Extended gravity theories from dynamical noncommutativity

Abstract

In this paper we couple noncommutative vielbein gravity to scalar fields. Noncommutativity is encoded in a $\star$-product between forms, given by an abelian twist (a twist with commuting vector fields). A geometric generalization of the Seiberg–Witten map for abelian twists yields an extended theory of gravity coupled to scalars, where all fields are ordinary (commutative) fields. The vectors defining the twist can be related to the scalar fields and their derivatives, and hence acquire dynamics. Higher derivative corrections to the classical Einstein–Hilbert and Klein–Gordon actions are organized in successive powers of the noncommutativity parameter $\theta ^{AB}$.

Appendix: Gamma matrices in \(D=4\)

We summarize in this appendix our gamma matrix conventions in $D=4$.

$$\begin{aligned}&\eta _{ab} =(1,-1,-1,-1),~~~\{\gamma _a,\gamma _b\}=2 \eta _{ab},~~~[\gamma _a,\gamma _b]=2 \gamma _{ab}, \end{aligned}$$

(9.1)

$$\begin{aligned}&\gamma _5 \equiv i \gamma _0\gamma _1\gamma _2\gamma _3,~~~\gamma _5 \gamma _5 = 1,~~~\varepsilon _{0123} = - \varepsilon ^{0123}=1, \end{aligned}$$

(9.2)

$$\begin{aligned}&\gamma _a^\dagger = \gamma _0 \gamma _a \gamma _0, ~~~\gamma _5^\dagger = \gamma _5 \end{aligned}$$

(9.3)

$$\begin{aligned}&\gamma _a^T = - C \gamma _a C^{-1},~~~\gamma _5^T = C \gamma _5 C^{-1}, ~~~C^2 =-1,~~~C^\dagger =C^T =-C \end{aligned}$$

(9.4)

1.1 Useful identities

$$\begin{aligned} \gamma _a\gamma _b&= \gamma _{ab}+\eta _{ab}\end{aligned}$$

(9.5)

$$\begin{aligned} \gamma _{ab} \gamma _5&= {i \over 2} \epsilon _{abcd} \gamma ^{cd}\end{aligned}$$

(9.6)

$$\begin{aligned} \gamma _{ab} \gamma _c&= \eta _{bc} \gamma _a - \eta _{ac} \gamma _b -i \varepsilon _{abcd}\gamma _5 \gamma ^d\end{aligned}$$

(9.7)

$$\begin{aligned} \gamma _c \gamma _{ab}&= \eta _{ac} \gamma _b - \eta _{bc} \gamma _a -i \varepsilon _{abcd}\gamma _5 \gamma ^d\end{aligned}$$

(9.8)

$$\begin{aligned} \gamma _a\gamma _b\gamma _c&= \eta _{ab}\gamma _c + \eta _{bc} \gamma _a - \eta _{ac} \gamma _b -i \varepsilon _{abcd}\gamma _5 \gamma ^d\end{aligned}$$

(9.9)

$$\begin{aligned} \gamma ^{ab} \gamma _{cd}&= -i \varepsilon ^{ab}_{~~cd}\gamma _5 - 4 \delta ^{[a}_{[c} \gamma ^{b]}_{~~d]} - 2 \delta ^{ab}_{cd}\end{aligned}$$

(9.10)

$$\begin{aligned} Tr(\gamma _a \gamma ^{bc} \gamma _d)&= 8~ \delta ^{bc}_{ad} \end{aligned}$$

(9.11)

$$\begin{aligned} Tr(\gamma _5 \gamma _a \gamma _{bc} \gamma _d)&= -4 \;\!i \varepsilon _{abcd} \end{aligned}$$

(9.12)

$$\begin{aligned} Tr(\gamma ^{rs} \gamma _a \gamma _{bc} \gamma _d)&= 4(-2 \delta ^{rs}_{cd} \eta _{ab} + 2 \delta ^{rs}_{bd} \eta _{ac} - 3! \delta ^{rse}_{abc} \eta _{ed}) \end{aligned}$$

(9.13)

$$\begin{aligned} Tr(\gamma _5 \gamma ^{rs} \gamma _a \gamma _{bc} \gamma _d)&= 4(-i \eta _{ab} \varepsilon ^{rs}_{~~cd} + i \eta _{ac} \varepsilon ^{rs}_{~~bd} + 2i \varepsilon _{abc}^{~~~e} \delta ^{rs}_{ed}) \end{aligned}$$

(9.14)

where $\delta ^{ab}_{cd} \equiv \frac{1}{2}(\delta ^a_c\delta ^b_d-\delta ^b_c\delta ^a_d)$, $\delta ^{rse}_{abc} \equiv {1 \over 3!} (\delta ^r_a \delta ^s_b \delta ^e_c$ + 5 terms), and indices antisymmetrization in square brackets has total weight $1$.