2 The Green–Schwarz Superstring: A Brief Motivation

The purpose of this section is to briefly review the Green–Schwarz (GS) formulation of the superstring. This is not done in a self-contained way, but rather as a very swift presentation of the features that will turn out to be universal in the formulation of brane effective actions.

There exist two distinct formulations for the (super)string:

  1. The worldsheet supersymmetry formulation, called the Ramond–Neveu–Schwarz (RNS) formulation2, where supersymmetry in 1 + 1 dimensions is manifest [432, 404].
  2. The GS formulation, where spacetime supersymmetry is manifest [256, 257, 258].

The RNS formulation describes a 1 + 1 dimensional supersymmetric field theory with degrees of freedom transforming under certain representations of some internal symmetry group. After quantisation, its spectrum turns out to be arranged into supersymmetry multiplets of the internal manifold, which is identified with spacetime itself. This formulation has two main disadvantages: the symmetry in the spectrum is not manifest and its extension to curved spacetime backgrounds is not obvious due to the lack of spacetime covariance.

The GS formulation is based on spacetime supersymmetry as its guiding symmetry principle. It allows a covariant extension to curved backgrounds through the existence of an extra fermionic gauge symmetry, kappa symmetry, that is universally linked to spacetime covariance and supersymmetry, as I will review below and in Sections 3 and 4. Unfortunately, its quantisation is much more challenging. The first volume of the Green, Schwarz and Witten book [260Jump To The Next Citation Point] provides an excellent presentation of both these formulations. Below, I just review its bosonic truncation, construct its supersymmetric extension in Minkowski spacetime, and conclude with an extension to curved backgrounds.

Bosonic string:
The bosonic GS string action is an extension of the covariant particle action describing geodesic propagation in a fixed curved spacetime with metric gmn

∫ ∘ ---------------- Sparticle = − m dτ − XΛ™m Λ™Xngmn (X ). (1 )
The latter is a one-dimensional diffeomorphic invariant action equaling the physical length of the particle trajectory times its mass m. Its degrees of freedom Xm (τ) are the set of maps describing the embedding of the trajectory with affine parameter τ into spacetime, i.e., the local coordinates m x of the spacetime manifold become dynamical fields m X (τ) on the world line. Diffeomorphisms correspond to the physical freedom in reparameterising the trajectory. The bosonic string action equals its tension Tf times its area
∫ √ -------- Sstring = − Tf d2σ − det𝒒. (2 )
This is the Nambu–Goto (NG) action [402, 249]: a 1 + 1 dimensional field theory with coordinates σμ μ = 0,1 describing the propagation of a Lorentzian worldsheet, through the set of embeddings Xm (σ) m = 0,1 ...d − 1, in a fixed d-dimensional Lorentzian spacetime with metric gmn (X ). Notice, this is achieved by computing the determinant of the pullback 𝒒μν of the spacetime metric into the worldsheet
𝒒 = ∂ Xm ∂ Xn g (X ). (3 ) μν μ ν mn
Thus, it is a nonlinear interacting theory in 1 + 1 dimensions. Furthermore, it is spacetime covariant, invariant under two-dimensional diffeomorphisms and its degrees of freedom {Xm } are scalars in two dimensions, but transform as a vector in d-dimensions.

Just as point particles can be charged under gauge fields, strings can be charged under 2-forms. The coupling to this extra field is minimal, as corresponds to an electrically-charged object, and is described by a Wess–Zumino (WZ) term

∫ S = Qf ℬ(2), (4 )
where the charge density Qf was introduced and ℬ stands for the pullback of the d-dimensional bulk 2-form B (2), i.e.,
1- m n μ ν ℬ(2) = 2∂μX ∂νX Bmn (X )dσ ∧ dσ . (5 )
Thus, the total bosonic action is:
∫ 2 √ -------- ∫ Sstring = − Tf d σ − det𝒒 + Qf ℬ (2). (6 )
Notice the extra coupling preserves worldsheet diffeomorphism invariance and spacetime covariance. In the string theory context, this effective action describes the propagation of a bosonic string in a closed string background made of a condensate of massless modes (gravitons and Neveu–Schwarz Neveu–Schwarz (NS-NS) 2-form B (X ) 2). In that case,
Tf = Qf = --1--= --1-, (7 ) 2πα ′ 2πβ„“2s
where β„“s stands for the length of the fundamental string.

For completeness, let me stress that at the classical level, the dynamics of the background fields (couplings) is not specified. Quantum mechanically, the consistency of the interacting theory defined in Eq. (6View Equation) requires the vanishing of the beta functions of the general nonlinear sigma models obtained by expanding the action around a classical configuration when dealing with the quantum path integral. The vanishing of these beta functions requires the background to solve a set of equations that are equivalent to Einstein’s equations coupled to an antisymmetric tensor3. This is illustrated in Figure 2View Image.

Supersymmetric extension:
The addition of extra internal degrees of freedom to overcome the existence of a tachyon and the absence of fermions in the bosonic string spectrum leads to supersymmetry. Thus, besides the spacetime vector m {X }, a set of 1 + 1 scalars fields α πœƒ transforming as a spinor under the bulk (internal) Lorentz symmetry SO (1,d − 1) is included. Instead of providing the answer directly, it is instructive to go over the explicit construction, following [260Jump To The Next Citation Point]. Motivated by the structure appearing in supersymmetric field theories, one looks for an action invariant under the supersymmetry transformations

δ πœƒA = πœ–A , δXm = ¯πœ–AΓ m πœƒA , (8 )
where A πœ– is a constant spacetime spinor, A At ¯πœ– = πœ– C with C the charge conjugation matrix and the label A counts the amount of independent supersymmetries A = 1,2,...𝒩. It is important to stress that both the dimension d of the spacetime and the spinor representation are arbitrary at this stage.

In analogy with the covariant superparticle [118], consider the action

Tf ∫ √ -- S1 = − --- d2σ h hμνΠμ ⋅ Π ν. (9 ) 2
This uses the Polyakov form of the action4 involving an auxiliary two-dimensional metric hμν. Π μ stands for the components of the supersymmetric invariant 1-forms
m m ¯A m A Π = dX + πœƒ Γ dπœƒ , (10 )
whereas Πμ ⋅ Π ν ≡ Πmμ Πnνηmn.

Even though the constructed action is supersymmetric and 2d diffeomorphic invariant, the number of on-shell bosonic and fermionic degrees of freedom does not generically match. To reproduce the supersymmetry in the spectrum derived from the quantisation of the RNS formulation, one must achieve such matching.

The current standard resolution to this situation is the addition of an extra term to the action while still preserving supersymmetry. This extra term can be viewed as an extension of the bosonic WZ coupling (4View Equation), a point I shall return to when geometrically reinterpreting the action so obtained [294]. Following [260Jump To The Next Citation Point], it turns out the extra term is

∫ ( ( ) ) S = T d2σ − πœ–μν∂ Xm ¯πœƒ1Γ ∂ πœƒ1 − πœƒ¯2Γ ∂ πœƒ2 + πœ–μν¯πœƒ1Γ m∂ πœƒ1 ¯πœƒ2Γ ∂ πœƒ2 . (11 ) 2 f μ m ν m ν μ m ν
Invariance under global supersymmetry requires, up to total derivatives, the identity
¯ m δπœ–S2 = 0 ⇐ ⇒ 2 ¯πœ–Γ mψ [1ψ2 Γ ψ3 ] = 0 , (12 )
for (ψ1, ψ2, ψ3) = (πœƒ, πœƒ′ = ∂ πœƒβˆ•∂σ1, Λ™πœƒ = ∂πœƒβˆ•∂ σ0) . This condition restricts the number of spacetime dimensions d and the spinor representation to be

Let us focus on the last case, which is well known to match the superspace formulation of 𝒩 = 2 type IIA/B5 Despite having matched the spacetime dimension and the spinor representation by the requirement of spacetime supersymmetry under the addition of the extra action term (11View Equation), the number of on-shell bosonic and fermionic degrees of freedom remains unequal. Indeed, Majorana–Weyl fermions in d = 10 have 16 real components, which get reduced to 8 on-shell components by Dirac’s equation. The extra 𝒩 = 2 gives rise to a total of 16 on-shell fermionic degrees of freedom, differing from the 8 bosonic ones coming from the 10-dimensional vector representation after gauge-fixing worldsheet reparameterisations.

The missing ingredient in the above discussion is the existence of an additional fermionic gauge symmetry, kappa symmetry, responsible for the removal of half of the fermionic degrees of freedom.6 This feature fixes the fermionic nature of the local parameter κ(σ) and requires πœƒ to transform by some projector operator

2 δκπœƒ = (πŸ™ + Γ κ)κ, with Γκ = πŸ™. (13 )
Here Γ κ is a Clifford-valued matrix depending non-trivially on {Xm, πœƒ}. The existence of such transformation is proven in [260].

The purpose of going over this explicit construction is to reinterpret the final action in terms of a more geometrical structure that will be playing an important role in Section 3.1. In more modern language, one interprets S1 + S2 as the action describing a superstring propagating in super-Poincaré [259]. The latter is an example of a supermanifold with local coordinates ZM = {Xm, πœƒα}. It uses the analogue of the superfield formalism in global supersymmetric field theories but in supergravity, i.e., with local supersymmetry. The superstring couples to two of these superfields, the supervielbein A E M(Z ) and the NS-NS 2-form superfield BAC, where the index M stands for curved superspace indices, i.e., M = {m, α}, and the index A for tangent flat superspace indices, i.e., A = {a, α}7.

In the case of super-Poincaré, the components EAM are explicitly given by

( ) Eam = δam , Eαα= δαα , Emα-= 0, Eaα = ¯πœƒΓ a δαα. (14 ) α-
These objects allow us to reinterpret the action S1 + S2 in terms of the pullbacks of these bulk objects into the worldsheet extending the bosonic construction
𝒒 = Π ⋅ Π = ∂ ZM Ea (Z)∂ ZN Eb (Z)η , μν μ ν μ M ν N ab ℬμν = ∂ μZM EAM (Z)∂νZN ECN (Z) BAC (Z). (15 )
Notice this allows us to write both Eqs. (9View Equation) and (11View Equation) in terms of the couplings defined in Eq. (15View Equation). This geometric reinterpretation is reassuring. If we work in standard supergravity components, Minkowski is an on-shell solution with metric gmn = ηmn, constant dilaton and vanishing gauge potentials, dilatino and gravitino. If we work in superspace, super-Poincaré is a solution to the superspace constraints having non-trivial fermionic components. The ones appearing in the NS-NS 2-form gauge potential are the ones responsible for the WZ term, as it should for an object, the superstring, that is minimally coupled to this bulk massless field.

It is also remarkable to point out that contrary to the bosonic string, where there was no a priori reason why the string tension Tf should be equal to the charge density Qf, its supersymmetric and kappa invariant extension fixes the relation Tf = Qf. This will turn out to be a general feature in supersymmetric effective actions describing the dynamics of supersymmetric states in string theory.

View Image

Figure 2: Different superstring formulations require curved backgrounds to be on-shell.

Curved background extension:
One of the spins of the superspace reinterpretation in Eq. (15View Equation) is that it allows its formal extension to any 𝒩 = 2 type IIA/B curved background [263]

--1--∫ 2 ∘ --------- -1---∫ S = − 2π α′ d σ − det𝒒 μν + 2πα ′ ℬ (2). (16 )
The dependence on the background is encoded both in the superfields A EM and BAC. The counting of degrees of freedom is not different from the one done for super-Poincaré. Thus, the GS superstring (16View Equation) still requires to be kappa symmetry invariant to have an on-shell matching of bosonic and fermionic degrees of freedom. It was shown in [89] that the effective action (16View Equation) is kappa invariant only when the 𝒩 = 2 d = 10 type IIA/B background is on-shell8. In other words, superstrings can only propagate in properly on-shell backgrounds in the same theory.

It is important to stress that in the GS formulation, kappa symmetry invariance requires the background fields to be on-shell, whereas in the RNS formulation, it is quantum Weyl invariance that ensures this self-consistency condition, as illustrated in Figure 2View Image.

The purpose of Section 3.1 is to explain how these ideas and necessary symmetry structures to achieve a manifestly spacetime covariant and supersymmetric invariant formulation extend to different half-BPS branes in string theory. More precisely, to M2-branes, M5-branes and D-branes.

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