# (0,2) dualities and the 4-simplex

- 38 Downloads

## Abstract

We propose that a simple, Lagrangian 2d \( \mathcal{N} \) = (0, 2) duality interface between the 3d \( \mathcal{N} \) = 2 XYZ model and 3d \( \mathcal{N} \) = 2 SQED can be associated to the simplest triangulated 4-manifold: the 4-simplex. We then begin to flesh out a dictionary between more general triangulated 4-manifolds with boundary and 2d \( \mathcal{N} \) = (0, 2) interfaces. In particular, we identify IR dualities of interfaces associated to local changes of 4d triangulation, governed by the (3), (2, 4), and (2, 4) Pachner moves. We check these dualities using supersymmetric half-indices. We also describe how to produce stand-alone 2d theories (as opposed to interfaces) capturing the geometry of 4-simplices and Pachner moves by making additional field-theoretic choices, and find in this context that the Pachner moves recover abelian \( \mathcal{N} \) = (0, 2) trialities of Gadde-Gukov-Putrov. Our work provides new, explicit tools to explore the interplay between 2d dualities and 4-manifold geometry that has been developed in recent years.

## Keywords

Duality in Gauge Field Theories Supersymmetric Gauge Theory Supersymmetry and Duality## Notes

### **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

## References

- [1]D. Gaiotto,
*N*= 2*dualities*,*JHEP***08**(2012) 034 [arXiv:0904.2715] [INSPIRE].ADSCrossRefGoogle Scholar - [2]D. Gaiotto, G.W. Moore and A. Neitzke,
*Four-dimensional wall-crossing via three-dimensional field theory*,*Commun. Math. Phys.***299**(2010) 163 [arXiv:0807.4723] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [3]D. Gaiotto, G.W. Moore and A. Neitzke,
*Wall-crossing, Hitchin systems and the WKB approximation*, arXiv:0907.3987 [INSPIRE]. - [4]T. Dimofte, S. Gukov and L. Hollands,
*Vortex counting and Lagrangian 3-manifolds*,*Lett. Math. Phys.***98**(2011) 225 [arXiv:1006.0977] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [5]Y. Terashima and M. Yamazaki, SL(2, ℝ)
*Chern-Simons, Liouville and gauge theory on duality walls*,*JHEP***08**(2011) 135 [arXiv:1103.5748] [INSPIRE].ADSCrossRefGoogle Scholar - [6]T. Dimofte, D. Gaiotto and S. Gukov,
*Gauge theories labelled by three-manifolds*,*Commun. Math. Phys.***325**(2014) 367 [arXiv:1108.4389] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [7]
- [8]T. Dimofte, D. Gaiotto and S. Gukov,
*3-manifolds and*3*d indices*,*Adv. Theor. Math. Phys.***17**(2013) 975 [arXiv:1112.5179] [INSPIRE].MathSciNetCrossRefGoogle Scholar - [9]S. Gukov, P. Putrov and C. Vafa,
*Fivebranes and*3*-manifold homology*,*JHEP***07**(2017) 071 [arXiv:1602.05302] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [10]S. Gukov, D. Pei, P. Putrov and C. Vafa,
*BPS spectra and*3*-manifold invariants*, arXiv:1701.06567 [INSPIRE]. - [11]S. Gukov, A.S. Schwarz and C. Vafa,
*Khovanov-Rozansky homology and topological strings*,*Lett. Math. Phys.***74**(2005) 53 [hep-th/0412243] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [12]
- [13]
- [14]B. Assel, S. Schäfer-Nameki and J.-M. Wong, M 5
*-branes on S*^{2}×*M*_{4}*: Nahm*’*s equations and*4*d topological σ-models*,*JHEP***09**(2016) 120 [arXiv:1604.03606] [INSPIRE].ADSCrossRefGoogle Scholar - [15]
- [16]M. Dedushenko, S. Gukov and P. Putrov,
*Vertex algebras and*4*-manifold invariants*, arXiv:1705.01645 [INSPIRE]. - [17]O. Aharony et al.,
*Aspects of N*= 2*supersymmetric gauge theories in three-dimensions*,*Nucl. Phys.***B 499**(1997) 67 [hep-th/9703110] [INSPIRE]. - [18]T. Dimofte, D. Gaiotto and R. van der Veen,
*RG domain walls and hybrid triangulations*,*Adv. Theor. Math. Phys.***19**(2015) 137 [arXiv:1304.6721] [INSPIRE]. - [19]T. Dimofte, D. Gaiotto and N.M. Paquette,
*Dual boundary conditions in*3*d SCFT*’*s*,*JHEP***05**(2018) 060 [arXiv:1712.07654] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [20]A. Gadde, S. Gukov and P. Putrov,
*Walls, lines and spectral dualities in*3*d gauge theories*,*JHEP***05**(2014) 047 [arXiv:1302.0015] [INSPIRE].ADSCrossRefGoogle Scholar - [21]C. Beem, T. Dimofte and S. Pasquetti,
*Holomorphic blocks in three dimensions*,*JHEP***12**(2014) 177 [arXiv:1211.1986] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [22]H.-J. Chung, T. Dimofte, S. Gukov and P. Sulkowski, 3
*d-*3*d correspondence revisited*,*JHEP***04**(2016) 140 [arXiv:1405.3663] [INSPIRE]. - [23]K. Costello, T. Dimofte and D. Gaiotto,
*Boundary chiral algebras*, to appear.Google Scholar - [24]M. Bullimore and A. Ferrari,
*Twisted Hilbert spaces of*3*d supersymmetric gauge theories*,*JHEP***08**(2018) 018 [arXiv:1802.10120] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [25]
- [26]B. Assel and S. Schäfer-Nameki,
*Six-dimensional origin of*\( \mathcal{N} \) = 4*SYM with duality defects*,*JHEP***12**(2016) 058 [arXiv:1610.03663] [INSPIRE]. - [27]C. Lawrie, S. Schäfer-Nameki and T. Weigand,
*Chiral*2*d theories from N*= 4*SYM with varying coupling*,*JHEP***04**(2017) 111 [arXiv:1612.05640] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [28]C. Lawrie, D. Martelli and S. Schäfer-Nameki,
*Theories of class F and anomalies*,*JHEP***10**(2018) 090 [arXiv:1806.06066] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [29]D. Gaiotto and T. Okazaki,
*Dualities of corner configurations and supersymmetric indices*, arXiv:1902.05175 [INSPIRE]. - [30]E.E. Moise,
*Affine structures in 3-manifolds: V. the triangulation theorem and hauptvermutung*,*Ann. Masth.***56**(1952) 96.MathSciNetCrossRefGoogle Scholar - [31]T. Radó,
*Uber den begriff der riemannschen fläche*,*Acta Litt. Sci. Szeged***2**(1925) 101.zbMATHGoogle Scholar - [32]M.H. Freedman et al.,
*The topology of four-dimensional manifolds*,*J. Diff. Geom.***17**(1982) 357.MathSciNetCrossRefGoogle Scholar - [33]R.C. Kirby, L. Siebenmann and L. Siebenmann,
*Foundational essays on topological manifolds, smoothings, and triangulations*. Princeton University Press, Princeton U.S.A. (1977).CrossRefGoogle Scholar - [34]J. Milnor et al.,
*Differential topology forty-six years later*,*Not. A.M.S.***58**(2011) 804.MathSciNetzbMATHGoogle Scholar - [35]C. Manolescu,
*Pin (2)-equivariant Seiberg-Witten floer homology and the triangulation conjecture*,*J. Amer. Math. Soc.***29**(2016) 147 [arXiv:1303.2354].MathSciNetCrossRefGoogle Scholar - [36]U. Pachner,
*Plhomeomorphic manifolds are equivalent by elementary shellings*,*Eur. J. Comb.***12**(1991) 129.CrossRefGoogle Scholar - [37]J.S. Carter, L.H. Kauffman and M. Saito,
*Structures and diagrammatics of four dimensional topological lattice field theories*,*Adv. Math.***146**(1999) 39 [math/9806023]. - [38]R. Kashaev,
*On realizations of Pachner moves in 4D*,*Journal of Knot Theory and Its Ramifications***24**(2015) 1541002 [arXiv:1504.01979] [INSPIRE].MathSciNetCrossRefGoogle Scholar - [39]
- [40]W.P. Thurston,
*The geometry and topology of three-manifolds*, Princeton University Princeton, Princeton U.S.A. (1979).Google Scholar - [41]J.S. Carter and M. Saito,
*Knotted surfaces and their diagrams*, Mathematical Surveys and Monographs volume 55, American Mathematical Society, Providence U.S.A. (1998).Google Scholar - [42]J.A. Hillman,
*Four-manifolds, geometries and knots*, Geometry & Topology Monographs volume 5. Geometry & Topology Publications, Coventry U.K. (2002).Google Scholar - [43]J.G. Ratcliffe and S.T. Tschantz,
*The volume spectrum of hyperbolic*4*-manifolds*,*Exp. Math.***9**(2000) 101.MathSciNetCrossRefGoogle Scholar - [44]D.D. Long and A.W. Reid,
*On the geometric boundaries of hyperbolic 4-manifolds*, math/0007197. - [45]A. Kolpakov and B. Martelli,
*Hyperbolic four-manifolds with one cusp*,*Geom. Funct. Anal.***23**(2013) 1903.MathSciNetCrossRefGoogle Scholar - [46]B. Martelli,
*Hyperbolic four-manifolds*, arXiv:1512.03661. - [47]R. Budney, B. A. Burton and J. Hillman,
*Triangulating a Cappell-Shaneson knot complement*, arXiv:1109.3899. - [48]A. Issa,
*Triangulating cappell-shaneson homotopy 4-spheres*, Master Thesis, University of Melbourne, Melbourne, Australia (2017).Google Scholar - [49]S. Matveev,
*Algorithmic topology and classification of 3-manifolds*, 2^{nd}edition, Algorithms and Computation in Mathematics volume 3, Springer, Berlin Germny (2007).Google Scholar - [50]R. Piergallini,
*Standard moves for standard polyhedra and spines*,*Rend. Circ. Mat. Palermo Suppl.*(1988) 391.Google Scholar - [51]R. Benedetti and C. Petronio,
*A finite graphic calculus for*3*-manifolds*,*Manuscr. Math.***8**(1995) 291.MathSciNetCrossRefGoogle Scholar - [52]G. Amendola,
*A calculus for ideal triangulations of three-manifolds with embedded arcs*,*Math. Nachr.***278**(2005) 975.MathSciNetCrossRefGoogle Scholar - [53]J.H. Rubinstein, H. Segerman and S. Tillmann,
*Traversing three-manifold triangulations and spines*, [arXiv:1812.02806]. - [54]N. Seiberg and E. Witten,
*Electric-magnetic duality, monopole condensation and confinement in N*= 2*supersymmetric Yang-Mills theory*,*Nucl. Phys.***B 426**(1994) 19 [*Erratum ibid.***B 430**(1994) 485] [hep-th/9407087] [INSPIRE]. - [55]S. Cecotti, A. Neitzke and C. Vafa,
*R-twisting and*4*d*/2*d correspondences*, arXiv:1006.3435 [INSPIRE]. - [56]R. Dijkgraaf and E. Witten,
*Topological gauge theories and group cohomology*,*Commun. Math. Phys.***129**(1990) 393 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [57]I. Brunner, J. Schulz and A. Tabler,
*Boundaries and supercurrent multiplets in*3*D Landau-Ginzburg models*,*JHEP***06**(2019) 046 [arXiv:1904.07258] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [58]M. Roček, K. Roumpedakis and S. Seifnashri, 3
*D dualities and supersymmetry enhancement from domain walls*, arXiv:1904.02722 [INSPIRE]. - [59]H. Jockers and P. Mayr,
*A*3*D gauge theory/quantum k-theory correspondence*, arXiv:1808.02040 [INSPIRE]. - [60]N.P. Warner,
*Supersymmetric, integrable boundary field theories*,*Nucl. Phys. Proc. Suppl.***45A**(1996) 154 [hep-th/9512183] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [61]E. Witten, SL(2, ℤ)
*action on three-dimensional conformal field theories with Abelian symmetry*, hep-th/0307041 [INSPIRE]. - [62]Y. Yoshida and K. Sugiyama,
*Localization of*3*d*\( \mathcal{N} \) = 2*supersymmetric theories on S*^{1}×*D*^{2}, arXiv:1409.6713 [INSPIRE]. - [63]Y. Imamura and S. Yokoyama,
*Index for three dimensional superconformal field theories with general R-charge assignments*,*JHEP***04**(2011) 007 [arXiv:1101.0557] [INSPIRE]. - [64]A. Kapustin and B. Willett,
*Generalized superconformal index for three dimensional field theories*, arXiv:1106.2484 [INSPIRE]. - [65]F. Benini, R. Eager, K. Hori and Y. Tachikawa,
*Elliptic genera of two-dimensional N*= 2*gauge theories with rank-one gauge groups*,*Lett. Math. Phys.***104**(2014) 465 [arXiv:1305.0533] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [66]F. Benini, R. Eager, K. Hori and Y. Tachikawa,
*Elliptic genera of*2*d*\( \mathcal{N} \) = 2*gauge theories*,*Commun. Math. Phys.***333**(2015) 1241 [arXiv:1308.4896] [INSPIRE]. - [67]W. Jaco and J.H. Rubinstein,
*Layered-triangulations of 3-manifolds*, math/0603601. - [68]C. Vafa and E. Witten,
*A strong coupling test of S duality*,*Nucl. Phys.***B 431**(1994) 3 [hep-th/9408074] [INSPIRE]. - [69]
- [70]H. Nakajima,
*Instantons and affine Lie algebras*,*Nucl. Phys. Proc. Suppl.***46**(1996) 154 [alg-geom/9510003] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [71]I. Grojnowski,
*Instantons and affine algebras I: the Hilbert scheme and vertex operators*, alg-geom/9506020 [INSPIRE]. - [72]T. Dimofte, M. Gabella and A.B. Goncharov,
*K-decompositions and*3*d gauge theories*,*JHEP***11**(2016) 151 [arXiv:1301.0192] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [73]D. Gang and K. Yonekura,
*Symmetry enhancement and closing of knots in*3*d/*3*d correspondence*,*JHEP***07**(2018) 145 [arXiv:1803.04009] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar - [74]A. Gadde and S. Gukov, 2
*d index and surface operators*,*JHEP***03**(2014) 080 [arXiv:1305.0266] [INSPIRE].