## Some progress towards something

On Thursday night I was reading Polyakov’s contribution to the book *50 years of Yang-Mills theory*. This is my version of a bed time story. ðŸ˜‰

Polyakov mentions some of his earlier work and how he solved different problems. I actually got tired after a while, and decided to work and solve my own problem. I had been sort of running away from it with feelings of overwhelming difficulty. There is also a chance that reading this post on the (now dead) string coffee table provided some motivation. In any case, here we go…

Let us consider the following metric in 5-dimensional spacetime:

We have coordinates and we will work in a static gauge where a worldsheet will be parametrized by the two spacetime coordinates . We will set .

Then taking derivatives of the coordinate vector we have:

and

Then, the ingredients that go into the Nambu-Goto lagrangian are:

and

The cross term goes as:

The determinant of the induced metric will be denoted by:

We see that since we have:

and

we get some cancellations in . In the end we can write

We can write the lagrangian as:

with

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