A Lightning Introduction to String Theory

TL;DR

This lightning tour surveys critical string theory, tracing the path from the 26-dimensional bosonic string to ten-dimensional superstrings and heterotic strings, and then to open strings with D-branes and orientifolds. It emphasizes two-dimensional conformal field theory, BRST quantization, and modular invariance as the core tools, and it outlines the spectrum, consistency conditions, and anomaly cancellations across the major vacua (Type IIA/IIB, SO(32) and E8×E8 heterotic, 0A/0B, and non-supersymmetric realizations). The discussion shows how D-branes and orientifolds introduce non-perturbative ingredients and gauge sectors, enabling tadpole cancellation and rich interconnections among vacua, including dualities and orbifold constructions. Overall, the work frames string theory as a unified framework for gravity and gauge interactions, while sketching the landscape of ten-dimensional vacua and the techniques needed to build consistent, quantum-consistent backgrounds.

Abstract

We give a lightning introduction to critical string theory, including the 26-dimensional bosonic string, the 10-dimensional superstrings and heterotic strings with and without spacetime supersymmetry. We also discuss open strings and D-branes, as well as the orientifold constructions, in ten dimensions.

Figures (6)

• Figure 1: The left figure shows the one-loop vacuum diagram of an open string which propagates for a (vertical) proper time $\tau_2$; its end-points trace the two boundaries of an annulus. The right figure illustrates the tree-level propagation of a closed string bouncing between the two boundaries; the proper time $\ell$ now flows horizontally. Both figures also display the double covering torus with modulus $i\tau_2/2$.
• Figure 2: D-branes and their open strings. The open strings stretching between the yellow and green branes include a massive vector that becomes massless when the relative distance $\delta$ goes to zero. The stack of the blue branes yields a non-Abelian gauge group.
• Figure 3: The left figure shows the one-loop vacuum diagram of a closed string which propagates for a (vertical) proper time $\tau_2$ and flips its orientation. The right figure illustrates the tree-level (horizontal) propagation of a closed string bouncing between the two cross-caps. Both figures also display the double covering torus with modulus $2i\tau_2$.
• Figure 4: The cross-cap is obtained by identifying antipodal points on the sphere. The sides of the fundamental domain are identified following the direction of the arrows, so that the surface has no boundaries and is unoriented.
• Figure 5: The left figure shows the one-loop vacuum diagram of an open string which propagates for a (vertical) proper time $\tau_2$ and flips its orientation; the end-points trace the single boundary of the Möbius strip composed by the red and green vertical lines. The right figure shows the tree-level (horizontal) propagation of a closed string bouncing between a boundary and a cross-cap. Both figures also display the double covering torus with modulus $\frac{1}{2}+\frac{i\tau_2}{2}$.