# Multiplying terms with index notation

I am trying to expand the flat-space action

$$S_{BI} = -T_p \int{d^{p+1}} \sigma \ \mathrm{Tr}\left( e^{-\phi} \sqrt{ -\det(\eta_{ab} + 4\pi^2\alpha^2 \partial_a\Phi^i\partial_b\Phi^i + 2\pi \alpha F_{ab}) \det(Q^{i}_{j}) } \right).\tag{1}$$

After some manipulation, I want to use the power series expansion of the natural log to expand the term

$$\text{ln}[\delta^{c}_{b} + \lambda^2\eta^{cd} \partial_d\Phi^i\partial_b\Phi^i ]\tag{2}$$ to the fourth power in $$\eta^{cd} \partial_d\Phi^i\partial_b\Phi^i.$$

My problem is, I don't understand how to compute powers of $$\eta^{cd} \partial_d\Phi^i\partial_b\Phi^i$$, i.e. $$(\eta^{cd} \partial_d\Phi^i\partial_b\Phi^i)^2$$. How do I treat the indices?

• Does it help if you write out the summation explicitly? – d_b Sep 28 '20 at 5:28
• @d_b I tried that at first but there were so many indices I kept confusing myself and I couldn't convince myself I was doing anything correctly – Hannah Sep 28 '20 at 5:50
• maybe get some practice with a two-dimensional case and then you get the picture – Andrew Steane Sep 28 '20 at 8:00

If you want to expand the Born-Infeld action$$^1$$ using $$\det(\mathbb{1}+M)~=~\exp({\rm tr}\ln(\mathbb{1}+M)),\tag{A}$$ then you need the logarithm of a whole matrix $$L~:=~\ln(\mathbb{1}+M)~=~-\sum_{n=1}^{\infty}\frac{1}{n} (-M)^n,\tag{B}$$ as opposed to the logarithm of a single matrix element (2). Then $$L^a{}_b~=~M^a{}_b -\frac{1}{2}M^a{}_cM^c{}_b +{\cal O}(M^3),\tag{C}$$ and so forth. Here $$a,b,c$$ are world-volume indices.
$$^1$$Additional complications arise from the (symmetrized) color-trace $${\rm Tr}$$ in a non-abelian Dirac-Born-Infeld action (1). We will here assume for simplicity an abelian gauge group $$U(1)$$.