Skip to main content
SpringerLink
Log in

Supersymmetric Wilson Loops, Instantons, and Deformed \(\mathcal{W}\)-Algebras

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Let \({\mathfrak {g}}\) be a simple Lie algebra. We study 1/2-BPS Wilson loops of supersymmetric 5d \({\mathfrak {g}}\)-type quiver gauge theories on a circle, in a non-trivial instanton background. The Wilson loops are codimension 4 defects of the gauge theory, and their interaction with self-dual instantons is captured by a modified 1d ADHM quantum mechanics. We compute the partition function as its Witten index. This index is a “qq-character” of a finite-dimensional irreducible representation of the quantum affine algebra \(U_q({\widehat{{\mathfrak {g}}}})\). Using gauge/vortex duality, we can understand the 5d physics in 3d gauge theory terms. Namely, we reinterpret the 5d theory with vortex flux from the point of view of the vortices themselves. This vortex perspective has an advantage: it has yet another dual description in terms of deformed \({\mathfrak {g}}\)-type Toda Theory on a cylinder, in free field formalism. We show that the gauge theory partition function is equal to a chiral correlator of the deformed Toda Theory, with stress tensor and higher spin operator insertions. We derive all the above results from type IIB string theory, compactified on a resolved ADE singularity times a cylinder with punctures, with various branes wrapping the blown-up 2-cycles.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

Similar content being viewed by others

Notes

  1. Quivers labeled by a simple Lie algebra are a subset of fractional quivers, which can have an arbitrary high lacing number r. These are mathematically well-defined, but physics imposes some restrictions on which theories are allowed: we can have \(r=1\) (simply-laced case), \(r=2\) (\(B_N, C_N, F_4\)), or \(r=3\) (\(G_2\)). Certain twisted affine algebras can arise as well in Physics, and they can perfectly be studied using the formalism we develop in this paper, though we leave the explicit analysis to future work.

  2. In a related context, a two-dimensional qq-character was defined in [17], again related to our three-dimensional partition function (in the simply-laced case) by a circle reduction. However, we want to keep the circle size finite here, since the 3d perspective has a crucial feature: as we will see, it has a dual description in terms of observables in a so-called deformed \({{\mathcal {W}}}\)-algebra theory on the cylinder.

  3. \({{\mathcal {W}}}({\mathfrak {g}})\)-algebras are also labeled by a choice of a nilpotent orbit, which in this paper will always be the maximal one.

  4. The expression “spin” is technically not correct here, as conformal symmetry is broken in \({{\mathcal {W}}}_{q,t}({\mathfrak {g}})\). Nevertheless, due to a lack of standard terminology in the literature, we decide here to label the generating currents of the deformed algebra by spin, just like their undeformed counterparts in for usual \({{\mathcal {W}}}({\mathfrak {g}})\)-algebras. We hope our labeling of the generators will not confuse the reader.

  5. Five-dimensional non simply-laced quiver gauge theories with 8 supercharges were recently studied in [28], and defined algebraically in the same context in [16]. Compactifying the setup on a torus \(T^2\), see also the related work in three dimensions [29], which defines an index in the same way we use here. Still in three dimensions, a non-stringy construction was proposed in [30]. More recently, a mathematical definition of the Coulomb branch of the theories was proposed in [31], as generalized slices in the affine Grassmanian when the Cartan matrix is of BCFG type.

  6. Choosing the other complex line \({\mathbb {C}}_t\) will in fact result in distinct physics. It would be interesting to investigate this further. We will encounter again the choice of a preferred line when discussing deformed \({\mathcal {W}}\)-algebra labeled by a non-simply Lie algebra.

  7. Alternatively, the net non-compact D5 brane charge is labeled by a weight in the Langlands dual algebra \(^L{\mathfrak {g}}\), and the net compact D5 brane charge is labeled by a root in \(^L{\mathfrak {g}}\). The Langlands dual algebra of \({\mathfrak {g}}\) is defined as the Lie algebra with the transpose Cartan matrix of \({\mathfrak {g}}\). See [28] for details when only D5 branes are present, and [29] for a discussion where a configuration of D3 branes is studied.

  8. The analysis in [13] predicts a quiver with the shape of an affine Dynkin diagram. However, in the little string limit, the gauge theory really becomes labeled by a finite quiver [34]: given an affine quiver, one linear combination of all nodes represents a D3 brane probing the ADE surface. The mass of this D3 brane is equal to the sum of inverse gauge couplings for all the nodes of the affine quiver, and this sum is equal to \(1/g_s\). Then, in the \(g_s\rightarrow 0\) limit, the D3 brane becomes non-dynamical and the gauge theory on it becomes a flavor symmetry. Therefore, the affine node effectively decouples of the quiver gauge theory in the little string context. We thank Mina Aganagic for pointing out this argument to us.

  9. In what follows, when we talk about a \({\mathcal {N}}=(0,2)\) or \({\mathcal {N}}=(0,4)\) multiplet in the context of the quantum mechanics, what we really mean is the reduction of a 2d \({\mathcal {N}}=(0,2)\) or \({\mathcal {N}}=(0,4)\) multiplet to 1d.

  10. In this paper, whenever the quiver is non simply-laced, what we mean by a supersymmetric “multiplet” is the folding of proper multiplets of a simply-laced theory, following the discussion in Sect. 2.1.2.

  11. The quantization of the above strings actually gives more than what we have described, as it is the ADHM quantum mechanics of the so-called 5d \({\mathcal {N}}=1^*\) SYM, with an extra hypermultiplet in the adjoint representation. In this paper, we decouple the adjoint hypermultiplet by sending its mass to infinity.

  12. For instance, in the case \({\mathfrak {g}}=A_1\), \(T^{5d}\) has a UV fixed point if \(N_f + 2 \,|k_{CS, bare}| \le 2\, n\). We take \(k_{CS, bare}=0\) here, and allow for all theories that still satisfy the inequality. When we later make contact with 3d physics, we will choose to saturate this inequality to simplify the analysis. Such inequalities exist for all \({\mathfrak {g}}\)-type quivers.

  13. Which tensor product representation exactly is dictated by a 1d Chern–Simons level \(K^{(a)}\) we are allowed to turn on for the fermion field action on each node, and which acts as a Lagrange multiplier term in the path integral (2.18). This corresponds to a background of \(K^{(a)}\) units of electric charge localized on the defect D1 branes, meaning there are \(K^{(a)}\) fundamental strings stretching between the D5 and D1 branes. One still needs to distribute the \(K^{(a)}\) units of string charge among the \(L^{(a)}\) D1 branes; in other words, one needs to choose a partition \((K^{(a)}_1, \ldots , K^{(a)}_{L^{(a)}})\) of \(L^{(a)}\). This in turn specifies a representation \({\mathbf{R }}^{(a)}={\mathbf{K }}^{(a)}_1\otimes \ldots \otimes {\mathbf{K }}^{(a)}_{L^{(a)}}\) of \(SU(n^{(a)})\), where each \({\mathbf{K }}^{(a)}_\rho \) is a fundamental representation of \(SU(n^{(a)})\). In this paper, we do not include such a 1d Chern Simons term; the partition function then becomes a sum over all possible tensor products of \(SU(n^{(a)})\) fundamental representations.

  14. As written, there are instanton corrections contained in \(X_{\mathbf{Q }}(\{z^{(a)}_\rho \})\). By sending such corrections to 0, \(X_{\mathbf{Q }}(\{z^{(a)}_\rho \})\) literally becomes the character of the representation \(\mathbf{Q }\). These instanton corrections have important physics of their own: they are related after two T-dualities to the monopole bubbling of ’t Hooft loops in 4d \(U(L^{(a)})\) SYM [43, 44].

  15. The literature on the representation theory of quantum affine algebras is quite rich, and remains an active subject of research to this day. For details on finite dimensional representations of quantum affine algebras, there are two main presentations, one due to Jimbo [45], and the other due to Drinfeld [46]. In our context, it is the latter presentation that is relevant. See also the works [47, 48]. Characters of representations of quantum affine algebras appeared in the literature under the name q-characters, in the work of Frenkel and Reshetikhin [49]. (q, t)-characters are a generalization of those characters, defined by the same authors [19] as Ward identities satisfied by deformed \({\mathcal {W}}_{q,t}\)-algebras. In the context of the BPS/CFT correspondence, those objects have recently appeared in a higher dimensional gauge theory context as qq-characters [14, 15], where “qq” stands for the two equivariant parameters q and t of the \(\Omega \)-background. In particular, one recovers the usual q-characters in the so-called Nekrasov–Shatashvili limit \(\epsilon _1\rightarrow 0\) [50, 51]. For related work on t-analogues of q-characters, see [52, 53].

  16. In the context of integrable systems, it is well known that XXZ spin chains have quantum affine symmetry. The fact that such algebras appear in the study of five dimensional theories on \({\mathbb {C}}^2\times S^1({\widehat{R}})\) is expected from the gauge/Bethe correspondence [54]. This will be true again in three dimensions, by construction, as we will see explicitly in the next section.

  17. An example where such a derivative term can appear is the \(D_4\) partition function \(\left[ \chi ^{{\mathfrak {g}}}\right] _{(0,1,0,0)}^{5d}(z^{(2)}_1)\), meaning there is only one D1 brane wrapping the non-compact 2-cycle \(S^*_2\) in a resolved \(D_4\) singularity. The partition function is then a sum of 29 terms, one of which involves derivatives of \(Y^{(a)}_{5d}\) operators. The attentive reader may wonder why there are 29 terms in the first place, since the second fundamental representation of \(D_4\) is 28-dimensional. However, finite dimensional irreducible representations of quantum affine algebras are in general bigger than their non affine counterpart. Indeed, the second fundamental representation \(V(\lambda _2)\) of \(U_q(\widehat{D_4})\) decomposes into irreducible representations of \(U_q(D_4)\) as \(V(\lambda _2) = \mathbf{28} \oplus \mathbf{1 }\): one necessarily needs to add the trivial representation 1 (an extra null weight) to the 28 to obtain an irreducible representation of \(U_q(\widehat{D_4})\).

  18. A classification of D5 brane defects subject to (3.1) was carried out in [55, 56]. It is shown there that the constraint has an important connection to nilpotent orbits. Namely, the Coulomb branch of the low energy theory on D5 branes, in the CFT limit \(m_s\rightarrow \infty \), is a nilpotent orbit, with Bala-Carter label directly readable from the Dynkin labels of \([S^*]\).

  19. The “position” of the defect on \({\mathcal {C}}\) is then the center of mass of the \(N_f\) D5 branes. Since we are setting \(g_s\rightarrow 0\), this is in fact a codimension 2 defect of the (2, 0) little string on \({\mathcal {C}}\).

  20. Once again, note that the quiver is really the one corresponding to \({\mathfrak {g}}\), and not to \({\widehat{{\mathfrak {g}}}}\), since we are taking the \(g_s\rightarrow 0\) limit, which decouples the affine node.

  21. Originally, the duality was phrased between a 4d \({\mathcal {N}}=2\) gauge theory placed in 2d \(\Omega \)-background, and a two-dimensional theory with \({{\mathcal {N}}}=(2,2)\) supersymmetry, living on the vortices of the former. The first evidence of this relation was given in [58], and further developed in [59], there it was recognized that the BPS spectra of the vortices living in the parent 4d theory matched the BPS spectrum of the 2d theory. This relation was also verified at the level partition functions. After specializing the 4d Coulomb parameters, it was observed that the superpotentials of both theories were the same [60, 61].

  22. This is sometimes called 3d Coulomb branch localization.

  23. It is also referred to as a holomorphic block [64]. Sometimes, this index is also defined on the manifold \(S^1({\widehat{R}})\times D\), where D is the disk. Because this new manifold has a boundary, it becomes important to specify the boundary conditions of the various fields at the edge of the disk. In particular, what we call antichiral multiplets are now understood as 3d chiral multiplets with Dirichlet boundary conditions, while our chiral multiplets are really chiral multiplets with Neumann boundary conditions. The gauge fields have Neumann boundary conditions, and the appearance of theta functions in the 3d vector multiplet is understood as the contribution of the 2d elliptic genus on the boundary torus. For details, see [65,66,67], and the discussion on boundary conditions in our context in [29, 68].

  24. Since we are interested in computing the partition function on the D3 branes, one may wonder why we should care about D1/D5 strings in the first place. The answer is that they quantize the fermions we called \(\chi \) in the action term (3.15). Those strings also played a role in 5d, as we discussed in Sect. 2.3.

  25. Throughout this discussion, we are assuming \(|q|, |t| < 1\) and \(y^{(a)}_{i}/y^{(a)}_{j}\le 1\) when \(i\le j\).

  26. For more details on the meaning of this constant, see [21].

  27. The way we write the deformed Cartan matrix follows the conventions of [19]. This corresponds to setting the bifundamental masses to be \(\mu ^{(a)}_{bif}=v^{(ab)}\) in the corresponding \({\mathfrak {g}}\)-shaped 3d quiver gauge theory, see Sect. 3.4.1. As we pointed out there, if one wishes, one can leave the bifundamental masses arbitrary instead; the price to pay is to slightly modify the definition of the deformed Cartan matrix in this section, which will now contain explicit bifundamental masses in its off-diagonal entries. For details, we refer the reader to [15, 16].

  28. The screening operators we write down are called “magnetic” in [19], and are defined with respect to the parameter q. Another “electric” set of screening and vertex operators can be constructed using the parameter t instead, which rotates the complementary line \({\mathbb {C}}_t\subset {\mathbb {C}}_q\times {\mathbb {C}}_t\), but these will not enter our discussion.

  29. In gauge theory terms, this was a “conformality” condition for the quiver \(T^{5d}\), and the vanishing of the Chern Simons levels in \(G^{3d}\).

  30. When the vacuum is \(|\psi \rangle \) instead of \(|0\rangle \), as it is in the correlator, this two-point is also responsible for a relative shift of one unit of 3d F.I. parameter between the various terms making up the generating current \(W^{(s)}(z)\). This happens because the zero mode in the \({\mathcal {Y}}\)-operator (4.20) acts nontrivially on the vacuum \(|\psi>\).

  31. As an example, by the JK-residue prescription, the locus \(\phi _I-M-\epsilon _+=0\) from (5.7) must lie inside the contour, while the locus \(\phi _I-a_i-\epsilon _+=0\) from (5.2) must lie outside. These poles could therefore pinch the contour, which would result in a singularity at \(a_i=M\); however, the numerator of (5.5) guarantees that this locus is in fact regular. The same pinching contour argument can be given for all the a priori problematic pairs of poles, with the same conclusion.

  32. In the actual correlator, the vacuum is labeled by \(|\psi \rangle \) instead of \(|0\rangle \), resulting in a relative shift of \({\widetilde{{\mathfrak {q}}}}\) between the two terms.

  33. In our conventions, node 2 is the central trivalent node of the \(E_6\) Dynkin diagram, and node 1 is the single node connected to it.

References

  1. Atiyah, M., Hitchin, N.J., Drinfeld, V., Manin, Y.: Construction of instantons. Phys. Lett. A 65, 185–187 (1978). https://doi.org/10.1016/0375-9601(78)90141-X

    Article  MathSciNet  MATH  ADS  Google Scholar 

  2. Tong, D.: The holographic dual of \(AdS_{3} \times S^{3} \times S^{3} \times S^{1}\). JHEP 04, 193 (2014). https://doi.org/10.1007/JHEP04(2014)193

    Article  Google Scholar 

  3. Tong, D., Wong, K.: Instantons, Wilson lines, and D-branes. Phys. Rev. D 91, 026007 (2015). https://doi.org/10.1103/PhysRevD.91.026007

    Article  ADS  Google Scholar 

  4. Douglas, M.R.: Branes within branes. NATO Sci. Ser. C 520, 267–275 (1999)

    MathSciNet  MATH  Google Scholar 

  5. Maldacena, J.M.: Wilson loops in large N field theories. Phys. Rev. Lett. 80, 4859–4862 (1998). https://doi.org/10.1103/PhysRevLett.80.4859

    Article  MathSciNet  MATH  ADS  Google Scholar 

  6. Rey, S.-J., Yee, J.-T.: Macroscopic strings as heavy quarks in large N gauge theory and anti-de Sitter supergravity. Eur. Phys. J. C 22, 379–394 (2001). https://doi.org/10.1007/s100520100799

    Article  MathSciNet  MATH  ADS  Google Scholar 

  7. Drukker, N., Fiol, B.: All-genus calculation of Wilson loops using D-branes. JHEP 02, 010 (2005). https://doi.org/10.1088/1126-6708/2005/02/010

    Article  MathSciNet  ADS  Google Scholar 

  8. Gomis, J., Passerini, F.: Holographic Wilson loops. JHEP 08, 074 (2006). https://doi.org/10.1088/1126-6708/2006/08/074

    Article  MathSciNet  ADS  Google Scholar 

  9. Yamaguchi, S.: Wilson loops of anti-symmetric representation and D5-branes. JHEP 05, 037 (2006). https://doi.org/10.1088/1126-6708/2006/05/037

    Article  MathSciNet  ADS  Google Scholar 

  10. Kim, H.-C.: Line defects and 5d instanton partition functions. JHEP 03, 199 (2016). https://doi.org/10.1007/JHEP03(2016)199

    Article  ADS  Google Scholar 

  11. Assel, B., Sciarappa, A.: Wilson loops in 5d \(\cal{N}=1\) theories and S-duality. JHEP 10, 082 (2018). https://doi.org/10.1007/JHEP10(2018)082

    Article  MATH  Google Scholar 

  12. Chang, C.-M., Ganor, O., Oh, J.: An index for ray operators in 5d \(E_{n}\) SCFTs. JHEP 02, 018 (2017). https://doi.org/10.1007/JHEP02(2017)018

    Article  MATH  Google Scholar 

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

  14. Nekrasov, N.: BPS/CFT correspondence: non-perturbative Dyson–Schwinger equations and qq-characters. JHEP 03, 181 (2016). https://doi.org/10.1007/JHEP03(2016)181

    Article  MATH  ADS  Google Scholar 

  15. Kimura, T., Pestun, V.: Quiver W-algebras. arXiv:1512.08533

  16. Kimura, T., Pestun, V.: Fractional quiver W-algebras. arXiv:1705.04410

  17. Nekrasov, N.: BPS/CFT correspondence IV: sigma models and defects in gauge theory. Lett. Math. Phys. 109, 579–622 (2019). https://doi.org/10.1007/s11005-018-1115-7

    Article  MathSciNet  MATH  ADS  Google Scholar 

  18. Alday, L.F., Gaiotto, D., Tachikawa, Y.: Liouville correlation functions from four-dimensional gauge theories. Lett. Math. Phys. 91, 167–197 (2010). https://doi.org/10.1007/s11005-010-0369-5

    Article  MathSciNet  MATH  ADS  Google Scholar 

  19. Frenkel, E., Reshetikhin, N.: Deformations of \(\cal{W}\)-algebras associated to simple Lie algebras. Commun. Math. Phys. 197, 1–32 (1998)

    MathSciNet  MATH  Google Scholar 

  20. Bouwknegt, P., Pilch, K.: On deformed W algebras and quantum affine algebras. Adv. Theor. Math. Phys. 2, 357–397 (1998). https://doi.org/10.4310/ATMP.1998.v2.n2.a6

    Article  MathSciNet  MATH  Google Scholar 

  21. Aganagic, M., Haouzi, N., Kozcaz, C., Shakirov, S.: Gauge/liouville triality. arXiv:1309.1687

  22. Aganagic, M., Haouzi, N.: ADE little string theory on a Riemann surface (and triality). arXiv:1506.04183

  23. Seiberg, N.: New theories in six-dimensions and Matrix description of M theory on \(T^5\) and \(T^5 / \mathbb{Z}_2\). Phys. Lett. B 408, 98–104 (1997). https://doi.org/10.1016/S0370-2693(97)00805-8

    Article  MathSciNet  Google Scholar 

  24. Losev, A., Moore, G.W., Shatashvili, S.L.: M & m’s. Nucl. Phys. B 522, 105–124 (1998). https://doi.org/10.1016/S0550-3213(98)00262-4

  25. Aharony, O.: A Brief review of ‘little string theories’. Class. Quant. Grav. 17, 929–938 (2000). https://doi.org/10.1088/0264-9381/17/5/302

  26. Agarwal, P., Kim, J., Kim, S., Sciarappa, A.: Wilson surfaces in M5-branes. JHEP 08, 119 (2018). https://doi.org/10.1007/JHEP08(2018)119

    Article  MathSciNet  MATH  ADS  Google Scholar 

  27. Nekrasov, N., Prabhakar, N.S.: Spiked instantons from intersecting D-branes. Nucl. Phys. B 914, 257–300 (2017). https://doi.org/10.1016/j.nuclphysb.2016.11.014

    Article  MATH  ADS  Google Scholar 

  28. Haouzi, N., Kozçaz, C.: The abcdefg of little strings. arXiv:1711.11065

  29. Aganagic, M., Frenkel, E., Okounkov, A.: Quantum q-langlands correspondence. arXiv:1701.03146

  30. Dey, A., Hanany, A., Koroteev, P., Mekareeya, N.: On three-dimensional quiver gauge theories of type B. JHEP 09, 067 (2017). https://doi.org/10.1007/JHEP09(2017)067

    Article  MathSciNet  MATH  ADS  Google Scholar 

  31. Nakajima, H., Weekes, A.: Coulomb branches of quiver gauge theories with symmetrizers. arXiv:1907.06552

  32. Aspinwall, P.S., Gross, M.: The SO(32) heterotic string on a K3 surface. Phys. Lett. B 387, 735–742 (1996). https://doi.org/10.1016/0370-2693(96)01095-7

    Article  MathSciNet  ADS  Google Scholar 

  33. Bershadsky, M., Intriligator, K.A., Kachru, S., Morrison, D.R., Sadov, V., Vafa, C.: Geometric singularities and enhanced gauge symmetries. Nucl. Phys. B 481, 215–252 (1996). https://doi.org/10.1016/S0550-3213(96)90131-5

    Article  MathSciNet  MATH  ADS  Google Scholar 

  34. Cachazo, F., Fiol, B., Intriligator, K.A., Katz, S., Vafa, C.: A Geometric unification of dualities. Nucl. Phys. B 628, 3–78 (2002). https://doi.org/10.1016/S0550-3213(02)00078-0

    Article  MathSciNet  MATH  ADS  Google Scholar 

  35. Nekrasov, N.A.: Seiberg–Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7, 831–864 (2003). https://doi.org/10.4310/ATMP.2003.v7.n5.a4

    Article  MathSciNet  MATH  Google Scholar 

  36. Banks, T., Seiberg, N., Silverstein, E.: Zero and one-dimensional probes with N=8 supersymmetry. Phys. Lett. B 401, 30–37 (1997). https://doi.org/10.1016/S0370-2693(97)00366-3

    Article  MathSciNet  ADS  Google Scholar 

  37. Jeffrey, L., Kirwan, F.: Localization for nonabelian group actions. arXiv:9307001

  38. Hwang, C., Kim, J., Kim, S., Park, J.: General instanton counting and 5d SCFT. JHEP 07, 063 (2015). https://doi.org/10.1007/JHEP07(2015)063, https://doi.org/10.1007/JHEP04(2016)094

  39. Cordova, C., Shao, S.-H.: An index formula for supersymmetric quantum mechanics. arXiv:1406.7853

  40. Hori, K., Kim, H., Yi, P.: Witten index and wall crossing. JHEP 01, 124 (2015). https://doi.org/10.1007/JHEP01(2015)124

    Article  MathSciNet  MATH  ADS  Google Scholar 

  41. Benini, F., Eager, R., Hori, K., Tachikawa, Y.: Elliptic genera of 2d \({\cal{N}}\) = 2 gauge theories. Commun. Math. Phys. 333, 1241–1286 (2015). https://doi.org/10.1007/s00220-014-2210-y

    Article  MathSciNet  MATH  Google Scholar 

  42. Nekrasov, N., Okounkov, A.: Seiberg–Witten theory and random partitions. Prog. Math. 244, 525–596 (2006). https://doi.org/10.1007/0-8176-4467-9_15

    Article  MathSciNet  MATH  Google Scholar 

  43. Brennan, T.D., Dey, A., Moore, G.W.: On ’t Hooft defects, monopole bubbling and supersymmetric quantum mechanics. JHEP 09, 014 (2018). https://doi.org/10.1007/JHEP09(2018)014

  44. Assel, B., Sciarappa, A.: On monopole bubbling contributions to ’t Hooft loops. JHEP 05, 180 (2019). https://doi.org/10.1007/JHEP05(2019)180

  45. Jimbo, M.: A q-analogue of U(g[(N+1)), Hecke algebra, and the Yang–Baxter equation. Lett. Math. Phys. 11, 247 (1986). https://doi.org/10.1007/BF00400222

    Article  MathSciNet  MATH  ADS  Google Scholar 

  46. Drinfeld, V.G.: A New realization of Yangians and quantized affine algebras. Sov. Math. Dokl. 36, 212–216 (1988)

    Google Scholar 

  47. Chari, V., Pressley, A.: Quantum affine algebras and their representations. arXiv:hep-th/9411145

  48. Chari, V.: Minimal affinizations of representations of quantum groups: the U-q(g) module structure. arXiv:hep-th/9411144

  49. Frenkel, E., Reshetikhin, N.: The q-characters of representations of quantum affine algebras and deformations of w-algebras. Contemp. Math. 248 (2000)

  50. Nekrasov, N., Pestun, V., Shatashvili, S.: Quantum geometry and quiver gauge theories. Commun. Math. Phys. 357, 519–567 (2018). https://doi.org/10.1007/s00220-017-3071-y

    Article  MathSciNet  MATH  ADS  Google Scholar 

  51. Bullimore, M., Kim, H.-C., Koroteev, P.: Defects and quantum Seiberg–Witten geometry. JHEP 05, 095 (2015). https://doi.org/10.1007/JHEP05(2015)095

    Article  MathSciNet  MATH  ADS  Google Scholar 

  52. Nakajima, H.: t-analogue of the q-characters of finite dimensional representations of quantum affine algebras. Phys. Combin. (2000). https://doi.org/10.1142/9789812810007_0009

    Article  Google Scholar 

  53. Frenkel, E., Hernandez, D.: Langlands duality for finite-dimensional representations of quantum affine algebras. Lett. Math. Phys. 96, 217–261 (2011). https://doi.org/10.1007/s11005-010-0426-0

    Article  MathSciNet  MATH  ADS  Google Scholar 

  54. Nekrasov, N.A., Shatashvili, S.L.: Quantization of integrable systems and four dimensional gauge theories. In: Proceedings, 16th International Congress on Mathematical Physics (ICMP09): Prague, Czech Republic, Aug 3–8, 2009, pp. 265–289 (2009). https://doi.org/10.1142/9789814304634_0015

  55. Haouzi, N., Schmid, C.: Little string origin of surface defects. JHEP 05, 082 (2017). https://doi.org/10.1007/JHEP05(2017)082

    Article  MathSciNet  MATH  ADS  Google Scholar 

  56. Haouzi, N., Schmid, C.: Little string defects and Bala–Carter theory. arXiv:1612.02008

  57. Assel, B., Gomis, J.: Mirror symmetry and loop operators. JHEP 11, 055 (2015). https://doi.org/10.1007/JHEP11(2015)055

    Article  MathSciNet  MATH  Google Scholar 

  58. Dorey, N.: The BPS spectra of two-dimensional supersymmetric gauge theories with twisted mass terms. JHEP 11, 005 (1998). https://doi.org/10.1088/1126-6708/1998/11/005

    Article  MathSciNet  MATH  ADS  Google Scholar 

  59. Dorey, N., Hollowood, T.J., Tong, D.: The BPS spectra of gauge theories in two-dimensions and four-dimensions. JHEP 05, 006 (1999). https://doi.org/10.1088/1126-6708/1999/05/006

    Article  ADS  Google Scholar 

  60. Dorey, N., Lee, S., Hollowood, T.J.: Quantization of integrable systems and a 2d/4d duality. JHEP 10, 077 (2011). https://doi.org/10.1007/JHEP10(2011)077

    Article  MathSciNet  MATH  ADS  Google Scholar 

  61. Chen, H.-Y., Dorey, N., Hollowood, T.J., Lee, S.: A new 2d/4d duality via integrability. JHEP 09, 040 (2011). https://doi.org/10.1007/JHEP09(2011)040

    Article  MathSciNet  MATH  ADS  Google Scholar 

  62. Nekrasov, N., Witten, E.: The omega deformation, branes, integrability, and liouville theory. JHEP 09, 092 (2010). https://doi.org/10.1007/JHEP09(2010)092

    Article  MathSciNet  MATH  ADS  Google Scholar 

  63. Hwang, C., Yi, P., Yoshida, Y.: Fundamental vortices, wall-crossing, and particle-vortex duality. JHEP 05, 099 (2017). https://doi.org/10.1007/JHEP05(2017)099

    Article  MathSciNet  MATH  ADS  Google Scholar 

  64. Beem, C., Dimofte, T., Pasquetti, S.: Holomorphic blocks in three dimensions. JHEP 12, 177 (2014). https://doi.org/10.1007/JHEP12(2014)177

    Article  MathSciNet  MATH  ADS  Google Scholar 

  65. Gadde, A., Gukov, S., Putrov, P.: Walls, lines, and spectral dualities in 3d gauge theories. JHEP 05, 047 (2014). https://doi.org/10.1007/JHEP05(2014)047

    Article  ADS  Google Scholar 

  66. Yoshida, Y., Sugiyama, K.: Localization of 3d \(\cal N\it =2\) supersymmetric theories on \(S^1 \times D^2\). arXiv:1409.6713

  67. Dimofte, T., Gaiotto, D., Paquette, N.M.: Dual boundary conditions in 3d SCFT’s. JHEP 05, 060 (2018). https://doi.org/10.1007/JHEP05(2018)060

    Article  MathSciNet  MATH  ADS  Google Scholar 

  68. Aganagic, M., Okounkov, A.: Elliptic stable envelope. arXiv:1604.00423

  69. Dotsenko, V.S., Fateev, V.A.: Conformal algebra and multipoint correlation functions in two-dimensional statistical models. Nucl. Phys. B 240, 312 (1984). https://doi.org/10.1016/0550-3213(84)90269-4

    Article  ADS  Google Scholar 

  70. Dijkgraaf, R., Vafa, C.: Toda theories, matrix models, topological strings, and N=2 gauge systems. arXiv:0909.2453

  71. Itoyama, H., Maruyoshi, K., Oota, T.: The quiver matrix model and 2d–4d conformal connection. Prog. Theor. Phys. 123, 957–987 (2010). https://doi.org/10.1143/PTP.123.957

    Article  MATH  ADS  Google Scholar 

  72. Mironov, A., Morozov, A., Shakirov, S.: Conformal blocks as Dotsenko–Fateev integral discriminants. Int. J. Mod. Phys. A 25, 3173–3207 (2010). https://doi.org/10.1142/S0217751X10049141

    Article  MathSciNet  MATH  ADS  Google Scholar 

  73. Morozov, A., Shakirov, S.: The matrix model version of AGT conjecture and CIV-DV prepotential. JHEP 08, 066 (2010). https://doi.org/10.1007/JHEP08(2010)066

    Article  MathSciNet  MATH  ADS  Google Scholar 

  74. Maruyoshi, K.: \(\beta \)-Deformed matrix models and 2d/4d correspondence. arXiv:1412.7124

  75. Bouwknegt, P., Schoutens, K.: W symmetry in conformal field theory. Phys. Rep. 223, 183–276 (1993). https://doi.org/10.1016/0370-1573(93)90111-P

    Article  MathSciNet  ADS  Google Scholar 

  76. Fateev, V.A., Litvinov, A.V.: Correlation functions in conformal Toda field theory. I. JHEP 11, 002 (2007). https://doi.org/10.1088/1126-6708/2007/11/002

    Article  MathSciNet  MATH  ADS  Google Scholar 

  77. 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). https://doi.org/10.1007/BF00398297

    Article  MathSciNet  MATH  ADS  Google Scholar 

  78. Feigin, B., Frenkel, E.: Quantum W algebras and elliptic algebras. Commun. Math. Phys. 178, 653–678 (1996). https://doi.org/10.1007/BF02108819

    Article  MathSciNet  MATH  ADS  Google Scholar 

  79. Awata, H., Kubo, H., Odake, S., Shiraishi, J.: Quantum W(N) algebras and Macdonald polynomials. Commun. Math. Phys. 179, 401–416 (1996). https://doi.org/10.1007/BF02102595

    Article  MathSciNet  MATH  ADS  Google Scholar 

  80. Ding, J., Iohara, K.: Generalization and deformation of drinfeld quantum affine algebras. arXiv:q-alg/9608002

  81. Miki, K.: A \((q, \gamma )\) analog of the \(w_{1+\infty }\) algebra. J. Math. Phys. (2007). https://doi.org/10.1063/1.2823979

    Article  MATH  Google Scholar 

  82. Mironov, A., Morozov, A., Zenkevich, Y.: Ding–Iohara–Miki symmetry of network matrix models. Phys. Lett. B 762, 196–208 (2016). https://doi.org/10.1016/j.physletb.2016.09.033

    Article  MathSciNet  MATH  ADS  Google Scholar 

  83. Aganagic, M., Haouzi, N., Shakirov, S.: \(A_n\)-Triality. arXiv:1403.3657

  84. Kimura, T., Mori, H., Sugimoto, Y.: Refined geometric transition and \(qq\)-characters. JHEP 01, 025 (2018). https://doi.org/10.1007/JHEP01(2018)025

    Article  MathSciNet  MATH  Google Scholar 

  85. Finkelberg, M., Tsymbaliuk, A.: Multiplicative slices, relativistic Toda and shifted quantum affine algebras. arXiv e-prints (Aug 2017). arXiv:1708.01795

Download references

Acknowledgements

We thank Alexei Morozov and Yegor Zenkevich for their patience in explaining to us their computation of the deformed stress tensor expectation value in q-Liouville. We thank Mina Aganagic, Nikita Nekrasov and Luigi Tizzano for discussions and comments at various stages of this project. The research of N. H. is supported by the Simons Center for Geometry and Physics.

Author information

Authors and Affiliations

  1. Institute for Advanced Study, Princeton, NJ, 08540, USA

    Nathan Haouzi

  2. Department of Physics, Boğaziçi University, Istanbul, Turkey

    Can Kozçaz

Authors
  1. Nathan Haouzi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Can Kozçaz
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nathan Haouzi.

Additional information

Communicated by V. Pestun.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Haouzi, N., Kozçaz, C. Supersymmetric Wilson Loops, Instantons, and Deformed \(\mathcal{W}\)-Algebras. Commun. Math. Phys. 393, 669–779 (2022). https://doi.org/10.1007/s00220-022-04375-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-022-04375-0

Search

Navigation