| Title:
|
Mathematical structures behind supersymmetric dualities (English) |
| Author:
|
Gahramanov, Ilmar |
| Language:
|
English |
| Journal:
|
Archivum Mathematicum |
| ISSN:
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0044-8753 (print) |
| ISSN:
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1212-5059 (online) |
| Volume:
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51 |
| Issue:
|
5 |
| Year:
|
2015 |
| Pages:
|
273-286 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
elliptic hypergeometric function |
| Keyword:
|
hypergeometric series on root systems |
| Keyword:
|
basic hypergeometric integrals |
| Keyword:
|
hyperbolic hypergeometric integrals |
| Keyword:
|
superconformal index |
| Keyword:
|
supersymmetric duality |
| Keyword:
|
Seiberg duality |
| Keyword:
|
mirror symmetry |
| MSC:
|
33D60 |
| MSC:
|
33D90 |
| MSC:
|
33E20 |
| MSC:
|
39A13 |
| MSC:
|
81T60 |
| idZBL:
|
Zbl 06537730 |
| idMR:
|
MR3449108 |
| DOI:
|
10.5817/AM2015-5-273 |
| . |
| Date available:
|
2016-01-11T10:07:38Z |
| Last updated:
|
2017-02-02 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/144770 |
| . |
| Reference:
|
[1] Aharony, O., Hanany, A., Intriligator, K.A., Seiberg, N., Strassler, M.: Aspects of $N=2$ supersymmetric gauge theories in three-dimensions.Nuclear Phys. B 499 (1997), 67–99, http://arxiv.org/abs/hep-th/9703110, arXiv:hep-th/9703110 [hep-th]. Zbl 0934.81063, MR 1468698, 10.1016/S0550-3213(97)00323-4 |
| Reference:
|
[2] Al-Salam, W.A., Ismail, M.E.H.: $q$-beta integrals and the $q$-Hermite polynomials.Pacific J. Math. 135 2) (1988), 209–221, http://dx.doi.org/10.2140/pjm.1988.135.209. Zbl 0658.33002, MR 0968609, 10.2140/pjm.1988.135.209 |
| Reference:
|
[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]. |
| Reference:
|
[4] Askey, R.: Ramanujan’s extensions of the gamma and beta functions.Amer. Math. Monthly 87 (5) (1980), 346–359, http://dx.doi.org/10.2307/2321202. Zbl 0437.33001, MR 0567718, 10.2307/2321202 |
| Reference:
|
[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. Zbl 0572.33012, MR 0783216 |
| Reference:
|
[6] Benini, F., Cremonesi, S.: Partition functions of ${\mathcal{N}=(2,2)}$ gauge theories on S$^{2}$ and vortices.Comm. Math. Phys. 334 (3) (2015), 62 pages + 16 of appendices, http://arxiv.org/abs/1206.2356, arXiv:1206.2356 [hep-th]. Zbl 1308.81131, MR 3312441, 10.1007/s00220-014-2112-z |
| Reference:
|
[7] Bhattacharya, J., Minwalla, S.: Superconformal indices for $N = 6$ Chern Simons theories.JHEP 0901 (014) (2009), 13 pp., http://dx.doi.org/10.1088/1126-6708/2009/01/014. MR 2480370, 10.1088/1126-6708/2009/01/014 |
| Reference:
|
[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]. MR 1477860 |
| Reference:
|
[9] Dimofte, T., Gaiotto, D.: An E7 Surprise.JHEP 1210 (129) (2012), 37pp., arXiv:1209.1404 [hep-th]. |
| Reference:
|
[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]. MR 2518083 |
| Reference:
|
[11] Dolan, F., Spiridonov, V., Vartanov, G.: From 4d superconformal indices to 3d partition functions.Phys. Lett. B 704 (2011), 234–241, http://arxiv.org/abs/1104.1787, arXiv:1104.1787 [hep-th]. MR 2843616, 10.1016/j.physletb.2011.09.007 |
| Reference:
|
[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]. Zbl 1342.81573, MR 3080568 |
| Reference:
|
[13] Felder, G., Varchenko, A.: The elliptic gamma function and $SL(3,{\mathbf{Z}})\ltimes{\mathbf{Z}}^3$.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. MR 1800253, 10.1006/aima.2000.1951 |
| Reference:
|
[14] Frenkel, I.B., Turaev, V.G.: Elliptic solutions of the Yang-Baxter equation and modular hypergeometric functions.The Arnold-Gelfand Mathematical Seminars, 1997, http://dx.doi.org/10.1007/978-1-4612-4122-5_9, pp. 171–204. Zbl 0974.17016, MR 1429892, 10.1007/978-1-4612-4122-5_9 |
| Reference:
|
[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]. MR 3045303 |
| Reference:
|
[16] Gahramanov, I., Rosengren, H.: . |
| Reference:
|
[17] Gahramanov, I., Rosengren, H.: Integral pentagon relations for 3d superconformal indices.http://arxiv.org/abs/1412.2926, arXiv:1412.2926 [hep-th]. |
| Reference:
|
[18] Gahramanov, I., Rosengren, H.: A new pentagon identity for the tetrahedron index.JHEP 1311 (2013), http://dx.doi.org/10.1007/JHEP11(2013)128, http://arxiv.org/abs/1309.2195, arXiv:1309.2195 [hep-th]. 10.1007/JHEP11(2013)128 |
| Reference:
|
[19] Gahramanov, I., Spiridonov, V.P.: The star-triangle relation and 3d superconformal indices.JHEP 1508 040 (2015), http://dx.doi.org/10.1007/JHEP08(2015)040. MR 3402125, 10.1007/JHEP08(2015)040 |
| Reference:
|
[20] Gahramanov, I., Vartanov, G.: Extended global symmetries for 4D $N = 1$ SQCD theories.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]. MR 3083462, 10.1088/1751-8113/46/28/285403 |
| Reference:
|
[21] Gahramanov, I.B., Vartanov, G.S.: Superconformal indices and partition functions for supersymmetric field theories.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]. MR 3204521, 10.1142/9789814449243_0076 |
| Reference:
|
[22] Gasper, G., Rahman, M.: Basic hypergeometric series.second ed., Cambridge University Press, 2004. Zbl 1129.33005, MR 2128719 |
| Reference:
|
[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]. MR 2889831 |
| Reference:
|
[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]. Zbl 1250.81107, MR 2833291 |
| Reference:
|
[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]. Zbl 1009.81573, MR 1352405 |
| Reference:
|
[26] Intriligator, K.A., Seiberg, N.: Mirror symmetry in three-dimensional gauge theories.Phys. Lett. B 387 (1996), 513–519, http://arxiv.org/abs/hep-th/9607207, arXiv:hep-th/9607207 [hep-th]. MR 1413696, 10.1016/0370-2693(96)01088-X |
| Reference:
|
[27] Kapustin, A., Willet, B.: Generalized superconformal index for three dimensional field theories.http://arxiv.org/abs/1106.2484, arXiv:1106.2484 [hep-th]. |
| Reference:
|
[28] Kim, S.: The complete superconformal index for $N=6$ Chern-Simons theory.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. MR 2562335, 10.1016/j.nuclphysb.2009.06.025 |
| Reference:
|
[29] Kinney, J., Maldacena, J.M., Minwalla, S., Raju, S.: An index for 4 dimensional super conformal theories.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]. Zbl 1122.81070, MR 2335774, 10.1007/s00220-007-0258-7 |
| Reference:
|
[30] Krattenthaler, C., Spiridonov, V., Vartanov, G.: Superconformal indices of three-dimensional theories related by mirror symmetry.JHEP 1106 (008) (2011), http://dx.doi.org/10.1007/JHEP06(2011)008, http://arxiv.org/abs/1103.4075, arXiv:1103.4075 [hep-th]. Zbl 1298.81186, MR 2870861, 10.1007/JHEP06(2011)008 |
| Reference:
|
[31] Narukawa, A.: The modular properties and the integral representations of the multiple elliptic gamma functions.Adv. Math. 189 (2) (2004), 247–267, http://arxiv.org/abs/math/0306164, arXiv:math/0306164. Zbl 1077.33024, MR 2101221, 10.1016/j.aim.2003.11.009 |
| Reference:
|
[32] Nassrallah, B., Rahman, M.: Projection formulas, a reproducing Kernel and a generating function for $q$-Wilson polynomials.SIAM J. Math. Anal. 16 (1) (1985), 186–197, http://dx.doi.org/10.1137/0516014. Zbl 0564.33009, MR 0772878, 10.1137/0516014 |
| Reference:
|
[33] Nishizawa, M.: An elliptic analogue of the multiple gamma function.J. Phys. A 34 (36) (2001), 7411–7421. Zbl 0993.33016, MR 1862776, 10.1088/0305-4470/34/36/320 |
| Reference:
|
[34] Pestun, V.: Localization of gauge theory on a four-sphere and supersymmetric Wilson loops.Comm. Math. Phys. 313 (71) (2012), 63 pp., http://arxiv.org/abs/0712.2824, arXiv:0712.2824 [hep-th]. Zbl 1257.81056, MR 2928219, 10.1007/s00220-012-1485-0 |
| Reference:
|
[35] Rahman, M.: An integral representation of a $_{10}\varphi _9$ and continuous bi-orthogonal $_{10}\varphi _9$.Canad. J. Math. 38 3) (1986), 605–618, http://dx.doi.org/10.4153/CJM-1986-030-6. MR 0845667, 10.4153/CJM-1986-030-6 |
| Reference:
|
[36] Rains, E.M.: Transformations of elliptic hypergeometric integrals.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. Zbl 1209.33014, MR 2630038, 10.4007/annals.2010.171.169 |
| Reference:
|
[37] Romelsberger, C.: Calculating the superconformal index and Seiberg duality.http://arxiv.org/abs/0707.3702, arXiv:0707.3702 [hep-th]. |
| Reference:
|
[38] Romelsberger, C.: Counting chiral primaries in $N = 1$, $d=4$ superconformal field theories.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]. MR 2241553, 10.1016/j.nuclphysb.2006.03.037 |
| Reference:
|
[39] Rosengren, H.: Elliptic hypergeometric series on root systems.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. Zbl 1066.33017, MR 2026866, 10.1016/S0001-8708(03)00071-9 |
| Reference:
|
[40] Rosengren, H.: Felder’s elliptic quantum group and elliptic hypergeometric series on the root system $A_n$.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. Zbl 1243.17010, MR 2817682, 10.1093/imrn/rnq184 |
| Reference:
|
[41] Ruijsenaars, S.N.M.: First order analytic difference equations and integrable quantum systems.J. Math. Phys. 38 (2) (1997), 1069–1146, http://dx.doi.org/10.1063/1.531809. Zbl 0877.39002, MR 1434226, 10.1063/1.531809 |
| Reference:
|
[42] Seiberg, N.: Electric-magnetic duality in supersymmetric non-Abelian gauge theories.Nuclear Phys. B 435 (1995), 129–146, http://dx.doi.org/10.1016/0550-3213(94)00023-8. Zbl 1020.81912, MR 1314365, 10.1016/0550-3213(94)00023-8 |
| Reference:
|
[43] Spiridonov, V.: Modified elliptic gamma functions and 6d superconformal indices.http://arxiv.org/abs/1211.2703, arXiv:1211.2703 [hep-th]. Zbl 1303.81195, MR 3177989 |
| Reference:
|
[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]. Zbl 1285.81064, MR 3148094 |
| Reference:
|
[45] Spiridonov, V., Vartanov, G.: Superconformal indices for $N = 1$ theories with multiple duals.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]. MR 2556156, 10.1016/j.nuclphysb.2009.08.022 |
| Reference:
|
[46] Spiridonov, V., Vartanov, G.: Supersymmetric dualities beyond the conformal window.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]. MR 2673041, 10.1103/PhysRevLett.105.061603 |
| Reference:
|
[47] Spiridonov, V., Vartanov, G.: Elliptic hypergeometry of supersymmetric dualities.Comm. Math. Phys. 304 (2011), 797–874, http://dx.doi.org/10.1007/s00220-011-1218-9. Zbl 1225.81137, MR 2794548, 10.1007/s00220-011-1218-9 |
| Reference:
|
[48] Spiridonov, V., Vartanov, G.: Elliptic hypergeometric integrals and $^{\prime }t$ Hooft anomaly matching conditions.JHEP 1206 (016) (2012), 18 pp., http://dx.doi.org/10.1007/s00220-011-1218-9. MR 3006892, 10.1007/s00220-011-1218-9 |
| Reference:
|
[49] Spiridonov, V.P.: On the elliptic beta function.Russian Math. Surveys 56 (1) (2001), 185–186, http://dx.doi.org/10.1070/RM2001v056n01ABEH000374. Zbl 0997.33009, MR 1846786, 10.1070/RM2001v056n01ABEH000374 |
| Reference:
|
[50] Spiridonov, V.P.: Theta hypergeometric integrals.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. MR 2044635, 10.1090/S1061-0022-04-00839-8 |
| Reference:
|
[51] Spiridonov, V.P.: Essays on the theory of elliptic hypergeometric functions.Russian Math. Surveys 63 (3) (2008), 405–472, http://dx.doi.org/10.1070/RM2008v063n03ABEH004533. Zbl 1173.33017, MR 2479997, 10.1070/RM2008v063n03ABEH004533 |
| Reference:
|
[52] Stokman, J.V.: Hyperbolic beta integrals.Adv. Math. 190 (1) (2005), 119–160, http://arxiv.org/abs/math/0303178, arXiv:math/0303178. Zbl 1072.33012, MR 2104907, 10.1016/j.aim.2003.12.003 |
| Reference:
|
[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]. |
| Reference:
|
[54] van de Bult, F.J.: Hyperbolic hypergeometric functions.Ph.D. thesis, University of Amsterdam, 2007. |
| Reference:
|
[55] van de Bult, F.J.: An elliptic hypergeometric integral with $W(F_4)$ symmetry.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. MR 2787288, 10.1007/s11139-010-9273-y |
| Reference:
|
[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. |
| Reference:
|
[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. |
| Reference:
|
[58] van de Bult, F.J., Rains, E.M.: Limits of elliptic hypergeometric biorthogonal functions.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. MR 3324567, 10.1016/j.jat.2014.06.009 |
| Reference:
|
[59] Witten, E.: Constraints on supersymmetry breaking.Nuclear Phys. B 202 (1982), 253–316, http://dx.doi.org/10.1016/0550-3213(82)90071-2. MR 0668987, 10.1016/0550-3213(82)90071-2 |
| Reference:
|
[60] Yamazaki, M.: Four-dimensional superconformal index reloaded.Theoret. and Math. Phys. 174 (1) (2013), 154–166, http://dx.doi.org/10.1007/s11232-013-0012-6. Zbl 1280.81103, MR 3172959, 10.1007/s11232-013-0012-6 |
| Reference:
|
[61] Zwiebel, B.I.: Charging the superconformal index.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]. Zbl 1306.81135, 10.1007/JHEP01(2012)116 |
| . |
