
[19] DOI: 10.1016/j.jalgebra.2011.05.005 · Zbl 1250.14028 · doi:10.1016/j.jalgebra.2011.05.005
[20] DOI: 10.1007/s00029-008-0057-9 · Zbl 1204.16008 · doi:10.1007/s00029-008-0057-9
[21] DOI: 10.1090/S0894-0347-10-00662-4 · Zbl 1208.16017 · doi:10.1090/S0894-0347-10-00662-4
[22] DOI: 10.1016/S0021-9800(68)80072-9 · Zbl 0247.31003 · doi:10.1016/S0021-9800(68)80072-9
[23] DOI: 10.4310/ATMP.2008.v12.n3.a2 · Zbl 1144.81501 · doi:10.4310/ATMP.2008.v12.n3.a2
[24] DOI: 10.1103/PhysRev.124.1664 · Zbl 0105.22403 · doi:10.1103/PhysRev.124.1664
[25] Fock
[26] Fock V. V. Goncharov A. B. , ’Cluster ensembles quantization and the dilogarithm II: the intertwiner’, Algebra, Arithmetic, and Geometry, vol. I (eds Y. Tschinkel and Y. Zarhin) (Progress in Mathematics 269, Springer, Boston, 2009) 655–673. · Zbl 1225.53070
[27] DOI: 10.1090/S0894-0347-01-00385-X · Zbl 1021.16017 · doi:10.1090/S0894-0347-01-00385-X
[28] DOI: 10.1007/s00222-003-0302-y · Zbl 1054.17024 · doi:10.1007/s00222-003-0302-y
[29] Franco, Bipartite field theories: from D-brane probes to scattering amplitudes, J. High Energy Phys. 2012 pp 1– (2012) · Zbl 1397.81239 · doi:10.1007/JHEP11(2012)141
[30] Franco, Brane dimers and quiver gauge theories, J. High Energy Phys. 2006 pp 096– (2006) · doi:10.1088/1126-6708/2006/01/096
[31] Fulton W. , Introduction to toric varieties, vol. 131 (Princeton University Press, Princeton, 1993). · Zbl 0813.14039 · doi:10.1515/9781400882526
[32] Gelfand I. M. , Kapranov M. Zelevinsky A. , Discriminants, resultants, and multidimensional determinants (Birkhauser, Boston, MA, 1994). · Zbl 0827.14036 · doi:10.1007/978-0-8176-4771-1
[33] Ginzburg
[34] Goncharov, Dimers and cluster integrable systems, Ann. Sci. École Norm. Sup. 46 pp 747– (2013) · Zbl 1288.37025 · doi:10.24033/asens.2201
[35] Gross M. , Tropical geometry and mirror symmetry, vol. 114 (American Mathematical Society, Providence, RI, 2011). · Zbl 1215.14061 · doi:10.1090/cbms/114
[36] Gulotta, Properly ordered dimers, r-charges, and an efficient inverse algorithm, J. High Energy Phys. 2008 pp 014– (2008) · Zbl 1245.81091 · doi:10.1088/1126-6708/2008/10/014
[37] Hanany, Brane tilings and specular duality, J. High Energy Phys. 2012 pp 1– (2012) · Zbl 1397.81369 · doi:10.1007/JHEP08(2012)107
[38] Hille, Fourier–Mukai transforms, London Math. Soc. Lecture Note Ser. 332 pp 147– (2007)
[39] Huybrechts D. , Fourier–Mukai transforms (Oxford University Press, USA, 2006). · Zbl 1095.14002 · doi:10.1093/acprof:oso/9780199296866.001.0001
[40] Igusa K.. , Higher Franz–Reidemeister torsion, vol. 31 (American Mathematical Society, Providence, RI, 2002). · doi:10.1090/amsip/031
[41] DOI: 10.1016/S0550-3213(03)00459-0 · Zbl 1059.81602 · doi:10.1016/S0550-3213(03)00459-0
[42] Ishii
[43] Ishii
[44] Ishii A. Ueda K. , A note on consistency conditions on dimer models, RIMS Kôkyûroku Bessatsu B24 (Research Institute for Mathematical Sciences, Kyoto, Japan, 2011) 143–164. · Zbl 1270.16012
[45] DOI: 10.1007/BF03026748 · Zbl 0876.14017 · doi:10.1007/BF03026748
[46] DOI: 10.3792/pjaa.72.135 · Zbl 0881.14002 · doi:10.3792/pjaa.72.135
[47] DOI: 10.1007/s00222-013-0491-y · Zbl 1308.14007 · doi:10.1007/s00222-013-0491-y
[48] Kasteleyn P. W. , ’Graph theory and crystal physics’, Graph Theory and Theoretical Physics (ed. F. Harary) (Academic Press, London, 1967) 43–110. · Zbl 0205.28402
[49] Kato, Zonotopes and four-dimensional superconformal field theories, J. High Energy Phys. 2007 pp 037– (2007) · doi:10.1088/1126-6708/2007/06/037
[50] Keller, Derived categories and tilting, London Math. Soc. Lecture Note Ser. 332 pp 49– (2007)
[51] Kenyon
[52] DOI: 10.1215/S0012-7094-06-13134-4 · Zbl 1100.14047 · doi:10.1215/S0012-7094-06-13134-4
[53] Kenyon R. , Okounkov A. Sheffield S. , ’Dimers and amoebae’, Ann. of Math. (2006) 1019–1056. · Zbl 1154.82007 · doi:10.4007/annals.2006.163.1019
[54] DOI: 10.1090/S0002-9947-04-03545-7 · Zbl 1062.05045 · doi:10.1090/S0002-9947-04-03545-7
[55] Kenyon, Trees and matchings, J. Combin. 7 pp R25– (2001)
[56] DOI: 10.1093/qmath/45.4.515 · Zbl 0837.16005 · doi:10.1093/qmath/45.4.515
[57] Kollár J. Matsusaka T. , ’Flip and flop’, Proc. ICM, Kyoto (1990) 709–714.
[58] Kontsevich · Zbl 0980.57006
[59] Le Bruyn L. , Noncommutative geometry and Cayley–Smooth orders (CRC Press, Boca Raton, 2007). · Zbl 1131.14006 · doi:10.1201/9781420064230
[60] Lee, Comments on anomalies and charges of toric-quiver duals, J. High Energy Phys. 2006 pp 068– (2006) · Zbl 1226.81209 · doi:10.1088/1126-6708/2006/03/068
[61] DOI: 10.1007/s00220-006-0087-0 · Zbl 1190.53041 · doi:10.1007/s00220-006-0087-0
[62] DOI: 10.1007/s002200000348 · Zbl 1043.82005 · doi:10.1007/s002200000348
[63] DOI: 10.2307/121119 · Zbl 1073.14555 · doi:10.2307/121119
[64] Mikhalkin G. , ’Amoebas of algebraic varieties and tropical geometry’, Differ. Faces Geom. (Springer, New York, 2004) 257–300. · Zbl 1072.14013
[65] Mikhalkin
[66] DOI: 10.1155/S107379280100023X · Zbl 0994.14032 · doi:10.1155/S107379280100023X
[67] Morita, Duality for modules and its applications to the theory of rings with minimum condition, Sci. Rep. Tokyo Kyoiku Daigaku. Sect. A 6 pp 83– (1958)
[68] Mozgovoy
[69] DOI: 10.1016/j.aim.2009.10.001 · Zbl 1191.14008 · doi:10.1016/j.aim.2009.10.001
[70] Nakamura, Hilbert schemes of abelian group orbits, J. Algebraic Geom. 10 pp 757– (2001)
[71] Oda, Convex bodies and algebraic geometry: an introduction to the theory of toric varieties. translated from the Japanese, Ergeb. Math. Grenzgeb. 3 pp 15– (1988) · Zbl 0628.52002
[72] Postnikov · Zbl 1180.55004
[73] Reid, Minimal models of canonical 3-folds, Adv. Stud. Pure Math. 1 pp 131– (1983) · Zbl 0558.14028
[74] Rickard, Morita theory for derived categories, J. London Math. Soc. (2) 39 pp 436– (1989) · Zbl 0642.16034 · doi:10.1112/jlms/s2-39.3.436
[75] Sheffield
[76] Stienstra
[77] DOI: 10.1016/0550-3213(96)00434-8 · Zbl 0896.14024 · doi:10.1016/0550-3213(96)00434-8
[78] DOI: 10.1080/14786436108243366 · Zbl 0126.25102 · doi:10.1080/14786436108243366
[79] DOI: 10.1006/jctb.1999.1941 · Zbl 1025.05052 · doi:10.1006/jctb.1999.1941
[80] Thurston
[81] Ueda, Homological mirror symmetry for toric orbifolds of toric del pezzo surfaces, J. Reine Angew. Math. 2013 pp 1– (2013) · Zbl 1285.53076 · doi:10.1515/crelle.2012.031
[82] Van den Bergh M. , ’Non-commutative crepant resolutions’, The legacy of Niels Henrik Abel (Springer, Berlin, 2004) 749–770. · Zbl 1082.14005 · doi:10.1007/978-3-642-18908-1_26
[83] DOI: 10.1007/s00029-014-0166-6 · Zbl 1378.16016 · doi:10.1007/s00029-014-0166-6
[84] Viro O. , ’Dequantization of real algebraic geometry on logarithmic paper’, European Congress of Mathematics (Springer, Barcelona, 2000) 135–146. · Zbl 1024.14026
[85] DOI: 10.1016/j.jalgebra.2008.11.012 · Zbl 1183.16015 · doi:10.1016/j.jalgebra.2008.11.012
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.