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Cheng–Duncan–Harvey umbral moonshine module conjecture

Let XX be a Niemeier root system, let m:=mXm:=m^X be its Coxeter number, and let GXG^X be its umbral group. Let IXI^X denote the index set used for the components of the vector-valued mock modular form HgX=(Hg,rX)H^X_g=(H^X_{g,r}). A bi-graded infinite-dimensional GXG^X-module is a decomposition

KˇX=⨁r∈IX⨁D∈Z,D≤0,\D=r2(mod4m)Kˇr,−D/4mX.\check{K}^X=\bigoplus_{r\in I^X}\bigoplus_{\substack{D\in\mathbb Z,\;D\leq 0,\D=r^2\pmod{4m}}}\check{K}^X_{r,-D/4m}.

Umbral moonshine module conjecture. There is a naturally defined module KˇX\check{K}^X such that, for g∈GXg\in G^X and r∈IXr\in I^X,

Hg,rX(τ)=−2q−1/4mδr,1+∑D∈Z,D≤0,\D=r2(mod4m)tr⁡(g∣Kˇr,−D/4mX)q−D/4m,H^X_{g,r}(\tau)=-2q^{-1/4m}\delta_{r,1}+\sum_{\substack{D\in\mathbb Z,\;D\leq 0,\D=r^2\pmod{4m}}}\operatorname{tr}(g\mid\check{K}^X_{r,-D/4m})q^{-D/4m},

so that HgXH^X_g is a McKay–Thompson series for KˇX\check{K}^X. The conjecture generalizes the Mathieu moonshine case X=A124X=A_1^{24}, for which GX≃M24G^X\simeq M_{24}. The paper proves the existence of the umbral moonshine modules in the remaining 22 cases, so this conjecture is solved; the statement also records a correction concerning the X=A83X=A_8^3 normalization.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. Cheng--Duncan--Harvey umbral moonshine module conjecture

    Let XX be a Niemeier root system, let GXG^X be its umbral group, let IXI^X and mm be the associated index set and Coxeter-number parameter, and let HgX=(Hg,rX)H^X_g=(H^X_{g,r}) be the vector-valued umbral McKay--Thompson series. Cheng--Duncan--Harvey conjecture. For each Niemeier root system XX, there is a bi-graded GXG^X-module

    KˇX=⨁r∈IX⨁D∈Z\D=r2(mod4m)Kˇr,−D/4mX,\check K^X=\bigoplus_{r\in I^X}\bigoplus_{\substack{D\in\mathbb{Z}\D=r^2\pmod{4m}}}\check K^X_{r,-D/4m},

    such that, for r∈IXr\in I^X,

    Hg,rX(τ)=−2q−1/4mδr,1+∑D∈Z\D=r2(mod4m)tr⁡(g∣Kˇr,−D/4mX)q−D/4m.H^X_{g,r}(\tau)=-2q^{-1/4m}\delta_{r,1}+\sum_{\substack{D\in\mathbb{Z}\D=r^2\pmod{4m}}}\operatorname{tr}(g\mid\check K^X_{r,-D/4m})q^{-D/4m}.

    This conjecture is the module-theoretic counterpart of the Rademacher-sum formulation of umbral moonshine; the source discusses a special A83A_8^3 qualification and states that the broader Rademacher-sum conjecture was known only in the Mathieu case at the time.

    source: John F. R. Duncan, Michael J. Griffin and Ken Ono, “Moonshine”, arXiv:1411.6571 (2015).

Sources & referencesView supporting material

Primary source

John F. R. Duncan, Michael J. Griffin and Ken Ono, “Proof of the Umbral Moonshine Conjecture”, arXiv:1503.01472 (2015).

finite-groupsmock-modular-formsrepresentation-theoryumbral-moonshine

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