Abstract
We study the hemisphere partition function of a three-dimensional \( \mathcal{N} \) = 4 supersymmetric U(N) gauge theory with one adjoint and one fundamental hypermultiplet — the ADHM quiver theory. In particular, we propose a distinguished set of UV boundary conditions which yield Verma modules of the quantised chiral rings of the Higgs and Coulomb branches. In line with a recent proposal by two of the authors in collaboration with M. Bullimore, we show explicitly that the hemisphere partition functions recover the characters of these modules in two limits, and realise blocks gluing exactly to the partition functions of the theory on closed three-manifolds. We study the geometry of the vortex moduli space and investigate the interpretation of the vortex partition functions as equivariant indices of quasimaps to the Hilbert scheme of points in ℂ2. We also investigate half indices of the ADHM quiver gauge theory in the presence of a line operator and discuss their geometric interpretation. Along the way we find interesting relations between our hemisphere blocks and related quantities in topological string theory and equivariant quantum K-theory.
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M. Bullimore, T. Dimofte, D. Gaiotto, J. Hilburn and H.-C. Kim, Vortices and Vermas, Adv. Theor. Math. Phys. 22 (2018) 803 [arXiv:1609.04406] [INSPIRE].
M. Aganagic, E. Frenkel and A. Okounkov, Quantum q-Langlands Correspondence, Trans. Moscow Math. Soc. 79 (2018) 1 [arXiv:1701.03146] [INSPIRE].
M. Bullimore, T. Dimofte and D. Gaiotto, The Coulomb Branch of 3d \( \mathcal{N} \) = 4 Theories, Commun. Math. Phys. 354 (2017) 671 [arXiv:1503.04817] [INSPIRE].
L. Rozansky and E. Witten, HyperKähler geometry and invariants of three manifolds, Selecta Math. 3 (1997) 401 [hep-th/9612216] [INSPIRE].
H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional \( \mathcal{N} \) = 4 gauge theories, I, Adv. Theor. Math. Phys. 20 (2016) 595 [arXiv:1503.03676] [INSPIRE].
A. Braverman, M. Finkelberg and H. Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional \( \mathcal{N} \) = 4 gauge theories, II, Adv. Theor. Math. Phys. 22 (2018) 1071 [arXiv:1601.03586] [INSPIRE].
S. Cremonesi, A. Hanany and A. Zaffaroni, Monopole operators and Hilbert series of Coulomb branches of 3d \( \mathcal{N} \) = 4 gauge theories, JHEP 01 (2014) 005 [arXiv:1309.2657] [INSPIRE].
M. Bullimore, A. Ferrari and H. Kim, Twisted Indices of 3d \( \mathcal{N} \) = 4 Gauge Theories and Enumerative Geometry of Quasi-Maps, JHEP 07 (2019) 014 [arXiv:1812.05567] [INSPIRE].
M. Bullimore, A.E.V. Ferrari and H. Kim, The 3d Twisted Index and Wall-Crossing, arXiv:1912.09591 [INSPIRE].
K. Costello, T. Creutzig and D. Gaiotto, Higgs and Coulomb branches from vertex operator algebras, JHEP 03 (2019) 066 [arXiv:1811.03958] [INSPIRE].
A. Kapustin, B. Willett and I. Yaakov, Nonperturbative Tests of Three-Dimensional Dualities, JHEP 10 (2010) 013 [arXiv:1003.5694] [INSPIRE].
D. Gang, E. Koh, K. Lee and J. Park, ABCD of 3d \( \mathcal{N} \) = 8 and 4 Superconformal Field Theories, arXiv:1108.3647 [INSPIRE].
F. Benini, K. Hristov and A. Zaffaroni, Black hole microstates in AdS4 from supersymmetric localization, JHEP 05 (2016) 054 [arXiv:1511.04085] [INSPIRE].
F. Benini, K. Hristov and A. Zaffaroni, Exact microstate counting for dyonic black holes in AdS4, Phys. Lett. B 771 (2017) 462 [arXiv:1608.07294] [INSPIRE].
S. Choi and C. Hwang, Universal 3d Cardy Block and Black Hole Entropy, JHEP 03 (2020) 068 [arXiv:1911.01448] [INSPIRE].
S. Choi, C. Hwang and S. Kim, Quantum vortices, M2-branes and black holes, arXiv:1908.02470 [INSPIRE].
H. Nakajima, Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994) 365 [INSPIRE].
T. Braden, A. Licata, N. Proudfoot and B. Webster, Quantizations of conical symplectic resolutions II: category O and symplectic duality, arXiv:1407.0964 [INSPIRE].
T. Braden, A. Licata, N. Proudfoot and B. Webster, Quantizations of conical symplectic resolutions, Astérisque (2016) iii–iv.
H. Nakajima, Quiver varieties and kac-moody algebras, Duke Math. J. 91 (1998) 515.
M. Aganagic and A. Okounkov, Quasimap counts and Bethe eigenfunctions, Moscow Math. J. 17 (2017) 565 [arXiv:1704.08746] [INSPIRE].
A. Smirnov, Quantum difference equations for quiver varieties, Ph.D. Thesis, Columbia U. (2016) [DOI] [INSPIRE].
P. Koroteev, P.P. Pushkar, A. Smirnov and A.M. Zeitlin, Quantum k-theory of Quiver Varieties and Many-Body Systems, arXiv:1705.10419 [INSPIRE].
C. Beem, T. Dimofte and S. Pasquetti, Holomorphic Blocks in Three Dimensions, JHEP 12 (2014) 177 [arXiv:1211.1986] [INSPIRE].
S. Pasquetti, Factorisation of N = 2 Theories on the Squashed 3-Sphere, JHEP 04 (2012) 120 [arXiv:1111.6905] [INSPIRE].
C. Hwang, H.-C. Kim and J. Park, Factorization of the 3d superconformal index, JHEP 08 (2014) 018 [arXiv:1211.6023] [INSPIRE].
A. Cabo-Bizet, Factorising the 3D Topological ly Twisted Index, JHEP 04 (2017) 115 [arXiv:1606.06341] [INSPIRE].
S. Crew, N. Dorey and D. Zhang, Factorisation of 3d \( \mathcal{N} \) = 4 twisted indices and the geometry of vortex moduli space, JHEP 08 (2020) 015 [arXiv:2002.04573] [INSPIRE].
F. Benini and W. Peelaers, Higgs branch localization in three dimensions, JHEP 05 (2014) 030 [arXiv:1312.6078] [INSPIRE].
M. Fujitsuka, M. Honda and Y. Yoshida, Higgs branch localization of 3d \( \mathcal{N} \) = 2 theories, PTEP 2014 (2014) 123B02 [arXiv:1312.3627] [INSPIRE].
M. Bullimore, S. Crew and D. Zhang, Boundaries, Vermas, and Factorisation, arXiv:2010.09741 [INSPIRE].
M. Bullimore, T. Dimofte, D. Gaiotto and J. Hilburn, Boundaries, Mirror Symmetry, and Symplectic Duality in 3d \( \mathcal{N} \) = 4 Gauge Theory, JHEP 10 (2016) 108 [arXiv:1603.08382] [INSPIRE].
M.F. Atiyah, N.J. Hitchin, V.G. Drinfeld and Y.I. Manin, Construction of Instantons, Phys. Lett. A 65 (1978) 185 [INSPIRE].
N. Nekrasov and A.S. Schwarz, Instantons on noncommutative R4 and (2,0) superconformal six-dimensional theory, Commun. Math. Phys. 198 (1998) 689 [hep-th/9802068] [INSPIRE].
O. Aharony, O. Bergman, D.L. Jafferis and J. Maldacena, N = 6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals, JHEP 10 (2008) 091 [arXiv:0806.1218] [INSPIRE].
K. Costello, Holography and Koszul duality: the example of the M2 brane, arXiv:1705.02500 [INSPIRE].
R. Kodera and H. Nakajima, Quantized Coulomb branches of Jordan quiver gauge theories and cyclotomic rational Cherednik algebras, Proc. Symp. Pure Math. 98 (2018) 49 [arXiv:1608.00875] [INSPIRE].
Y. Yoshida and K. Sugiyama, Localization of three-dimensional \( \mathcal{N} \) = 2 supersymmetric theories on S1 × D2, PTEP 2020 (2020) 113B02 [arXiv:1409.6713] [INSPIRE].
H. Nakajima, Lectures on Hilbert Schemes of Points on Surfaces, University Lecture Series, American Mathematical Society (1999).
A. Smirnov, Elliptic stable envelope for Hilbert scheme of points in the plane, arXiv:1804.08779 [INSPIRE].
I.G. Macdonald, Symmetric functions and Hall polynomials, Oxford University Press (1998).
J. de Boer, K. Hori, H. Ooguri and Y. Oz, Mirror symmetry in three-dimensional gauge theories, quivers and D-branes, Nucl. Phys. B 493 (1997) 101 [hep-th/9611063] [INSPIRE].
M. Porrati and A. Zaffaroni, M theory origin of mirror symmetry in three-dimensional gauge theories, Nucl. Phys. B 490 (1997) 107 [hep-th/9611201] [INSPIRE].
K.A. Intriligator and N. Seiberg, Mirror symmetry in three-dimensional gauge theories, Phys. Lett. B 387 (1996) 513 [hep-th/9607207] [INSPIRE].
N. Bobev, M. Bullimore and H.-C. Kim, Supersymmetric Casimir Energy and the Anomaly Polynomial, JHEP 09 (2015) 142 [arXiv:1507.08553] [INSPIRE].
T. Dimofte, D. Gaiotto and N.M. Paquette, Dual boundary conditions in 3d SCFT’s, JHEP 05 (2018) 060 [arXiv:1712.07654] [INSPIRE].
H. Dinkins and A. Smirnov, Characters of tangent spaces at torus fixed points and 3d-mirror symmetry, arXiv:1908.01199 [INSPIRE].
S. Crew, H. Dinkins and D. Zhang, Hemisphere Blocks and Mirror Symmetry of Twisted Indices, in preparation.
A. Okounkov, Lectures on K-theoretic computations in enumerative geometry, arXiv:1512.07363 [INSPIRE].
I. Ciocan-Fontanine, M. Konvalinka and I. Pak, Quantum cohomology of Hilbn (ℂ2) and the weighted hook walk on young diagrams, J. Algebra 349 (2012) 268.
A. Smirnov, Rationality of capped descendent vertex in K-theory, arXiv:1612.01048 [INSPIRE].
A. Okounkov and A. Smirnov, Quantum difference equation for Nakajima varieties, arXiv:1602.09007 [INSPIRE].
A. Okounkov, On the crossroads of enumerative geometry and geometric representation theory, arXiv:1801.09818 [INSPIRE].
Y. Kononov, A. Okounkov and A. Osinenko, The 2-leg vertex in K-theoretic DT theory, arXiv:1905.01523 [INSPIRE].
A. Iqbal, C. Kozcaz and C. Vafa, The Refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [INSPIRE].
G. Bonelli, A. Sciarappa, A. Tanzini and P. Vasko, Vortex partition functions, wall crossing and equivariant Gromov-Witten invariants, Commun. Math. Phys. 333 (2015) 717 [arXiv:1307.5997] [INSPIRE].
G. Bonelli, A. Sciarappa, A. Tanzini and P. Vasko, The Stringy Instanton Partition Function, JHEP 01 (2014) 038 [arXiv:1306.0432] [INSPIRE].
D. Gaiotto and J. Oh, Aspects of Ω-deformed M-theory, arXiv:1907.06495 [INSPIRE].
D. Gaiotto and T. Okazaki, Sphere correlation functions and Verma modules, JHEP 02 (2020) 133 [arXiv:1911.11126] [INSPIRE].
A. Braverman, M. Finkelberg and J. Shiraishi, Macdonald polynomials, Laumon spaces and perverse coherent sheaves, arXiv:1206.3131.
E.R. Gansner, The hil lman-grassl correspondence and the enumeration of reverse plane partitions, J. Combin. Theor. A 30 (1981) 71.
H. Awata and H. Kanno, Refined BPS state counting from Nekrasov’s formula and Macdonald functions, Int. J. Mod. Phys. A 24 (2009) 2253 [arXiv:0805.0191] [INSPIRE].
M. Aganagic, A. Klemm, M. Mariño and C. Vafa, The Topological vertex, Commun. Math. Phys. 254 (2005) 425 [hep-th/0305132] [INSPIRE].
A. Iqbal and A.-K. Kashani-Poor, The Vertex on a strip, Adv. Theor. Math. Phys. 10 (2006) 317 [hep-th/0410174] [INSPIRE].
M. Taki, Refined Topological Vertex and Instanton Counting, JHEP 03 (2008) 048 [arXiv:0710.1776] [INSPIRE].
C. Closset and H. Kim, Comments on twisted indices in 3d supersymmetric gauge theories, JHEP 08 (2016) 059 [arXiv:1605.06531] [INSPIRE].
F. Benini and A. Zaffaroni, A topological ly twisted index for three-dimensional supersymmetric theories, JHEP 07 (2015) 127 [arXiv:1504.03698] [INSPIRE].
T. Dimofte and S. Gukov, Refined, Motivic, and Quantum, Lett. Math. Phys. 91 (2010) 1 [arXiv:0904.1420] [INSPIRE].
M. Bershadsky, C. Vafa and V. Sadov, D-branes and topological field theories, Nucl. Phys. B 463 (1996) 420 [hep-th/9511222] [INSPIRE].
S.M. Hosseini and N. Mekareeya, Large N topological ly twisted index: necklace quivers, dualities, and Sasaki-Einstein spaces, JHEP 08 (2016) 089 [arXiv:1604.03397] [INSPIRE].
A.P. Polychronakos, Quantum Hall states as matrix Chern-Simons theory, JHEP 04 (2001) 011 [hep-th/0103013] [INSPIRE].
N. Dorey, D. Tong and C. Turner, Matrix model for non-Abelian quantum Hal l states, Phys. Rev. B 94 (2016) 085114 [arXiv:1603.09688] [INSPIRE].
N. Dorey, D. Tong and C. Turner, A Matrix Model for WZW, JHEP 08 (2016) 007 [arXiv:1604.05711] [INSPIRE].
A. Hanany and D. Tong, Vortices, instantons and branes, JHEP 07 (2003) 037 [hep-th/0306150] [INSPIRE].
H. Nakajima, Handsaw quiver varieties and finite W-algebras, Moscow Math. J. 12 (2012) 633 [arXiv:1107.5073] [INSPIRE].
V. Pestun, Review of localization in geometry, J. Phys. A 50 (2017) 443002 [arXiv:1608.02954] [INSPIRE].
T. Ekholm, P. Kucharski and P. Longhi, Physics and geometry of knots-quivers correspondence, Commun. Math. Phys. 379 (2020) 361 [arXiv:1811.03110] [INSPIRE].
H. Awata, S. Odake and J. Shiraishi, Integral representations of the Macdonald symmetric functions, Commun. Math. Phys. 179 (1996) 647 [q-alg/9506006] [INSPIRE].
P. Di Francesco and R. Kedem, Difference equations for graded characters from quantum cluster algebra, Transform. Groups 23 (2018) 391.
A.N. Kirillov and M. Noumi, Affine hecke algebras and raising operators for macdonald polynomials, q-alg/9605004.
A. Nedelin, S. Pasquetti and Y. Zenkevich, T[SU(N)] duality webs: mirror symmetry, spectral duality and gauge/CFT correspondences, JHEP 02 (2019) 176 [arXiv:1712.08140] [INSPIRE].
N. Dorey and D. Zhang, Superconformal quantum mechanics on Kähler cones, JHEP 05 (2020) 115 [arXiv:1911.06787] [INSPIRE].
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Crew, S., Dorey, N. & Zhang, D. Blocks and vortices in the 3d ADHM quiver gauge theory. J. High Energ. Phys. 2021, 234 (2021). https://doi.org/10.1007/JHEP03(2021)234
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DOI: https://doi.org/10.1007/JHEP03(2021)234