M-theory is the conjectured non-perturbative completion of string theory unifying all consistent versions of superstring theory, and a key open problem remains to understand this completion.
Since its inception, M-theory has greatly contributed to the understanding of string theory, and its basic ideas have found their way into the textbooks. Nevertheless, a proper definition or rather a formulation of M-theory as a coherent theory, and, consequently, a formulation of a full non-perturbative completion of string theory, remains an open problem.
A key issue in formulating M-theory is that its underlying principles have remained unclear. If available hints are anything to go by, M-theory is not simply going to be defined by a Lagrangian, a scattering matrix or any other traditional structure of quantum physics.
However, one set of ideas that has been at the forefront from the early days of string theory is the appearance of what nowadays is encapsulated under the notion of higher structures. This is essentially short for higher homotopy structures (as in higher homotopy groups). Such structures manifest themselves in the higher degrees that are carried by string theoretic objects as compared to their field theoretic counterparts, notably the higher degrees of the flux fields, and the resulting higher order gauge-of-gauge transformations. Generally, the appearance of homotopy theory is simply the consequence of the gauge principle in physics.
Once we switch to the second quantisation of the string, known as string field theory, higher algebraic structures become ubiquitous. In particular, the Hilbert space of closed string field theory carries a higher algebraic structure known as an L∞-algebra. Whilst bosonic open string field theory has a Chern—Simons like formulation, its supersymmetric completion also requires the introduction of a higher homotopy algebra known as an A∞-algebra.
We develop and analyse these higher structures, such as higher differential geometry and homotopy algebras, with an aim of getting a better understanding of the objects and structures appearing in string and M-theory.