Abstract
We study the moduli spaces of k U(N ) vortices which are realized by the Higgs branch of a U(k) supersymmetric gauge theory. The theory has 4 supercharges and lives on k D1-branes in a N D3- and NS5-brane background. We realize the vortex moduli space as a \( {\mathbb{C}}^{*} \) projection of the vortex master space. The Hilbert series is calculated in order to characterize the algebraic structure of the vortex master space and to identify the precise \( {\mathbb{C}}^{*} \) projection. As a result, we are able to fully classify the moduli spaces up to 3 vortices.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A.A. Abrikosov, The magnetic properties of superconducting alloys, J. Phys. Chem. Solid 2 (1957) 199.
H.B. Nielsen and P. Olesen, Vortex Line Models for Dual Strings, Nucl. Phys. B 61 (1973) 45 [INSPIRE].
A. Hanany and D. Tong, Vortices, instantons and branes, JHEP 07 (2003) 037 [hep-th/0306150] [INSPIRE].
R. Auzzi, M. Shifman and A. Yung, Composite non-Abelian flux tubes in N = 2 SQCD, Phys. Rev. D 73 (2006) 105012 [Erratum ibid. D 76 (2007) 109901] [hep-th/0511150] [INSPIRE].
K. Hashimoto and D. Tong, Reconnection of non-Abelian cosmic strings, JCAP 09 (2005) 004 [hep-th/0506022] [INSPIRE].
M. Eto et al., Non-Abelian Vortices of Higher Winding Numbers, Phys. Rev. D 74 (2006) 065021 [hep-th/0607070] [INSPIRE].
M. Eto et al., Group Theory of Non-Abelian Vortices, JHEP 11 (2010) 042 [arXiv:1009.4794] [INSPIRE].
A. Hanany and K. Hashimoto, Reconnection of colliding cosmic strings, JHEP 06 (2005) 021 [hep-th/0501031] [INSPIRE].
A. Hanany and D. Tong, Vortex strings and four-dimensional gauge dynamics, JHEP 04 (2004) 066 [hep-th/0403158] [INSPIRE].
M. Eto et al., On the moduli space of semilocal strings and lumps, Phys. Rev. D 76 (2007) 105002 [arXiv:0704.2218] [INSPIRE].
R. Auzzi, S. Bolognesi, J. Evslin, K. Konishi and A. Yung, NonAbelian superconductors: Vortices and confinement in N = 2 SQCD, Nucl. Phys. B 673 (2003) 187 [hep-th/0307287] [INSPIRE].
M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Moduli space of non-Abelian vortices, Phys. Rev. Lett. 96 (2006) 161601 [hep-th/0511088] [INSPIRE].
M. Eto, Y. Isozumi, M. Nitta, K. Ohashi and N. Sakai, Solitons in the Higgs phase: The moduli matrix approach, J. Phys. A 39 (2006) R315 [hep-th/0602170] [INSPIRE].
A. Hanany and A. Zaffaroni, The master space of supersymmetric gauge theories, Adv. High Energy Phys. 2010 (2010) 427891.
D. Forcella, A. Hanany, Y.-H. He and A. Zaffaroni, The Master Space of N = 1 Gauge Theories, JHEP 08 (2008) 012 [arXiv:0801.1585] [INSPIRE].
A. Zaffaroni, The master space of N = 1 quiver gauge theories: Counting BPS operators, prepared for 8th Workshop on Continuous Advances in QCD (CAQCD-08), Minneapolis, Minnesota U.S.A., 15-18 May 2008.
D. Forcella, Master Space and Hilbert Series for N = 1 Field Theories, arXiv:0902.2109 [INSPIRE].
A. Butti, D. Forcella, A. Hanany, D. Vegh and A. Zaffaroni, Counting Chiral Operators in Quiver Gauge Theories, JHEP 11 (2007) 092 [arXiv:0705.2771] [INSPIRE].
S. Benvenuti, A. Hanany and N. Mekareeya, The Hilbert Series of the One Instanton Moduli Space, JHEP 06 (2010) 100 [arXiv:1005.3026] [INSPIRE].
A. Hanany, N. Mekareeya and S.S. Razamat, Hilbert Series for Moduli Spaces of Two Instantons, JHEP 01 (2013) 070 [arXiv:1205.4741] [INSPIRE].
A. Dey, A. Hanany, N. Mekareeya, D. Rodríguez-Gómez, and R.-K. Seong, Hilbert Series for Moduli Spaces of Instantons on \( {\mathbb{C}}^2/{\mathbb{Z}}_n \), JHEP 01 (2014) 182 [arXiv:1309.0812] [INSPIRE].
M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Y. Manin, Construction of Instantons, Phys. Lett. A 65 (1978) 185 [INSPIRE].
S. Benvenuti, B. Feng, A. Hanany and Y.-H. He, Counting BPS Operators in Gauge Theories: Quivers, Syzygies and Plethystics, JHEP 11 (2007) 050 [hep-th/0608050] [INSPIRE].
A. Hanany and C. Romelsberger, Counting BPS operators in the chiral ring of N = 2 supersymmetric gauge theories or N = 2 braine surgery, Adv. Theor. Math. Phys. 11 (2007) 1091 [hep-th/0611346] [INSPIRE].
B. Feng, A. Hanany and Y.-H. He, Counting gauge invariants: The Plethystic program, JHEP 03 (2007) 090 [hep-th/0701063] [INSPIRE].
A. Hanany, N. Mekareeya and G. Torri, The Hilbert Series of Adjoint SQCD, Nucl. Phys. B 825 (2010) 52 [arXiv:0812.2315] [INSPIRE].
A. Hanany and R.-K. Seong, Brane Tilings and Reflexive Polygons, Fortsch. Phys. 60 (2012) 695 [arXiv:1201.2614] [INSPIRE].
A. Hanany and R.-K. Seong, Brane Tilings and Specular Duality, JHEP 08 (2012) 107 [arXiv:1206.2386] [INSPIRE].
S. Cremonesi, A. Hanany and R.-K. Seong, Double Handled Brane Tilings, JHEP 10 (2013) 001 [arXiv:1305.3607] [INSPIRE].
E. Getzler and M.M. Kapranov, Modular operads, dg-ga/9408003.
A. Hanany and E. Witten, Type IIB superstrings, BPS monopoles and three-dimensional gauge dynamics, Nucl. Phys. B 492 (1997) 152 [hep-th/9611230] [INSPIRE].
C.-j. Kim, K.-M. Lee and S.-H. Yi, Tales of D0 on D6-branes: Matrix mechanics of identical particles, Phys. Lett. B 543 (2002) 107 [hep-th/0204109] [INSPIRE].
J.-P. Magnot, Ambrose-Singer theorem on diffeological bundles and complete integrability of the KP equation, Int. J. Geom. Meth. Mod. Phys. 10 (2013) 1350043 [INSPIRE].
A. Hanany and R. Kalveks, in preparation.
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1403.4950
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Hanany, A., Seong, RK. Hilbert series and moduli spaces of k U(N ) vortices. J. High Energ. Phys. 2015, 12 (2015). https://doi.org/10.1007/JHEP02(2015)012
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP02(2015)012