{"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":"f9fdc551-10a8-56eb-8378-2de96c8e034c","task_key":"train--Challenge~5f3~5fmain","task_revision_id":"3","upstream_id":"Challenge_3_main","short_description":"In the AdS$_3$/BCFT$_2$ correspondence, consider a setup where the bulk black…","config":"","split":"train","body":"{\"code_template\":\"import sympy as sp\\n\\nm, r0, eta = sp.symbols('m r0 eta', positive=True)\\n\\ndef answer(m, eta, r0):\\n    r\\\"\\\"\\\"\\n    Return expression of the one-point function\\n\\n    Inputs\\n    ----------\\n    m: sympy.Symbol, mass of the buld field, $m$\\n    eta: sympy.Symbol, the brane tension, $\\\\eta$\\n    r0: sympy.Symbol, the black hole radius, $r_0$\\n\\n    Output\\n    ----------\\n    O_x: sympy.Expr, the one-point function, $\\\\langle \\\\mathcal{O}(x) \\\\rangle$\\n    \\\"\\\"\\\"\\n\\n    # ------------------ FILL IN YOUR RESULT BELOW ------------------\\n    O_x = ...   # a SymPy expression of inputs\\n    # ---------------------------------------------------------------\\n\\n    return O_x\",\"problem_description\":\"# Problem setup:\\nIn the AdS$_3$/BCFT$_2$ correspondence, consider a setup where the bulk black hole geometry—characterized by an inverse temperature $\\\\beta = \\\\frac{2\\\\pi}{r_0}$—is given by\\n\\\\begin{equation}\\nds^2=f(r)d\\\\tau_E^2+\\\\frac{dr^2}{f(r)}+r^2 d\\\\phi^2, \\\\qquad f(r)=r^2-r_0^2,\\n\\\\end{equation}\\nwith $\\\\phi \\\\sim \\\\phi + 2\\\\pi$ and $\\\\tau_E \\\\sim \\\\tau_E + \\\\beta = \\\\tau_E + \\\\frac{2\\\\pi}{r_0}$, and this geometry is terminated by a spherically symmetric brane of tension $\\\\eta$, where $0<\\\\eta<1$, behind the horizon. Let $\\\\mathcal{O}(x)$ be a scalar primary operator in the BCFT, whose bulk dual is a massive scalar field of mass $m$. For simplicity, assume the operator is inserted at the Euclidean time-reflection symmetric slice $\\\\tau_E = 0$.\\n\\n# Main problem:\\n\\nUsing the geodesic approximation, determine the form of the one-point function $\\\\langle \\\\mathcal{O}(x) \\\\rangle$. How does the result depend on the mass $m$ of the bulk field, the black hole radius $r_0$ and the brane tension $\\\\eta$?\"}","display_format":"text","language":"","answer_status":"published","assets":[],"source_url":"https://critpt.com/","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}