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Background Field Method for D=3, N=3 Supersymmetric Gauge

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Background Field Method for D=3, N=3 Supersymmetric Gauge BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIES I.L. Buchbinder TSPU, Tomsk, Russia The VIII International Workshop "Supersymmtries and Quantum Symmetries", Dubna, July 29 { August 3,2009 I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 1 / 24 Aims Formulation of gauge invariant effective action for d = 3, N = 3 supersymmetric gauge theories, formulated in d = 3, N = 3 harmonic superspace. Construction of manifestly N = 3 supersymmetric effective action in terms of off-shell d = 3, N = 3 superfields. d = 3, N = 3 harmonic supergraph calculations. Talk is based on the results obtained in collaboration with E. Ivanov, O. Lechtenfeld, N. Pletnev, I. Samsonov, B. Zupnik I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 2 / 24 Motivations Studing a quantum structure of recently proposed Bagger-Lambert-Gustavsson (BLG) theory (J. Bagger, N. Lambert, Phys. Rev. D75 (2007) 045020; D77 (2008) 065008; JHEP 0802 (2008) 105; A. Gustavsson, JHEP 0804 (2008) 083) and Aharony-Bergman-Jefferis-Maldacena (ABJM) theory (O. Aharony, O. Bergman, D.L. Jafferis, J. Maldacena, JHEP 0810 (2008) 091). Both theories related to low-energy description of M2-branes and contain the Chern-Simons fields and some number of scalar and spinor fields. These theories possess N = 8 supersymmetry N = 6 supersymmetry resepectively. One can show that BLG and ABJM theories can be formulated in terms of d = 3, N = 3 harmonic superspace (I.L.B, E.A. Ivanov, O. Lechtenfeld, N.G. Pletnev, I.B. Samsonov, B.M. Zupnik, JHEP 03 (2009) 096). This formulation provides 3 manifest (off-shell) supersymmetries and 5 (BLG) or 3 (ABJM) hidden (on-shell) supersymmetries of classical action. Developing a general procedure for calculating the effective action for such theories preserving manifest gauge invariance and three manifest supersymmetries of classical action. I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 3 / 24 Contents Gauge and matter theories in d = 3, N = 3 harmonic superspace. d = 3, N = 3 background field method Structure of superfield propagators in background field d = 3, N = 3 non-renormaization theorem Summary and prospects I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 4 / 24 Gauge and matter theories in d = 3, N = 3 harmonic superspace. Standard N = 3 superspace: µ (ij) fx ; θα g; i; j = 1; 2 (indices of SU(2)): Flat superspace derivatives: kj @ kj β @ Dα = α + iθ @αβ;@αβ = αβ @θkj @x Gauge covariant derivatives: ij ij ij rα = Dα + Vα ; rαβ = @αβ + Vαβ : Basic superfield strength W ij: 1 frij; rklg = ir ("ik"jl + "il"jk) − " ("ikW jl + "ilW jk + "jlW ik + "jkW il) α β αβ 2 αβ Basic constraint: (ij kl) rα W = 0 I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 5 / 24 Harmonic N = 3 superspace: µ ++ −− 0 ± Coordinates: fx ; θα ; θα ; θα; ui g; ± +i + −i − +i − Harmonics: ui 2 SU(2); u ui = 0; u ui = 0; u ui = 1. ++ −− 0 ij Here θα ; θα ; θα are harmonic projections of θα : ij ++ ij + + −− ij − − 0 ij + − θα −! θα = θα ui uj ; θα = θα ui uj ; θα = θα ui uj Harmonic derivarives: D++; D−−; D0 = [D++; D−−]: ++ + @ −− − @ D = ui − + :::; D = ui + + :::; @ui @ui ++ −− 0 Grassmann derivatives: Dα ;Dα ;Dα: I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 6 / 24 Analytic superfields: ++ µ ++ 0 Dα ΦA = 0 =) ΦA = ΦA(xA; θα ; θα; u); αβ αβ m αβ α++ β−− β++ α−− where xA = (γm) xA = x + i(θ θ + θ θ ). + µ ++ 0 ± q-hypermultiplet: q (xA; θα ; θα; ui ) + i ¯ i ¯ q : ff ; fi; α; iαg; i = 1; 2: ++ µ ++ 0 ± Vector superfield: V (xA; θα ; θα; ui ) ++ (ij) (ij) V : fAµ; φ ; λα; λα g: I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 7 / 24 Gauge and matter theories in d = 3, N = 3 harmonic superspace. d = 3, N = 3 Superfiled actions N = 3 Chern-Simons action: 1 n Z ++ ++ ik X (−1) 3 6 V (z; u1) :::V (z; un) SCS = tr d xd θdu1 : : : dun ; 4π n (u+u+) ::: (u+u+) n=2 1 2 n 1 Hypermultiplet action: Z (−4) + ++ ++ + SH = dζ q¯ (D + V )q : Gauge transformations: ++ ++ ++ δΛV = −D Λ − [V ; Λ]: + + δΛq = Λq : I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 8 / 24 Gauge and matter theories in d = 3, N = 3 harmonic superspace. Yang-Mills action: 1 Z S = − tr dζ(−4) (W ++)2 ; [g] = 1=2: SY M g2 Superfield strength: 1 W ++ = − D++αD++V −−; D++W ++ + [V ++;W ++] = 0: 4 α 1 Z ++ ++ ++ −− X n V (z; u1)V (z; u2) :::V (z; un) V (z; u) = (−1) du1 : : : dun : (u+u+)(u+u+) ::: (u+u+) n=1 1 1 2 n ++ ++ δΛW = [Λ;W ] I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 9 / 24 d = 3, N = 3 background field method The background field method is a tool for studying the general structure of effective actions in gauge theories with preservation of classical gauge invariance in quantum theory on all steps of calculations (Yang-Mills theory, quantum gravity, B.S. DeWitt, 1965, 1967). Basic idea: splitting the initial fields into classical and quantum fields and fixing the gauge symmetry only for quantum fields. Aims: Formulation of the background field method for the N = 3, d = 3 Chern-Simons-matter theory with the action ++ + + ++ S = SCS[V ] + Sq[¯q ; q ;V ] Path integral representation for the effective action, propagators, vertices. I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 10 / 24 Path integral for Yang-Mills type theories Gauge theory is given by: Set of fields Φi Action S0[Φ] i i α Gauge transformations δΦ = Rα[Φ]ξ Quantum theory is described by action: S[Φ; c;¯ c] = S0[Φ] + SGF [Φ] + SGH [Φ; c;¯ c] α α Gauge fixing action SGF : F F α β Ghost action SGH : c¯αMβ [Φ]c Background-quantum splitting: Φ −! Φ + φ Gauge fixing function F α depends both on background and quantum fields, it fixes only quantum field gauge transformation so that the corresponding SGF is invariant under background field gauge transformations. Effective action constructed in terms of such a gauge fixing function will be gauge invariant under background field gauge transformations. I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 11 / 24 Background field method Background{quantum splitting V ++ −! V ++ + κv++ ; Realization of initial gauge transformations: (i) Background transformations δV ++ = −D++λ − [V ++; λ] = −∇++λ ; δv++ = [λ, v++] (ii) Quantum transformations 1 δV ++ = 0 ; δv++ = − r++λ − [v++; λ] κ Covariant harmonic derivative r++ is constructed on the base of background field V ++. I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 12 / 24 Background field method Structure of Chern-Simons action after background-quantum splitting: 1 Z S [V +++v++] = S [V ++]− tr dζ(−4) v++W ++(V ++)+∆S [V ++; v++] CS CS κ CS Here 1 n n−2 Z ++ ++ ++ ++ X (−1) κ 9 vτ (z; u1) : : : vτ (z; un) ∆SCS[V ; v ] = tr d zdu1 : : : dun n (u+u+) ::: (u+u+) n=2 1 2 n 1 Background filed dependent gauge fixing function: F (4) = r++v++ The corresponding Faddev-Popov determinant: ++ ++ ++ ++ ++ ∆FP [V ; v ] = Det r (r + κv ) I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 13 / 24 ++ Result for effective action ΓCS[V ] ++ ++ ei(ΓCS [V ]−SCS [V ]) = Z ++ ++ ++ ++ 1=2 ^ ++ i(S2[v ,b,c,φ,V ]+Sint[v ;b;c;V ]) = (Det(4;0)∆) Dv DbDcDφe 1 Z Z S [v++; b; c; φ, V ++] = tr dζ(−4)v++∆^ v++ + tr dζ(−4)b(r++)2c 2 2 1 Z + tr dζ(−4)φ(r++)2φ, 2 1 n n−2 Z ++ ++ ++ ++ X (−1) κ 9 v (z; u1) : : : v (z; un) S [v ; b; c; V ] = tr d zdu : : : du τ τ int n 1 n + + + + n=3 (u1 u2 ) ::: (un u1 ) Z −κtr dζ(−4)r++b[v++; c] 1 ∆^ = (D++)2(r−−)2 8 I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 14 / 24 Hypermultiplet in background field Hypermultiplet action after background-quantum splitting of V ++ Z (−4) + ++ ++ + Sq = dζ q¯ (r + κv )q ; Background-quantum splitting for hypermultiplet q+ −! q+ + q+ ; q¯+ −! q¯+ + q¯+ : Transformation of hypermultiplet action after background-quantum splitting of both V ++ and q+ + + ++ Sq −! Sq[¯q ; q ;V ] + Slin + S2 + Sint Z (−4) + ++ + + ++ + + ++ + S2 = dζ (q¯ r q + q¯ κv q +q ¯ κv q ) Z (−4) + ++ + Sint = κ dζ q¯ v q I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 15 / 24 Total effective action ++ + + Effective action Γq[V ; q¯ ; q ] corresponding to hypermultiplet: ++ ++ + + ++ + + Z ei(Γq [V +v ;q¯ ;q ]−Sq [V ;q¯ ;q ]) = D¯q+Dq+ei(S2+Sint) Total effective action: ++ + + ++ + + Z ei(Γ[V ;q¯ ;q ]−S[V ;q¯ ;q ]) = D(QF )D(GH)ei(S2[QF;GH;BF ]+Sint[QF;GH;BF ]) S2[QF; GH; BF ]: Propagators for quantum fields and ghosts in background fields. Sint[QF; GH; BF ]: Vertices for quantum fields and ghosts in background fields. Total effective action is manifestly gauge invariant and N = 3 supersymmetric by construction The loop contributions to effective action are given by harmonic supergraphs, form of which is defined by the above propagators and vertices I.L.
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