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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.

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

problem description

# 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.

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.

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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: 1
config / split: / train

Official source

initial import