Instanton Counting, Quantum Geometry and Algebra

The aim of this memoir for “Habilitation à Diriger des Recherches” is to present quantum geometric and algebraic aspects of supersymmetric gauge theory, which emerge from non-perturbative nature of the vacuum structure induced by instantons. We start with a brief summary of the equivariant localization of the instanton moduli space, and show how to obtain the instanton partition function and its generalization to quiver gauge theory and supergroup gauge theory in three ways: the equivariant index formula, the contour integral formula, and the combinatorial formula. We then explore the geometric description of N = 2 gauge theory based on Seiberg–Witten geometry together with its string/M-theory perspective. Through its relation to integrable systems, we show how to quantize such a geometric structure via the Ω-deformation of gauge theory. We also discuss the underlying quantum algebraic structure arising from the supersymmetric vacua. We introduce the notion of quiver W-algebra constructed through double quantization of Seiberg–Witten geometry, and show its specific features: affine quiver W-algebras, fractional quiver W-algebras, and their elliptic deformations.

[1]  Yuji Tachikawa.  N=2 Supersymmetric Dynamics for Pedestrians , 2013, 1312.2684.

[2]  N. Nekrasov.  BPS/CFT Correspondence III: Gauge Origami Partition Function and qq-Characters , 2017, 1701.00189.

[3]  Webs of (p,q) 5-branes, five dimensional field theories and grid diagrams , 1997, hep-th/9710116.

[4]  Andrei Okounkov,et al.  Quantum Calabi-Yau and Classical Crystals , 2003 .

[5]  Taro Kimura,et al.  Quantum integrability from non-simply laced quiver gauge theory , 2018, Journal of High Energy Physics.

[6]  R. Dijkgraaf,et al.  Quantum Curves and D-Modules , 2008, 0810.4157.

[7]  J. Gomis,et al.  M2-brane surface operators and gauge theory dualities in Toda , 2014, 1407.1852.

[8]  M. Zabzine,et al.  Elliptic modular double and 4d partition functions , 2017, 1703.04614.

[9]  Richard J. Szabo.  Equivariant Cohomology and Localization of Path Integrals , 2000 .

[10]  A. Kapustin,et al.  Exact results for Wilson loops in superconformal Chern-Simons theories with matter , 2009, 0909.4559.

[11]  黒川 信重.  Multiple Sine Functions , 1993 .

[12]  Shlomo S. Razamat,et al.  Bootstrapping the superconformal index with surface defects , 2012, 1207.3577.

[13]  B. Feigin,et al.  Plane partitions with a “pit”: generating functions and representation theory , 2015, 1512.08779.

[14]  S. Rey,et al.  Five-dimensional gauge theories from shifted web diagrams , 2018, Physical Review D.

[15]  E. Witten.  Monopoles and four-manifolds , 1994, hep-th/9411102.

[16]  D. Gaiotto.  Asymptotically free = 2 theories and irregular conformal blocks , 2009, 0908.0307.

[17]  E. Witten,et al.  Supersymmetric Yang-Mills theory and integrable systems , 1995, hep-th/9510101.

[18]  G. Bonelli,et al.  Gauge Theories on ALE Space and Super Liouville Correlation Functions , 2011, 1107.4609.

[19]  N. Drukker,et al.  A supermatrix model for N = 6 super Chern-Simons-matter theory , 2010 .

[20]  T. Okuda,et al.  Exact results for boundaries and domain walls in 2d supersymmetric theories , 2013, 1308.2217.

[21]  A. Morozov,et al.  Generation of matrix models by Ŵ-operators , 2009 .

[22]  Ghost D-branes , 2006, hep-th/0601024.

[23]  X. Yin,et al.  Notes on superconformal Chern-Simons-Matter theories , 2007, 0704.3740.

[24]  J. Duistermaat,et al.  On the variation in the cohomology of the symplectic form of the reduced phase space , 1982 .

[25]  Fabrizio Nieri.  An elliptic Virasoro symmetry in 6d , 2015, Letters in Mathematical Physics.

[26]  Lectures on Seiberg-Witten Invariants , 1995, alg-geom/9510012.

[27]  Taro Kimura.  Integrating over quiver variety and BPS/CFT correspondence , 2019, Letters in Mathematical Physics.

[28]  C. Vafa,et al.  Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions , 2007, 0709.4446.

[29]  S. Rey,et al.  Dual little strings and their partition functions , 2017, 1710.02455.

[30]  On the K-theory of the cyclic quiver variety , 1999, math/9902091.

[31]  O. Schiffmann,et al.  The elliptic Hall algebra, Cherednik Hecke algebras and Macdonald polynomials , 2008, Compositio Mathematica.

[32]  H. Osborn,et al.  Applications of the superconformal index for protected operators and q-hypergeometric identities to N=1 dual theories , 2008, 0801.4947.

[33]  L. Alday,et al.  Liouville Correlation Functions from Four-Dimensional Gauge Theories , 2009, 0906.3219.

[34]  V. Pestun,et al.  Quantum Geometry and Quiver Gauge Theories , 2013, 1312.6689.

[35]  L. Rastelli,et al.  Infinite Chiral Symmetry in Four Dimensions , 2013, 1312.5344.

[36]  H. Awata,et al.  Five-dimensional AGT conjecture and the deformed Virasoro algebra , 2009, 0910.4431.

[37]  M. Jimbo,et al.  Solitons and Infinite Dimensional Lie Algebras , 1983 .

[38]  Geometric engineering of quantum field theories , 1996, hep-th/9609239.

[39]  On the quantum moduli space of vacua of N = 2 supersymmetric SU(Nc) gauge theories , 1995, hep-th/9505075.

[40]  O. Schiffmann,et al.  The elliptic Hall algebra and the equivariant K-theory of the Hilbert scheme of $\mathbb{A}^2$ , 2009, 0905.2555.

[41]  B.Feigin,et al.  Super Liouville conformal blocks from N=2 SU(2) quiver gauge theories , 2011, 1105.5800.

[42]  Taro Kimura,et al.  Refined geometric transition and qq-characters , 2017, 1705.03467.

[43]  Yuji Tachikawa.  A brief review of the 2d/4d correspondences , 2016, 1608.02964.

[44]  O. W. Greenberg.  Spin and Unitary Spin Independence in a Paraquark Model of Baryons and Mesons , 1964 .

[45]  S. Donaldson.  Geometry of four-manifolds , 1990 .

[46]  Rvovv Utivama.  Invariant Theoretical Interpretation of Interaction , 2011 .

[47]  M. Flohr,et al.  Conformal Field Theory , 2006 .

[48]  Shlomo S. Razamat,et al.  The 4d Superconformal Index from q-deformed 2d Yang-Mills , 2011, 1104.3850.

[49]  田中 正,et al.  SUPERSTRING THEORY , 1989, The Lancet.

[50]  D. Gaiotto.  Preprint Typeset in Jhep Style -hyper Version N = 2 Dualities , 2022 .

[51]  Andrei Okounkov,et al.  Seiberg-Witten theory and random partitions , 2003, hep-th/0306238.

[52]  E. Witten,et al.  Branes and Supergroups , 2014, 1410.1175.

[53]  Edward Witten,et al.  Topological sigma models , 1988 .

[54]  V. Pestun.  Localization of Gauge Theory on a Four-Sphere and Supersymmetric Wilson Loops , 2007, 0712.2824.

[55]  M. Gell-Mann.  THE EIGHTFOLD WAY: A THEORY OF STRONG INTERACTION SYMMETRY , 1961 .

[56]  Taro Kimura,et al.  Fractional quiver W-algebras , 2017, Letters in Mathematical Physics.

[57]  Two-dimensional gauge theories revisited , 1992, hep-th/9204083.

[58]  Nathan Haouzi.  Quantum geometry and θ-angle in five-dimensional super Yang-Mills , 2020, Journal of High Energy Physics.

[59]  T. Quella,et al.  Superspace conformal field theory , 2013, 1307.7724.

[60]  Nathan Seiberg,et al.  String theory and noncommutative geometry , 1999 .

[61]  Alex Weekes,et al.  Coulomb branches of quiver gauge theories with symmetrizers , 2019, Journal of the European Mathematical Society.

[62]  O. Aharony,et al.  Fractional M2-branes , 2008, 0807.4924.

[63]  J. Manschot,et al.  Generalized quiver mutations and single-centered indices , 2013, 1309.7053.

[64]  S. Rey,et al.  Dual little strings from F-theory and flop transitions , 2016, 1610.07916.

[65]  O. Schiffmann.  Drinfeld realization of the elliptic Hall algebra , 2010, 1004.2575.

[66]  Lakshya Bhardwaj.  Classification of 6d N=(1,0) gauge theories , 2015, 1502.06594.

[67]  B. Feigin,et al.  Yangians and cohomology rings of Laumon spaces , 2008, 0812.4656.

[68]  J. L. Lopes,et al.  A model for leptons , 1977 .

[69]  L. Clavelli,et al.  Pomeron factorization in general dual models , 1973 .

[70]  M. Atiyah,et al.  Construction of Instantons , 1978 .

[71]  A Combinatorial Study on Quiver Varieties , 2005, math/0510455.

[72]  M. Zabzine,et al.  q-Virasoro Modular Double and 3d Partition Functions , 2016, Communications in Mathematical Physics.

[73]  Tom Rudelius,et al.  F-theory and the Classification of Little Strings , 2015, 1511.05565.

[74]  S. Rey,et al.  Beyond triality: dual quiver gauge theories and little string theories , 2018, Journal of High Energy Physics.

[75]  H. Nakajima,et al.  Yang-Mills instantons on ALE gravitational instantons , 1990 .

[76]  N. Nekrasov,et al.  Magnificent Four with Colors , 2018, Communications in Mathematical Physics.

[77]  B. Pioline,et al.  Wall-crossing, Rogers dilogarithm, and the QK/HK correspondence , 2011, 1110.0466.

[78]  P. Cai,et al.  Reliable Perturbative Results for Strong Interactions ? , 2011 .

[79]  B. Feigin,et al.  A commutative algebra on degenerate CP^1 and Macdonald polynomials , 2009, 0904.2291.

[80]  E. Witten,et al.  Monopoles, duality and chiral symmetry breaking in N=2 supersymmetric QCD , 1994, hep-th/9408099.

[81]  C. Keller,et al.  From SO/Sp instantons to W-algebra blocks , 2010, 1012.4468.

[82]  B. Eynard,et al.  Invariants of algebraic curves and topological expansion , 2007, math-ph/0702045.

[83]  D-Particle Bound States and Generalized Instantons , 1998, hep-th/9803265.

[84]  Noppadol Mekareeya,et al.  On three-dimensional quiver gauge theories of type B , 2016, Journal of High Energy Physics.

[85]  E. Frenkel,et al.  Langlands Duality for Finite-Dimensional Representations of Quantum Affine Algebras , 2009, 0902.0447.

[86]  Ron Y. Donagi.  Seiberg-Witten integrable systems , 1997 .

[87]  Type IIB Superstrings, BPS Monopoles, And Three-Dimensional Gauge Dynamics , 1996, hep-th/9611230.

[88]  T. Nakano,et al.  Charge Independence for V-particles , 1953 .

[89]  N. Nekrasov.  BPS/CFT correspondence: non-perturbative Dyson-Schwinger equations and qq-characters , 2015, 1512.05388.

[90]  M. Finkelberg,et al.  Quantization of Drinfeld Zastava in type A , 2010, 1306.5427.

[91]  A. Shapere,et al.  Coulomb Phase of N=2 Supersymmetric QCD. , 1995, Physical review letters.

[92]  N. Dorey,et al.  Quantization of integrable systems and a 2d/4d duality , 2011, 1103.5726.

[93]  M. HAx.  Three-Triplet Model with Double SU ( 3 ) Symmetry * , 2011 .

[94]  M. Mariño,et al.  Exact results in ABJM theory from topological strings , 2009, 0912.3074.

[95]  H. Nakajima.  Quiver varieties and Kac-Moody algebras , 1998 .

[96]  G. Bonelli,et al.  Vertices, vortices & interacting surface operators , 2011, 1102.0184.

[97]  Y. Zenkevich,et al.  Quiver W ϵ 1 , ϵ 2 algebras of 4D N = 2 gauge theories , 2019, 1912.09969.

[98]  V. Dotsenko,et al.  Four-point correlation functions and the operator algebra in 2D conformal invariant theories with central charge C≤1 , 1984 .

[99]  S. Glashow.  Partial Symmetries of Weak Interactions , 1961 .

[100]  M. Zabzine,et al.  3D mirror symmetry fromS-duality , 2018, Physical Review D.

[101]  Kota Yoshioka,et al.  Instanton counting on blowup. I. 4-dimensional pure gauge theory , 2003, math/0306198.

[102]  H. Knight.  Spectra of Tensor Products of Finite Dimensional Representations of Yangians , 1995 .

[103]  K. Schoutens,et al.  W symmetry in conformal field theory , 1992, hep-th/9210010.

[104]  Instanton counting, Macdonald function and the moduli space of D-branes , 2005, hep-th/0502061.

[105]  Taro Kimura,et al.  Vortex counting from field theory , 2012, 1204.1968.

[106]  M. Atiyah,et al.  Deformations of instantons. , 1977, Proceedings of the National Academy of Sciences of the United States of America.

[107]  N. Dorey,et al.  A new 2d/4d duality via integrability , 2011, 1104.3021.

[109]  N. Nekrasov,et al.  ABCD of Instantons , 2004, hep-th/0404225.

[110]  Kantaro Ohmori,et al.  2-Group Symmetries of 6D Little String Theories and T-Duality , 2021, Annales Henri Poincare.

[111]  Sayed Fawad Hassan,et al.  Introduction to S-Duality in N = 2 Supersymmetric Gauge Theories , 1997 .

[112]  K. Miki.  A (q,γ) analog of the W1+∞ algebra , 2007 .

[113]  Michael Atiyah,et al.  The moment map and equivariant cohomology , 1984 .

[114]  Joonho Kim,et al.  Wilson surfaces in M5-branes , 2018, Journal of High Energy Physics.

[115]  Denis Bernard,et al.  Introduction to classical integrable systems , 2003 .

[116]  N. Reshetikhin,et al.  Towards the classification of completely integrable quantum field theories (the Bethe-Ansatz associated with dynkin diagrams and their automorphisms) , 1987 .

[117]  C. Vafa,et al.  Negative branes, supergroups and the signature of spacetime , 2016, 1603.05665.

[118]  S. Hawking,et al.  Gravitational Multi - Instantons , 1978 .

[119]  Raoul Bott,et al.  The Yang-Mills equations over Riemann surfaces , 1983, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[120]  N. Nekrasov.  BPS/CFT correspondence IV: sigma models and defects in gauge theory , 2017, Letters in Mathematical Physics.

[121]  Small instantons in string theory , 1995, hep-th/9511030.

[122]  Taro Kimura,et al.  Web construction of ABCDEFG and affine quiver gauge theories , 2019, Journal of High Energy Physics.

[123]  P. Kronheimer.  The construction of ALE spaces as hyper-Kähler quotients , 1989 .

[124]  L. C. Jeffrey,et al.  Localization for nonabelian group actions , 1993 .

[125]  I. Bernstein,et al.  DIFFERENCE AND DIFFERENTIAL EQUATIONS FOR THE COLORED JONES FUNCTION , 2007 .

[126]  The q-characters of representations of quantum affine algebras and deformations of W-algebras , 1998, math/9810055.

[127]  Hee-Cheol Kim.  Line defects and 5d instanton partition functions , 2016, 1601.06841.

[128]  A. Zajac,et al.  Ungauging schemes and Coulomb branches of non-simply laced quiver theories , 2020, Journal of High Energy Physics.

[129]  Mirror symmetry and exact solution of 4D $N = 2$ gauge theories: I , 1997, hep-th/9706110.

[130]  Taro Kimura.  β-Ensembles for Toric Orbifold Partition Function , 2011, 1109.0004.

[131]  B. Feigin,et al.  AFFINE KAC-MOODY ALGEBRAS AT THE CRITICAL LEVEL AND GELFAND-DIKII ALGEBRAS , 1992 .

[132]  Issues on orientifolds: on the brane construction of gauge theories with SO(2n) global symmetry , 1999, hep-th/9903242.

[133]  J. Polynomials.  SUPERANALOGS OF THE CALOGERO OPERATORS AND JACK POLYNOMIALS , 2000 .

[134]  Edward Witten,et al.  Topological quantum field theory , 1988 .

[135]  Yalong Cao,et al.  Zero-dimensional Donaldson–Thomas invariants of Calabi–Yau 4-folds , 2017, Advances in Mathematics.

[136]  M. Vergne,et al.  Toric reduction and a conjecture of Batyrev and Materov , 2003, math/0306311.

[137]  Nikita A.Nekrasov,et al.  Supersymmetric vacua and Bethe ansatz , 2009, 0901.4744.

[138]  NS5-Branes, T-Duality and Worldsheet Instantons , 2002, hep-th/0204186.

[139]  J. Zhou,et al.  On ω-Lie superalgebras , 2017, Journal of Algebra and Its Applications.

[140]  Taro Kimura,et al.  Band spectrum is D-brane , 2015, 1509.04676.

[141]  Yuji Tachikawa,et al.  Instanton counting with a surface operator and the chain-saw quiver , 2011, 1105.0357.

[142]  N. Nekrasov,et al.  Extended Seiberg-Witten theory and integrable hierarchy , 2006, hep-th/0612019.

[143]  Michael B. Green,et al.  Anomaly-free chiral theories in six dimensions , 1985 .

[144]  Yalong Cao,et al.  K-Theoretic DT/PT Correspondence for Toric Calabi–Yau 4-Folds , 2019, Communications in Mathematical Physics.

[145]  Elliptic quantum groups , 1994, hep-th/9412207.

[146]  H. Konno,et al.  Elliptic Algebra U_{q,p}(g^) and Quantum Z-algebras , 2014, 1404.1738.

[147]  H. Nakajima.  Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras , 1994 .

[148]  M. Atiyah,et al.  Self-duality in four-dimensional Riemannian geometry , 1978, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[149]  V. Pestun.  Review of localization in geometry , 2016, 1608.02954.

[150]  Satoshi X. Nakamura.  On the Jeffrey–Kirwan residue of BCD-instantons , 2015, 1502.04188.

[151]  Shlomo S. Razamat,et al.  Localization techniques in quantum field theories , 2016, 1608.02952.

[152]  R. Poghossian.  Deforming SW curve , 2010, 1006.4822.

[153]  Nikita Nekrasov,et al.  Issues in topological gauge theory , 1997, hep-th/9711108.

[154]  Five dimensional gauge theories and relativistic integrable systems , 1996, hep-th/9609219.

[155]  Lectures on instanton counting , 2003, math/0311058.

[156]  Jeffrey A. Harvey,et al.  Unwinding strings and t duality of Kaluza-Klein and h monopoles , 1997 .

[157]  A. Hanson,et al.  Asymptotically flat self-dual solutions to euclidean gravity , 1978 .

[158]  R. L. Mills,et al.  Conservation of Isotopic Spin and Isotopic Gauge Invariance , 1954 .

[159]  Julius F. Grimminger,et al.  Magnetic lattices for orthosymplectic quivers , 2020, Journal of High Energy Physics.

[160]  C. Vafa,et al.  Matrix models, topological strings, and supersymmetric gauge theories , 2002, hep-th/0206255.

[161]  S. Donaldson.  An application of gauge theory to four-dimensional topology , 1983 .

[162]  A. Okounkov,et al.  Quantum $q$-Langlands Correspondence , 2017, Transactions of the Moscow Mathematical Society.

[163]  O. Schiffmann,et al.  On the Hall algebra of an elliptic curve, I , 2005, math/0505148.

[164]  S. Rey,et al.  Triality in Little String Theories , 2017, 1711.07921.

[165]  Introduction to Seiberg-Witten Theory and its Stringy Origin , 1997 .

[166]  L. Alday,et al.  Affine SL(2) Conformal Blocks from 4d Gauge Theories , 2010, Letters in Mathematical Physics.

[167]  F. Kirwan,et al.  Localization and the quantization conjecture , 1997 .

[168]  J. F. Morales,et al.  Gauge theories on Ω-backgrounds from non commutative Seiberg-Witten curves , 2011, 1103.4495.

[169]  A. Polyakov.  Interaction of Goldstone Particles in Two-Dimensions. Applications to Ferromagnets and Massive Yang-Mills Fields , 1975 .

[170]  S. Govindarajan.  Introduction to Conformal Field Theory , 1993 .

[171]  Taro Kimura,et al.  Quiver elliptic W-algebras , 2016, Letters in Mathematical Physics.

[172]  Lectures on the Langlands program and conformal field theory , 2005, hep-th/0512172.

[173]  Michele Vergne,et al.  Arrangement of hyperplanes. I. Rational functions and Jeffrey-Kirwan residue , 1999 .

[174]  L. Faddeev.  Discrete Heisenberg-Weyl Group and modular group , 1995 .

[175]  A. Kirillov.  Quiver Representations and Quiver Varieties , 2016 .

[176]  I. G. MacDonald,et al.  Symmetric functions and Hall polynomials , 1979 .

[177]  Tomoki Nakanishi,et al.  T-systems and Y-systems in integrable systems , 2010, 1010.1344.

[178]  On Deformed W-algebras and Quantum Affine Algebras , 1998, math/9801112.

[179]  Taro Kimura,et al.  Quiver W-algebras , 2015, 1512.08533.

[180]  Taro Kimura,et al.  2d partition function in Ω-background and vortex/instanton correspondence , 2015, 1509.08630.

[181]  M. GELL-MAI.  The Interpretation of the New Particles as Displaced Charge Multiplets , 2007 .

[182]  L. Alday,et al.  Surface defects, the superconformal index and q-deformed Yang-Mills , 2013, 1303.4460.

[183]  Geoffrey Mason,et al.  The Santa Cruz Conference on Finite Groups , 1981 .

[184]  E. Witten.  Supersymmetry and Morse theory , 1982 .

[185]  C. Vafa,et al.  Matrix models, geometric engineering and elliptic genera , 2003, hep-th/0310272.

[186]  S. Gukov,et al.  Vortex Counting and Lagrangian 3-Manifolds , 2010, 1006.0977.

[187]  P. Richmond,et al.  The superconformal index and an elliptic algebra of surface defects , 2014, 1401.3379.

[188]  The Calculus of many instantons , 2002, hep-th/0206063.

[189]  Taro Kimura,et al.  Quantum elliptic Calogero-Moser systems from gauge origami , 2019, 1908.04928.

[190]  B. Feigin,et al.  Quantum-algebras and elliptic algebras , 1995, q-alg/9508009.

[191]  J. Morgan.  The Seiberg-Witten equations and applications to the topology of smooth four-manifolds , 1995 .

[192]  Deformations of W-algebras associated to simple Lie algebras , 1997, q-alg/9708006.

[193]  INTEGRABLE SYSTEMS AND SUPERSYMMETRIC GAUGE THEORY , 1995, hep-th/9509161.

[194]  An Index for 4 Dimensional Super Conformal Theories , 2005, hep-th/0510251.

[195]  H. Osborn.  Solutions of the Dirac equation for general instanton solutions , 1978 .

[196]  M. Jimbo,et al.  Quantum toroidal gl 1 ?> and Bethe ansatz , 2015, 1502.07194.

[197]  B. L. Floch,et al.  A slow review of the AGT correspondence , 2020, 2006.14025.

[198]  Joonho Kim,et al.  General instanton counting and 5d SCFT , 2014, 1406.6793.

[199]  E. Witten,et al.  Rigid Surface Operators , 2008, 0804.1561.

[200]  M. Mariño.  Instantons and Large N , 2015 .

[201]  N. Wyllard.  A(N-1) conformal Toda field theory correlation functions from conformal N = 2 SU(N) quiver gauge theories , 2009, 0907.2189.

[202]  H. Awata,et al.  Five-Dimensional AGT Relation and the Deformed β-Ensemble , 2010, 1004.5122.

[203]  G. ’t Hooft,et al.  Computation of the quantum effects due to a four-dimensional pseudoparticle , 2011 .

[204]  Karen K. Uhlenbeck,et al.  Instantons and Four-Manifolds , 1984 .

[205]  V. Dotsenko,et al.  Conformal Algebra and Multipoint Correlation Functions in 2d Statistical Models , 1996 .

[206]  A. Mironov,et al.  Spectral duality in elliptic systems, six-dimensional gauge theories and topological strings , 2016, 1603.00304.

[207]  A quantum deformation of the Virasoro algebra and the Macdonald symmetric functions , 1995, q-alg/9507034.

[208]  Michèle Vergne,et al.  Heat Kernels and Dirac Operators: Grundlehren 298 , 1992 .

[209]  E. D'hoker,et al.  Lectures on supersymmetric Yang-Mills theory and integrable systems , 1999, hep-th/9912271.

[210]  V. Varadarajan.  Supersymmetry for Mathematicians: An Introduction , 2004 .

[211]  M. Gell-Mann,et al.  Advantages of the color octet gluon picture , 1973 .

[212]  K. Nishijima.  Charge Independence Theory of V Particles , 1955 .

[213]  Topological Strings and Integrable Hierarchies , 2003, hep-th/0312085.

[214]  J. Maldacena,et al.  N=6 superconformal Chern-Simons-matter theories, M2-branes and their gravity duals , 2008, 0806.1218.

[215]  Integrating over Higgs Branches , 2000 .

[216]  V. Jones.  A polynomial invariant for knots via von Neumann algebras , 1985 .

[217]  C. Vafa,et al.  The Topological Vertex , 2003, hep-th/0305132.

[218]  The refined topological vertex , 2007, hep-th/0701156.