2019年12月
Tensorial generalization of characters
Journal of High Energy Physics
- ,
- ,
- 巻
- 2019
- 号
- 12
- 記述言語
- 英語
- 掲載種別
- 研究論文(学術雑誌)
- DOI
- 10.1007/jhep12(2019)127
- 出版者・発行元
- Springer Science and Business Media LLC
<title>A<sc>bstract</sc>
</title>
In rainbow tensor models, which generalize rectangular complex matrix model (RCM) and possess a huge gauge symmetry U(<italic>N</italic>
1) × … × U(<italic>N</italic>
<italic>r</italic>
), we introduce a new sub-basis in the linear space of gauge invariant operators, which is a redundant basis in the space of operators with non-zero Gaussian averages. Its elements are labeled by r-tuples of Young diagrams of a given size equal to the power of tensor field. Their tensor model averages are just products of dimensions: <inline-formula>
<alternatives>
<tex-math>$$ \left\langle \chi {R}_1,\dots, {R}_r\right\rangle \sim {C}_{R_1},\dots {,}_{R_r}\left({N}_1\right)\dots {D}_{R_r}\left({N}_r\right) $$</tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mfenced>
<mml:mrow>
<mml:mi>χ</mml:mi>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:mrow>
<mml:mo>…</mml:mo>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>r</mml:mi>
</mml:msub>
</mml:mfenced>
<mml:mo>∼</mml:mo>
<mml:msub>
<mml:mi>C</mml:mi>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mo>…</mml:mo>
<mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>r</mml:mi>
</mml:msub>
</mml:msub>
<mml:mfenced>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:mfenced>
<mml:mo>…</mml:mo>
<mml:msub>
<mml:mi>D</mml:mi>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>r</mml:mi>
</mml:msub>
</mml:msub>
<mml:mfenced>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mi>r</mml:mi>
</mml:msub>
</mml:mfenced>
</mml:math>
</alternatives>
</inline-formula> of representations <italic>R</italic>
<italic>i</italic>
of the linear group SL(<italic>N</italic>
<italic>i</italic>
), with<inline-formula>
<alternatives>
<tex-math>$$ {C}_{R_1},\dots {,}_{R_r} $$</tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mspace />
<mml:msub>
<mml:mi>C</mml:mi>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mo>…</mml:mo>
<mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>r</mml:mi>
</mml:msub>
</mml:msub>
</mml:math>
</alternatives>
</inline-formula>
<italic>,</italic> made of the ClebschGordan coefficients of representations <italic>R</italic>
<italic>i</italic>
of the symmetric group. Moreover, not only the averages, but the operators <inline-formula>
<alternatives>
<tex-math>$$ {\chi}_{\overrightarrow{R } } $$</tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>χ</mml:mi>
<mml:mover>
<mml:mi>R</mml:mi>
<mml:mo>→</mml:mo>
</mml:mover>
</mml:msub>
</mml:math>
</alternatives>
</inline-formula> themselves exist only when these <inline-formula>
<alternatives>
<tex-math>$$ {C}_{\overrightarrow{R } } $$</tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>C</mml:mi>
<mml:mover>
<mml:mi>R</mml:mi>
<mml:mo>→</mml:mo>
</mml:mover>
</mml:msub>
</mml:math>
</alternatives>
</inline-formula> are non-vanishing. This sub-basis is much similar to the basis of characters (Schur functions) in matrix models, which is distinguished by the property \character) ~ <italic>character,</italic> which opens a way to lift the notion and the theory of characters (Schur functions) from matrices to tensors. In particular, operators <inline-formula>
<alternatives>
<tex-math>$$ {\chi}_{\overrightarrow{R } } $$</tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mspace />
<mml:msub>
<mml:mi>χ</mml:mi>
<mml:mover>
<mml:mi>R</mml:mi>
<mml:mo>→</mml:mo>
</mml:mover>
</mml:msub>
</mml:math>
</alternatives>
</inline-formula> are eigenfunctions of operators which generalize the usual cut-andjoin operators <inline-formula>
<alternatives>
<tex-math>$$ \hat{W} $$</tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mspace />
<mml:mover>
<mml:mi>W</mml:mi>
<mml:mo>̂</mml:mo>
</mml:mover>
</mml:math>
</alternatives>
</inline-formula>
<italic>;</italic> they satisfy orthogonality conditions similar to the standard characters, but they do not form a <italic>full</italic> linear basis for all gauge-invariant operators, only for those which have non-vanishing Gaussian averages.
</title>
In rainbow tensor models, which generalize rectangular complex matrix model (RCM) and possess a huge gauge symmetry U(<italic>N</italic>
1) × … × U(<italic>N</italic>
<italic>r</italic>
), we introduce a new sub-basis in the linear space of gauge invariant operators, which is a redundant basis in the space of operators with non-zero Gaussian averages. Its elements are labeled by r-tuples of Young diagrams of a given size equal to the power of tensor field. Their tensor model averages are just products of dimensions: <inline-formula>
<alternatives>
<tex-math>$$ \left\langle \chi {R}_1,\dots, {R}_r\right\rangle \sim {C}_{R_1},\dots {,}_{R_r}\left({N}_1\right)\dots {D}_{R_r}\left({N}_r\right) $$</tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mfenced>
<mml:mrow>
<mml:mi>χ</mml:mi>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:mrow>
<mml:mo>…</mml:mo>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>r</mml:mi>
</mml:msub>
</mml:mfenced>
<mml:mo>∼</mml:mo>
<mml:msub>
<mml:mi>C</mml:mi>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mo>…</mml:mo>
<mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>r</mml:mi>
</mml:msub>
</mml:msub>
<mml:mfenced>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:mfenced>
<mml:mo>…</mml:mo>
<mml:msub>
<mml:mi>D</mml:mi>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>r</mml:mi>
</mml:msub>
</mml:msub>
<mml:mfenced>
<mml:msub>
<mml:mi>N</mml:mi>
<mml:mi>r</mml:mi>
</mml:msub>
</mml:mfenced>
</mml:math>
</alternatives>
</inline-formula> of representations <italic>R</italic>
<italic>i</italic>
of the linear group SL(<italic>N</italic>
<italic>i</italic>
), with<inline-formula>
<alternatives>
<tex-math>$$ {C}_{R_1},\dots {,}_{R_r} $$</tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mspace />
<mml:msub>
<mml:mi>C</mml:mi>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mn>1</mml:mn>
</mml:msub>
</mml:msub>
<mml:mo>,</mml:mo>
<mml:mo>…</mml:mo>
<mml:msub>
<mml:mo>,</mml:mo>
<mml:msub>
<mml:mi>R</mml:mi>
<mml:mi>r</mml:mi>
</mml:msub>
</mml:msub>
</mml:math>
</alternatives>
</inline-formula>
<italic>,</italic> made of the ClebschGordan coefficients of representations <italic>R</italic>
<italic>i</italic>
of the symmetric group. Moreover, not only the averages, but the operators <inline-formula>
<alternatives>
<tex-math>$$ {\chi}_{\overrightarrow{R } } $$</tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>χ</mml:mi>
<mml:mover>
<mml:mi>R</mml:mi>
<mml:mo>→</mml:mo>
</mml:mover>
</mml:msub>
</mml:math>
</alternatives>
</inline-formula> themselves exist only when these <inline-formula>
<alternatives>
<tex-math>$$ {C}_{\overrightarrow{R } } $$</tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>C</mml:mi>
<mml:mover>
<mml:mi>R</mml:mi>
<mml:mo>→</mml:mo>
</mml:mover>
</mml:msub>
</mml:math>
</alternatives>
</inline-formula> are non-vanishing. This sub-basis is much similar to the basis of characters (Schur functions) in matrix models, which is distinguished by the property \character) ~ <italic>character,</italic> which opens a way to lift the notion and the theory of characters (Schur functions) from matrices to tensors. In particular, operators <inline-formula>
<alternatives>
<tex-math>$$ {\chi}_{\overrightarrow{R } } $$</tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mspace />
<mml:msub>
<mml:mi>χ</mml:mi>
<mml:mover>
<mml:mi>R</mml:mi>
<mml:mo>→</mml:mo>
</mml:mover>
</mml:msub>
</mml:math>
</alternatives>
</inline-formula> are eigenfunctions of operators which generalize the usual cut-andjoin operators <inline-formula>
<alternatives>
<tex-math>$$ \hat{W} $$</tex-math>
<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mspace />
<mml:mover>
<mml:mi>W</mml:mi>
<mml:mo>̂</mml:mo>
</mml:mover>
</mml:math>
</alternatives>
</inline-formula>
<italic>;</italic> they satisfy orthogonality conditions similar to the standard characters, but they do not form a <italic>full</italic> linear basis for all gauge-invariant operators, only for those which have non-vanishing Gaussian averages.
- リンク情報
- ID情報
-
- DOI : 10.1007/jhep12(2019)127
- eISSN : 1029-8479