Some Recent Developments in String Theory

• In 1984 K. Kikkawa and M. Yamasaki of Osaka University demonstrated that if you ‘curl up’ one of the extra dimensions into a circle with radius R, the applicable theory is the same as if we curl up this dimension with radius 1/R.
• If we apply this T-duality to various superstrings, we reduce five types of string theory down to three types.
• In the 1990s Juan Maldacena showed that string theory that included gravity in five dimensions could be seen as being equivalent to a four-dimensional quantum field theory in four dimensions.
• The consequent ‘AdS/CFT’ could provide a superior way to deal with gravity by connecting it to quantum field theory.
• However, in 1995 Edward Witten showed that the string theories known as type I, type IIA and type IIB, and the two heterotic string theories (SO(32) and E₈×E₈), could be reduced to a single theory, M-theory.
• In nine dimensions type IIA and IIB strings are identical, and so are E₈×E₈ and SO(32) strings.
• S-duality and U-duality have helped define the duality between the perturbative and non-perturbative parts of string theory.
• In 1998 Alain Connes, Michael R. Douglas and Albert Schwarz made some significant contributions to the relation between matrix models and M-theory by using a noncom-mutative quantum field theory.
• Edward Witten (with Paul Townsend) has also demonstrated a duality between ten-dimensional type IIA strings and 11-dimensional supergravity.
• Cumrun Vafa and Andrew Strominger have shown that the Bekenstein–Hawking entropy of a black hole is accounted for by solitonic states of superstring theory.
• Vafa is also partially responsible for the Gopakumar–Vafa conjecture which suggests that the Gromov–Witten invariants of a Calabi–Yau 3-fold can be canonically expressed in terms of integer invariants (which we can call BPS numbers).