M-strings. (English) Zbl 1393.81031
Summary: M2 branes suspended between adjacent parallel M5 branes lead to light strings, the ‘M-strings’. In this paper we compute the elliptic genus of M-strings, twisted by maximally allowed symmetries that preserve 2d \((2, 0)\) supersymmetry. In a codimension one subspace of parameters this reduces to the elliptic genus of the \((4,4)\) supersymmetric \(A_{n-1}\) quiver theory in 2d. We contrast the elliptic genus of \(N\) M-strings with the \((4,4)\) sigma model on the \(N\)-fold symmetric product of \(\mathbb R^4\). For \(N=1\) they are the same, but for \(N>1\) they are close, but not identical. Instead the elliptic genus of \((4, 4)\) \(N\) M-strings is the same as the elliptic genus of \((4,0)\) sigma models on the \(N\)-fold symmetric product of \(\mathbb R^4\), but where the right-moving fermions couple to a modification of the tangent bundle. This construction arises from a dual \(A_{n-1}\) quiver 6d gauge theory with \(U(1)\) gauge groups. Moreover, we compute the elliptic genus of domain walls which separate different numbers of M2 branes on the two sides of the wall.
MSC:
| 81T30 | String and superstring theories; other extended objects (e.g., branes) in quantum field theory |
References:
| [1] | Witten, E.: On the conformal field theory of the Higgs branch. JHEP 9707, 003 (1997). hep-th/9707093 · Zbl 0949.81056 |
| [2] | Berman, D.S., Perry, M.J., Sezgin, E., Thompson, D.C.: Boundary Conditions for interacting membranes. JHEP 1004, 025 (2010). arXiv:0912.3504 [hep-th] · Zbl 1272.81140 |
| [3] | Lin, H., Lunin, O., Maldacena, J.M.: Bubbling AdS space and 1/2 BPS geometries. JHEP 0410, 025 (2004). hep-th/0409174 |
| [4] | Gomis, J., Rodriguez-Gomez, D., Van Raamsdonk, M., Verlinde, H.: A Massive study of M2-brane proposals. JHEP 0809, 113 (2008). arXiv:0807.1074 [hep-th] · Zbl 1245.81167 |
| [5] | Kim, H.-C., Kim, S.: Supersymmetric vacua of mass-deformed M2-brane theory. Nucl. Phys. B 839, 96 (2010). arXiv:1001.3153 [hep-th] · Zbl 1206.81103 |
| [6] | Benini, F., Eager, R., Hori, K., Tachikawa Y.: Elliptic genera of two-dimensional N=2 gauge theories with rank-one gauge groups. Lett. Math. Phys. 104, 465-493 (2014). arXiv:1305.0533 [hep-th] · Zbl 1312.58008 |
| [7] | Gadde, A., Gukov, S.: 2d Index and surface operators. J. High. Energy Phys. 2014, 80 (2014). arXiv:1305.0266 [hep-th] · Zbl 1333.81399 |
| [8] | Gopakumar, R., Vafa, C.: M theory and topological strings. 1. hep-th/9809187 · Zbl 0922.32015 |
| [9] | Gopakumar, R., Vafa, C.: M theory and topological strings. 2. hep-th/9812127 · Zbl 0922.32015 |
| [10] | Hollowood, T.J., Iqbal, A., Vafa, C.: Matrix models, geometric engineering and elliptic genera. JHEP 0803, 069 (2008). hep-th/0310272 |
| [11] | Iqbal, A., Kozcaz, C., Vafa, C.: The refined topological vertex. JHEP 0910, 069 (2009). hep-th/0701156 · Zbl 1207.81123 |
| [12] | Lockhart, G., Vafa C.: Superconformal partition functions and non-perturbative topological strings. arXiv:1210.5909 [hep-th] · Zbl 1402.81248 |
| [13] | Kim, H.-C., Kim, J., Kim, S.: Instantons on the 5-sphere and M5-branes. arXiv:1211.0144 [hep-th] · Zbl 1342.81442 |
| [14] | Witten, E.: Solutions of four-dimensional field theories via M theory. Nucl. Phys. B 500, 3 (1997). hep-th/9703166 · Zbl 0934.81066 |
| [15] | Aganagic, M., Klemm, A., Marino, M., Vafa, C.: The topological vertex. Commun. Math. Phys. 254, 425 (2005). hep-th/0305132 · Zbl 1114.81076 |
| [16] | Leung, N.C., Vafa, C.: Branes and toric geometry. Adv. Theor. Math. Phys. 2, 91 (1998). hep-th/9711013 · Zbl 0914.14024 |
| [17] | Aharony, O., Hanany, A., Kol, B.: Webs of (p,q) five-branes, five-dimensional field theories and grid diagrams. JHEP 9801, 002 (1998). hep-th/9710116 · Zbl 0625.57008 |
| [18] | Dijkgraaf, R., Vafa, C., Verlinde, E.: M-theory and a topological string duality. hep-th/0602087 |
| [19] | Nekrasov, N.A.; Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831 (2004). hep-th/0206161 · Zbl 1008.81062 |
| [20] | Vafa, C., Witten, E.: A strong coupling test of S duality. Nucl. Phys. B 431, 3 (1994). hep-th/9408074 · Zbl 0964.81522 |
| [21] | Witten, E.: Some comments on string dynamics. In: Los Angeles 1995, Future perspectives in string theory, pp. 501-523. hep-th/9507121 |
| [22] | Ooguri, H., Vafa, C.: Two-dimensional black hole and singularities of CY manifolds. Nucl. Phys. B 463, 55 (1996). hep-th/9511164 · Zbl 1003.83511 |
| [23] | Minahan, J.A., Nemeschansky, D., Vafa, C., Warner, N.P.: E strings and N=4 topological Yang-Mills theories. Nucl. Phys. B 527, 581 (1998). hep-th/9802168 · Zbl 0951.81025 |
| [24] | Douglas, M.R., Moore, G.W.: D-branes, quivers, and ALE instantons. hep-th/9603167 |
| [25] | Aharony, O., Berkooz, M., Kutasov, D., Seiberg, N.: Linear dilatons, NS five-branes and holography. JHEP 9810, 004 (1998). hep-th/9808149 · Zbl 0955.81040 |
| [26] | Aharony, O.: A brief review of ‘little string theories’. Class. Quant. Grav. 17, 929 (2000). hep-th/9911147 · Zbl 0952.81040 |
| [27] | Aganagic, M., Shakirov, S.: Refined Chern-Simons theory and topological string. arXiv:1210.2733 [hep-th] · Zbl 1322.81069 |
| [28] | Iqbal, A., Kozcaz, C., Shabbir, K.: Refined topological vertex, cylindric partitions and the U(1) adjoint theory. Nucl. Phys. B 838, 422 (2010). arXiv:0803.2260 [hep-th] · Zbl 1206.81083 |
| [29] | Dorey, N., Hollowood, T.J., Khoze, V.V., Mattis, M.P.: The Calculus of many instantons. Phys. Rept. 371, 231 (2002). hep-th/0206063 · Zbl 0999.81058 |
| [30] | Hollowood, T.J., Kingaby, T.: A Comment on the chi(y) genus and supersymmetric quantum mechanics. Phys. Lett. B 566, 258 (2003). hep-th/0303018 · Zbl 1051.81513 |
| [31] | Nakajima, H.: Lectures on Hilbert schemes of points on surfaces, vol. 18. American Mathematical Society, Providence (1999) · Zbl 0949.14001 |
| [32] | Nakajima, H., Yoshioka, K.: Instanton counting on blowup. I. 4-dimensional pure gauge theory. Inventiones Mathematicae, vol. 162, pp. 313-355 (2005). arXiv:math/0306198 · Zbl 1100.14009 |
| [33] | Kim, H.-C., Kim, S., Koh, E., Lee, K., Lee, S.: On instantons as Kaluza-Klein modes of M5-branes. JHEP 1112, 031 (2011). arXiv:1110.2175 [hep-th] · Zbl 1306.81118 |
| [34] | Hosono, S., Saito, M.H., Takahashi, A.: Holomorphic anomaly equation and BPS state counting of rational elliptic surface. Adv. Theor. Math. Phys. 3, 177 (1999). hep-th/9901151 · Zbl 1062.14504 |
| [35] | Klemm, A., Manschot, J., Wotschke, T.: Quantum geometry of elliptic Calabi-Yau manifolds. arXiv:1205.1795 [hep-th] · Zbl 1270.81180 |
| [36] | Hosono, S.: Counting BPS states via holomorphic anomaly equations. Fields Inst.Commun. (Submitted). hep-th/0206206 · Zbl 1046.81086 |
| [37] | Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311 (1994). hep-th/9309140 · Zbl 0815.53082 |
| [38] | Huang, M.-x., Klemm, A.: Direct integration for general Ω backgrounds. Adv. Theor. Math. Phys. 16(3), 805 (2012). arXiv:1009.1126 [hep-th] · Zbl 1276.81098 |
| [39] | Krefl, D., Walcher, J.: Extended holomorphic anomaly in gauge theory. Lett. Math. Phys. 95, 67 (2011). arXiv:1007.0263 [hep-th] · Zbl 1205.81118 |
| [40] | Katz, S., Mayr, P., Vafa, C.: Mirror symmetry and exact solution of 4-D N=2 gauge theories: 1. Adv. Theor. Math. Phys. 1, 53 (1998). hep-th/9706110 · Zbl 0912.32016 |
| [41] | Samuel, B.: Chern classes of the tangent bundle on the Hilbert scheme of points on the affine plane. J. Algebraic Geom. 14, 761-787 (2005). arXiv:math/0410458 · Zbl 1120.14002 |
| [42] | Gritsenko, V.: Complex vector bundles and Jacobi forms. math/9906191 [math.AG] · Zbl 1015.11015 |
| [43] | Smirnov, A.: On the instanton R-matrix. arXiv:1302.0799 · Zbl 1345.81056 |
| [44] | Carlsson, E., Okounkov, A.: Exts and vertex operators. Duke Math. J. 161(9), 1797-1815 (2012). arXiv:0801.2565 · Zbl 1256.14010 |
| [45] | Katz, S.H., Sharpe, E.: D-branes, open string vertex operators, and Ext groups. Adv. Theor. Math. Phys. 6, 979 (2003). hep-th/0208104 |
| [46] | Berman, D.S., Harvey, J.A.: The Self-dual string and anomalies in the M5-brane. JHEP 0411, 015 (2004). hep-th/0408198 |
| [47] | Witten E.: Elliptic genera and quantum field theory. Commun. Math. Phys. 109, 525 (1987) · Zbl 0625.57008 · doi:10.1007/BF01208956 |
| [48] | Witten, E.: Phases of N=2 theories in two-dimensions. Nucl. Phys. B 403, 159 (1993). hep-th/9301042 · Zbl 0910.14020 |
| [49] | Kawai, T., Mohri, K.: Geometry of (0,2) Landau-Ginzburg orbifolds. Nucl. Phys. B 425, 191 (1994). hep-th/9402148 · Zbl 1049.58502 |
| [50] | Iqbal, A., Vafa, C.: BPS degeneracies and superconformal index in diverse dimensions. arXiv:1210.3605 [hep-th] |
| [51] | Pestun, V.: Localization of gauge theory on a four-sphere and supersymmetric Wilson loops. Commun. Math. Phys. 313, 71 (2012). arXiv:0712.2824 [hep-th] · Zbl 1257.81056 |
| [52] | Alday, L.F., Gaiotto, D., Tachikawa, Y.: Liouville correlation functions from four-dimensional Gauge theories. Lett. Math. Phys. 91, 167 (2010). arXiv:0906.3219 [hep-th] · Zbl 1185.81111 |
| [53] | Awata, H., Yamada, Y.: Five-dimensional AGT relation and the deformed beta-ensemble. Prog. Theor. Phys. 124, 227 (2010). arXiv:1004.5122 [hep-th] · Zbl 1201.81074 |
| [54] | Nieri, F., Pasquetti, S., Passerini, F.: 3d & 5d gauge theory partition functions as q-deformed CFT correlators. arXiv:1303.2626 [hep-th] · Zbl 1305.81112 |
| [55] | Hama, N., Hosomichi, K., Lee, S.: SUSY Gauge theories on squashed three-spheres. JHEP 1105, 014 (2011). arXiv:1102.4716 [hep-th] · Zbl 1296.81061 |
| [56] | Imamura, Y., Yokoyama, D.: N=2 supersymmetric theories on squashed three-sphere. Phys. Rev. D 85, 025015 (2012). arXiv:1109.4734 [hep-th] |
| [57] | Imamura, Y.: Perturbative partition function for squashed S5. Prog. Theor. Exp. Phys. 073B01 (2013). arXiv:1210.6308 [hep-th] |
| [58] | Nekrasov, N., Pestun, V.: Seiberg-Witten geometry of four dimensional N=2 quiver gauge theories. arXiv:1211.2240 [hep-th] |
| [59] | Aharony, O., Bergman, O., Jafferis D.L., Maldacena, J.: N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals. JHEP 0810, 091 (2008). arXiv:0806.1218 [hep-th] · Zbl 1245.81130 |
| [60] | Seiberg, N., Witten, E.: Comments on string dynamics in six-dimensions. Nucl. Phys. B 471, 121 (1996). hep-th/9603003 · Zbl 1003.81535 |
| [61] | Ganor, O.J., Hanany, A.: Small E(8) instantons and tensionless noncritical strings. Nucl. Phys. B 474, 122 (1996). hep-th/9602120 · Zbl 0925.81170 |
| [62] | Sakai, K.: Seiberg-Witten prepotential for E-string theory and random partitions. JHEP 1206, 027 (2012). arXiv:1203.2921 [hep-th] · Zbl 1397.81283 |
| [63] | Macdonald, I.G.: Symmetric functions and Hall polynomials, 2nd edn. Oxford Mathematical Monographs. Oxford Science Publications, Oxford (1995) · Zbl 0824.05059 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
