# Central charges in the supersymmetry algebra of the $d = 10$ Type-IIA theory

According to Witten's paper, ''String Theory Dynamics in Various Dimensions'' [arXiv:hep-th/9503124], the supersymmetry algebra for the Type-IIA theory in 10 dimensions has the structure $\{Q, Q\} \sim \{Q', Q'\} \sim P$, where $Q_\alpha$ and $Q'_{\dot{\alpha}}$ denote the supersymmetry generators of the two chiralities, and that a central charge can appear in the anticommutator $\{Q_{\alpha}, Q_{\dot{\alpha}}'\} \sim \delta_{\alpha\dot{\alpha}}W$ (all this appears on page 5 of the paper).

However, very naively, this seems to be slightly at variance with Chris Hull's paper, ''Gravitational Duality, Branes and Charges'' [arXiv:hep-th/9705162v3]. Specifically, in Hull's paper, equation (2.2) gives the $\{Q, Q\}$ anticommutator (in Witten's language), which in general also includes central charges. Why can't the central chage ''W'' of Witten could in principle enter in equation (2.2) of Hull?

In $\mathcal{N} > 1$ supersymmetry in $d = 4$, one finds that $\{Q, Q'\} \sim P$ whereas $\{Q, Q\} \sim \text{central charges}$. This pattern changes in $d = 10$ for Type-IIA: what is the moral to be drawn from this?

Edit: It is clear to me that in 11D, the SUSY algebra has the ''normal'' form, $\{Q, Q\} \sim P$, and compactifying on a circle, the $11^{th}$ component of $P$ yields what is called the central charge $W$ in Witten's paper. My question is about the apparent discrepancy in the two sources above, and about getting this result without compactifying down from eleven dimensions.

• They are using different notations. Witten is writing the algebra in terms of two 16-component Majorana-Weyl spinors of so(9,1), while Hull is writing it in terms of one 32-component Majorana spinor of so(10,1). As you said, the IIA algebra can be obtained by reducing the 11d algebra. Typically one writes the reduced algebra in terms of the minimal spinor representation in that dimension. Hull is not bothering to do so for whatever reason. – Elliot Schneider Aug 24 '16 at 14:03
• The moral for writing down the most general supersymmetry algebra is to include all terms on the RHS which can appear in the Clebsch-Gordon decomposition of the spinor product on the LHS which are consistent with the symmetry/anti-symmetry of the indices. These are the many "central charges" which appear on the RHS of Hull's algebra. Only the scalar $Z$ is actually central; the others transform under Lorentz transformations and are therefore not central, so the name is imprecise but conventional. Witten only writes down the genuine central charge $Z$/$W$, and suppresses the others. – Elliot Schneider Aug 24 '16 at 14:10