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Gauge/Vortex Duality and AGT

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New Dualities of Supersymmetric Gauge Theories

Part of the book series: Mathematical Physics Studies ((MPST))

Abstract

AGT correspondence relates a class of 4d gauge theories in four dimensions to conformal blocks of Liouville CFT. There is a simple proof of the correspondence when the conformal blocks admit a free field representation. In those cases, vortex defects of the gauge theory play a crucial role, extending the correspondence to a triality. This makes use of a duality between 4d gauge theories in a certain background, and the theories on their vortices. The gauge/vortex duality is a physical realization of large N duality of topological string which was conjectured in Dijkgraaf and Vafa (Toda theories, matrix models, topological strings, and N \(=\) 2 gauge systems [1]) to provide an explanation for AGT correspondence. This paper is a review of Aganagic et al. (Gauge/Liouville triality [2]), written for the special volume edited by J. Teschner.

A citation of the form [V:x] refers to article number x in this volume.

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Notes

  1. 1.

    For early studies leading to [8–10], see [11–15].

  2. 2.

    Proofs of (some aspects of) AGT correspondence using different ideas appeared in [20–24].

  3. 3.

    This follows by compactifying M-theory with M5 brane on \(\Sigma \) on a \(T^2\) transverse to the M5 brane. Since the \(T^2\) is transverse to the branes, it does not change the low energy physics. By shrinking one of the cycles of the \(T^2\) first, we go to down to IIA string with an NS5 brane wrapping \(\Sigma \). T-dualizing on the remaining compact transverse circle, we obtain IIB on \(Y_{\Sigma }\).

  4. 4.

    The full topological string partition function in the presence of branes is given by the matrix integral in (3.6) and (3.7), describing open strings, times a purely closed topological string partition function of \(Y_S\). This will be relevant later on.

  5. 5.

    It may be useful to summarize what the large N asymptotic regime is, on each side of the correspondence. On the B-model side, it is sending \(g_s\) to zero while keeping the combination \(N g_s\) fixed. On the gauge theory side, it is sending \(\epsilon _1 = -\epsilon _2\) to zero while keeping the Coulomb parameters fixed. On the Liouville side, it is sending all the momenta as well as the number N of screening insertions to infinity, while keeping their ratios fixed.

  6. 6.

    For general \(\epsilon _{1,2}\) the background does not simply decouple into a product of a Calabi-Yau manifold times the \(\Omega \) background where the gauge theory lives. Turning on arbitrary \(\Omega \) background requires the theory to have an \(U(1)\in SU(2)_R\) R-symmetry to preserve supersymmetry. This requires the target Calabi-Yau manifold to admit a U(1) action; this U(1) action is used in constructing the background.

  7. 7.

    One should not confuse the vortices here with surface operators in the gauge theory, studied for example in [35–37]. The surface operators are solutions on the Coulomb branch, with infinite tension. From the M5 brane perspective, surface operators are semi-infinite M2 branes ending on M5’s.

  8. 8.

    Usually, the gauge theories on M5 branes wrapping Riemann surfaces are said to be of special unitary type, rather than unitary type. There is no contradiction; the U(1) centers of the gauge groups that arise on branes are typically massive by Green-Schwarz mechanism. This does not affect the BPS tension of the solutions, see e.g. discussion in [38].

  9. 9.

    In [40] one proves that any flat gauge field on the punctured disk preserves supersymmetry of the \(\Omega \) background.

  10. 10.

    We thank Cumrun Vafa for discussion relating to this point.

  11. 11.

    See [42] for a highly nontrivial example.

  12. 12.

    This 3d background was used in [6, 25, 40, 43] as a natural path to defining the 2d \(\Omega \)-background. For a review see [44].

  13. 13.

    At very short distances there is a UV fixed point corresponding to it, which is a strongly coupled theory, accessible via its string or M-theory embedding [48, 49].

  14. 14.

    The second four-dimensional limit gives the 4d \(\mathcal{N}=2\) \(U(\ell )\) gauge theory with \(2\ell \) fundamental hypermultiplets by [17, 32]. In the Seiberg-Witten curve, one writes \(f_{i}\) as \(f_i = e^{R \mu _i}\), and takes R to zero keeping x / R, \(e^p R\), \(e^{\zeta }R\) and the \(\mu \)’s fixed in the limit. The effect of this is that the 4d curve has the same form as (5.4), but with Q and P replaced by polynomials of the same degree, but in x, rather than \(e^x\).

  15. 15.

    By varying the Coulomb branch and the mass parameters, the real mass m of the 5d hypermultiplet can go through zero. This exchanges the fundamental hypermultiplet of mass m for an anti-fundamental of mass \(-m\), while at the same time the 5d Chern-Simons level jumps by 1 [53]. A relation between the anti-fundamental and the fundamental hypermultiplet contributions to the partition function reflects this, see [2] for details.

  16. 16.

    This partition function is the index studied in [54–56] with application to knot theory; see also [57]. The index is a chiral building block of the \(S^3\) or \(S^2\times S^1\) partition functions [58–64], deformed by t, the fugacity of a very particular flavor symmetry.

  17. 17.

    The contours of integration are the same as in the undeformed case—encircling the segments \([0, z_a]\). The q deformation affects the operators and the algebra, but not the contours. It is important to emphasize that these contours agree with the alternative approach [68] where the free field integrals are replaced by Jackson q-integrals: in our picture, the latter are the residue sums for the former.

  18. 18.

    The shift by v is due to the \(\Omega \) background. It is natural that the partition function becomes singular at the point where the two branches meet; this determines the shift.

  19. 19.

    See [2] for a proof, and [52, 69] for earlier work making use of this.

References

  1. Dijkgraaf, R., Vafa, C.: Toda Theories, Matrix Models, Topological Strings, and N \(=\) 2 Gauge Systems. arXiv:0909.2453 [hep-th]

  2. Aganagic, M., Haouzi, N., Kozcaz, C., Shakirov, S.: Gauge/Liouville Triality. arXiv:1309.1687 [hep-th]

  3. Gopakumar, R., Vafa, C.: On the gauge theory/geometry correspondence. Adv. Theor. Math. Phys. 3, 1415–1443 (1999). arXiv:hep-th/9811131

    Google Scholar 

  4. Dijkgraaf, R., Vafa, C.: Matrix models, topological strings, and supersymmetric gauge theories. Nucl. Phys. B644, 3–20 (2002). doi:10.1016/S0550-3213(02)00766-6. arXiv:hep-th/0206255 [hep-th]

    Google Scholar 

  5. Aganagic, M., Dijkgraaf, R., Klemm, A., Marino, M., Vafa, C.: Topological strings and integrable hierarchies. Commun. Math. Phys. 261, 451–516 (2006). doi:10.1007/s00220-005-1448-9. arXiv:hep-th/0312085 [hep-th]

    Google Scholar 

  6. Nekrasov, N.A.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2004). arXiv:hep-th/0206161 [hep-th]

    Google Scholar 

  7. Neitzke, A., Vafa, C.: Topological strings and their physical applications. arXiv:hep-th/0410178 [hep-th]

  8. Chen, H.-Y., Dorey, N., Hollowood, T.J., Lee, S.: A New 2d/4d Duality via Integrability. JHEP 1109, 040 (2011). arXiv:1104.3021 [hep-th]

  9. Dorey, N., Lee, S., Hollowood, T.J.: Quantization of integrable systems and a 2d/4d duality. JHEP 1110, 077 (2011). arXiv:1103.5726 [hep-th]

  10. Aganagic, M.: M-theory, large \(N\) duality and the dynamics of vortices. Talk at 11th Simons Summer Workshop at SCGP (2013)

    Google Scholar 

  11. Dorey, N.: The BPS spectra of two-dimensional supersymmetric gauge theories with twisted mass terms. JHEP 9811, 005 (1998). arXiv:hep-th/9806056 [hep-th]

    Google Scholar 

  12. Dorey, N., Hollowood T.J., Tong, D.: The BPS spectra of gauge theories in two-dimensions and four-dimensions. JHEP 9905, 006 (1999). arXiv:hep-th/9902134 [hep-th]

  13. Hanany, A., Tong, D.: Vortices, instantons and branes. JHEP 0307, 037 (2003). arXiv:hep-th/0306150 [hep-th]

    Google Scholar 

  14. Hanany, A., Tong, D.: Vortex strings and four-dimensional gauge dynamics. JHEP 0404, 066 (2004). arXiv:hep-th/0403158 [hep-th]

    Google Scholar 

  15. Shifman, M., Yung, A.: NonAbelian string junctions as confined monopoles. Phys. Rev. D70, 045004 (2004). arXiv:hep-th/0403149 [hep-th]

  16. Alday, L.F., Gaiotto, D., Tachikawa, Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91, 167–197 (2010). arXiv:0906.3219 [hep-th]

    Google Scholar 

  17. Gaiotto, D.: N \(=\) 2 dualities. JHEP 1208, 034 (2012). arXiv:0904.2715 [hep-th]

  18. Gaiotto, D., Moore, G.W., Neitzke, A.: Wall-crossing, Hitchin systems, and the WKB approximation. arXiv:0907.3987 [hep-th]

  19. Aganagic, M., Haouzi N., Shakirov, S.: \(A_n\)-Triality. arXiv:1403.3657 [hep-th]

  20. Fateev, V., Litvinov, A.: On AGT conjecture. JHEP 1002, 014 (2010). arXiv:0912.0504 [hep-th]

  21. Alba, V.A., Fateev, V.A., Litvinov, A.V., Tarnopolskiy, G.M.: On combinatorial expansion of the conformal blocks arising from AGT conjecture. Lett. Math. Phys. 98, 33–64 (2011). arXiv:1012.1312 [hep-th]

    Google Scholar 

  22. Mironov, A., Morozov, A., Shakirov S.: A direct proof of AGT conjecture at beta \(=\) 1. JHEP 1102, 067 (2011). arXiv:1012.3137 [hep-th]

  23. Morozov, A. Smirnov, A.: Finalizing the proof of AGT relations with the help of the generalized Jack polynomials. arXiv:1307.2576

  24. Braverman, A., Finkelberg, M., Nakajima, H.: Instanton moduli spaces and W-algebras. arXiv:1406.2381 [math.QA]

  25. Losev, A.S., Marshakov, A., Nekrasov, N.A.: Small instantons, little strings and free fermions. arXiv:hep-th/0302191 [hep-th]

  26. Hollowood, T.J., Iqbal, A., Vafa, C.: Matrix models, geometric engineering and elliptic genera. JHEP 0803, 069 (2008). arXiv:hep-th/0310272 [hep-th]

    Google Scholar 

  27. Bershadsky, M., Cecotti, S., Ooguri, H., Vafa, C.: Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes. Commun. Math. Phys. 165, 311–428 (1994). arXiv:hep-th/9309140 [hep-th]

    Google Scholar 

  28. Dotsenko, V., Fateev, V.: Conformal algebra and multipoint correlation functions in two-dimensional statistical models. Nucl. Phys. B 240, 312 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  29. Dotsenko, V., Fateev, V.: Four point correlation functions and the operator algebra in the two-dimensional conformal invariant theories with the central charge c! ‘1". Nucl. Phys. B251, 691 (1985)

    Google Scholar 

  30. Goulian, M., Li, M.: Correlation functions in Liouville theory. Phys. Rev. Lett. 66, 2051–2055 (1991)

    Google Scholar 

  31. Strominger, A.: Massless black holes and conifolds in string theory. Nucl. Phys. B451, 96–108 (1995). arXiv:hep-th/9504090 [hep-th]

    Google Scholar 

  32. Witten, E.: Solutions of four-dimensional field theories via M theory. Nucl. Phys. B500, 3–42 (1997). arXiv:hep-th/9703166 [hep-th]

    Google Scholar 

  33. Greene, B.R., Morrison, D.R., Vafa, C.: A geometric realization of confinement. Nucl. Phys. B481, 513–538 (1996). arXiv:hep-th/9608039 [hep-th]

    Google Scholar 

  34. Hori, K., Ooguri, H., Vafa, C.: NonAbelian conifold transitions and N \(=\) 4 dualities in three-dimensions. Nucl. Phys. B504, 147–174 (1997). arXiv:hep-th/9705220 [hep-th]

  35. Gukov, S., Witten, E.: Branes and quantization. Adv. Theor. Math. Phys. 13 (2009). arXiv:0809.0305 [hep-th]

    Google Scholar 

  36. Alday, L.F., Gaiotto, D., Gukov, S., Tachikawa, Y., Verlinde, H.: Loop and surface operators in N \(=\) 2 gauge theory and Liouville modular geometry. JHEP 1001, 113 (2010). arXiv:0909.0945 [hep-th]

  37. Dimofte, T., Gukov S., Hollands, L.: Vortex counting and Lagrangian 3-manifolds. Lett. Math. Phys. 98, 225–287 (2011). arXiv:1006.0977 [hep-th]

    Google Scholar 

  38. Douglas, M.R., Moore, G.W.: D-branes, quivers, and ALE instantons. arXiv:hep-th/9603167 [hep-th]

  39. Nekrasov, N.A., Shatashvili, S.L.: Quantization of integrable systems and four dimensional gauge theories. arXiv:0908.4052 [hep-th]

  40. Nekrasov, N., Witten, E.: The omega deformation, branes, integrability, and Liouville theory. JHEP 1009, 092 (2010). arXiv:1002.0888 [hep-th]

  41. Aganagic, M., Cheng, M.C.N., Dijkgraaf, R., Krefl, D., Vafa, C.: Quantum geometry of refined topological strings. arXiv:1105.0630 [hep-th]

  42. Aganagic, M., Shakirov, S.: Refined Chern-Simons theory and topological string. arXiv:1210.2733 [hep-th]

  43. Nekrasov, N., Okounkov, A.: Seiberg-Witten theory and random partitions. arXiv:hep-th/0306238 [hep-th]

  44. Nekrasov, N., Shadchin, S.: ABCD of instantons. Commun. Math. Phys. 252 359–391 (2004). arXiv:hep-th/0404225 [hep-th]

    Google Scholar 

  45. Shiraishi, J., Kubo, H., Awata, H., Odake, S.: A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions. Lett. Math. Phys. 38, 33–51 (1996). arXiv:q-alg/9507034 [q-alg]

    Google Scholar 

  46. Awata, H., Kubo, H., Morita, Y., Odake, S., Shiraishi, J.: Vertex operators of the q Virasoro algebra: defining relations, adjoint actions and four point functions. Lett. Math. Phys. 41, 65–78 (1997). arXiv:q-alg/9604023 [q-alg]

  47. Frenkel, E., Reshetikhin, N.: Deformations of W-algebras associated to simple Lie algebras. In eprint arXiv:q-alg/9708006, p. 8006. August (1997)

  48. Seiberg, N.: Five-dimensional SUSY field theories, nontrivial fixed points and string dynamics. Phys. Lett. B388, 753–760 (1996). arXiv:hep-th/9608111 [hep-th]

    Google Scholar 

  49. Intriligator, K.A., Morrison, D.R., Seiberg, N.: Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces. Nucl. Phys. B497, 56–100 (1997). arXiv:hep-th/9702198 [hep-th]

    Google Scholar 

  50. Nekrasov, N.: Five dimensional gauge theories and relativistic integrable systems. Nucl. Phys. B531, 323–344 (1998). arXiv:hep-th/9609219 [hep-th]

    Google Scholar 

  51. Faddeev, L., Kashaev, R.: Quantum dilogarithm. Mod. Phys. Lett. A9 427–434 (1994). arXiv:hep-th/9310070 [hep-th]

    Google Scholar 

  52. Awata, H., Kanno, H.: Refined BPS state counting from Nekrasov’s formula and Macdonald functions. Int. J. Mod. Phys. A24, 2253–2306 (2009). arXiv:0805.0191 [hep-th]

    Google Scholar 

  53. Witten, E.: Phase transitions in M theory and F theory. Nucl. Phys. B471, 195–216 (1996). arXiv:hep-th/9603150 [hep-th]

    Google Scholar 

  54. Aganagic, M., Shakirov, S.: Knot homology from refined Chern-Simons theory. arXiv:1105.5117 [hep-th]

  55. Aganagic, M., Shakirov, S.: Refined Chern-Simons theory and knot homology. arXiv:1202.2489 [hep-th]

  56. Aganagic, M., Schaeffer, K.: Orientifolds and the refined topological string. JHEP 1209, 084 (2012). arXiv:1202.4456 [hep-th]

  57. Fuji, H., Gukov S., Sulkowski, P.: Super-A-polynomial for knots and BPS states. Nucl. Phys. B867, 506–546 (2013). arXiv:1205.1515 [hep-th]

    Google Scholar 

  58. Hama, N., Hosomichi, K., Lee, S.: Notes on SUSY Gauge theories on three-sphere. JHEP 1103, 127 (2011). arXiv:1012.3512 [hep-th]

  59. Kapustin, A., Willett, B.: Generalized superconformal index for three dimensional field theories. arXiv:1106.2484 [hep-th]

  60. Hama, N., Hosomichi, K., Lee, S.: SUSY Gauge theories on squashed three-spheres. JHEP 1105, 014 (2011). arXiv:1102.4716 [hep-th]

  61. Pasquetti, S.: Factorisation of N \(=\) 2 theories on the squashed 3-sphere. JHEP 1204, 120 (2012). arXiv:1111.6905 [hep-th]

  62. Nieri, F., Pasquetti, S., Passerini, F.: 3d and 5d gauge theory partition functions as q-deformed CFT correlators. arXiv:1303.2626 [hep-th]

  63. Beem, C., Dimofte, T., Pasquetti, S.: Holomorphic blocks in three dimensions. arXiv:1211.1986 [hep-th]

  64. Taki, M.: Holomorphic blocks for 3d non-abelian partition functions. arXiv:1303.5915 [hep-th]

  65. Cheng, M.C.N., Dijkgraaf, R., Vafa, C.: Non-perturbative topological strings and conformal blocks. JHEP 09, 022 (2011). arXiv:1010.4573 [hep-th]

  66. Aganagic, M., Klemm, A., Marino, M., Vafa, C.: Matrix model as a mirror of Chern-Simons theory. JHEP 0402, 010 (2004). arXiv:hep-th/0211098 [hep-th]

    Google Scholar 

  67. Awata, H., Yamada, Y.: Five-dimensional AGT relation and the deformed beta-ensemble. Prog. Theor. Phys. 124, 227–262 (2010). arXiv:1004.5122 [hep-th]

  68. Mironov, A., Morozov, A., Shakirov, S., Smirnov, A.: Proving AGT conjecture as HS duality: extension to five dimensions. Nucl. Phys. B855, 128–151 (2012). arXiv:1105.0948 [hep-th]

    Google Scholar 

  69. Dimofte, T., Gukov, S., Hollands, L.: Vortex counting and Lagrangian 3-manifolds. Lett. Math. Phys. 98, 225–287 (2011). arXiv:1006.0977 [hep-th]

    Google Scholar 

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Authors and Affiliations

  1. Center for Theoretical Physics, University of California, Berkeley, USA

    Mina Aganagic & Shamil Shakirov

  2. Department of Mathematics, University of California, Berkeley, USA

    Mina Aganagic & Shamil Shakirov

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  1. Mina Aganagic
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  2. Shamil Shakirov
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Correspondence to Mina Aganagic .

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Editors and Affiliations

  1. String Theory Group, DESY String Theory Group, Hamburg, Hamburg, Germany

    Jörg Teschner

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Aganagic, M., Shakirov, S. (2016). Gauge/Vortex Duality and AGT. In: Teschner, J. (eds) New Dualities of Supersymmetric Gauge Theories. Mathematical Physics Studies. Springer, Cham. https://doi.org/10.1007/978-3-319-18769-3_13

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