Department of Mathematics  

Duke University

\[ f = \frac1{2L}\sqrt{\frac T\mu} \] \[ \newcommand{\normord}[1]{:\mathrel{#1}:} \def\ff#1#2{{\textstyle\frac{#1}{#2}}} \begin{split} Q &= \sum_n\left(L^{(\alpha)}_{-n} + L^{(a)}_{-n}\right)c_n + \sum_r G^{(\alpha a)}_{-r}\gamma_r - \ff12\sum_{m,n}(m-n)\normord{c_{-m}c_{-n}b_{m+n}}\\ &\hspace{10mm}+ \sum_{m,r}\left(\ff32m+r\right)\normord{c_{-m}\beta_{-r}\gamma_{m+r}} - \sum_{r,s}\gamma_{-r}\gamma_{-s}b_{r+s} - ac_0, \end{split} \]

Quantum Mechanics and String Theory
(Math 590-2)

Fall 2022

Instructor: Paul Aspinwall

Credits: 1.00, Hours: 03.0

Time: MW 1:45PM - 3:00PM

Location: Physics 119


Office Hours

  • 2:00 to 3:00pm each Tuesday
  • 10:00 to 11:00am each Friday


Math 212/219/222 and Math 221, or consent from me.


is weekly. See gradescope in Sakai.


There will be a final exam.


  • See "Resources" in Sakai.


A rough outline is as follows
  • Classical Mechanics (a quick review)
    • Newtonian
    • Lagrangian
    • Hamiltonian
  • A String made of particles
    • Classical analysis of vibrations of a violin string
    • Open and Closed Strings
  • Quantum Mechanics
    • Hilbert spaces and operators
    • Canonical quantization
    • Quantization of the vibrating string
  • Symmetries in Quantum mechanics
    • Groups and Lie groups
    • Projective representations and central extensions
    • Wigners Theorem
    • Gauge symmetries
    • Cohomology and BRST quantization
  • Quantization of a fundamental string
    • The Virasoro algebra
    • BRST quantization and the criticial dimension
    • The string spectrum
  • Supersymmetry
    • The superstring
    • Critical dimension and the GSO projection
  • Modular Invariance
    • Bosons
    • Fermions
    • A circle
    • Tori
    • The Heterotic String
    • The 2-torus and Mirror Symmetry


This course will not be based on a textbook as there is none which is suitable. Two main references (at a more advanced level) are
  • M. B. Green, J. H. Schwarz and E. Witten, Superstring Theory, (Volumes 1 and 2), Cambridge 1987
  • J. Polchinski, String Theory, (Volumes 1 and 2), Cambridge 1998.
There is also a lower level book, but this takes a quite different route than this course:
  • B. Zweibach, A First Course in String Theory, Cambridge 2009.

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