Introduction to String Theory

Annotation

The course is an introduction to string theory, mostly dedicated to the bosonic string. Topics covered include quantization in the lightcone and conformal gauges, Virasoro algebra, BRST quantization, vertex operators, worldsheet methods (beta-function, calculation of Weyl anomaly), analysis of the spectrum and the calculation of tree amplitudes (Veneziano, Shapiro-Virasoro).

Course plan

1. Historical introduction to string theory. Effective theories of strong interactions. Veneziano amplitude. Higher spin resonances.
2. Classical relativistic particle. Action with an additional `monad’ field. The massless limit. Various gauges (static, lightcone, `covariant’).
3. Nambu-Goto action as a system with constraints. Lightcone gauge for the string. Worldsheet Hamiltonian, conserved charges, dispersion relation for the excited string.
4. Polyakov action for the string. Its classical diffeomorphism and Weyl invariance. Infinitesimal form of the gauge transformations. Energy-momentum tensor and Virasoro constraints.
5. Virasoro constraints as integrals of motion, their Poisson algebra. Conformal gauge. Conformal symmetry as residual gauge symmetry.
6. Quantization of string modes. Computation of the central extension in the Virasoro algebra using soft cutoff regularization. Spectrum in lightcone gauge, mass level, level matching. Exponential growth of the number of states. Lorentz algebra anomalies, critical dimension.
7. Quantization in conformal gauge. Analogy with Gupta-Bleuler quantization in electrodynamics. Analogy with symplectic reduction.
8. Equivalence of string spectra in lightcone vs. conformal gauges  (at low levels). BRST quantization of the string sigma model, diffeomorphism ghosts, their conformal dimensions.
9. BRST-charge for the string sigma model. Its nilpotence and relation to critical dimension. Cancellation of the anomaly in the Virasoro algebra. Physical states via cohomology. 
10. Calculation of the Weyl anomaly from correlation functions of energy-momentum tensor, relation to central charge. Strings in curved target space, calculation of the beta function. 
11. State/operator correspondence applied to the string sigma model. Vertex operators of physical states. Genus expansion of the string sigma model. The disc/sphere as tree level worldsheets.
12. Scattering amplitudes as correlation functions of vertex operators. Tree level amplitudes (Veneziano, Virasoro-Shapiro). Introduction to loop calculations in string theory. Zero modes of ghost fields, their relation to conformal Killing vectors and Teichmüller space.

Literature

1. M.Green, J.Schwarz, E.Witten “Superstring theory”, Volume 1, Cambridge University Press (2012)
2. R.Blumenhagen, D.Lüst, S.Theisen “Basic concepts of string theory”, Springer (2013)
3. E.Kiritsis “String theory in a nutshell”, Princeton University Press (2019)
4. J.Polchinski “String theory, volume 1”, Cambridge University Press (1998)
5. A.Cappelli, E.Castellani, F.Colomo, P. Di Vecchia “The birth of string theory”, Cambridge University Press (2012)
6. K.Becker, M.Becker, J.Schwarz “String theory and M-theory: a modern introduction”, Cambridge University Press (2007)
7. B.Zwiebach “A first course in string theory”, Cambridge University Press (2009)