{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"critpt","formal_name":"CritPt","introduction":"CritPt evaluates scientific understanding and multi-step reasoning and computation on research-level physics problems. Its public dataset contains 70 challenges with problem descriptions and code templates.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://critpt.com/","indexing_mode":"noindex","profile":{"resources":[],"task_format":"","scoring":"","metric":"","size":"","answer_access":"","license":"","citation":"","maintainer":"","released":"","why_hard":"","related":[]}},"task_id":"c7e23935-3ba2-5aff-9303-552b61ba41ce","task_key":"train--Challenge~5f1~5fmain","task_revision_id":"2","upstream_id":"Challenge_1_main","short_description":"Consider a quantum field theory with holographic dual. Under a Weyl…","config":"","split":"train","body":"{\"code_template\":\"def answer():\\n    r\\\"\\\"\\\"\\n    Return coefficients of the terms.\\n\\n    Input\\n    ----------\\n    None\\n\\n    Output\\n    ----------\\n    coeffs: list[float], the coefficients of terms in $X^{(4)}$, in the order given in the problem\\n    \\\"\\\"\\\"\\n    # ------------------ FILL IN YOUR RESULTS BELOW ------------------\\n    coeffs = ...\\n    # ---------------------------------------------------------------\\n\\n    return coeffs\",\"problem_description\":\"\\n\\n# Problem setup:\\nConsider a quantum field theory with holographic dual. Under a Weyl transformation, the boundary metric transforms as $\\\\gamma_{\\\\mu\\\\nu}^{(0)}\\\\to{\\\\cal B}^{-2}(x)\\\\gamma_{\\\\mu\\\\nu}^{(0)}$. The Weyl anomaly ${\\\\cal A}_k$ of the theory in $2k$ dimensions appears in the transformation of the partition function:\\n\\\\begin{align}\\nZ[\\\\gamma_{\\\\mu\\\\nu}^{(0)}]\\\\to e^{-{\\\\cal A}_k}Z[{\\\\cal B}(x)^{-2}\\\\gamma^{(0)}].\\n\\\\end{align}\\nThis can be computed by evaluating the on-shell action of the bulk gravitational theory.\\n\\nThe holographic Weyl anomaly in $d\\\\leqslant8$ can be express using the following quantities:\\n\\\\begin{align}\\nP_{\\\\mu\\\\nu}={}&R^{(0)}_{\\\\mu\\\\nu}-\\\\frac{R^{(0)}}{2(d-1)}\\\\gamma_{\\\\mu\\\\nu}^{(0)}\\\\,,\\\\\\\\\\nB_{\\\\mu\\\\nu}={}&\\\\frac{1}{d-2}\\\\big(\\\\nabla^{(0)}_\\\\rho\\\\nabla_{(0)}^\\\\rho  P_{\\\\mu\\\\nu}-\\\\nabla^{(0)}_\\\\rho\\\\nabla^{(0)}_{\\\\nu} P_{\\\\mu}{}^{\\\\rho}- W^{(0)}_{\\\\rho\\\\nu\\\\mu\\\\sigma} P^{\\\\sigma\\\\rho}\\\\big)\\\\,,\\\\\\\\\\nO_{\\\\mu\\\\nu}={}&\\\\nabla_{(0)}^\\\\lambda\\\\nabla^{(0)}_\\\\lambda B_{\\\\mu\\\\nu}-2W^{(0)}_{\\\\rho\\\\nu\\\\mu\\\\lambda}B^{\\\\lambda\\\\rho}-\\\\frac{4}{d-2}B_{\\\\mu\\\\nu}P^\\\\mu{}_\\\\mu+\\\\frac{2(d-4)}{(d-2)^2}\\\\big(2P^{\\\\rho\\\\lambda}\\\\nabla^{(0)}_\\\\lambda C_{(\\\\mu\\\\nu)\\\\rho}\\\\\\\\\\n&+\\\\nabla^{(0)}_\\\\lambda PC_{(\\\\mu\\\\nu)}{}^\\\\lambda-C^{\\\\rho}{}_{\\\\mu}{}^{\\\\lambda}C_{\\\\lambda\\\\nu\\\\rho}+ \\\\nabla_{(0)}^\\\\lambda P^\\\\rho{}_{(\\\\mu}C_{\\\\nu)\\\\rho\\\\lambda}-W^{(0)}_{\\\\rho\\\\mu\\\\nu\\\\lambda}P^{\\\\lambda}{}_\\\\sigma P^{\\\\sigma\\\\rho}\\\\big)\\\\,,\\\\\\\\\\n\\\\Omega_{\\\\mu\\\\nu}={}&\\\\nabla_{(0)}^\\\\lambda\\\\nabla^{(0)}_\\\\lambda B_{\\\\mu\\\\nu}-2W^{(0)}_{\\\\rho\\\\nu\\\\mu\\\\lambda}B^{\\\\lambda\\\\rho}-4B_{\\\\mu\\\\nu}P^\\\\mu{}_\\\\mu+2(d-4)\\\\big(2P^{\\\\rho\\\\lambda}\\\\nabla^{(0)}_\\\\lambda C_{(\\\\mu\\\\nu)\\\\rho}\\\\\\\\\\n&+\\\\nabla^{(0)}_\\\\lambda PC_{(\\\\mu\\\\nu)}{}^\\\\lambda-C^{\\\\rho}{}_{\\\\mu}{}^{\\\\lambda}C_{\\\\lambda\\\\nu\\\\rho}+ \\\\nabla_{(0)}^\\\\lambda P^\\\\rho{}_{(\\\\mu}C_{\\\\nu)\\\\rho\\\\lambda}-W^{(0)}_{\\\\rho\\\\mu\\\\nu\\\\lambda}P^{\\\\lambda}{}_\\\\sigma P^{\\\\sigma\\\\rho}\\\\big)+P_{\\\\mu\\\\rho}P^{\\\\rho\\\\sigma}P_{\\\\sigma\\\\nu},\\n\\\\end{align}\\nwhere $R^{(0)}_{\\\\mu\\\\nu}$ is the Ricci tensor for the boundary metric $\\\\gamma_{\\\\mu\\\\nu}^{(0)}$, $\\\\nabla^{(0)}_\\\\mu$ is the covariant derivative associated with $\\\\gamma^{(0)}_{ij}$ on the boundary, $W^{(0)}_{\\\\rho\\\\nu\\\\mu\\\\sigma}$ is the Weyl tensor on the boundary, and $C_{\\\\mu\\\\nu\\\\rho}=\\\\nabla^{(0)}_\\\\rho P_{\\\\mu\\\\nu}-\\\\nabla^{(0)}_\\\\nu P_{\\\\mu\\\\rho}$.\\n\\nThe final expression will have the form\\n\\\\begin{align}\\n{\\\\cal A}_4=&-\\\\frac{L^7}{8\\\\pi G}\\\\int d^8x\\\\sqrt{-\\\\det\\\\gamma^{(0)}}X^{(4)}\\\\ln {\\\\cal B},\\n\\\\end{align}\\nwhere $X^{(4)}$ may contain the following terms: $\\\\text{tr}(P^4)$, $\\\\text{tr}(P^3)$, $\\\\text{tr}(P^3)\\\\text{tr}(P)$, $\\\\text{tr}(BP)$, $\\\\text{tr}(BP^2)$, $\\\\text{tr}(B^2)$, $\\\\text{tr}(B^2P)$, $\\\\text{tr}(OP)$, $\\\\text{tr}(OP^2)$, $\\\\text{tr}(\\\\Omega)$, $\\\\text{tr}(\\\\Omega P)$.\\n\\n# Main problem:\\nDetermine the coefficients of these terms in $X^{(4)}$.\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}