Mathematical structures behind supersymmetric dualities

Ilmar Gahramanov

Archivum Mathematicum (2015)

  • Volume: 051, Issue: 5, page 273-286
  • ISSN: 0044-8753

Abstract

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The purpose of these notes is to give a short survey of an interesting connection between partition functions of supersymmetric gauge theories and hypergeometric functions and to present the recent progress in this direction.

How to cite

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Gahramanov, Ilmar. "Mathematical structures behind supersymmetric dualities." Archivum Mathematicum 051.5 (2015): 273-286. <http://eudml.org/doc/276206>.

@article{Gahramanov2015,
abstract = {The purpose of these notes is to give a short survey of an interesting connection between partition functions of supersymmetric gauge theories and hypergeometric functions and to present the recent progress in this direction.},
author = {Gahramanov, Ilmar},
journal = {Archivum Mathematicum},
keywords = {elliptic hypergeometric function; hypergeometric series on root systems; basic hypergeometric integrals; hyperbolic hypergeometric integrals; superconformal index; supersymmetric duality; Seiberg duality; mirror symmetry; elliptic hypergeometric function; hypergeometric series on root systems; basic hypergeometric integrals; hyperbolic hypergeometric integrals; superconformal index; supersymmetric duality; Seiberg duality; mirror symmetry},
language = {eng},
number = {5},
pages = {273-286},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Mathematical structures behind supersymmetric dualities},
url = {http://eudml.org/doc/276206},
volume = {051},
year = {2015},
}

TY - JOUR
AU - Gahramanov, Ilmar
TI - Mathematical structures behind supersymmetric dualities
JO - Archivum Mathematicum
PY - 2015
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 051
IS - 5
SP - 273
EP - 286
AB - The purpose of these notes is to give a short survey of an interesting connection between partition functions of supersymmetric gauge theories and hypergeometric functions and to present the recent progress in this direction.
LA - eng
KW - elliptic hypergeometric function; hypergeometric series on root systems; basic hypergeometric integrals; hyperbolic hypergeometric integrals; superconformal index; supersymmetric duality; Seiberg duality; mirror symmetry; elliptic hypergeometric function; hypergeometric series on root systems; basic hypergeometric integrals; hyperbolic hypergeometric integrals; superconformal index; supersymmetric duality; Seiberg duality; mirror symmetry
UR - http://eudml.org/doc/276206
ER -

References

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  1. Aharony, O., Hanany, A., Intriligator, K.A., Seiberg, N., Strassler, M., 10.1016/S0550-3213(97)00323-4, Nuclear Phys. B 499 (1997), 67–99, http://arxiv.org/abs/hep-th/9703110, arXiv:hep-th/9703110 [hep-th]. (1997) Zbl0934.81063MR1468698DOI10.1016/S0550-3213(97)00323-4
  2. Al-Salam, W.A., Ismail, M.E.H., 10.2140/pjm.1988.135.209, Pacific J. Math. 135 2) (1988), 209–221, http://dx.doi.org/10.2140/pjm.1988.135.209. (1988) Zbl0658.33002MR0968609DOI10.2140/pjm.1988.135.209
  3. Amariti, A., Klare, C., A journey to 3d: exact relations for adjoint SQCD from dimensional reduction, http://arxiv.org/abs/1106.2484, arXiv:1409.8623 [hep-th]. 
  4. Askey, R., 10.2307/2321202, Amer. Math. Monthly 87 (5) (1980), 346–359, http://dx.doi.org/10.2307/2321202. (1980) Zbl0437.33001MR0567718DOI10.2307/2321202
  5. Askey, R., Wilson, J.A., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, Mem. Amer. Math. Soc. 54 (319) (1985), iv+55 pp. (1985) Zbl0572.33012MR0783216
  6. Benini, F., Cremonesi, S., 10.1007/s00220-014-2112-z, Comm. Math. Phys. 334 (3) (2015), 62 pages + 16 of appendices, http://arxiv.org/abs/1206.2356, arXiv:1206.2356 [hep-th]. (2015) Zbl1308.81131MR3312441DOI10.1007/s00220-014-2112-z
  7. Bhattacharya, J., Minwalla, S., 10.1088/1126-6708/2009/01/014, JHEP 0901 (014) (2009), 13 pp., http://dx.doi.org/10.1088/1126-6708/2009/01/014. (2009) MR2480370DOI10.1088/1126-6708/2009/01/014
  8. de Boer, J., Hori, K., Oz, Y., Yin, Z., Branes and mirror symmetry in N = 2 supersymmetric gauge theories in three-dimensions, Nuclear Phys. B 502 (1997), 107–124, http://arxiv.org/abs/hep-th/9702154, arXiv:hep-th/9702154 [hep-th]. (1997) MR1477860
  9. Dimofte, T., Gaiotto, D., An E7 Surprise, JHEP 1210 (129) (2012), 37pp., arXiv:1209.1404 [hep-th]. (2012) 
  10. Dolan, F., Osborn, H., Applications of the superconformal index for protected operators and q -hypergeometric identities to N = 1 dual theories, Nuclear Phys. B 818 (2009), 137–178, http://arxiv.org/abs/0801.4947, arXiv:0801.4947 [hep-th]. (2009) MR2518083
  11. Dolan, F., Spiridonov, V., Vartanov, G., 10.1016/j.physletb.2011.09.007, Phys. Lett. B 704 (2011), 234–241, http://arxiv.org/abs/1104.1787, arXiv:1104.1787 [hep-th]. (2011) MR2843616DOI10.1016/j.physletb.2011.09.007
  12. Doroud, N., Gomis, J., Le Floch, B., Lee, S., Exact Results in D=2 Supersymmetric Gauge Theories, JHEP 1305 (093) (2013), http://arxiv.org/abs/1206.2606, arXiv:1206.2606 [hep-th]. (2013) Zbl1342.81573MR3080568
  13. Felder, G., Varchenko, A., 10.1006/aima.2000.1951, Adv. Math 156 (1) (2000), 44–76, http://dx.doi.org/http://dx.doi.org/10.1006/aima.2000.1951, http://arxiv.org/abs/math/9907061, arXiv:math/9907061. (2000) MR1800253DOI10.1006/aima.2000.1951
  14. Frenkel, I.B., Turaev, V.G., 10.1007/978-1-4612-4122-5_9, The Arnold-Gelfand Mathematical Seminars, 1997, http://dx.doi.org/10.1007/978-1-4612-4122-5_9, pp. 171–204. (1997) Zbl0974.17016MR1429892DOI10.1007/978-1-4612-4122-5_9
  15. Gadde, A., Yan, W., Reducing the 4d index to the S 3 partition function, JHEP 1212 (003) (2012), 12 pp., http://arxiv.org/abs/1104.2592, arXiv:1104.2592 [hep-th]. (2012) MR3045303
  16. Gahramanov, I., Rosengren, H., [unknown] 
  17. Gahramanov, I., Rosengren, H., Integral pentagon relations for 3d superconformal indices, http://arxiv.org/abs/1412.2926, arXiv:1412.2926 [hep-th]. 
  18. Gahramanov, I., Rosengren, H., 10.1007/JHEP11(2013)128, JHEP 1311 (2013), http://dx.doi.org/10.1007/JHEP11(2013)128, http://arxiv.org/abs/1309.2195, arXiv:1309.2195 [hep-th]. (2013) DOI10.1007/JHEP11(2013)128
  19. Gahramanov, I., Spiridonov, V.P., 10.1007/JHEP08(2015)040, JHEP 1508 040 (2015), http://dx.doi.org/10.1007/JHEP08(2015)040. (2015) MR3402125DOI10.1007/JHEP08(2015)040
  20. Gahramanov, I., Vartanov, G., 10.1088/1751-8113/46/28/285403, J. Phys. A 46 (2013), 285403, http://dx.doi.org/10.1088/1751-8113/46/28/285403, http://arxiv.org/abs/1303.1443, arXiv:1303.1443 [hep-th]. (2013) MR3083462DOI10.1088/1751-8113/46/28/285403
  21. Gahramanov, I.B., Vartanov, G.S., 10.1142/9789814449243_0076, XVIIth Intern. Cong. Math. Phys. (2013), 695–703, http://dx.doi.org/10.1142/9789814449243_0076, http://arxiv.org/abs/1310.8507, arXiv:1310.8507 [hep-th]. (2013) MR3204521DOI10.1142/9789814449243_0076
  22. Gasper, G., Rahman, M., Basic hypergeometric series, second ed., Cambridge University Press, 2004. (2004) Zbl1129.33005MR2128719
  23. Imamura, Y., Relation between the 4d superconformal index and the S 3 partition function, JHEP 1109 (133) (2011), 20 pp., http://arxiv.org/abs/1104.4482, arXiv:1104.4482 [hep-th]. (2011) MR2889831
  24. Imamura, Y., Yokoyama, S., Index for three dimensional superconformal field theories with general R-charge assignments, JHEP 1104 (2011), 22 pp., http://arxiv.org/abs/1101.0557, arXiv:1101.0557 [hep-th]. (2011) Zbl1250.81107MR2833291
  25. Intriligator, K.A., New RG fixed points and duality in supersymmetric SP(N(c)) and SO(N(c)) gauge theories, Nuclear Phys. B 448 (1995), 187–198, http://arxiv.org/abs/hep-th/9505051, arXiv:hep-th/9505051 [hep-th]. (1995) Zbl1009.81573MR1352405
  26. Intriligator, K.A., Seiberg, N., 10.1016/0370-2693(96)01088-X, Phys. Lett. B 387 (1996), 513–519, http://arxiv.org/abs/hep-th/9607207, arXiv:hep-th/9607207 [hep-th]. (1996) MR1413696DOI10.1016/0370-2693(96)01088-X
  27. Kapustin, A., Willet, B., Generalized superconformal index for three dimensional field theories, http://arxiv.org/abs/1106.2484, arXiv:1106.2484 [hep-th]. 
  28. Kim, S., 10.1016/j.nuclphysb.2009.06.025, Nuclear Phys. B 821 (2009), 241–284, http://dx.doi.org/10.1016/j.nuclphysb.2012.07.015, 10.1016/j.nuclphysb.2009.06.025. (2009) MR2562335DOI10.1016/j.nuclphysb.2009.06.025
  29. Kinney, J., Maldacena, J.M., Minwalla, S., Raju, S., 10.1007/s00220-007-0258-7, Comm. Math. Phys. 275 (2007), 209–254, http://dx.doi.org/10.1007/s00220-007-0258-7, http://arxiv.org/abs/hep-th/0510251, arXiv:hep-th/0510251 [hep-th]. (2007) Zbl1122.81070MR2335774DOI10.1007/s00220-007-0258-7
  30. Krattenthaler, C., Spiridonov, V., Vartanov, G., 10.1007/JHEP06(2011)008, JHEP 1106 (008) (2011), http://dx.doi.org/10.1007/JHEP06(2011)008, http://arxiv.org/abs/1103.4075, arXiv:1103.4075 [hep-th]. (2011) Zbl1298.81186MR2870861DOI10.1007/JHEP06(2011)008
  31. Narukawa, A., 10.1016/j.aim.2003.11.009, Adv. Math. 189 (2) (2004), 247–267, http://arxiv.org/abs/math/0306164, arXiv:math/0306164. (2004) Zbl1077.33024MR2101221DOI10.1016/j.aim.2003.11.009
  32. Nassrallah, B., Rahman, M., 10.1137/0516014, SIAM J. Math. Anal. 16 (1) (1985), 186–197, http://dx.doi.org/10.1137/0516014. (1985) Zbl0564.33009MR0772878DOI10.1137/0516014
  33. Nishizawa, M., 10.1088/0305-4470/34/36/320, J. Phys. A 34 (36) (2001), 7411–7421. (2001) Zbl0993.33016MR1862776DOI10.1088/0305-4470/34/36/320
  34. Pestun, V., 10.1007/s00220-012-1485-0, Comm. Math. Phys. 313 (71) (2012), 63 pp., http://arxiv.org/abs/0712.2824, arXiv:0712.2824 [hep-th]. (2012) Zbl1257.81056MR2928219DOI10.1007/s00220-012-1485-0
  35. Rahman, M., 10.4153/CJM-1986-030-6, Canad. J. Math. 38 3) (1986), 605–618, http://dx.doi.org/10.4153/CJM-1986-030-6. (1986) MR0845667DOI10.4153/CJM-1986-030-6
  36. Rains, E.M., 10.4007/annals.2010.171.169, Ann. Math. 171 (1) (2010), 169–243, http://dx.doi.org/10.4007/annals.2010.171.169, http://arxiv.org/abs/math/0309252, arXiv:math/0309252. (2010) Zbl1209.33014MR2630038DOI10.4007/annals.2010.171.169
  37. Romelsberger, C., Calculating the superconformal index and Seiberg duality, http://arxiv.org/abs/0707.3702, arXiv:0707.3702 [hep-th]. 
  38. Romelsberger, C., 10.1016/j.nuclphysb.2006.03.037, Nuclear Phys. B 747 (2006), 329–353, http://dx.doi.org/10.1016/j.nuclphysb.2006.03.037, http://arxiv.org/abs/hep-th/0510060, arXiv:hep-th/0510060 [hep-th]. (2006) MR2241553DOI10.1016/j.nuclphysb.2006.03.037
  39. Rosengren, H., 10.1016/S0001-8708(03)00071-9, Adv. Math. 181 (2) (2004), 417–447, http://dx.doi.org/10.1016/S0001-8708(03)00071-9, http://arxiv.org/abs/math/0207046, arXiv:math/0207046. (2004) Zbl1066.33017MR2026866DOI10.1016/S0001-8708(03)00071-9
  40. Rosengren, H., 10.1093/imrn/rnq184, Int. Math. Res. Not. 2011 (2011), 2861–2920, http://dx.doi.org/10.1093/imrn/rnq184, http://arxiv.org/abs/1003.3730, arXiv:1003.3730. (2011) Zbl1243.17010MR2817682DOI10.1093/imrn/rnq184
  41. Ruijsenaars, S.N.M., 10.1063/1.531809, J. Math. Phys. 38 (2) (1997), 1069–1146, http://dx.doi.org/10.1063/1.531809. (1997) Zbl0877.39002MR1434226DOI10.1063/1.531809
  42. Seiberg, N., 10.1016/0550-3213(94)00023-8, Nuclear Phys. B 435 (1995), 129–146, http://dx.doi.org/10.1016/0550-3213(94)00023-8. (1995) Zbl1020.81912MR1314365DOI10.1016/0550-3213(94)00023-8
  43. Spiridonov, V., Modified elliptic gamma functions and 6d superconformal indices, http://arxiv.org/abs/1211.2703, arXiv:1211.2703 [hep-th]. Zbl1303.81195MR3177989
  44. Spiridonov, V., Vartanov, G., Elliptic hypergeometry of supersymmetric dualities II. Orthogonal groups, knots, and vortices, http://arxiv.org/abs/1107.5788, arXiv:1107.5788 [hep-th]. Zbl1285.81064MR3148094
  45. Spiridonov, V., Vartanov, G., 10.1016/j.nuclphysb.2009.08.022, Nuclear Phys. B 824 (2010), 192–216, http://dx.doi.org/10.1016/j.nuclphysb.2009.08.022, http://arxiv.org/abs/0811.1909, arXiv:0811.1909 [hep-th]. (2010) MR2556156DOI10.1016/j.nuclphysb.2009.08.022
  46. Spiridonov, V., Vartanov, G., 10.1103/PhysRevLett.105.061603, Phys. Rev. Lett. 105 (2010), 061603, http://dx.doi.org/10.1103/PhysRevLett.105.061603, http://arxiv.org/abs/1003.6109, arXiv:1003.6109 [hep-th]. (2010) MR2673041DOI10.1103/PhysRevLett.105.061603
  47. Spiridonov, V., Vartanov, G., 10.1007/s00220-011-1218-9, Comm. Math. Phys. 304 (2011), 797–874, http://dx.doi.org/10.1007/s00220-011-1218-9. (2011) Zbl1225.81137MR2794548DOI10.1007/s00220-011-1218-9
  48. Spiridonov, V., Vartanov, G., 10.1007/s00220-011-1218-9, JHEP 1206 (016) (2012), 18 pp., http://dx.doi.org/10.1007/s00220-011-1218-9. (2012) MR3006892DOI10.1007/s00220-011-1218-9
  49. Spiridonov, V.P., 10.1070/RM2001v056n01ABEH000374, Russian Math. Surveys 56 (1) (2001), 185–186, http://dx.doi.org/10.1070/RM2001v056n01ABEH000374. (2001) Zbl0997.33009MR1846786DOI10.1070/RM2001v056n01ABEH000374
  50. Spiridonov, V.P., 10.1090/S1061-0022-04-00839-8, Algebra i Analiz 15 (6) (2003), 161–215, http://dx.doi.org/10.1090/S1061-0022-04-00839-8, http://arxiv.org/abs/arXiv:math/0303205, arXiv:math/0303205. (2003) MR2044635DOI10.1090/S1061-0022-04-00839-8
  51. Spiridonov, V.P., 10.1070/RM2008v063n03ABEH004533, Russian Math. Surveys 63 (3) (2008), 405–472, http://dx.doi.org/10.1070/RM2008v063n03ABEH004533. (2008) Zbl1173.33017MR2479997DOI10.1070/RM2008v063n03ABEH004533
  52. Stokman, J.V., 10.1016/j.aim.2003.12.003, Adv. Math. 190 (1) (2005), 119–160, http://arxiv.org/abs/math/0303178, arXiv:math/0303178. (2005) Zbl1072.33012MR2104907DOI10.1016/j.aim.2003.12.003
  53. Tizzano, L., Winding, J., Multiple sine, multiple elliptic gamma functions and rational cones, http://arxiv.org/abs/1502.05996, arXiv:1502.05996 [math.CA]. 
  54. van de Bult, F.J., Hyperbolic hypergeometric functions, Ph.D. thesis, University of Amsterdam, 2007. (2007) 
  55. van de Bult, F.J., 10.1007/s11139-010-9273-y, Ramanujan J. 25 (1) (2011), 1–20, http://dx.doi.org/10.1007/s11139-010-9273-y, http://arxiv.org/abs/0909.4793, arXiv:0909.4793. (2011) MR2787288DOI10.1007/s11139-010-9273-y
  56. van de Bult, F.J., Rains, E.M., Limits of multivariate elliptic beta integrals and related bilinear forms, http://arxiv.org/abs/1110.1460, arXiv:1110.1460. 
  57. van de Bult, F.J., Rains, E.M., Limits of multivariate elliptic hypergeometric biorthogonal functions, http://arxiv.org/abs/1110.1458, arXiv:1110.1458. 
  58. van de Bult, F.J., Rains, E.M., 10.1016/j.jat.2014.06.009, J. Approx. Theory 193 (0) (2015), 128–163, http://dx.doi.org/http://dx.doi.org/10.1016/j.jat.2014.06.009, http://arxiv.org/abs/1110.1456, arXiv:1110.1456. (2015) MR3324567DOI10.1016/j.jat.2014.06.009
  59. Witten, E., 10.1016/0550-3213(82)90071-2, Nuclear Phys. B 202 (1982), 253–316, http://dx.doi.org/10.1016/0550-3213(82)90071-2. (1982) MR0668987DOI10.1016/0550-3213(82)90071-2
  60. Yamazaki, M., 10.1007/s11232-013-0012-6, Theoret. and Math. Phys. 174 (1) (2013), 154–166, http://dx.doi.org/10.1007/s11232-013-0012-6. (2013) Zbl1280.81103MR3172959DOI10.1007/s11232-013-0012-6
  61. Zwiebel, B.I., 10.1007/JHEP01(2012)116, JHEP 1201 (116) (2012), 31 pp., http://dx.doi.org/10.1007/JHEP01(2012)116, http://arxiv.org/abs/1111.1773, arXiv:1111.1773 [hep-th]. (2012) Zbl1306.81135DOI10.1007/JHEP01(2012)116

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