In this letter, we will study the Chern-Simons-matter theory in Harmonic
superspace. It will be shown that this superspace is well suited to write
theories with high amount of supersymmetry. This will be done using harmonic
variables. The harmonic superspace will have N=3 supersymmetry. It will be argued that it will
be possible to analyse this theory in non-anticommutative superspace. The
non-anticommutative superspace for this theory will be explicitly constructed.
References
[1]
Galperin, A.S., Ivanov, E.A., Ogievetsky, V.I. and Sokatchev, E.S. (2001) Harmonic Superspace. Cambridge University Press, Cambridge. http://dx.doi.org/10.1017/CBO9780511535109
[2]
Galperin, A., Ivanov, E., Ogievetsky, V. and Sokatchev, E. (1984) Harmonic Superspace: Key to N = 2
Supersymmetry Theories. JETP Letters,
40.
[3]
Zupnik, B.M. (1998) Supersymmetries and Quantum Symmetries. In: Wess, J. and
Ivanov, E., Eds., Springer Lect. Notes in Phys, 524, 116.
[4]
Zupnik, B.M. and Hetselius, D.V. (1988) Three-Dimensional Extended Supersymmetry in Harmonic
Superspace. Sov. J. Nucl. Phys. (Engl. Transl.) (United
States), 47.
Galperin,
A., Ivanov, E., Kalitzin, S., Ogievetsky, V. and Sokatchev, E. (1984) Unconstrained N = 2 Matter, Yang-Mills and Supergravity Theories in Harmonic Superspace. Classical and Quantum Gravity, 1, 469. http://dx.doi.org/10.1088/0264-9381/1/5/004
[11]
Buchbinder, I.L., Ivanov, E.A., Lechtenfeld,
O., Pletnev, N.G., Samsonov, I.B. and Zupnik, B.M. (2009) ABJM Models in Script N = 3
Harmonic Superspace. Journal
of High Energy Physics, 2009,
096. http://dx.doi.org/10.1088/1126-6708/2009/03/096
[12]
Soloviev, M.A. (2013) Algebras with
Convergent Star Products and Their Representations in Hilbert Spaces. Journal of Mathematical Physics, 54, Article
ID: 073517. http://dx.doi.org/10.1063/1.4815996
You, Y. and Fradkin, E. (2013) Field Theory
of Nematicity in the Spontaneous Quantum Anomalous Hall Effect. Physical
Review B, 88, Article ID: 235124. http://dx.doi.org/10.1103/PhysRevB.88.235124
[16]
Faizal, M. (2014) Multiverse in the Third
Quantized Formalism. Communications in Theoretical Physics, 62,
697.
[17]
Piatek, M. (2014) Classical Torus Conformal Block, N = 2* Twisted
Super-Potential and the Accessory Parameter of Lamé Equation. Journal
of High Energy Physics, 2014, 124. http://dx.doi.org/10.1007/JHEP03(2014)124
[18]
Faizal, M. and Kruglov, S.I. (2014)
Deformation of the Dirac Equation. arXiv:1406.2653
Papenbrock, T. and Weidenmüller, H.A. (2014)
Effective Field Theory for Finite Systems with Spontaneously Broken Symmetry. Physical Review C, 89, Article ID: 014334. http://dx.doi.org/10.1103/PhysRevC.89.014334
Faizal, M. Deformation of Second and Third Quantization. arXiv:1503.04797
[25]
Ali,
A.F., Faizal, M. and Majumder, B. (2015) Absence of an Effective Horizon for Black Holes in Gravity’s
Rainbow. EPL (Europhysics Letters), 109, Article ID: 20001. http://dx.doi.org/10.1209/0295-5075/109/20001
Witten, E. and Homology, K. (2011) Khovanov Homology and Gauge Theory. arXiv:1108.3103
[32]
Faizal, M. and
Khan, M. (2011) A Superspace Formulation of the BV Action for Higher Derivative
Theories. The European Physical Journal
C-Particles and Fields, 71, 1-5. http://dx.doi.org/10.1140/epjc/s10052-011-1603-8
[33]
Faizal, M. (2014) Deformation of the Wheeler-DeWitt
Equation. International Journal of Modern
Physics A, 29, Article ID: 1450106. http://dx.doi.org/10.1142/S0217751X14501061
Witten, E. (2009) Geometric Langlands from
Six Dimensions. arXiv:0905.2720
[36]
Faizal, M., Ali, A.F. and Nassar, A. (2014)
AdS/CFT Correspondence beyond Its Supergravity Approximation.
arXiv:1405.4519
[37]
Witten, E. 2010) Analytic
Continuation of Chern-Simons Theory. arXiv:1001.2933
[38]
Witten, E. (2008) The
Problem of Gauge Theory. arXiv:0812.4512
[39]
Faizal, M. (2014) Consequences of
Deformation of the Heisenberg Algebra. International Journal of Geometric Methods in Modern Physics, 12, Article ID: 1550022.
Witten, E. (2004) Parity Invariance for
String in Twistor Space. Advances in
Theoretical and Mathematical Physics, 8, 799-796.
[51]
Witten, E. (2004) Perturbative Gauge Theory
as a String Theory in Twistor Space. Communications in Mathematical Physics, 252, 189-258. http://dx.doi.org/10.1007/s00220-004-1187-3
[52]
Seiberg, N. (2003) Noncommutative Superspace,
Script N =
1/2 Supersymmetry, Field Theory and String Theory. Journal of High Energy Physics, 2003,
010. http://dx.doi.org/10.1088/1126-6708/2003/06/010
[53]
Faizal, M. (2011) Spontaneous Breaking of
Lorentz Symmetry by Ghost Condensation in Perturbative Quantum Gravity. Journal of Physics A: Mathematical and Theoretical, 44, Article ID: 402001. http://dx.doi.org/10.1088/1751-8113/44/40/402001
Cachazo, F., Seiberg, N. and Witten, E.
(2003) Chiral Rings and Phases of Supersymmetric Gauge Theories. Journal of High Energy Physics, 2003,
018. http://dx.doi.org/10.1088/1126-6708/2003/04/018
Faizal, M. (2012) Multiverse in the Third
Quantized Horava-Lifshitz Theory of Gravity. Modern Physics Letters A, 27, Article ID: 1250007. http://dx.doi.org/10.1142/S0217732312500071
[62]
Witten, E. (2003) Chiral Ring of Sp(N) and
SO(N) Supersymmetric Gauge Theory in Four Dimensions. Chinese Annals
of Mathematics, 24,
403. http://dx.doi.org/10.1142/S0252959903000402
Witten, E. (2002) Singularities in String
Theory. Proceedings of the ICM, Vol.
1, Beijing,
2002, 495-504.
[65]
Witten,
E. (2002) Comments on String Theory. arXiv:hepth/0212247
[66]
Faizal, M. and Smith, D.J.
(2012) Supersymmetric Chern-Simons Theory in the Presence of a Boundary. Physical Review D, 85, Article ID:
105007. http://dx.doi.org/10.1103/PhysRevD.85.105007
[67]
Faizal, M. (2012) Noncommutativity and
Non-Anticommutativity Perturbative Quantum Gravity. Modern Physics Letters A, 27, Article ID: 1250075. http://dx.doi.org/10.1142/S0217732312500757
[68]
Friedmann, T. and Witten, E. (2003) Unification Scale, Proton Decay, and Manifolds of G2 Holonomy. Advances in Theoretical and Mathematical
Physics, 7, 577-617. http://dx.doi.org/10.4310/ATMP.2003.v7.n4.a1
Witten, E. (2002) Quest for Unification. arXiv:hep-ph/0207124
[71]
Witten, E. (2002)
Deconstruction, G2 Holonomy, and Doublet-Triplet Splitting. arXiv:hep-ph/0201018
[72]
Faizal,
M. (2012) Gauge and Supersymmetric Invariance of a Boundary Bagger-Lambert-Gustavsson
Theory. Journal of High Energy Physics,
2012, 17. http://dx.doi.org/10.1007/JHEP04(2012)017
[73]
Witten, E. (2001) Multi-Trace Operators,
Boundary Conditions, and AdS/CFT Correspondence.
arXiv:hep-th/0112258
Faizal, M. (2012) The BV Formalization of
Chern-Simons
Theory on Deformed Superspace. Communications
in Theoretical Physics, 58, 704. http://dx.doi.org/10.1088/0253-6102/58/5/14
Faizal, M. (2014) Absence of Black Holes
Information Paradox in Group Field Cosmology. International Journal of Geometric Methods in Modern Physics, 11, Article ID: 1450010. http://dx.doi.org/10.1142/S0219887814500108
[79]
Nazaryan, V. and Carlson, C.E. (2005) Field
Theory in Noncommutative Minkowski Superspace. Physical Review D, 71, Article ID: 025019. http://dx.doi.org/10.1103/PhysRevD.71.025019
[80]
Nazaryan, V. and Carlson, C.E. (2005) A Field
Theoretical Model in Noncommutative Minkowski Superspace. International Journal of Modern Physics A, 20, 3495-3501. http://dx.doi.org/10.1142/S0217751X05026820
Faizal, M. (2013) Deformed Super-Yang-Mills
in Batalin-Vilkovisky Formalism. International
Journal of Theoretical Physics, 52, 392-403. http://dx.doi.org/10.1007/s10773-012-1344-y
Kobayashi, Y. and Sasaki, S. (2005) Nonlocal
Wess-Zumino Model on Nilpotent Noncommutative Superspace. Physical Review D, 72, Article ID: 065015. http://dx.doi.org/10.1103/PhysRevD.72.065015
[88]
Faizal, M., Mandal, B.P. and Upadhyay, S.
(2013) Finite BRST Transformations for the Bagger-Lambert-Gustavsson Theory. Physics Letters B, 721, 159-163. http://dx.doi.org/10.1016/j.physletb.2013.02.057
[89]
Kruglov, S.I. and Faizal, M. (2014) Wave
Function of the Universe from a Matrix Valued First-Order Formalism.
arXiv:1408.3794
Awad, A. and Ali, A.F. (2014) Minimal Length,
Friedmann Equations and Maximum Density. Journal
of High Energy Physics, 2014, 93. http://dx.doi.org/10.1007/JHEP06(2014)093
[93]
Witten, E. (2000) Supersymmetric index in
Four-Dimensional Gauge Theories. Advances
in Theoretical and Mathematical Physics, 5, 841-907.
Faizal, M. Deformation of Second and Third
Quantization. arXiv:1503.04797
[97]
Cook,
J.S. (2006) Gauged Wess-Zumino Model in Noncommutative Minkowski
Superspace. Journal of Mathematical Physics,
47, Article ID: 012304. http://dx.doi.org/10.1063/1.2162330
[98]
Chang-Young, E., Kim,
H. and Nakajima, H. (2008) Noncommutative Superspace and Super Heisenberg
Group. Jour- nal
of High Energy Physics, 2008,
004. http://dx.doi.org/10.1088/1126-6708/2008/04/004
[99]
Faizal, M. and Awad, A. (2015) Four
Dimensional Supersymmetric Theories in Presence of a Boundary.
arXiv:1502.07717
[100]
Das, S., Robbins, M.P. and Walton, M.A. (2014) Generalized
Uncertainty Principle Corrections to the Simple Harmonic Oscillator in Phase
Space. arXiv:1412.6467
[101]
Balasubramanian, V., Das, S. and Vagenas, E.C. (2014)
Generalized Uncertainty Principle and Self-Adjoint Operators.
arXiv:1404.3962
[102]
Garattini, R. Vacuum
Energy Estimates in Quantum Gravity and the Wheeler-DeWitt Equation. arXiv:gr-qc/9604004
[103]
Majumder,
B. (2013) Quantum Rainbow Cosmological Model with Perfect Fluid. International Journal of Modern Physics D,
22, Article ID:
1350079. http://dx.doi.org/10.1142/S021827181350079X
[104]
Faizal, M., Khalil, M.M. and Das, S. (2014)
Time Crystals from Minimum Time Uncertainty. arXiv:1501.03111
[105]
Gangopadhyay,
S., Dutta, A. and Faizal, M. (2015) Constraints on the Generalized Uncertainty
Principle from Black Hole Thermodynamics. arXiv:1501.01482
[106]
Faizal, M. and
Tsun, T.S. (2014) Supersymmetric Duality in Superloop Space. arXiv:1412.7594
[107]
Faizal,
M., Ali, A.F. and Das, S. (2014) Discreteness of Time in the Evolution of the
Universe. arXiv:1411.5675
[108]
Pramanik, S., Faizal, M., Moussa, M. and Ali, A.F.
(2014) The Path Integral Quantization Corresponding to the Deformed Heisenberg
Algebra. arXiv:1411.4979
[109]
Faizal, M. and Khalil, M.M. (2014) GUP-Corrected
Thermodynamics for All Black Objects and the Existence of Remnants.
arXiv:1411.4042
[110]
Ali, A.F., Faizal, M. and Khalil, M.M. (2014) Absence of
Black Holes at LHC Due to Gravity’s Rainbow.
arXiv:1410.4765
[111]
Ali, A.F., Faizal, M. and
Khalil, M.M. (2014) Remnants of Black Rings from Gravity’s Rainbow. Journal of High Energy Physics, 2014, 159. http://dx.doi.org/10.1007/JHEP12(2014)159
[112]
Ali, A.F., Faizal, M. and Khalil, M.M. (2014)
Remnant for All Black Objects Due to Gravity’s Rainbow.
arXiv:1410.5706
[113]
Majumder,
B. and Sen, S. (2012) Do the Modified Uncertainty Principle and Polymer Quantization
Predict Same Physics? Physics Letters B,
717, 291-294. http://dx.doi.org/10.1016/j.physletb.2012.09.035