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) {x , θα }, 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 ∇α = Dα + Vα , ∇αβ = ∂αβ + Vαβ .

Basic superﬁeld strength W ij: 1 {∇ij, ∇kl} = i∇ (εikεjl + εilεjk) − ε (εikW jl + εilW jk + εjlW ik + εjkW il) α β αβ 2 αβ Basic constraint: (ij kl) ∇α 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: {x , θα , θα , θα, ui },

± +i + −i − +i − Harmonics: ui ∈ 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 : {f , fi, ψα, ψiα}, i = 1, 2.

++ µ ++ 0 ± Vector superﬁeld: V (xA, θα , θα, ui )

++ (ij) (ij) V : {Aµ, φ , λα, λα }.

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:

∞ 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. SYM g2 Superﬁeld strength: 1 W ++ = − D++αD++V −−, D++W ++ + [V ++,W ++] = 0. 4 α

∞ 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++ = − ∇++λ − [v++, λ] κ Covariant harmonic derivative ∇++ 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

∞ 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) = ∇++v++

The corresponding Faddev-Popov determinant:

++ ++ ++ ++ ++ ∆FP [V , v ] = Det ∇ (∇ + 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 ˆ ++ 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++∆ˆ v++ + tr dζ(−4)b(∇++)2c 2 2 1 Z + tr dζ(−4)φ(∇++)2φ, 2

∞ 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)∇++b[v++, c]

1 ∆ˆ = (D++)2(∇−−)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¯ (∇ + 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¯ ∇ 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. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 16 / 24 Gauge and hypermultiplet propagators in background ﬁeld V ++

Equations of motion for propagators:

++ ++ (2,2) ˆ (2,2) (2,2) hv (1)v (2)i = G (1|2) : ∆G (1|2) = δA (1|2)

+ + (1,1) ++ (1,1) (3,1) hq¯ (1)q (2)i = G (1|2) : ∇ G (1|2) = δA (1|2)

Analytic delta-function: 1 δ(4−q,q)(1|2) = − D++αD++ δ9(z − z )δ(−q,q)(u , u ) A 4 (1) (1)α 1 2 1 2 Solutions to the equations for propagators is given in terms of covariant superﬁeld operators, acting on analytic superﬁelds as follows: ˆ 2 ∆ = + background superstrength and covariant derivative terms. ˆ = + background superstrength and covariant derivative terms.

I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 17 / 24 Propagators

Any loop contribution to effective action is expressed in terms of superstrengths and their covariant derivatives. For vanishing background filed V ++ the propagators are defined in terms of Grassmann delta-functions and spinor derivatives such a way that allows to obtain any loop contribution to effective action in form of integral over full d = 3, N = 3 superspace. Free propagators:

(2,2) 1 0 2 (2,2) G0 (1|2) = (D ) δA (1|2) , 1 δ9(z − z ) G(1,1)(1|2) = − (D0 )2(D++)2(D++)2 1 2 , 0 (1) (1) (2) + + 3 16 (u1 u2 )

I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 18 / 24 Supergraphs

Simple consequence of the N = 3 supergraph technique:

All tadpole one-loop supergraphs as well as one-loop hypermultiplet self energy supergraph vanish only due to propagator structure.

I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 19 / 24 The N = 3, d = 3 non-renormalization theorem

General statement: The effective action in the N = 3 Chern-Simons model with arbitrary number of hypermultiplets in some representation of gauge group is completely finite in the sense that there are no any UV quantum divergences in harmonic supergraphs contributing to the effective action.

Divergences in Chern-Simons type theories: N = 0 (non-supersymmetric) theories: β-function for Chern-Simons coupling in an arbitrary Chern-Simons-matter theory is trivial (Kapustin, Pronin, 1993, 1994), the divergences may occur only in the sector of matter ﬁelds. N = 1 and N = 2 supersymmetric theories: in general case such theories with scale-invariant superpotentials are not free of UV divergences, but the divergence cancellation may occur for some particular superpotentials (Avdeev, Grigoryev, Kazakov, Kondrashuk, 1992, 1993; Gates, Nishino, 1992). Aim: Calculation of superﬁcial degree of divergences in N = 3 theories.

I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 20 / 24 Power counting

Consider an arbitrary background filed dependent supergraph G with L loops, P propagators, Nmat external matter lines. To evaluate the superficial degree of divergences ω(G) it is sufficient to know only the free superfield propagators. The result is: 1 1 ω(G) = 3L − 2P + (2P − N − 3L) − N = −N − N mat 2 D mat 2 mat

3L is a contribution of loop momenta −1 −2P comes from the factors in the propagators +2P comes from the factors (D++)2(D0)2 in hypermultiplet propagators ++ 2 −Nmat arises due to the fact that one factor (D ) in each external matter line is used to restore full N = 3 measure in hypermultiplet vertices. −3L comes since we can apply the identity

6 ++ 2 ++ 2 0 2 9 + + 4 9 δ (θ1 − θ2)(D1 ) (D2 ) (D2) δ (z2 − z1) = 16(u1 u2 ) δ (z1 − z2) in each loop to shrink the loop into dot in θ space. 1 − 2 ND comes from ND covariant spinor derivatives which can be transfered on the external lines with help of integration by parts

I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 21 / 24 Power counting

As a result: Any supergraph with external matter lines are automatically ﬁnite.

The only dangerous supergraphs are ones with ND = 0. Advantage of background field method. Due to manifest gauge invariance of effective action, it means the loop supergraphs for effective action must be expressed in terms of superfield strengths and their covariant derivatives constructed from background superfield V ++. Therefore some number of covariant derivatives must be taken from propagators and transfered on external V ++ lines.

ND > 0 and hence ω(G) < 0. Any loop supergraph in the theory under consideration is UV ﬁnite.

I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 22 / 24 Summary and prospects

Basic results:

The background field method for the general N = 3 Chern-Simons-matter theory, which allows, in principle, to compute the effective action preserving manifest gauge invariance and N = 3 supersymmetry on all steps of quantum calculations. Background field dependent propagators and vertices.Possibility to compute any supergraphs. Non-renormalization theorem. The theory under consideration is UV finite. Only IR singularities can appear in the massless hypermultiplet theory, which can be avoided by using either the massive hypermultiplets or by doing all the calculations with non-zero background field where all the propagators are effectively massive. The operator ∆ˆ

I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 23 / 24 Summary and prospects

Open problems:

Structure of low energy effective action in vector multiplet and hypermultiplet sectors. Techniques for calculating the one-loop effective action. Problem of Det∆ˆ . Applications to the N = 6 and N = 8 supersymmetric ABJM and BLG theories which describe the M2 branes in superstring theory. Study the effective actions in these models and possible relations to the dynamics of M2 branes with the quantum corrections taken into account. Study the composite operators for the hypermultiplet superfields in the ABJM theory. Such operators are interesting from the point of view of the AdS/CFT correspondence for the three-dimensional field models.

I.L. Buchbinder (Tomsk) BAKGROUND FIELD METHOD FOR d = 4, N = 3 SUPERGAUGE THERIESDubna, 2009 24 / 24