nLab D-brane geometry

Idea

Regarding the quantum string as a 2-spectral triple, it defines a spectral geometry (typically but not necessarily a noncommutative geometry) which is the effective spacetime as seen by this quantum string (as read of from, notably, its energy spectrum). For the open string the most prominent aspect of its 2d SCFT worldsheet theory are its boundary conditions. In the spectral interpretation these correspond to the presence of D-branes in the effective target spacetime. Much geometric information is contained in these D-brane states, and the resulting concept of (noncommutative) geometry has accordingly been called D-brane geometry or D-geometry for short.

Examples

Fuzzy spheres

The fuzzy spheres appear in D-brane geometry:

1. the fuzzy funnels of Dp-D(p+2)-brane intersections have fuzzy 2-sphere slices

2. the fuzzy funnels of Dp-D(p+4)-brane intersections have fuzzy 4-sphere slides

3. the supersymmetric classical solutions of the BMN matrix model are precisely fuzzy 2-sphere configurations (BMN 02 (5.4)).

References

• Michael Douglas, Two Lectures on D-Geometry and Noncommutative Geometry (arxiv:hep-th/9901146)

• Liang Kong, Conformal field theory and a new geometry, in Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory (arXiv:1107.3649)

• Chien-Hao Liu, Azumaya noncommutative geometry and D-branes - an origin of the master nature of D-branes (arXiv:1112.4317)

With emphasis on the IKKT matrix model and the noncommutative geometry of fuzzy spheres:

• Harold Steinacker, Emergent Geometry and Gravity from Matrix Models: an Introduction, Class. Quant. Grav.27: 133001, 2010 (arXiv:1003.4134)

• Harold Steinacker, Non-commutative geometry and matrix models, PoS QGQGS2011 (2011) 004 (arXiv:1109.5521)

• Badis Ydri, Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory (arXiv:1708.00734), published as: Matrix Models of String Theory, IOP 2018 (ISBN:978-0-7503-1726-9)