Distributional Geometry of Squashed Cones (1306.4000v2)

Abstract: A regularization procedure developed in [1] for the integral curvature invariants on manifolds with conical singularities is generalized to the case of squashed cones. In general, the squashed conical singularities do not have rotational O(2) symmetry in a subspace orthogonal to a singular surface $\Sigma$ so that the surface is allowed to have extrinsic curvatures. A new feature of the squashed conical singularities is that the surface terms in the integral invariants, in the limit of small angle deficit, now depend also on the extrinsic curvatures of $\Sigma$. A case of invariants which are quadratic polynomials of the Riemann curvature is elaborated in different dimensions and applied to several problems related to entanglement entropy. The results are in complete agreement with computations of the logarithmic terms in entanglement entropy of 4D conformal theories [2]. Among other applications of the suggested method are logarithmic terms in entanglement entropy of non-conformal theories and a holographic formula for entanglement entropy in theories with gravity duals.

Distributional Geometry of Squashed Cones: An Academic Review

The paper presents an extension of a regularization method for curvature integrals over manifolds with conical singularities to a broader context involving squashed cones. In contrast to symmetric cones, squashed cones lack rotational O(2) symmetry orthogonal to the singular surface Σ, allowing for non-zero extrinsic curvatures. This modification introduces an innovative dimension where the surface terms in integral invariants also become dependent on these extrinsic curvatures when the angle deficit is small.

Key Methods and Results

The authors employ geometrical techniques akin to those previously used in dealing with symmetric cones, however, adapted for squashed cones. Integral curvature invariants, particularly those quadratic in the Riemann curvature, are scrutinized across various dimensions. These computations are meticulously applied to entanglement entropy within different theoretical frameworks, aligning the findings with extant calculations of logarithmic terms in 4D conformal theories. A notable outcome is that surface integrals on squashed cones are not merely influenced by intrinsic curvature but also by extrinsic factors, with significant implications for entanglement entropy calculations in non-conformal theories or those with gravity duals.

Theoretical and Practical Implications

The implications of this generalization are significant for both the theoretical understanding of entanglement entropy in field theories and practical applications in holographic descriptions in theories with gravity duals. The paper offers a potential holographic formula for entanglement entropy in five-dimensional Gauss-Bonnet gravity, where non-minimal curvature corrections are standard. It suggests that the holographic relation adapts to include quadratic curvature terms, hence extending applicability beyond the conventional Einstein gravity description.

Future Directions in AI and Mathematical Physics

The exploration into squashed cones surfaces opens avenues for investigating geometric properties across various manifold configurations and applying these findings within quantum field theories and gravity models. Future research could expand these methodologies to encompass higher-order polynomial curvature terms or explore ramifications in other dimensions, providing broader impacts in AI applications concerning data representation on complex geometric structures.

Conclusion

This paper methodically advances the paper of integral curvature invariants on manifolds with singularities by addressing complexities presented by extrinsic curvatures. The regularization procedures deployed here facilitate nuanced understandings of entanglement entropy and bolster theoretical frameworks spanning gravitational entropy calculations. These insights present substantial contributions to mathematical physics literature, fostering continued scholarly inquiry into harmonizing quantum theories with geometric formalism.