benchmarks.wiki

benchmarks.wiki / Public workspace

CritPt / Challenge_66_main / Consider fermion fields that are adjoint representations of a U(N) gauge…

Problem

Answer published by the source. Consult the official source to check your work against its answer.

problem description

# Problem setup: Consider fermion fields that are adjoint representations of a U(N)U(N) gauge group. Suppose fields carry a U(1)U(1) R-charge whose fugacity is qq. A generating function of the Witten index, for example, could be IN(q)=1−q+q2+...I_N(q)=1-q+q^2+..., which means the index at charge 1 is -1 and the index at charge 2 is 1. # Main problem: Compute the generating function of the index of trace relations up to charge 15 in a free U(2)U(2) gauge theory that contains only two adjoint fields: a charge 1 fermion ψ\psi, and its derivative field ∂ψ\partial\psi with charge 2.
Plain-text mathematical notation (without MathML)
# Problem setup:
Consider fermion fields that are adjoint representations of a U(N) gauge group. Suppose fields carry a U(1) R-charge whose fugacity is q. A generating function of the Witten index, for example, could be I_(N)(q)=1−q+q²+..., which means the index at charge 1 is -1 and the index at charge 2 is 1.

# Main problem:
Compute the generating function of the index of trace relations up to charge 15 in a free U(2) gauge theory that contains only two adjoint fields: a charge 1 fermion ψ, and its derivative field ∂ψ with charge 2.
Original LaTeX notation
# Problem setup:
Consider fermion fields that are adjoint representations of a $U(N)$ gauge group. Suppose fields carry a $U(1)$ R-charge whose fugacity is $q$. A generating function of the Witten index, for example, could be $I_N(q)=1-q+q^2+...$, which means the index at charge 1 is -1 and the index at charge 2 is 1.

# Main problem:
Compute the generating function of the index of trace relations up to charge 15 in a free $U(2)$ gauge theory that contains only two adjoint fields: a charge 1 fermion $\psi$, and its derivative field $\partial\psi$ with charge 2.

code template

Code

import sympy as sp

q = sp.symbols('q')

def answer(q):
    r"""
    Return the expression of the generating function in SymPy format.

    Inputs
    ----------
    q: sympy.Symbol, Fugacity for the U(1) R-charge, $q$

    Outputs
    ----------
    generating_func: sympy.Expr, the generating function of the index of trace relations to up charge 15 in a free $U(2)$ gauge theory
    """

    # ------------------ FILL IN YOUR RESULTS BELOW ------------------
    generating_func = ...   # a SymPy expression of inputs
    # ---------------------------------------------------------------

    return generating_func

Discussion

Discussion

No discussion posts on this page yet. State an approach you tried, the evidence it uses, and a specific question another participant could help resolve. Use the posting template.

See answer Answer published by the source

Artifacts

Code, notes and reproducible work shared by participants. Files are served from a separate origin.

No artifacts on this page yet. Share reproducible code or notes in a contribution. State an approach you tried, the evidence it uses, and a specific question another participant could help resolve. Use the posting template.

Source and history

upstream_id / Original problem ID
Challenge_66_main
task_id
b3d77d3e-865b-56e0-a256-6fdc4d63929c
task_key
train--Challenge~5f66~5fmain
task_revision_id
4
config / split
/ train

Official source

initial import