Duality Symmetric String and M-Theory

Abstract: We review recent developments in duality symmetric string theory. We begin with the world sheet doubled formalism which describes strings in an extended space time with extra coordinates conjugate to winding modes. This formalism is T-duality symmetric and can accommodate non-geometric T-fold backgrounds which are beyond the scope of Riemannian geometry. Vanishing of the conformal anomaly of this theory can be interpreted as a set of spacetime equations for the background fields. These equations follow from an action principle that has been dubbed Double Field Theory (DFT). We review the aspects of generalised geometry relevant for DFT. We outline recent extensions of DFT and explain how, by relaxing the so-called strong constraint with a Scherk Schwarz ansatz, one can obtain backgrounds that simultaneously depend on both the regular and T-dual coordinates. This provides a purely geometric higher dimensional origin to gauged supergravities that arise from non-geometric compactification. We then turn to M-theory and describe recent progress in formulating an E_{n(n)} U-duality covariant description of the dynamics. We describe how spacetime may be extended to accommodate coordinates conjugate to brane wrapping modes and the construction of generalised metrics in this extend space that unite the bosonic fields of supergravity into a single object. We review the action principles for these theories and their novel gauge symmetries. We also describe how a Scherk Schwarz reduction can be applied in the M-theory context and the resulting relationship to the embedding tensor formulation of maximal gauged supergravities.

• The paper introduces a duality-symmetric formulation that unifies world-sheet doubling in string theory with extended spacetime in M-theory, manifesting both T- and U-dualities.
• The paper employs Double Field Theory to incorporate non-geometric backgrounds and a strong constraint that ensures gauge algebra closure.
• The paper outlines future directions to relax conventional constraints and explore fermionic sectors and extended duality symmetries beyond E7.

An Overview of "Duality Symmetric String and M-Theory" by David S. Berman and Daniel C. Thompson

The paper "Duality Symmetric String and M-Theory" by Berman and Thompson presents significant developments in the formulation of string and M-theory frameworks that make dualities manifest at the level of the action. These dualities, such as T-duality in string theory and U-duality in M-theory, are powerful symmetries that relate seemingly distinct configurations of these theories. The authors explore both world-sheet and spacetime approaches that incorporate duality as a central feature, presenting the concept of Duality Symmetric String Theory and Double Field Theory (DFT), as well as discussing generalizations toward M-theory.

Duality Symmetric String Theory and Double Field Theory (DFT)

The authors begin by discussing the world-sheet doubled formalism, which is a duality symmetric approach to string theory. In this framework, additional coordinates are introduced such that the string's target space is locally doubled. This allows the full T-duality group to act linearly, providing a symmetric treatment to momentum and winding modes of the string.

The spacetime counterpart is given by Double Field Theory (DFT), as developed in later works, which extends the target space by incorporating dual coordinates, leading to an $O(d,d)$ invariance for strings compactified on $T^d$. The DFT provides a coherent framework that accommodates non-geometric backgrounds (T-folds) and allows for an exploration of string theory solutions beyond traditional geometric descriptions.

Key to DFT is the imposition of a "strong constraint," a necessary condition for closure of the gauge algebra that severely restricts the coordinate dependence but allows for a consistent classical and quantum theory that reflects the dualities inherent in string theory.

M-Theory Extensions and U-Duality

In higher-dimensional theories like M-theory, analogous constructions are proposed through exceptional generalized geometry, where U-duality groups such as $E_n$ for various $n$ are made manifest. The idea is to extend the spacetime coordinates and reorganize the field content (including metric, three-form, and six-form fields in supergravity) into representations of these duality groups.

The concept of "extended spacetime" is introduced, where extra dimensions are included to linearly realize U-duality groups of M-theory. This leads to a formulation where the action is invariant under these duality transformations. The resulting theory is not just a toroidal compactification but a new structure that provides insights into non-trivial backgrounds and exotic branes, which do not have a geometric interpretation in conventional spacetime.

Implications and Future Directions

The formulations presented have profound implications both for theoretical explorations within string theory and practical applications, such as in cosmology and particle physics model building. The duality symmetric approaches may offer new insights into moduli stabilization and hints towards a deeper understanding of non-perturbative aspects of string/M-theory.

However, several challenges remain — most notably the strong constraint in DFT, which restricts dependencies to only half of the doubled coordinates, thus making the theory locally equivalent to standard supergravity. The search for ways to consistently relax this constraint to explore full dependencies could unlock new phases of string/M-theory.

Future developments aim to establish fully covariant formulations for fermionic sectors and extend beyond $E_7$ to potentially capture all aspects of eleven-dimensional supergravity. Moreover, a deeper investigation into the global and topological properties of these extended spaces could elucidate the nature of non-geometric backgrounds and exotic branes.

In summary, Berman and Thompson's paper articulates a well-rounded approach to duality symmetries in string and M-theory, offering an innovative pathway to understand the fabric of these theories through a unified formalism.