String Theory

Table of Contents

Notation

  • Greek letters, e.g. string_theory_8e5e29091f3d01e42203c44c1336fe194fa7113a.png, for variables or indices of variables
  • Std. leteters, e.g. string_theory_c92373d2b03b4a43413cfd5e8385039ba98ecf38.png, for indices of target space
  • string_theory_7cb462b24c54b3812020400b52d2bffe2f5324d4.png denotes how the Lagrangian changes wrt string_theory_de32b094129e4e03f562577e6b8b51603d8ca54e.png

Just notes

  • Gauge theories are used to obtain a "larger" space in which the symmetries (e.g. Lorentz invariance) becomes linear transforms
  • Start with action

    string_theory_bbed8a054a8b607696f97c1a80e21c1106acf8d8.png

    where string_theory_2e87ddd81a8341ac85c5b5d94608b54b9a890a8d.png is some constant (mass), and string_theory_e41d4d30c9391a3a1b561d4f81528f3b79b7d23d.png is the arclength

  • Nambu-Goto action:

    string_theory_d0a58f23e507778466c370dca759dc3b5d44e5b3.png

    where

    string_theory_faf6fe49ec4338635602f68822483bb424958f84.png

    with string_theory_72e80c974806e0f5f9a8076874cc374b413cf8d4.png, i.e. string_theory_c3afebdf19c0f850d677754a62c6612e5ce7cbd1.png a string_theory_a2f513d8afadeb432d8ac8d36ee3e627c1b219e6.png matrix.

    • We are pulling back the metric of the target space
  • Symmetries
    • Gauge symmetries:
      • Reparametrizations $δ σα = - ζα(z, σ)

        string_theory_d8a0788c89afbb9fe21784b16864f84c8a18568c.png

      • Std. parametrization:

        string_theory_532fb8cafcb137ec48dd4e2d09d9ab58342a9654.png

        where string_theory_f31d2cfc617e81f1af172250a127a63516b62c7c.png is called the string length, and string_theory_d1f6bb1b651fd7c16aa8c86a437c709203aeb4a3.png is called tension (energy of unit of length of the string)

    • Global symmetries:
      • string_theory_aa8a5558414eaaea0e5e8ab684a568e2123c624c.png (where string_theory_3f880c90642d2c8032988493c8ed3ea3e0d55367.png are the Killing coefficients)
  • Polyakov action:

    string_theory_6ca0bc8707caf42e82344d2348524a4cd4fa0b33.png

    where

    string_theory_1306c30fb7c2b8995fdd827ea98d9296d2246978.png

    and string_theory_5dc59e8e67581e3469d6fd42edc34acb3f86c387.png is a string_theory_a2f513d8afadeb432d8ac8d36ee3e627c1b219e6.png symmetric metric.

    1. No kinetic term implies the equations of motion are string_theory_9cd38798589869f07639fb1a2755653f279c2cb5.png, thus the symmetries become algebraic!
      • Note

        string_theory_51f5dc43c874d8bd6720717e6f3618d455bc20c1.png

        for some function string_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png, which we can solve for if we know the action, but in general it can be any string_theory_93369077affb352dbdb97e8b3182fd50784f2b14.png, and so this is an example of a non-trivial Gauge symmetry

    2. Symmetries of string_theory_d1ae5817b6cf14bdf3608e48944fb09debab5149.png
      1. Global: