String Theory Reveals New Connections Between Fields

Scientists are exploring novel geometric structures within open string field theory, potentially bridging the gap between open and closed string descriptions. Yichul Choi from the School of Natural Sciences, Institute for Advanced Study, Princeton, NJ, USA, alongside collaborators, define a 2-form connection within the space of classical solutions using the open string star product and integration. This work introduces new observables, higher holonomies and a 3-form curvature, invariant under the infinite-dimensional gauge algebra of the theory. The research, motivated by recent advances in higher Berry phases, suggests a possible identification of this connection with the Kalb-Ramond field in closed string backgrounds and discusses related sigma models, offering a significant step towards understanding the underlying geometry of string theory and its potential applications to quantum field theory.

This work draws a parallel with the Berry phase in quantum mechanics, extending the concept to a higher-dimensional framework and utilising the open string star product and integration.

The discovery builds upon recent advances in understanding higher Berry phases and offers a potential pathway to describing string geometry through boundary data. Researchers suggest this 2-form connection may correspond to the Kalb-Ramond B-field, a crucial component of closed string backgrounds that describes the interaction of strings with electromagnetic fields.

This research establishes a framework for encoding geometric information within the boundaries of a conformal field theory (CFT), potentially offering a new approach to understanding how spacetime emerges from quantum mechanics. The core achievement lies in defining a mathematical object, the 2-form connection, that links the properties of conformal boundary conditions to the geometry of the underlying string theory.

This connection is expressed through higher holonomies and a 3-form curvature, both of which represent measurable quantities independent of the infinite-dimensional gauge algebra governing open string field theory. The implications extend to the construction of sigma models, where the target space geometry is directly related to these boundary conditions and the associated B-field.

The study demonstrates that the full data of a rational conformal field theory can be determined by specifying a chiral algebra and a consistent solution to the boundary bootstrap axiom, effectively defining the theory through its boundaries. This approach leverages the theory of modular tensor categories and three-dimensional topological quantum field theories to reconstruct the complete CFT from boundary information.

Furthermore, the work builds upon recent findings indicating that the geometry of certain non-linear sigma models, including both the metric and the Kalb-Ramond B-field, can be fully encoded in boundary correlation functions. This suggests a profound relationship between boundary data and the fundamental properties of spacetime, potentially offering a new perspective on stringy geometry.

Defining Higher Holonomies via Open String Field Theory and Sigma Models

A central methodological component of this work involves defining a 2-form connection within the space of classical solutions to bosonic open string field theory, utilising the open string star product and integration techniques. This approach establishes a framework analogous to the Berry phase observed in quantum mechanics, extending the concept to a higher-dimensional context and drawing inspiration from recent investigations into higher Berry phases.

The selection of this mathematical construct allows for the identification of new observables, higher holonomies and a 3-form curvature, which remain invariant under the infinite-dimensional gauge algebra inherent to open string field theory. To further explore this connection, the research investigates sigma models, where the target space is defined by the moduli space of conformal boundary conditions of a two-dimensional conformal field theory (CFT).

These sigma models incorporate a 2-form connection related to the previously defined 2-form connection, facilitating the study of how boundary conditions influence the overall CFT and providing a means to probe the relationship between open and closed string theories. The study leverages the established theory of modular tensor categories (MTCs) and three-dimensional topological quantum field theories (TQFTs) to analyse rational conformal field theories (RCFTs).

Specifically, the research examines how a solution to the boundary bootstrap axiom on a rational conformal boundary gives rise to an algebra object within the MTC. This algebra object is then used to define a topological surface defect in the corresponding TQFT, effectively linking boundary conditions to a geometric representation within the TQFT framework. The dimensional reduction of this TQFT on an interval then recovers the RCFT, demonstrating a pathway to reconstruct the full CFT from boundary data.

Emergent higher curvature from open string solution space geometry

Calculations reveal a 2-form connection defined within the space of classical solutions to bosonic open string field theory, constructed using the open string star product and integration. Corresponding higher holonomies and a resultant 3-form curvature emerge as novel observables, remaining invariant under the infinite-dimensional gauge algebra inherent to open string field theory.

This definition draws parallels with the Berry phase in quantum mechanics and builds upon recent investigations into higher Berry phases. The research proposes that this 2-form connection corresponds to the Kalb-Ramond B-field of a closed string background, particularly under suitable conditions. The connection, Bij(λ), is expressed as an integral over classical solutions Ψ(λ) and their derivatives with respect to parameters λi and λj, where λ represents a point in an N-dimensional subspace X of all classical solutions.

This integral incorporates the wedge product of differentials and a symmetric operation, ensuring gauge invariance. Specifically, the connection is calculated as the integral of Ψ∗∂Ψ∂λi∗∂Ψ∂λj minus its exchange, providing a well-defined 2-form. The resulting expression serves as a foundational element for constructing higher holonomies and the associated 3-form curvature.

Consideration is given to sigma models where the target space corresponds to the moduli space of conformal boundary conditions within a two-dimensional conformal field theory. These models utilise a 2-form connection akin to the one defined in the open string field theory. The research suggests that, when the parameter space X can be interpreted as a moduli space of D-instantons, the calculated 2-form connection Bij(λ) should align with the background Kalb-Ramond B-field, offering new perspectives on how the non-commutative open string star algebra encapsulates information about massless closed string background fields when a geometric interpretation is available.

Geometric connections and invariant holonomies in open string field theory

Scientists have long sought a complete, self-consistent formulation of string theory, one that moves beyond perturbative approximations. This work offers a novel approach, defining a geometric structure, a 2-form connection, within the framework of open string field theory, analogous to the familiar Berry phase in quantum mechanics, but extended to the more complex realm of string interactions.

This new connection, and its associated higher holonomies, appear to be invariants, meaning they remain consistent regardless of how spacetime is deformed. This hints at a deeper, more robust structure underlying string interactions, a structure that could allow for calculations independent of arbitrary backgrounds. However, establishing a definitive link to the Kalb-Ramond field of closed string theory will require substantial further investigation.

The connection isn’t guaranteed and may only hold under specific conditions. Future work will likely focus on exploring the sigma models built upon this connection, and on determining whether it can be extended to incorporate more complex string configurations and interactions, potentially bridging the gap between abstract theory and observable phenomena.