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Hirosi Ooguri
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Introduction to String Theory, Part 1

Course Instructor: Hirosi Ooguri , the Fred Kavli Professor of Theoretical Physics and Mathematics and the Director of the Walter Burke Institute for Theoretical Physics.

  • Lectures: 9:00 - 10:30 on Mondays and Wednesdays, beginning on January 6 and ending on March 12.
  • Starting on February 12, the class will meet at 269 Lauritsen.
  • Office Hours: 10:30 - 11:00 on Wednesdays after the class. Students can also make appointments with me as needed.

Course Description:

Prerequisites: Ph 205 or equivalent.

  • Ph 250 A will cover the worldsheet formulation of string theory, including conformal field theory, supersymmetry, the emergence of gravity, scattering amplitudes, T-duality, and D-branes. We will also discuss how to build semi-realistic models of elementary particle physics from string theory.
  • Ph 250 B will cover advanced topics such as non-perturbative dualities, Calabi-Yau geometry and mirror symmetry, black holes, the holographic principle and its relation to quantum information theory, and constraints on gravitational theories.

Course Announcement:

  • The first class meeting will be on January 6.
  • I will miss the class on January 8 to participate in Strings 2025 in Abu Dhabi as a panelist.
  • If you are registered students, please make sure to sign up for your final presentations: Registration Form for Final Presentations

Course Diary:

  • January 6: discussed how quantum gravity is different from quantum field theory without gravity. gave a proof of the Weinberg-Witten theorem as an example.
  • January 13: discussed the large N gauge theory, the Z_N one-form symmetry and Wilson loops, and emergent string theory. discussed the first quantization of a point particle and the Nambu-Goto action for a string.
  • January 15: discussed the conformal gauge and the light-cone gauge of the Nambu-Goto action. started to discuss the physical spectrum of the worldsheet theory.
  • January 22: derived the critical dimensions D=26 from the consistency between the light-cone gauge and conformal gauge.
  • January 27: showed that the massless spectrum of a closed bosonic string contains the graviton, a rank-2 anti-symmetric tensor, and the dilaton. Mentioned Weinberg's theorem. Started discussion on conformal field theory. Showed that the conformal symmetry in two dimensions is infinite dimensional.
  • February 3: discussed the modular invariance of genus-1 amplitudes.
  • February 5: started to discuss the toroidal compactification. Show that string coordinates allow winding numbers, taking values in the lattice to define the torus, and momenta take are in its dual lattice. We started to discuss the duality that exchanges the lattice and dual lattice. There is a symmetry enhancement at the self-dual point.
  • February 10: Discussed the circle compactification at the self-dual radius R = 2^{1/2} and the free fermion radius R = 1.
  • February 12: Continued the discussion on free fermions. Discussed the distinction between the NS sector and the Remond sector, the modular invariance of the genus-one partition function, and the spin operator that interpolates the two sectors.
  • February 19: Discussed the Narain compactification. As an introduction to superstring, we started to discuss quantization of superparticle.
  • February 24: Introduced the worldsheet action for superstring theory in the NS-R formalism. Discussed the Hilbert space of the theory.
  • February 26: Discussed the spacetime supersymmetry in the NS-R formalism. There are two closed superstring theories, type IIA and IIB. They differ by relative chiralities of the left and right movers. We also discussed the spin operators that map the NS to R ground states.
  • March 3: Discussed the massless states in the R-R sector of type IIA and IIB theories. Began to discuss heterotic strings.
  • March 5: Discussed the symmetry and spectrum of heterotic string and introduced the Narain compactification.
  • March 10: Discussed simple orbifold models. Pointed out that the moduli space of the type II string on the T^4/Z_2 orbifold is similar to that of the Narain compactification of the heterotic string on T^4: both are O(4,20: Z)\O(4, 20)/O(4) x O(20). There was a final presentation by a group of students about the relation between the central charge of the Virasoro algebra and the Weyl anomaly. There was a question on the gauge fixing procedure. This reference may be helpful: Faddeev Paper

Lecture Videos: YouTube Playlist

Final Presentations:

Registered students should give final presentations in the last two class meetings. Each presentation should be done by a group of 2 - 3 students. Each group will have 20 minutes for the presentation with 5 additional minutes for Q&A.

You can use either the black board or the projector.

Please complete the spreadsheet for the final presentation schedule. Note that there are 3 slots available on March 10 and 3 slots available on March 12. Slots will be assigned on a first-come, first-served basis.

Registration Form for Final Presentations

The final presentations can be about anything you found interesting about string theory, but you can also choose from the following list:

  1. Read section 4 of Polchinski [0], describe the BRST quantization and explain the use of the BRST operator in the context of bosonic string. Show that the BRST operator is nilpotentent at the critical dimensions D = 26 and explain why the physical Hilbert space has a positive definite inner product.
  2. Read section 3.4 of Polchinski [0] and explain the relation between the central charge c of the Virasoro algebra and the Weyl anomaly. Calculate the central charge for the BRST ghost in the bosonic string.
  3. Read section 8.4 of Polchinski [0], [1], and [2]. Explain why the Narain moduli space for bosonic string on the torus T^D. is O(D,D:Z)\O(D,D:R)/O(D:R)xO(D:R) and why it is spanned by the deformation space of the metric G_{ij} and the antisymmetric field B_{ij}. (These papers are about heterotic strings, but you can adopt their results for bosonic string.)
  4. Read [3] and tell us what you found interesting about superstring in this paper.
  5. Read [4] and explain what Weinberg did in this paper.
  6. Read [5] and explain what is done in this paper. Explain possible loopholes in this no-go theorem.

References:

[0] "String theory Vol. 1: An introduction to the bosonic string," by Joseph Polchinski, DOI: 10.1017/CBO9780511816079.

[1] "New Heterotic String Theories in Uncompactified Dimensions < 10," K.S. Narain, DOI: 10.1016/0370-2693(86)90682-9.

[2] "A Note on Toroidal Compactification of Heterotic String Theory," by K.S. Narain, M.H. Sarmadi, Edward Witten, DOI: 10.1016/0550-3213(87)90001-0.

[3] "Conformal invariance, supersymmetry and string theory," by Daniel Friedan, Emil Martinec, Stephen Shenker, DOI: 10.1016/S0550-3213(86)80006-2.

[4] "Photons and Gravitons in Perturbation Theory: Derivation of Maxwell's and Einstein's Equations," Steven Weinberg, DOI: 10.1103/PhysRev.138.B988.

[5] "All Possible Symmetries of the 𝑆 Matrix," Sidney Coleman and Jeffrey Mandula, DOI: 10.1103/PhysRev.159.1251.

Textbook:

For the first couple of weeks, I will use my lecture notes: TASI Lectures on Perturbative String Theories

References:

String Theory

Conformal field theory

Topological field theory

AdS/CFT correspondence