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M2 Kappa Symmetry on a CJS Background

Physical scope

Conditional local classical invariance of the single M2 Green–Schwarz action for the coefficientwise smooth complex-Grassmann-valued field model and odd local gauge parameter defined in the item, on an on-shell D=11 superspace background with the explicit source-mapped torsion and four-form conventions; no quantum measure or M-theory existence claim.

Statement

Let $Σ_{3}$ be a compact oriented smooth three-manifold without boundary, and let $Z$ be a Green–Schwarz field map as defined in def-m2-brane-worldvolume-action, over the finite complex Grassmann algebra $A$ and complexified target sheaf specified there, with real embedded timelike body $X$ and nondegenerate Lorentzian body metric $g_{i j}^{(0)}$ . Assume its image lies in a target superchart on which $C_{3}$ is defined. The even induced metric has $A_{\overset{ˉ}{0}}$ -valued components $g_{i j} = E_{i}^{a} E_{j}^{b} η_{a b}$ , and $\sqrt{- \det g}$ is the positive-body square root fixed by that definition. Assume the local on-shell eleven-dimensional CJS superspace constraints

T_{α β}^{a} = - 2 i (C Γ^{a})_{α β}, G_{α β γ δ} = G_{α β γ a} = 0, G_{α β a b} = + 2 i (C Γ_{a b})_{α β}, T_{b α}^{a} = T_{β γ}^{α} = T_{b c}^{a} = G_{α a b c} = 0,

with $G_{4} = d C_{3}$ , $d G_{4} = 0$ , vector indices $a = 0, \dots, 10$ , real spinor indices $α = 1, \dots, 32$ , ${Γ_{a}, Γ_{b}} = 2 η_{a b}$ for mostly-plus $η$ , and an invertible real Majorana pairing $C^{T} = - C$ , $Γ_{a}^{T} = - C Γ_{a} C^{- 1}$ . Take $ℏ = c = 1$ , $T_{M 2} > 0$ with units of length $^{- 3}$ , and dimensionless worldvolume chart coordinates. The classical Green–Schwarz action is $S = - T_{M 2} \int_{Σ_{3}} d^{3} ξ \sqrt{- \det g} + T_{M 2} \int_{Σ_{3}} Z^{*} C_{3}$ , with $C_{3} = - 6 B_{3}$ relative to $B_{3} = \frac{1}{6} E^{A} \land E^{B} \land E^{C} B_{C B A}$ . For an open set $U \subseteq Σ_{3}$ , let $κ \in C_{c}^{\infty} (U, X^{*} S) \otimes_{R} A_{\overset{ˉ}{1}}$ be a coefficientwise smooth odd section of the pulled-back real Majorana bundle, of dimension length $^{1 / 2}$ , and extend it by zero outside $U$ . Define the even infinitesimal variation $δ_{κ} Z$ by

δ_{κ} Z^{M} E_{M}^{a} = 0, δ_{κ} Z^{M} E_{M}^{α} = Δ^{α}, Δ = (I + Γ_{κ}) κ, Γ_{κ} = \frac{ε^{i j k} γ_{i j k}}{3! \sqrt{- \det g}},

where $I$ is the identity on $X^{*} S \otimes A$ , $γ_{i} = E_{i}^{a} Γ_{a}$ , $γ_{i j k} = γ_{[i} γ_{j} γ_{k]}$ , and $ε^{012} = + 1$ . Fix a spin frame in the selected target superchart and use its induced local trivialization for these components. The Majorana pairing is extended complex-bilinearly, so $\overset{ˉ}{ψ} = ψ^{T} C$ does not conjugate coefficients. The Clifford matrices act $A_{\overset{ˉ}{0}}$ -linearly on $X^{*} S \otimes A$ ; the nonzero body of $- \det g$ makes the inverse and positive square root well-defined. This variation leaves the action invariant coefficientwise in $A_{\overset{ˉ}{0}}$ . The result is a conditional local classical gauge symmetry; it proves neither a quantum path-integral measure nor existence of a nonperturbative M-theory.

Statement references

CJS Superspace Constraints for the M2 Membrane

M2 Brane Worldvolume Action

Bergshoeff, Sezgin, and Townsend, Supermembranes and Eleven-Dimensional Supergravity, eqs. (3)–(20)

Bergshoeff, Sezgin, and Townsend, Properties of the Eleven-Dimensional Supermembrane Theory, §2, eqs. (2.2)–(2.11), (2.18)–(2.22)

Townsend, Four Lectures on M-theory, §2

Facts & Assumptions

Given: The compact oriented worldvolume without boundary, coefficientwise smooth parity-preserving Green–Schwarz field map with embedded timelike body, nondegenerate body metric, local CJS superspace background, and even variation with odd parameter as displayed in the Statement.

[F1]

The supersymmetric action is the Nambu–Goto term plus the $+ C_{3}$ Wess–Zumino term, with the stated induced metric and closed-worldvolume boundary condition (M2 Brane Worldvolume Action).

[F2]

In the $+ C_{3}$ convention the dimension-zero constraints are $T_{α β}^{a} = - 2 i (C Γ^{a})_{α β}$ and $G_{α β a b} = + 2 i (C Γ_{a b})_{α β}$ ; the other components and conventional torsions used here are specified in the same supplier (CJS Superspace Constraints for the M2 Membrane).

[F3]

For an oriented Lorentzian three-plane in a 32-dimensional real Clifford module, $Γ_{κ}^{2} = I$ , $Γ_{κ}$ commutes with every tangent $γ_{i}$ , $ε^{i j k} γ_{j k} = 2 \sqrt{- g} γ^{i} Γ_{κ}$ , and the body of $(I \pm Γ_{κ}) / 2$ has rank $16$ (M2 Worldvolume Clifford Projector). These identities hold also for the nilpotent even deformations used here: in each real frame they are real analytic identities in the entries $E_{i}^{a}$ on the open Lorentzian locus. Their finite Taylor substitutions into the complex nilpotent parts preserve products, inverses and the selected square root, so the zero identities remain zero. This justifies the scalar extension for the actual deformed three-plane, rather than only for constant real matrices. Only the body-rank assertion is used; no ordinary real rank of a Grassmann module is asserted.

[F4]

The background is the adopted on-shell CJS formulation with its component action and conventions (Eleven-Dimensional CJS Supergravity Bosonic Action).

[A1]

The parameter $κ$ is odd and $Δ = (I + Γ_{κ}) κ$ , so $Δ$ is odd and $δ$ is an even derivation. Contraction $i_{δ}$ has form degree $- 1$ and intrinsic parity zero, with $i_{δ} E^{a} = v^{a} : = δ Z^{M} E_{M}^{a}$ and $i_{δ} E^{α} = Δ^{α}$ . It obeys the form-degree Leibniz rule, and $δ = d i_{δ} + i_{δ} d$ on pulled-back forms. In the row-connection convention of [F2], put $D_{i} v^{a} = \partial_{i} v^{a} - v^{b} (Z^{*} Ω_{b}^{a})_{i}$ . The covariant variation is $δ_{c o v} E_{i}^{a} : = δ E_{i}^{a} - E_{i}^{b} Ω_{b}^{a} (δ)$ ; the subtracted term is an infinitesimal Lorentz rotation and has zero contribution to $δ g_{i j}$ since $Ω$ preserves $η$ . The real Majorana matrices obey the stated transpose identities and are extended complex-bilinearly. The reversed-index form components and Koszul rule are those of [F2].

Proof

technique · direct variation

1.1givenA1F1F2algebra

Applying the Cartan identity to $T^{a} = d E^{a} + E^{b} \land Ω_{b}^{a}$ gives $δ_{c o v} E_{i}^{a} = D_{i} v^{a} + (Z^{*} i_{δ} T^{a})_{i}$ . Here $v^{a} = 0$ . The constraints in [F2] leave only $\frac{1}{2} E^{β} \land E^{α} T_{α β}^{a}$ . Contracting gives $- E^{β} Δ^{α} T_{α β}^{a}$ , since $Δ^{β} E^{α} = - E^{α} Δ^{β}$ and the spinor slots are symmetric. Thus $δ_{c o v} E_{i}^{a} = + 2 i {\overset{ˉ}{E}}_{i} Γ^{a} Δ$ . The Lorentz rotation drops out of the metric by [A1], so $δ g_{i j} = 2 E_{(i ∣ a ∣} δ_{c o v} E_{j)}^{a}$ and

δ \sqrt{- g} = \frac{1}{2} \sqrt{- g} g^{i j} δ g_{i j} = + 2 i \sqrt{- g} {\overset{ˉ}{E}}_{i} γ^{i} Δ .

The determinant derivative follows by differentiating its finite polynomial and using the inverse of $g$ ; the positive-body square root is invertible. Consequently the Nambu–Goto variation is $- 2 i T_{M 2} \int d^{3} ξ \sqrt{- g} {\overset{ˉ}{E}}_{i} γ^{i} Δ$ .

1.2givenA1F1F2F3algebra

Cartan's identity gives $δ (Z^{*} C_{3}) = Z^{*} (i_{δ} G_{4}) + d Z^{*} (i_{δ} C_{3})$ . The exact term integrates to zero coefficientwise on the compact boundaryless worldvolume. In the reversed-index expansion of [F2], the two-spinor part is $\frac{1}{2! 2!} E^{b} \land E^{a} \land E^{β} \land E^{α} G_{α β a b}$ . The other components contract to zero because $v^{a} = 0$ and the three-spinor and one-spinor components vanish. Contracting the displayed part gives $- \frac{1}{2} E^{b} \land E^{a} \land E^{β} Δ^{α} G_{α β a b} = \frac{1}{2} E^{β} \land E^{a} \land E^{b} Δ^{α} G_{α β a b}$ . The constraints and the symmetric matrix $C Γ_{a b}$ therefore yield

δ S_{W Z} = + i T_{M 2} \int d^{3} ξ ε^{i j k} {\overset{ˉ}{E}}_{i} γ_{j k} Δ = + 2 i T_{M 2} \int d^{3} ξ \sqrt{- g} {\overset{ˉ}{E}}_{i} γ^{i} Γ_{κ} Δ .

Here $γ_{j k} = E_{j}^{a} E_{k}^{b} Γ_{a b}$ , and the last equality uses [F3]. Indeed $Γ_{a b}^{T} = - C Γ_{a b} C^{- 1}$ and $C^{T} = - C$ give $(C Γ_{a b})^{T} = C Γ_{a b}$ ; no conjugation of the odd coefficients is used. The source map $C_{3} = - 6 B_{3}$ fixes the normalization of $G_{α β a b}$ used in this contraction.

2.1givenA1F3step 1.1step 1.2algebra∎

Adding the two variations from steps 1.1 and 1.2 gives

δ_{κ} S_{M 2} = - 2 i T_{M 2} \int d^{3} ξ \sqrt{- g} {\overset{ˉ}{E}}_{i} γ^{i} (I - Γ_{κ}) Δ .

By the transformation in [A1], $Δ = (I + Γ_{κ}) κ$ , and [F3] gives $(I - Γ_{κ}) Δ = (I - Γ_{κ}) (I + Γ_{κ}) κ = (I - Γ_{κ}^{2}) κ = 0$ . Hence $δ_{κ} S_{M 2} = 0$ coefficientwise in $A_{\overset{ˉ}{0}}$ for every smooth local odd $κ$ . This is a classical local variation and uses no quantum measure.

M2 Kappa Symmetry on a CJS Background

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources