Geodesic
Four-dimensional superconformal index reloaded
Teoretičeskaâ i matematičeskaâ fizika, Tome 174 (2013) no. 1, pp. 177-192
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the four-dimensional $\mathcal N\ge 1$ superconformal index and its generalization to the lens space. We discuss reductions of the latter to the three-dimensional $\mathcal N\ge 2$ sphere partition function, the three-dimensional $\mathcal N\ge 2$ superconformal index, and the two-dimensional $\mathcal{N}\ge(2,2)$ sphere partition function. We apply these reductions to a class of four-dimensional $\mathcal N=1$ superconformal field theories dual to toric Calabi–Yau manifolds, and we find surprising connections with integrable spin chains and hyperbolic geometry. We comment on the problem of classifying infrared fixed points of four-dimensional and three-dimensional supersymmetric gauge theories.
Mots-clés : superconformal index
Keywords: elliptic gamma function, sphere partition function, dimensional reduction. 
@article{TMF_2013_174_1_a11,
     author = {M. Yamazaki},
     title = {Four-dimensional superconformal index reloaded},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {177--192},
     year = {2013},
     volume = {174},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2013_174_1_a11/}
}
TY  - JOUR
AU  - M. Yamazaki
TI  - Four-dimensional superconformal index reloaded
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2013
SP  - 177
EP  - 192
VL  - 174
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/TMF_2013_174_1_a11/
LA  - ru
ID  - TMF_2013_174_1_a11
ER  - 
%0 Journal Article
%A M. Yamazaki
%T Four-dimensional superconformal index reloaded
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2013
%P 177-192
%V 174
%N 1
%U http://geodesic.mathdoc.fr/item/TMF_2013_174_1_a11/
%G ru
%F TMF_2013_174_1_a11
M. Yamazaki. Four-dimensional superconformal index reloaded. Teoretičeskaâ i matematičeskaâ fizika, Tome 174 (2013) no. 1, pp. 177-192. http://geodesic.mathdoc.fr/item/TMF_2013_174_1_a11/

[1] C. Klare, A. Tomasiello, A. Zaffaroni, Supersymmetry on curved spaces and holography, arXiv: 1205.1062 | MR

[2] T. T. Dumitrescu, G. Festuccia, N. Seiberg, Exploring curved superspace, arXiv: 1205.1115 | MR

[3] C. Romelsberger, Nucl. Phys. B, 747:3 (2006), 329–353, arXiv: hep-th/0510060 | DOI | MR | Zbl

[4] J. Kinney, J. M. Maldacena, S. Minwalla, S. Raju, Commun. Math. Phys., 275:1 (2007), 209–254, arXiv: hep-th/0510251 | DOI | MR | Zbl

[5] A. Kapustin, B. Willett, I. Yaakov, JHEP, 03 (2010), 089, 29 pp., arXiv: 0909.4559 | DOI | MR

[6] D. L. Jafferis, The exact superconformal R-symmetry extremizes $Z$, arXiv: 1012.3210 | MR

[7] N. Hama, K. Hosomichi, S. Lee, JHEP, 03 (2011), 127, 14 pp., arXiv: 1012.3512 | DOI | MR

[8] S. Kim, Nucl. Phys. B, 821:1–2 (2009), 241–284, arXiv: 0903.4172 | DOI | MR | Zbl

[9] Y. Imamura, S. Yokoyama, JHEP, 1104 (2011), 007, 21 pp., arXiv: 1101.0557 | DOI | MR

[10] N. Seiberg, Nucl. Phys. B, 435:1–2 (1995), 129–146, arXiv: hep-th/9411149 | DOI | MR | Zbl

[11] K. A. Intriligator, N. Seiberg, Phys. Lett. B, 387:3 (1996), 513–519, arXiv: hep-th/9607207 | DOI | MR

[12] F. Benini, T. Nishioka, M. Yamazaki, Phys. Rev. D, 86:6 (2012), 065015, 10 pp., arXiv: 1109.0283 | DOI | MR

[13] E. Witten, Adv. Theor. Math. Phys., 5:5 (2002), 841–907, arXiv: hep-th/0006010 | DOI | MR

[14] V. P. Spiridonov, UMN, 63:3(381) (2008), 3–72 | DOI | DOI | MR | Zbl

[15] C. Romelsberger, Calculating the superconformal index and Seiberg duality, arXiv: 0707.3702

[16] V. Spiridonov, G. Vartanov, Commun. Math. Phys., 304 (2011), 797–874, arXiv: 0910.5944 | DOI | MR | Zbl

[17] V. P. Spiridonov, G. S. Vartanov, Elliptic hypergeometry of supersymmetric dualities II. Orthogonal groups, knots, and vortices, arXiv: 1107.5788 | MR

[18] F. A. H. Dolan, V. P. Spiridonov, G. S. Vartanov, Phys. Lett. B, 704:3 (2011), 234–241, arXiv: 1104.1787 | DOI | MR

[19] A. Gadde, W. Yan, Reducing the 4d index to the $S^3$ partition function, arXiv: 1104.2592 | MR

[20] Y. Imamura, JHEP, 09 (2011), 133, 19 pp., arXiv: 1104.4482 | DOI | MR

[21] T. Nishioka, Y. Tachikawa, M. Yamazaki, JHEP, 08 (2011), 003, 22 pp., arXiv: 1105.4390 | DOI | MR

[22] L. Faddeev, A. Y. Volkov, Phys. Lett. B, 315:3–4 (1993), 311–318, arXiv: hep-th/9307048 | DOI | MR | Zbl

[23] L. D. Faddeev, Lett. Math. Phys., 34:3 (1995), 249–254, arXiv: hep-th/9504111 | DOI | MR | Zbl

[24] L. D. Faddeev, R. M. Kashaev, Modern Phys. Lett. A, 9:5 (1994), 427–434, arXiv: hep-th/9310070 | DOI | MR | Zbl

[25] D. Gang, Chern–Simons theory on $L(p,q)$ lens spaces and localization, arXiv: 0912.4664

[26] J. Kallen, JHEP, 08 (2011), 008, 32 pp., arXiv: 1104.5353 | DOI | MR

[27] K. Ohta, Y. Yoshida, Non-Abelian localization for supersymmetric Yang–Mills–Chern–Simons theories on Seifert manifold, arXiv: 1205.0046

[28] L. F. Alday, M. Fluder, J. Sparks, The large $N$ limit of M2-branes on Lens spaces, arXiv: 1204.1280

[29] N. Hama, K. Hosomichi, S. Lee, SUSY Gauge theories on squashed three-spheres, JHEP, 05, 2011, 23 pp., arXiv: 1102.4716 | DOI | MR

[30] M. Yamazaki, JHEP, 05 (2012), 147, 50 pp., arXiv: 1203.5784 | DOI

[31] Y. Terashima, M. Yamazaki, Phys. Rev. Lett., 109:9 (2012), 091602, 4 pp., arXiv: 1203.5792 | DOI

[32] A. Hanany, D. Vegh, JHEP, 10 (2007), 029, 35 pp., arXiv: hep-th/0511063 | DOI | MR

[33] R. Baxter, Phil. Trans. Roy. Soc. London, 289:1359 (1978), 315–346 | DOI | MR

[34] V. V. Bazhanov, S. M. Sergeev, A master solution of the quantum Yang–Baxter equation and classical discrete integrable equations, arXiv: 1006.0651 | MR

[35] V. V. Bazhanov, S. M. Sergeev, Nucl. Phys. B, 856:2 (2012), 475–496, arXiv: 1106.5874 | DOI | MR | Zbl

[36] V. V. Bazhanov, V. V. Mangazeev, S. M. Sergeev, Nucl. Phys. B, 784:3 (2007), 234–258, arXiv: hep-th/0703041 | DOI | MR | Zbl

[37] D. Gaiotto, $N=2$ dualities, arXiv: 0904.2715 | MR

[38] A. Gadde, E. Pomoni, L. Rastelli, S. S. Razamat, JHEP, 03 (2010), 032, 22 pp., arXiv: 0910.2225 | DOI | MR

[39] A. Gadde, L. Rastelli, S. S. Razamat, W. Yan, Phys. Rev. Lett., 106 (2011), 241602, 4 pp., arXiv: 1104.3850 | DOI

[40] F. Benini, Y. Tachikawa, D. Xie, JHEP, 09 (2010), 063, 32 pp., arXiv: 1007.0992 | DOI | MR

[41] D. R. Gulotta, C. P. Herzog, S. S. Pufu, From necklace quivers to the $F$-theorem, operator counting, and $T(U(N))$, arXiv: 1105.2817 | MR

[42] S. Benvenuti, S. Pasquetti, 3D-partition functions on the sphere: exact evaluation and mirror symmetry, arXiv: 1105.2551 | MR

[43] S. Cecotti, C. Vafa, Classification of complete $N=2$ supersymmetric theories in 4 dimensions, arXiv: 1103.5832 | MR

[44] M. R. Douglas, Spaces of quantum field theories, arXiv: 1005.2779

[45] P. C. Argyres, N. Seiberg, JHEP, 12 (2007), 088, 26 pp., arXiv: 0711.0054 | DOI | MR | Zbl

[46] I. Bah, B. Wecht, New $N=1$ superconformal field theories in four dimensions, arXiv: 1111.3402 | MR

[47] Y. Terashima, M. Yamazaki, JHEP, 08 (2011), 135, 46 pp., arXiv: 1103.5748 | DOI

[48] T. Dimofte, D. Gaiotto, S. Gukov, Gauge theories labelled by three-manifolds, arXiv: 1108.4389 | MR

[49] S. Cecotti, C. Cordova, C. Vafa, Braids, walls, and mirrors, arXiv: 1110.2115

[50] T. Dimofte, D. Gaiotto, S. Gukov, 3-Manifolds and 3d indices, arXiv: 1112.5179 | MR

[51] N. Doroud, J. Gomis, B. Le Floch, S. Lee, Exact results in $D=2$ supersymmetric gauge theories, arXiv: 1206.2606 | MR

[52] F. Benini, S. Cremonesi, Partition functions of $N=(2,2)$ gauge theories on $S^2$ and vortices, arXiv: 1206.2356 | MR