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10 the Worldsheet Theory of Superstrings

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10 the Worldsheet Theory of Superstrings 10 The worldsheet theory of superstrings 10.1 The supersymmetric Polyakov action We begin by constructing a version of the Polyakov action with local (1, 1) supersymmetry 3 on the worldsheet. The metric field gab is accompanied by the spin 2 gravitino field χa↵, µ 1 whereas the embedding coordinate field X is accompanied by a set of spin 2 fermion fields µ ↵.TheEuclideanactionis 1 2 1 ab µ µ a b a µ 1 1 S = d σpg g @aX @bXµ + Γ @a µ (χaΓ Γ ) @bXµ χb µ 4⇡ ↵0 − p↵0 − 4 Z ✓ ◆(10.1) Here the 2D spinor convention is the Euclidean version of the convention described in section 3.4. The action (10.1) is subject to a number of gauge symmetries: di↵eomorphism, local supersymmetry (super-di↵eomorphism), Weyl symmetry, and super-Weyl symmetry. The local supersymmetry transformation takes the form i i δe a = "Γ χa,δgab = "(Γaχb +Γbχa), 1 δχ =2 spin" + (χ Γ Γbχ )(Γij"), a ra 4 a ij b µ µ (10.2) δX = p↵0" , 1 1 µ = @ Xµ χ µ Γa", p a − 2 a ✓ ↵0 ◆ where the supersymmetry generating parameter "(σ)isafermionicspinorfield,and spin is ra the spin connection defined in (3.68). µ 1 The Weyl transformation assigns weight 0 to X ,andweight 2 to χa and µ. Note that µ a the kinetic term for µ is Weyl invariant due to Γ µ =0.Thesuper-Weyltransformatoin i µ µ leaves e a, X ,and invariant, and transforms the gravitino by δχa =Γa⇣ (10.3) for a fermionic spinor field ⇣.Theinvarianceof(10.1)under(10.3)followsfromtheidentities b a b a µ ΓaΓ Γ =0and(ΓΓ )↵(Γb µ)β =0. We can fix the di↵eomorphism and Weyl gauge invariance by setting gab to a fiducial metricg ˆab as before. The super-di↵eomorphism can be fixed by the gauge condition b Γ Γaχb =0, (10.4) and finally χa subject to (10.4) can be set to zero by a super-Weyl transformation of the form (10.3). We are then left with simply the action of D free massless bosons Xµ and 155 µ 18 fermions on the worldsheet. If we choose the fiducial metric to be Euclidean,g ˆab = δab, the action can be written in complex coordinates as 1 2 S = d2z @Xµ@¯X + µ@¯ + µ@ . (10.5) 4⇡ ↵ µ µ µ Z ✓ 0 ◆ Recall that our convention for the measure is d2z =2d2σ. e e For later generalization, it will be useful to reformulate (10.5) in the language of (1, 1) superspace on the worldsheet, by extending the coordinates (z,z¯)toincludeformalGrass- mannian coordinates (✓, ✓¯), and packaging Xµ, µ, µ into a “real superfield”, µ µ ↵0 µ e ↵0 µ µ X (z,z,¯ ✓,✓¯) X (z,z¯)+i ✓ (z,z¯)+i ✓¯ (z,z¯)+✓✓¯F (z,z¯), (10.6) ⌘ r 2 r 2 where F µ will play the role of an auxiliary field that does note carry independent propagating degrees of freedom. The action (10.5) can be equivalently written as the superspace integral 1 2 2 µ S = d zd ✓D✓¯X D✓Xµ, (10.7) 2⇡↵ 0 Z where D✓ and D✓¯ are super-derivatives, defined as ¯ D✓ = @✓ + ✓@z,D✓¯ = @✓¯ + ✓@z¯. (10.8) The super-derivatives obey the following basic relations, 2 2 D = @ ,D= @¯ , D ,D¯ =0. (10.9) ✓ z ✓¯ z¯ { ✓ ✓} After expanding the integrand of (10.7) in the component fields Xµ, µ, µ,Fµ,andinte- grating out the Grassmannian variables (✓, ✓¯), (10.7) reduces to (10.5) plus a quadratic term in the auxiliary fields that does not a↵ect the dynamics. e The action (10.7) exhibits manifest 2D (1, 1) super-Poincar´esymmetry. The supersym- metry variation of the fields can be summarized as δ✏X =(✏Q✓ +¯✏Q✓¯) X, (10.10) where ✏ and✏ ¯ are Grassmannian parameters, Q✓ and Q✓¯ are defined as Q = @ ✓@ ,Q¯ = @¯ ✓@¯ . (10.11) ✓ ✓ − z ✓ ✓ − z¯ It is easy to see that Q✓,Q✓¯ anti-commute with D✓,D✓¯. Consequently, under (10.10), the variation of the action (10.7) vanishes as it is the superspace integral of a total derivative. 18Consequences of the gauge fixing at the quantum level will be considered in section 10.3 and 11.2. 156 10.2 =1superconformal algebra N The free field theory defined by the action (10.5) or (10.7) admits conformal symmetry generatd by the stress-energy tensor 1 µ 1 µ T = @X @Xµ @ µ (10.12) −↵0 − 2 3 of central charge c = 2 D, D being the target spacetime dimension, and its anti-holomorphic counterpart. The conformal symmetry does not commute with supersymmetry; together they generate the superconformal symmetry. In particular, (10.12) is related by supersymmetry 3 to a spin 2 holomorphic fermionic supercurrent 2 µ G = i @Xµ. (10.13) r↵0 Together T (z)andG(z)generate =1superconformalsymmetry. N More generally, the holomorphic =1superconformalsymmetryisgeneratedbya N stress-energy tensor T (z) that obeys Virasoro algebra of central charge c,togetherwitha 3 fermionic current G(z)thatisaweight 2 primary with respect to the Virasoro algebra. The OPE of G(z)withitselftakestheform 2c 2 G(z)G(0) + T (0). (10.14) ⇠ 3z3 z One of the coefficients on the RHS is fixed by the normalization convention for G,while the other coefficient can be fixed by applying conformal Ward identities to the correlation function G(z )G(z )T (z ) . Acting on any operator (0) at the origin, we can perform the h 1 2 3 i O Laurent expansion of G(z) G G(z)= r , r+ 3 (10.15) z 2 r Z+⌫ 2X where ⌫ =0if is in the holormophic Ramond sector, and ⌫ = 1 is is in the holomorphic O 2 O Neveu-Schwarz sector. The new commutation relations involving the Virasoro generators Ln and the superconformal generators Gr are n [Ln,Gr]= r Gn+r, 2 − (10.16) ⇣ ⌘ c 2 Gr,Gs =2Lr+s + (4r 1)δr, s. { } 12 − − Note that in the R sector, G and L c = G2 are the generators of the holomorphic part 0 0 − 24 0 of the =1super-Poincar´ealgebraonthecylinder. N 157 Likewise, we can speak of the anti-holomorphic = 1 superconformal algebra (SCA) N generated by T (¯z)andG(¯z). If a CFT admits both a holomorphic and an anti-holomorphic = 1 SCA, we say that it admits =(1, 1) SCA. N e e N Similarly to how 2D conformal symmetries obey the same group relations as holomorphic coordinate transformations z z0(z), the =1superconformalsymmetrygeneratedby 7! N T (z)andG(z) can be organized according to transformations of superspace coordinates (z,✓) (z0,✓0), which we refer to as “superconformal maps”, such that D transforms 7! ✓ covariantly, namely D✓ =(D✓✓0)D✓0 , (10.17) where D✓0 = @✓0 + ✓0@z0 . Let us expand z0(z,✓)=f(z)+✓⇣(z), (10.18) ✓0(z,✓)=g(z)+✓h(z), where ⇣ and g are Grassmannian valued functions of z. f,⇣,g,h are all holomorphic in z, but this is not enough to satisfy (10.17). Indeed, by definition, D✓ =(D✓✓0)@✓0 +(D✓z0)@z0 =(h + ✓@g)@✓0 +(⇣ + ✓@f)@z0 . (10.19) For the RHS to be proportional to D✓0 ,weneed ⇣ + ✓@f =(h + ✓@g)✓0, (10.20) which is equivalent to ⇣ = gh, @f =(@g)g + h2. (10.21) Thus, we have learned that a general superconformal map takes the form z0(z,✓)=f(z)+✓g(z)h(z), (10.22) ✓0(z,✓)=g(z)+✓h(z),h= @f + g@g. ± Note that while g and @g are both Grassmannian valued,p@f is invertible, and so the square root in the expression for h(z)canbedefi[email protected] Taylor series terminates if g depends on finitely many Grassmannian parameters. The choice of sign or branch of the square root is important. We will see in section 11.1 that this is related to the spin structure on a super Riemann surface. The infinitesimal version of (10.22) takes the form 1 1 δz = ✏ v(z)+ ✓⌘(z) ,δ✓= ✏ [⌘(z)+✓@v(z)] , (10.23) 2 2 158 where ✏ is a small bosonic parameter. We can also describe the infinitesimal superconformal transformation in terms of a super vector field V(z,✓)=v(z)+✓⌘(z), via 1 δz = ✏V ✓✓,δ✓ = ✏D✓V. (10.24) − 2 Under a general superconformal map (z,✓) (z0,✓0), V transforms into V0,with✏V0 = 7! δz0 + ✓0✓0.Ashortcalculationgives 2 V0(z0,✓0)=(D✓✓0) V(z,✓). (10.25) The bosonic stress-energy tensor and the fermionic supercurrent can be organized into the super stress tensor, 1 T(z,✓)= G(z)+✓T(z), (10.26) 2 ¯ 1 and its anti-holomorphic counterpart T(¯z,✓). Note that the factor 2 in front of G(z) on the RHS of (10.26) is fixed by consistency with (10.14). The Noether charge that corre- −LV sponds to the holomorphic superconformale symmetry generated by V acts on a state/operator (0) by O dzd✓ (0) = V(z,✓)T(z,✓) (0) LV ·O 2⇡i O I (10.27) dz 1 = v(z)T (z)+ ⌘(z)G(z) (0). 2⇡i 2 O I Here the superspace contour integral is defined simply as the contour integral in the bosonic variable together with the Grassmannian integral in the fermionic variable. Using either (10.25) or the OPE of supercurrents, one can compute the commutator of the superconformal charges, [ , ]= , LV LW L[V,W] 1 (10.28) [V, W]=V@W W@V + (D✓V)(D✓W). − 2 The last line gives the structure constant of the = 1 SCA, which will appear in the BRST N transformation of the superconformal ghosts in section 10.5.
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