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U(1) Charges and Moduli in the D1-D5 System

Finn Larsen, Emil Martinec

TL;DR

This work analyzes the decoupled D1-D5 system on $T^4\times S^1$ to understand a rich set of $U(1)$ charged excitations and their masses and entropy across moduli, linking BPS constraints to non-BPS spectra in the decoupled theory. It derives a detailed mass and moduli framework, including fixed scalars and singular loci, and shows how the global moduli identifications reduce to a diagonal lattice subgroup $\Gamma_0(N)$ acting on a canonical charge frame. The authors then connect these spacetime results to the symmetric orbifold CFT ${\sl Sym}^N(T^4)\times\tilde{T}^4$, providing a consistent map between BPS spectra, moduli dependences, and the structure of weak-coupling domains corresponding to different partitions of $N$. This establishes a coherent picture in which the orbifold point sits at a cusp of the moduli space, with other partitions realized in regions connected through strong coupling, and it clarifies how U(1) charges and moduli organize the dual CFT data and entropy counting. Overall, the paper advances the understanding of how moduli control U(1) charge spectra and how symmetric orbifold descriptions emerge and organize in the D1-D5 AdS/CFT system.

Abstract

The decoupling limit of the D1-D5 system compactified on T^4\times S^1 has a rich spectrum of U(1) charged excitations. Even though these states are not BPS in the limit, BPS considerations determine the mass and the semiclassical entropy for a given charge vector. The dependence of the mass formula on the compactification moduli situates the symmetric orbifold Sym^N(T^4) x T^4 conformal field theory in the moduli space. A detailed analysis of the global identifications of the moduli space yields a picture of multiple weak-coupling limits - one for each factorization of N into D1 and D5 charges d1 and d5=N/d1 - joined through regions of strong coupling in the CFT moduli space.

U(1) Charges and Moduli in the D1-D5 System

TL;DR

This work analyzes the decoupled D1-D5 system on to understand a rich set of charged excitations and their masses and entropy across moduli, linking BPS constraints to non-BPS spectra in the decoupled theory. It derives a detailed mass and moduli framework, including fixed scalars and singular loci, and shows how the global moduli identifications reduce to a diagonal lattice subgroup acting on a canonical charge frame. The authors then connect these spacetime results to the symmetric orbifold CFT , providing a consistent map between BPS spectra, moduli dependences, and the structure of weak-coupling domains corresponding to different partitions of . This establishes a coherent picture in which the orbifold point sits at a cusp of the moduli space, with other partitions realized in regions connected through strong coupling, and it clarifies how U(1) charges and moduli organize the dual CFT data and entropy counting. Overall, the paper advances the understanding of how moduli control U(1) charge spectra and how symmetric orbifold descriptions emerge and organize in the D1-D5 AdS/CFT system.

Abstract

The decoupling limit of the D1-D5 system compactified on T^4\times S^1 has a rich spectrum of U(1) charged excitations. Even though these states are not BPS in the limit, BPS considerations determine the mass and the semiclassical entropy for a given charge vector. The dependence of the mass formula on the compactification moduli situates the symmetric orbifold Sym^N(T^4) x T^4 conformal field theory in the moduli space. A detailed analysis of the global identifications of the moduli space yields a picture of multiple weak-coupling limits - one for each factorization of N into D1 and D5 charges d1 and d5=N/d1 - joined through regions of strong coupling in the CFT moduli space.

Paper Structure

This paper contains 20 sections, 65 equations, 1 figure.

Table of Contents

  1. Introduction and Summary
  2. The Setting
  3. The Scaling Limit:
  4. The Tensor Background:
  5. The Vector Excitations:
  6. Masses and Moduli
  7. The Tensor Fields:
  8. The Fixed Scalars:
  9. The Singularities in Moduli Space:
  10. The Vector Fields:
  11. Entropy and U(1) Charges
  12. A fundamental domain for the moduli space
  13. Comparison with the symmetric orbifold CFT
  14. The spectrum, BPS and otherwise
  15. Details for $X^4=T^4$:
  16. ...and 5 more sections

Figures (1)

  • Figure 1: Fundamental domain in $\tau=\chi+i/g_{s}$ of the D1-D5 CFT for $N=6$. Corresponding boundary segments are identified under elements of $\Gamma_0(6)$ (in this example, the identified segments are symmetric about the imaginary axis). Heavy dashed lines denote the subspaces where the spacetime CFT becomes singular.The line from $\tau=0$ to $i\infty$ is the singular locus for $(d_1,d_5)=(1,6)$ or $(6,1)$, with the lower end corresponding to the former charge assignment and the upper end corresponding to the latter. Similarly, the circular arc from $\tau=1/3$ to $1/2$ is the singular locus for charges $(3,2)$ and $(2,3)$, respectively. Each rational cusp corresponds to a different weak-coupling limit of the spacetime CFT.