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Green's Functions and Non-Singlet Glueballs on Deformed Conifolds

Silviu S. Pufu, Igor R. Klebanov, Thomas Klose, Jennifer Lin

TL;DR

This work analyzes Green's functions and non-singlet glueball spectra on Stenzel spaces, treating generalized deformed conifolds as Ricci-flat tangent bundles TS^{d-1} with SO(d) symmetry. By decomposing functions into SO(d) harmonics on the Stiefel base V_{d,2} and exploiting decoupling within fixed representations, the authors derive coupled radial ODEs for the Green's function and for bulk mode fluctuations, revealing mixing among same-quantum-number harmonics caused by the ε-deformation that breaks the would-be U(1) R-symmetry. They solve these systems numerically for d=4 and d=5 to obtain bound-state spectra of non-singlet glueballs in several SO(d) representations, and they complement the results with a WKB analysis to characterize the large-n behavior. The methods are generalized to arbitrary d (including d=5 relevant to the CGLP background) and extended to compute bound-state masses in the M-theory setup, illuminating how confinement-like spectra arise in these warped geometries and enabling systematic studies of non-singlet sectors in gauge/gravity duality.

Abstract

We study the Laplacian on Stenzel spaces (generalized deformed conifolds), which are tangent bundles of spheres endowed with Ricci flat metrics. The (2d-2)-dimensional Stenzel space has SO(d) symmetry and can be embedded in C^d through the equation \sum_{i = 1}^d {z_i^2} = ε^2. We discuss the Green's function with a source at a point on the S^{d-1} zero section of TS^{d-1}. Its calculation is complicated by mixing between different harmonics with the same SO(d) quantum numbers due to the explicit breaking by the ε-deformation of the U(1) symmetry that rotates z_i by a phase. A similar mixing affects the spectrum of normal modes of warped deformed conifolds that appear in gauge/gravity duality. We solve the mixing problem numerically to determine certain bound state spectra in various representations of SO(d) for the d=4 and d=5 examples.

Green's Functions and Non-Singlet Glueballs on Deformed Conifolds

TL;DR

This work analyzes Green's functions and non-singlet glueball spectra on Stenzel spaces, treating generalized deformed conifolds as Ricci-flat tangent bundles TS^{d-1} with SO(d) symmetry. By decomposing functions into SO(d) harmonics on the Stiefel base V_{d,2} and exploiting decoupling within fixed representations, the authors derive coupled radial ODEs for the Green's function and for bulk mode fluctuations, revealing mixing among same-quantum-number harmonics caused by the ε-deformation that breaks the would-be U(1) R-symmetry. They solve these systems numerically for d=4 and d=5 to obtain bound-state spectra of non-singlet glueballs in several SO(d) representations, and they complement the results with a WKB analysis to characterize the large-n behavior. The methods are generalized to arbitrary d (including d=5 relevant to the CGLP background) and extended to compute bound-state masses in the M-theory setup, illuminating how confinement-like spectra arise in these warped geometries and enabling systematic studies of non-singlet sectors in gauge/gravity duality.

Abstract

We study the Laplacian on Stenzel spaces (generalized deformed conifolds), which are tangent bundles of spheres endowed with Ricci flat metrics. The (2d-2)-dimensional Stenzel space has SO(d) symmetry and can be embedded in C^d through the equation \sum_{i = 1}^d {z_i^2} = ε^2. We discuss the Green's function with a source at a point on the S^{d-1} zero section of TS^{d-1}. Its calculation is complicated by mixing between different harmonics with the same SO(d) quantum numbers due to the explicit breaking by the ε-deformation of the U(1) symmetry that rotates z_i by a phase. A similar mixing affects the spectrum of normal modes of warped deformed conifolds that appear in gauge/gravity duality. We solve the mixing problem numerically to determine certain bound state spectra in various representations of SO(d) for the d=4 and d=5 examples.

Paper Structure

This paper contains 32 sections, 148 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Generalized deformed conifolds
  3. Coordinates and metrics
  4. Topology and group actions
  5. Normalizable functions on Stiefel manifolds
  6. Decoupling of differential equations
  7. The Laplacian on the deformed conifold
  8. Coordinates and Killing vectors
  9. The Laplacian
  10. Green's Function
  11. Normalization of the Green's function
  12. A coupled system of equations
  13. Expectation values of mesonic operators
  14. Basis functions on the deformed conifold
  15. Group actions on traceless polynomials
  16. ...and 17 more sections

Figures (3)

  • Figure 1: Non-singlet spin-2 glueball masses for the warped deformed conifold Klebanov:2000hb. Solid/dashed lines represent states with even/odd parity. The representations are ordered according to the mass of the lowest state. The numbers being plotted correspond to the values of $\hat{m}$ as defined in (\ref{['mhatdef']}).
  • Figure 2: The logarithm of $\lvert \det \gamma(m^2) \rvert$ in the range $m^2 \in [0, 80]$ for the even sector of the $[\frac{9}{2},\frac{9}{2}]$ representation. The glueball masses are located at the dips in the plot. The masses can be seen to organize into 5 towers.
  • Figure 3: Glueball masses for spin-2 excitations of the warped M-theory background Cvetic:2000db constructed from the 8-d Stenzel space. Solid/dashed lines represent states with even/odd parity. The representations are ordered according to the mass of the lowest state. The numbers being plotted correspond to $\hat{m}$ as defined in (\ref{['mhatDefM']}).