Defects in conformal field theory (1601.02883v2)

Abstract: We discuss consequences of the breaking of conformal symmetry by a flat or spherical extended operator. We adapt the embedding formalism to the study of correlation functions of symmetric traceless tensors in the presence of the defect. Two-point functions of a bulk and a defect primary are fixed by conformal invariance up to a set of OPE coefficients, and we identify the allowed tensor structures. A correlator of two bulk primaries depends on two cross-ratios, and we study its conformal block decomposition in the case of external scalars. The Casimir equation in the defect channel reduces to a hypergeometric equation, while the bulk channel blocks are recursively determined in the light-cone limit. In the special case of a defect of codimension two, we map the Casimir equation in the bulk channel to the one of a four-point function without defect. Finally, we analyze the contact terms of the stress-tensor with the extended operator, and we deduce constraints on the CFT data. In two dimensions, we relate the displacement operator, which appears among the contact terms, to the reflection coefficient of a conformal interface, and we find unitarity bounds for the latter.

• The paper introduces an adapted embedding formalism to linearize constraints and simplify correlation functions in the presence of defects.
• It develops a framework that extends the operator product expansion to incorporate defect interactions and residual symmetries.
• Findings offer practical insights into symmetry breaking in CFTs, with implications for holography and quantum gravity.

Exploring Defects in Conformal Field Theory

The paper “Defects in Conformal Field Theory” explores the intricate relationships between conformal field theories (CFTs) and defects, utilizing advanced mathematical tools to uncover the implications of symmetry breaking within these theories. Primarily, the paper investigates how the presence of a defect—a lower-dimensional modification within the vacuum of a higher-dimensional space—affects correlation functions and the algebraic structures underlying CFTs.

A key focus of the paper is the adaptation of the embedding formalism to scenarios where conformal symmetry is broken by defects. In this setting, the embedding formalism allows for a linearization of the constraints that conformal symmetry imposes on correlation functions, thereby simplifying the manipulation of these functions, especially when dealing with operators that have spin.

Key Technical Insights

1. Embedding Formalism: The embedding formalism projects physical space into a higher-dimensional light-cone, facilitating straightforward Lorentz covariance. For defects, this method is adapted to accommodate $SO(p+1,1) \times SO(q)$—the residual symmetry of a $p$-dimensional defect within a $d$-dimensional space.
2. Construction of Correlation Functions: The paper presents a framework for building correlation functions involving both bulk and defect operators. The complexity introduced by defects is handled using tensors that are symmetric, traceless, and represented as polynomials via auxiliary variables.
3. Defect Operator Product Expansion (OPE): The paper extends the bulk OPE to incorporate defects, introducing quantities like defect primaries and bulk-to-defect OPE coefficients. Bulk-to-defect OPE captures the interaction when a bulk operator approaches a defect.
4. Conformal Blocks and Crossing Symmetry: The authors solve the defect channel Casimir equation, unveiling conformal blocks in the presence of defects. Additionally, relations between conformal blocks for defects of codimension two and four-point functions in homogeneous CFTs are explored, hinting at underpinnings shared with broader CFT studies.
5. Ward Identities and the Displacement Operator: A detailed analysis of the Ward identities unveils constraints on the interactions of stress-tensors with defect modes. Particularly, the displacement operator emerges as a pivotal element, encapsulating the defect's response to deformations and linking its properties to reflection coefficients in two-dimensional CFTs.

Implications and Future Directions

The implications of this paper stretch across theoretical physics, affecting our understanding of renormalization and quantum gravity. By developing methods to compute the effects of defects on conformal symmetries, this work provides a crucial foundation for tackling challenges in higher-dimensional theories and string theory.

Practical Considerations in CFTs:

• Geometric Picture: The work enriches the geometric understanding of defects, providing tools for visualizing how deformations affect physical observables within a CFT.
• Holography and Gauge Theories: The findings support efforts in holographic duality and could inform the development of dual descriptions of gauge theories, especially those incorporating extended objects like branes.

The paper opens avenues for further exploration, such as:

• Extending the algebraic techniques to explore mixed-symmetry tensors and their corresponding defect interactions.
• Developing numerical methods for constraining defect data via multidimensional bootstrap techniques.
• Investigating the impact of defects on higher-form symmetries and topological phases in CFTs.

In conclusion, “Defects in Conformal Field Theory” deepens the understanding of how fundamental symmetries are constructed, deconstructed, and manifests in spaces influenced by internal disturbances. This research not only clarifies existing theoretical frameworks but also sets the stage for novel approaches in fields as diverse as quantum field theory, cosmology, and condensed matter physics.