This is not directly possible in C++. C++ is a compiled language, so the names of functions and variables are not present in the executable file - so there is no way for the code to associate your string with the name of a function. 
You can get a similar effect by using function pointers, but in your case you are trying to use a member function as well, which complicates matters a little bit. 
I will make a little example, but wanted to get an answer in before I spend 10 minutes to write code. 
Edit: Here's some code to show what I mean:
#include <algorithm>
#include <string>
#include <iostream>
#include <functional>
class modify_field 
{
public:
    std::string modify(std::string str)
        {
            return str;
        }
    std::string reverse(std::string str)
        {
            std::reverse(str.begin(), str.end());
            return str;
        }
};
typedef std::function<std::string(modify_field&, std::string)> funcptr;
funcptr fetch_function(std::string select)
{
    if (select == "forward")
        return &modify_field::modify;
    if (select == "reverse")
        return &modify_field::reverse;
    return 0;
}
int main()
{
    modify_field mf;
    std::string example = "CAT";
    funcptr fptr = fetch_function("forward");
    std::cout << "Normal: " << fptr(mf, example) << std::endl;
    fptr = fetch_function("reverse");
    std::cout << "Reverse: " << fptr(mf, example) << std::endl;
}
Of course, if you want to store the functions in a map<std::string, funcptr>, then that is entirely possible.