# What is the determinant of the induced metric $h_{ab}$ of the NG action?

In the introductory section of Polchinski String Theory: An introduction to the bosonic string we are given the induced metric which reads $$h_{ab} = \partial_a X^{\mu}\partial_b X_{\mu}\tag{1.2.8}$$ This is how we usually write the induced metric. My question is firstly, why do we write it like this, and secondly why we use the determinant in the action $$S_{NG}=-\frac{1}{2\pi\alpha'}\int{d\tau\;d\sigma\sqrt{-det(h_{ab})}}\tag{1.2.9}$$ since the determinant, when I compute it gives $$(\partial_\tau X^\mu)^2(\partial_\sigma X_\mu)^2-(\partial_\tau X)(\partial_\sigma X)(\partial_\sigma X)(\partial_\tau X)$$

which would give zero?

First of all I would recommend having a copy of "A First Course in String Theory" by Barton Zwiebach while working through Polchinski.

$$S_{NG}=-\frac{1}{2\pi\alpha'}\int d\tau\ d\sigma\ \sqrt{(\dot{X}\cdot X')^2-(\dot{X})^2(X')^2}.$$

The reasoning behind is quite simple. We want to construct an action for a string that is Lorentz invariant. Remember that the action for the point particle is given by the proper length of the world-line of our particle

$$S_{PP}=-m\int ds.$$

Now in the case of strings, the natural object to consider would be the world-sheet area. It is a straigthforward exercise to show that this is indeed what the Nambu-Goto action captures.

Now lets understand how the induced metric comes into play. The concept of an induced metric arises when we want to calculate the infinitessimal line element of a surface embedded in a bigger space. Talking in string lingo, we are interested in the dynamics of a 2 dimensional surface, the world-sheet living in a flat Minkowski space (as we advance in string theory we consider different background geometries.). The infinitessimall line element on the world-sheet is given by

$$ds^2 = g_{\mu\nu}dX^\mu dX^\nu.$$

But we want to understand the geometry of this surface without refferring to the extrinsic world. Lets parametrize the world-sheet using two coordinates $$\sigma^a=(\tau,\sigma)$$ for $$a=0,1$$. Now using the chain rule

$$ds^2 = g_{\mu\nu}(\partial_a X^\mu d\sigma^a)(\partial_b X^\nu d\sigma^b)$$ $$=(g_{\mu\nu}\partial_a X^\mu\partial_b X^\nu)d\sigma^ad\sigma^b$$ $$=h_{ab}d\sigma^ad\sigma^b,$$

where we defined $$h_{ab} = g_{\mu\nu}\partial_a X^\mu\partial_b X^\nu = \partial_a X \cdot\partial_b X$$. This should answer your question regarding what is an induced metric. It is simply a tool that enables us to compute line elements on a surface without going out off the surface.

Now back to the Nambu-Goto action. Lets write down the induced metric in a matrix form

$$h_{ab}= \begin{pmatrix} \dot{X}\cdot\dot{X} & \dot{X}\cdot X'\\ X'\cdot\dot{X} & X'\cdot X'\end{pmatrix},$$

taking the determinant

$$h=-(\dot{X}\cdot X')^2+(\dot{X})^2(X')^2,$$

therefore we can write the Nambu-Goto action in terms of the determinant of the induced metric,

$$S_{NG}=-\frac{1}{2\pi\alpha'}\int d\tau\ d\sigma\ \sqrt{-h}.$$

Now why all this hustle? Its because when we write the action in terms of the determinant of the induced metric, showing that the action is reparametrization invariant is just a trivial exercise.