Abstract
We explore the geometry behind the modular bootstrap and its image in the space of Taylor coefficients of the torus partition function. In the first part, we identify the geometry as an intersection of planes with the convex hull of moment curves on R+⊗ℤ, with boundaries characterized by the total positivity of generalized Hankel matrices. We phrase the Hankel constraints as a semi-definite program, which has several advantages, such as the validity of bounds irrespective of spin truncation. We derive bounds on the gap, twist-gap, and the space of Taylor coefficients themselves. We find that if the gap is above \( {\Delta }_{\textrm{gap}}^{\ast } \), where \( \frac{c-1}{12}<{\Delta}_{\textrm{gap}}^{\ast }<\frac{c}{12} \), all coefficients become bounded on both sides and kinks develop in the space. In the second part, we propose an analytic method of imposing the integrality condition for the degeneracy number in the spinless bootstrap, which leads to a non-convex geometry. We find that even at very low derivative order this condition rules out regions otherwise allowed by bootstraps at high derivative order.
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Acknowledgments
We thank Scott Collier, Shu-Heng Shao and Yuan Xin for discussions. LYC, YTH and HCW are supported by MoST Grant No. 109-2112-M-002 -020 -MY3 and 112-2811-M-002 -054 -MY2. This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 101025095. WL is supported by the US Department of Energy Office of Science under Award Number DE-SC0015845, and the Simons Collaboration on the Non-Perturbative Bootstrap. T.-C. H. is supported by the Simons Collaboration on Global Categorical Symmetries. This work was performed in part at the Aspen Center for Physics, which is supported by National Science Foundation grant PHY-2210452.
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Chiang, LY., Huang, TC., Huang, Yt. et al. The geometry of the modular bootstrap. J. High Energ. Phys. 2024, 209 (2024). https://doi.org/10.1007/JHEP02(2024)209
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DOI: https://doi.org/10.1007/JHEP02(2024)209