non-analytic

1/1 1.3 D-branes

We saw that string theory has two expansion parameters：lₛ(or rather the dimension-less lₛ/r，with r a curvature radius)，and eф，which when constant is also called gₛ.

In(1.1.21)，we wrote the expansion as a power series,but we wondered whether this

really captures the whole dependence.Recall that a function is called (real) analytic

if it coincides with its Taylor series around every point. A famous nonanalytic

function is

f(g)＝e⁻¹/g². (1.3.1)

Both f and all its derivatives vanish at g＝0，so its Taylor series vanishes identically，but f(g) ≠ 0. In physics，we call an effect with such a dependence on the coupling nonperturbαtiνe.In attempts at describing nonperturbative aspects of field theory，

two types of objects often come up：solitons， which are large and stable field

configurations，localized in space；and instantons，which occur in the Euclidean path

integral and are localized in time as well. In string theory，the most prominent objects

that play these roles are D-brαnes；this section is devoted to them.

1.3.1 Solitons and instantons