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# The Automorphisms of Affine Fusion Rings
Terry Gannon
Department of Mathematical Sciences, University of Alberta, Edmonton, Canada, T6G 2G1 e-mail: [email protected]
# 1. Introduction
Verlinde’s formula [33]
$$
V_{a^{1}\ldots a^{t}}^{(g)}=\sum_{b\in\Phi}(S_{0b})^{2(1-g)}\frac{S_{a^{1}b}}{S_{0b}}\cdot\cdot\cdot\frac{S_{a^{t}b}}{S_{0b}}
$$
arose first in rational conformal field theory (RCFT) as an extremely useful expression for the dimensions of conformal blocks on a genus $g$ surface with $t$ punctures. $\Phi$ here is the finite set of ‘primary fields’. The matrix $S$ comes from a representation of $\mathrm{SL_{2}}(\mathbb{Z})$ defined by the chiral characters of the theory. Contrary to appearances, these numbers $V_{\star\cdots\star}^{(g)}$ will always be nonnegative integers. See the excellent bibliography in [6] for references to the physics literature.
These numbers are remarkable for also arising in several other contexts: for example, as dimensions of spaces of generalised theta functions; as certain tensor product coefficients in quantum groups and Hecke algebras at roots of 1 and Chevalley groups for $\mathbb{F}_{p}$ ; as certain knot invariants for 3-manifolds; as composition laws of the superselection sectors in algebraic quantum field theories; as dimensions of spaces of intertwiners in vertex operator algebras (VOAs); in von Neumann algebras as “Connes’ fusion”; in quantum cohomology; and in Lusztig’s exotic Fourier transform. See for example [7,20,19,32,11,10,36,37,26], and references therein.
The more fundamental of these numbers are those corresponding to a sphere with three punctures. It is more convenient to write these in the form (called fusion coefficients)
$$
N_{a b}^{c}\,{\overset{\mathrm{def}}{=}}\,V_{a,b,C c}^{(0)}=\sum_{d\in\Phi}{\frac{S_{a d}S_{b d}S_{c d}^{*}}{S_{0d}}}
$$
where $C$ is a permutation of $\Phi$ called charge-conjugation and will be defined below. The fusion coefficients uniquely determine all other Verlinde dimensions (1.1a). The symmetries of the numbers (1.1b), i.e. the permutations $\pi$ of $\Phi$ obeying
$$
{\cal N}_{\pi a,\pi b}^{\pi c}={\cal N}_{a b}^{c}\ ,
$$
are precisely the symmetries of all numbers of the form (1.1a).
|
<html><body>
<h1 data-bbox="140 66 471 87">The Automorphisms of Affine Fusion Rings </h1>
<p data-bbox="260 115 351 129">Terry Gannon </p>
<p data-bbox="147 137 465 185">Department of Mathematical Sciences, University of Alberta, Edmonton, Canada, T6G 2G1 e-mail: [email protected] </p>
<h1 data-bbox="255 214 355 228">1. Introduction </h1>
<p data-bbox="93 243 216 258">Verlinde’s formula [33] </p>
<div class="equation" data-bbox="209 272 402 308">$$
V_{a^{1}\ldots a^{t}}^{(g)}=\sum_{b\in\Phi}(S_{0b})^{2(1-g)}\frac{S_{a^{1}b}}{S_{0b}}\cdot\cdot\cdot\frac{S_{a^{t}b}}{S_{0b}}
$$</div>
<p data-bbox="70 318 541 408">arose first in rational conformal field theory (RCFT) as an extremely useful expression for the dimensions of conformal blocks on a genus $g$ surface with $t$ punctures. $\Phi$ here is the finite set of ‘primary fields’. The matrix $S$ comes from a representation of $\mathrm{SL_{2}}(\mathbb{Z})$ defined by the chiral characters of the theory. Contrary to appearances, these numbers $V_{\star\cdots\star}^{(g)}$ will always be nonnegative integers. See the excellent bibliography in [6] for references to the physics literature. </p>
<p data-bbox="70 408 541 523">These numbers are remarkable for also arising in several other contexts: for example, as dimensions of spaces of generalised theta functions; as certain tensor product coefficients in quantum groups and Hecke algebras at roots of 1 and Chevalley groups for $\mathbb{F}_{p}$ ; as certain knot invariants for 3-manifolds; as composition laws of the superselection sectors in algebraic quantum field theories; as dimensions of spaces of intertwiners in vertex operator algebras (VOAs); in von Neumann algebras as “Connes’ fusion”; in quantum cohomology; and in Lusztig’s exotic Fourier transform. See for example [7,20,19,32,11,10,36,37,26], and references therein. </p>
<p data-bbox="70 523 541 552">The more fundamental of these numbers are those corresponding to a sphere with three punctures. It is more convenient to write these in the form (called fusion coefficients) </p>
<div class="equation" data-bbox="223 567 387 602">$$
N_{a b}^{c}\,{\overset{\mathrm{def}}{=}}\,V_{a,b,C c}^{(0)}=\sum_{d\in\Phi}{\frac{S_{a d}S_{b d}S_{c d}^{*}}{S_{0d}}}
$$</div>
<p data-bbox="70 613 541 657">where $C$ is a permutation of $\Phi$ called charge-conjugation and will be defined below. The fusion coefficients uniquely determine all other Verlinde dimensions (1.1a). The symmetries of the numbers (1.1b), i.e. the permutations $\pi$ of $\Phi$ obeying </p>
<div class="equation" data-bbox="266 673 344 689">$$
{\cal N}_{\pi a,\pi b}^{\pi c}={\cal N}_{a b}^{c}\ ,
$$</div>
<p data-bbox="70 700 399 715">are precisely the symmetries of all numbers of the form (1.1a). </p>
|
0002044v1
| 0 | 612 | 792 | 1,275 | 1,650 |
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[{"type": "title", "bbox": [140, 66, 471, 87], "content": "The Automorphisms of Affine Fusion Rings", "index": 0}, {"type": "text", "bbox": [260, 115, 351, 129], "content": "Terry Gannon", "index": 1}, {"type": "text", "bbox": [147, 137, 465, 185], "content": "Department of Mathematical Sciences, University of Alberta, Edmonton, Canada, T6G 2G1 e-mail: [email protected]", "index": 2}, {"type": "title", "bbox": [255, 214, 355, 228], "content": "1. Introduction", "index": 3}, {"type": "text", "bbox": [93, 243, 216, 258], "content": "Verlinde’s formula [33]", "index": 4}, {"type": "interline_equation", "bbox": [209, 272, 402, 308], "content": "", "index": 5}, {"type": "text", "bbox": [70, 318, 541, 408], "content": "arose first in rational conformal field theory (RCFT) as an extremely useful expression for the dimensions of conformal blocks on a genus surface with punctures. here is the finite set of ‘primary fields’. The matrix comes from a representation of defined by the chiral characters of the theory. Contrary to appearances, these numbers will always be nonnegative integers. See the excellent bibliography in [6] for references to the physics literature.", "index": 6}, {"type": "text", "bbox": [70, 408, 541, 523], "content": "These numbers are remarkable for also arising in several other contexts: for example, as dimensions of spaces of generalised theta functions; as certain tensor product coefficients in quantum groups and Hecke algebras at roots of 1 and Chevalley groups for ; as certain knot invariants for 3-manifolds; as composition laws of the superselection sectors in algebraic quantum field theories; as dimensions of spaces of intertwiners in vertex operator algebras (VOAs); in von Neumann algebras as “Connes’ fusion”; in quantum cohomology; and in Lusztig’s exotic Fourier transform. See for example [7,20,19,32,11,10,36,37,26], and references therein.", "index": 7}, {"type": "text", "bbox": [70, 523, 541, 552], "content": "The more fundamental of these numbers are those corresponding to a sphere with three punctures. It is more convenient to write these in the form (called fusion coefficients)", "index": 8}, {"type": "interline_equation", "bbox": [223, 567, 387, 602], "content": "", "index": 9}, {"type": "text", "bbox": [70, 613, 541, 657], "content": "where is a permutation of called charge-conjugation and will be defined below. The fusion coefficients uniquely determine all other Verlinde dimensions (1.1a). The symmetries of the numbers (1.1b), i.e. the permutations of obeying", "index": 10}, {"type": "interline_equation", "bbox": [266, 673, 344, 689], "content": "", "index": 11}, {"type": "text", "bbox": [70, 700, 399, 715], "content": "are precisely the symmetries of all numbers of the form (1.1a).", "index": 12}]
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[{"bbox": [141, 69, 469, 88], "content": "The Automorphisms of Affine Fusion Rings", "parent_index": 0, "line_index": 0}, {"bbox": [261, 117, 351, 131], "content": "Terry Gannon", "parent_index": 1, "line_index": 0}, {"bbox": [147, 140, 464, 153], "content": "Department of Mathematical Sciences, University of Alberta,", "parent_index": 2, "line_index": 0}, {"bbox": [227, 154, 386, 167], "content": "Edmonton, Canada, T6G 2G1", "parent_index": 2, "line_index": 1}, {"bbox": [215, 173, 396, 185], "content": "e-mail: [email protected]", "parent_index": 2, "line_index": 2}, {"bbox": [257, 216, 355, 229], "content": "1. Introduction", "parent_index": 3, "line_index": 0}, {"bbox": [95, 245, 213, 259], "content": "Verlinde’s formula [33]", "parent_index": 4, "line_index": 0}, {"bbox": [70, 322, 541, 338], "content": "arose first in rational conformal field theory (RCFT) as an extremely useful expression for", "parent_index": 6, "line_index": 0}, {"bbox": [70, 336, 541, 352], "content": "the dimensions of conformal blocks on a genus surface with punctures. here is the", "parent_index": 6, "line_index": 1}, {"bbox": [71, 352, 541, 366], "content": "finite set of ‘primary fields’. The matrix comes from a representation of defined", "parent_index": 6, "line_index": 2}, {"bbox": [68, 361, 545, 386], "content": "by the chiral characters of the theory. Contrary to appearances, these numbers will", "parent_index": 6, "line_index": 3}, {"bbox": [71, 382, 541, 396], "content": "always be nonnegative integers. See the excellent bibliography in [6] for references to the", "parent_index": 6, "line_index": 4}, {"bbox": [70, 397, 165, 410], "content": "physics literature.", "parent_index": 6, "line_index": 5}, {"bbox": [94, 410, 540, 425], "content": "These numbers are remarkable for also arising in several other contexts: for example,", "parent_index": 7, "line_index": 0}, {"bbox": [71, 426, 541, 439], "content": "as dimensions of spaces of generalised theta functions; as certain tensor product coefficients", "parent_index": 7, "line_index": 1}, {"bbox": [69, 439, 542, 455], "content": "in quantum groups and Hecke algebras at roots of 1 and Chevalley groups for ; as", "parent_index": 7, "line_index": 2}, {"bbox": [70, 454, 540, 467], "content": "certain knot invariants for 3-manifolds; as composition laws of the superselection sectors in", "parent_index": 7, "line_index": 3}, {"bbox": [72, 469, 540, 482], "content": "algebraic quantum field theories; as dimensions of spaces of intertwiners in vertex operator", "parent_index": 7, "line_index": 4}, {"bbox": [70, 482, 541, 497], "content": "algebras (VOAs); in von Neumann algebras as “Connes’ fusion”; in quantum cohomology;", "parent_index": 7, "line_index": 5}, {"bbox": [70, 496, 541, 511], "content": "and in Lusztig’s exotic Fourier transform. See for example [7,20,19,32,11,10,36,37,26], and", "parent_index": 7, "line_index": 6}, {"bbox": [71, 512, 167, 525], "content": "references therein.", "parent_index": 7, "line_index": 7}, {"bbox": [94, 525, 541, 540], "content": "The more fundamental of these numbers are those corresponding to a sphere with three", "parent_index": 8, "line_index": 0}, {"bbox": [70, 540, 520, 555], "content": "punctures. It is more convenient to write these in the form (called fusion coefficients)", "parent_index": 8, "line_index": 1}, {"bbox": [71, 616, 541, 631], "content": "where is a permutation of called charge-conjugation and will be defined below. The", "parent_index": 10, "line_index": 0}, {"bbox": [70, 630, 542, 645], "content": "fusion coefficients uniquely determine all other Verlinde dimensions (1.1a). The symmetries", "parent_index": 10, "line_index": 1}, {"bbox": [70, 643, 385, 660], "content": "of the numbers (1.1b), i.e. the permutations of obeying", "parent_index": 10, "line_index": 2}, {"bbox": [70, 703, 398, 717], "content": "are precisely the symmetries of all numbers of the form (1.1a).", "parent_index": 12, "line_index": 0}]
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[{"bbox": [209, 272, 402, 308], "content": "V_{a^{1}\\ldots a^{t}}^{(g)}=\\sum_{b\\in\\Phi}(S_{0b})^{2(1-g)}\\frac{S_{a^{1}b}}{S_{0b}}\\cdot\\cdot\\cdot\\frac{S_{a^{t}b}}{S_{0b}}", "parent_index": 5, "subtype": "interline"}, {"bbox": [322, 342, 328, 349], "content": "g", "parent_index": 6, "subtype": "inline"}, {"bbox": [401, 339, 405, 347], "content": "t", "parent_index": 6, "subtype": "inline"}, {"bbox": [471, 338, 479, 347], "content": "\\Phi", "parent_index": 6, "subtype": "inline"}, {"bbox": [285, 353, 293, 362], "content": "S", "parent_index": 6, "subtype": "inline"}, {"bbox": [461, 352, 498, 364], "content": "\\mathrm{SL_{2}}(\\mathbb{Z})", "parent_index": 6, "subtype": "inline"}, {"bbox": [491, 365, 516, 380], "content": "V_{\\star\\cdots\\star}^{(g)}", "parent_index": 6, "subtype": "inline"}, {"bbox": [506, 441, 519, 453], "content": "\\mathbb{F}_{p}", "parent_index": 7, "subtype": "inline"}, {"bbox": [223, 567, 387, 602], "content": "N_{a b}^{c}\\,{\\overset{\\mathrm{def}}{=}}\\,V_{a,b,C c}^{(0)}=\\sum_{d\\in\\Phi}{\\frac{S_{a d}S_{b d}S_{c d}^{*}}{S_{0d}}}", "parent_index": 9, "subtype": "interline"}, {"bbox": [106, 618, 116, 627], "content": "C", "parent_index": 10, "subtype": "inline"}, {"bbox": [226, 618, 234, 627], "content": "\\Phi", "parent_index": 10, "subtype": "inline"}, {"bbox": [305, 649, 313, 655], "content": "\\pi", "parent_index": 10, "subtype": "inline"}, {"bbox": [330, 646, 339, 655], "content": "\\Phi", "parent_index": 10, "subtype": "inline"}, {"bbox": [266, 673, 344, 689], "content": "{\\cal N}_{\\pi a,\\pi b}^{\\pi c}={\\cal N}_{a b}^{c}\\ ,", "parent_index": 11, "subtype": "interline"}]
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The point of introducing the $N_{a b}^{c}$ in (1.1b) is that they define an algebraic structure, the fusion ring. Consider all formal linear combinations of objects $\chi_{a}$ labelled by the $a\in\Phi$ ; the multiplication is defined to have structure constants $N_{a b}^{c}$ :
$$
\chi_{a}\chi_{b}=\sum_{c\in\Phi}N_{a b}^{c}\chi_{c}
$$
As an abstract ring, it is not so interesting (the fusion ring over $\mathbb{C}$ is isomorphic to $\mathbb{C}^{||\Phi||}$ with operations defined component-wise; over $\mathbb{Q}$ it will be a direct sum of number fields). This is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a polynomial ring. Of course it is important in both contexts that we have a preferred basis, namely $\{\chi_{a}\}$ , and so proper definitions of isomorphisms etc. must respect that.
The most important examples of fusion rings are associated to the affine algebras, and it is to these that this paper is devoted. Their automorphisms appear explicitly for instance in the classification of modular invariant (i.e. torus) partition functions [17,18], and also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For instance, fusion-automorphisms (more generally, -homomorphisms) generate large classes of nonnegative integer representations of the fusion-ring, each of which is associated to a boundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the coefficient matrix of the torus partition function is a permutation matrix (in which case the partition function is called an automorphism invariant), we get a fusion ring automorphism. However most torus partition functions are not automorphism invariants (although MooreSeiberg assert that there is a sense in which any torus partition function can be interpreted as one — see e.g. [3]), and most fusion ring automorphisms do not correspond to partition functions. Nevertheless, the two problems are related. The automorphism invariants for the affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1 below) involving q-dimensions will be very useful to us, and conversely the arguments in Section 4 of this paper could be used to considerably simplify the proofs of [17,18].
It is surprising that it is even possible to find all affine fusion automorphisms, and in fact the arguments turn out to be rather short. It is remarkable that the answer is so simple: with few exceptions, they correspond to the Dynkin diagram symmetries.
A related task is determining which affine fusion rings are isomorphic. We answer this in section 5 below; as expected most fusion rings with different names are nonisomorphic.
Acknowledgements. Most of this paper was written during a visit to Max-Planck in Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank Yi-Zhi Huang for clarifying an issue concerning [8,21].
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<html><body>
<p data-bbox="70 70 541 115">The point of introducing the $N_{a b}^{c}$ in (1.1b) is that they define an algebraic structure, the fusion ring. Consider all formal linear combinations of objects $\chi_{a}$ labelled by the $a\in\Phi$ ; the multiplication is defined to have structure constants $N_{a b}^{c}$ : </p>
<div class="equation" data-bbox="258 130 353 160">$$
\chi_{a}\chi_{b}=\sum_{c\in\Phi}N_{a b}^{c}\chi_{c}
$$</div>
<p data-bbox="70 173 540 245">As an abstract ring, it is not so interesting (the fusion ring over $\mathbb{C}$ is isomorphic to $\mathbb{C}^{||\Phi||}$ with operations defined component-wise; over $\mathbb{Q}$ it will be a direct sum of number fields). This is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a polynomial ring. Of course it is important in both contexts that we have a preferred basis, namely $\{\chi_{a}\}$ , and so proper definitions of isomorphisms etc. must respect that. </p>
<p data-bbox="71 246 541 475">The most important examples of fusion rings are associated to the affine algebras, and it is to these that this paper is devoted. Their automorphisms appear explicitly for instance in the classification of modular invariant (i.e. torus) partition functions [17,18], and also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For instance, fusion-automorphisms (more generally, -homomorphisms) generate large classes of nonnegative integer representations of the fusion-ring, each of which is associated to a boundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the coefficient matrix of the torus partition function is a permutation matrix (in which case the partition function is called an automorphism invariant), we get a fusion ring automorphism. However most torus partition functions are not automorphism invariants (although MooreSeiberg assert that there is a sense in which any torus partition function can be interpreted as one — see e.g. [3]), and most fusion ring automorphisms do not correspond to partition functions. Nevertheless, the two problems are related. The automorphism invariants for the affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1 below) involving q-dimensions will be very useful to us, and conversely the arguments in Section 4 of this paper could be used to considerably simplify the proofs of [17,18]. </p>
<p data-bbox="70 476 541 518">It is surprising that it is even possible to find all affine fusion automorphisms, and in fact the arguments turn out to be rather short. It is remarkable that the answer is so simple: with few exceptions, they correspond to the Dynkin diagram symmetries. </p>
<p data-bbox="71 518 541 547">A related task is determining which affine fusion rings are isomorphic. We answer this in section 5 below; as expected most fusion rings with different names are nonisomorphic. </p>
<p data-bbox="70 554 540 598">Acknowledgements. Most of this paper was written during a visit to Max-Planck in Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank Yi-Zhi Huang for clarifying an issue concerning [8,21]. </p>
</body></html>
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0002044v1
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{"preproc_blocks": [{"type": "text", "bbox": [70, 70, 541, 115], "lines": [{"bbox": [93, 73, 540, 89], "spans": [{"bbox": [93, 73, 249, 89], "score": 1.0, "content": "The point of introducing the ", "type": "text"}, {"bbox": [249, 75, 268, 88], "score": 0.93, "content": "N_{a b}^{c}", "type": "inline_equation", "height": 13, "width": 19}, {"bbox": [269, 73, 540, 89], "score": 1.0, "content": " in (1.1b) is that they define an algebraic structure,", "type": "text"}], "index": 0}, {"bbox": [70, 87, 541, 104], "spans": [{"bbox": [70, 87, 439, 104], "score": 1.0, "content": "the fusion ring. Consider all formal linear combinations of objects ", "type": "text"}, {"bbox": [440, 93, 453, 101], "score": 0.9, "content": "\\chi_{a}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [453, 87, 541, 104], "score": 1.0, "content": " labelled by the", "type": "text"}], "index": 1}, {"bbox": [71, 100, 430, 120], "spans": [{"bbox": [71, 104, 101, 114], "score": 0.91, "content": "a\\in\\Phi", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [101, 100, 405, 120], "score": 1.0, "content": "; the multiplication is defined to have structure constants ", "type": "text"}, {"bbox": [405, 104, 425, 116], "score": 0.93, "content": "N_{a b}^{c}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [425, 100, 430, 120], "score": 1.0, "content": ":", "type": "text"}], "index": 2}], "index": 1}, {"type": "interline_equation", "bbox": [258, 130, 353, 160], "lines": [{"bbox": [258, 130, 353, 160], "spans": [{"bbox": [258, 130, 353, 160], "score": 0.94, "content": "\\chi_{a}\\chi_{b}=\\sum_{c\\in\\Phi}N_{a b}^{c}\\chi_{c}", "type": "interline_equation"}], "index": 3}], "index": 3}, {"type": "text", "bbox": [70, 173, 540, 245], "lines": [{"bbox": [69, 174, 539, 192], "spans": [{"bbox": [69, 174, 412, 192], "score": 1.0, "content": "As an abstract ring, it is not so interesting (the fusion ring over ", "type": "text"}, {"bbox": [412, 178, 421, 187], "score": 0.9, "content": "\\mathbb{C}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [422, 174, 513, 192], "score": 1.0, "content": " is isomorphic to ", "type": "text"}, {"bbox": [513, 176, 539, 187], "score": 0.92, "content": "\\mathbb{C}^{||\\Phi||}", "type": "inline_equation", "height": 11, "width": 26}], "index": 4}, {"bbox": [70, 190, 540, 205], "spans": [{"bbox": [70, 190, 313, 205], "score": 1.0, "content": "with operations defined component-wise; over ", "type": "text"}, {"bbox": [314, 192, 324, 203], "score": 0.89, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [324, 190, 540, 205], "score": 1.0, "content": " it will be a direct sum of number fields).", "type": "text"}], "index": 5}, {"bbox": [69, 204, 542, 221], "spans": [{"bbox": [69, 204, 542, 221], "score": 1.0, "content": "This is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a", "type": "text"}], "index": 6}, {"bbox": [70, 218, 541, 235], "spans": [{"bbox": [70, 218, 541, 235], "score": 1.0, "content": "polynomial ring. Of course it is important in both contexts that we have a preferred basis,", "type": "text"}], "index": 7}, {"bbox": [70, 234, 487, 249], "spans": [{"bbox": [70, 234, 113, 249], "score": 1.0, "content": "namely ", "type": "text"}, {"bbox": [113, 235, 138, 247], "score": 0.93, "content": "\\{\\chi_{a}\\}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [138, 234, 487, 249], "score": 1.0, "content": ", and so proper definitions of isomorphisms etc. must respect that.", "type": "text"}], "index": 8}], "index": 6}, {"type": "text", "bbox": [71, 246, 541, 475], "lines": [{"bbox": [94, 247, 541, 263], "spans": [{"bbox": [94, 247, 541, 263], "score": 1.0, "content": "The most important examples of fusion rings are associated to the affine algebras,", "type": "text"}], "index": 9}, {"bbox": [70, 262, 541, 277], "spans": [{"bbox": [70, 262, 541, 277], "score": 1.0, "content": "and it is to these that this paper is devoted. Their automorphisms appear explicitly for", "type": "text"}], "index": 10}, {"bbox": [70, 277, 541, 291], "spans": [{"bbox": [70, 277, 541, 291], "score": 1.0, "content": "instance in the classification of modular invariant (i.e. torus) partition functions [17,18],", "type": "text"}], "index": 11}, {"bbox": [70, 290, 541, 306], "spans": [{"bbox": [70, 290, 541, 306], "score": 1.0, "content": "and also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For", "type": "text"}], "index": 12}, {"bbox": [70, 306, 541, 320], "spans": [{"bbox": [70, 306, 541, 320], "score": 1.0, "content": "instance, fusion-automorphisms (more generally, -homomorphisms) generate large classes", "type": "text"}], "index": 13}, {"bbox": [71, 319, 542, 334], "spans": [{"bbox": [71, 319, 542, 334], "score": 1.0, "content": "of nonnegative integer representations of the fusion-ring, each of which is associated to a", "type": "text"}], "index": 14}, {"bbox": [70, 334, 541, 349], "spans": [{"bbox": [70, 334, 541, 349], "score": 1.0, "content": "boundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the", "type": "text"}], "index": 15}, {"bbox": [71, 349, 541, 363], "spans": [{"bbox": [71, 349, 541, 363], "score": 1.0, "content": "coefficient matrix of the torus partition function is a permutation matrix (in which case the", "type": "text"}], "index": 16}, {"bbox": [70, 363, 540, 377], "spans": [{"bbox": [70, 363, 540, 377], "score": 1.0, "content": "partition function is called an automorphism invariant), we get a fusion ring automorphism.", "type": "text"}], "index": 17}, {"bbox": [70, 377, 540, 392], "spans": [{"bbox": [70, 377, 540, 392], "score": 1.0, "content": "However most torus partition functions are not automorphism invariants (although Moore-", "type": "text"}], "index": 18}, {"bbox": [71, 392, 540, 406], "spans": [{"bbox": [71, 392, 540, 406], "score": 1.0, "content": "Seiberg assert that there is a sense in which any torus partition function can be interpreted", "type": "text"}], "index": 19}, {"bbox": [71, 407, 540, 420], "spans": [{"bbox": [71, 407, 540, 420], "score": 1.0, "content": "as one — see e.g. [3]), and most fusion ring automorphisms do not correspond to partition", "type": "text"}], "index": 20}, {"bbox": [70, 419, 541, 434], "spans": [{"bbox": [70, 419, 541, 434], "score": 1.0, "content": "functions. Nevertheless, the two problems are related. The automorphism invariants for", "type": "text"}], "index": 21}, {"bbox": [71, 435, 541, 448], "spans": [{"bbox": [71, 435, 541, 448], "score": 1.0, "content": "the affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1", "type": "text"}], "index": 22}, {"bbox": [70, 448, 541, 464], "spans": [{"bbox": [70, 448, 541, 464], "score": 1.0, "content": "below) involving q-dimensions will be very useful to us, and conversely the arguments in", "type": "text"}], "index": 23}, {"bbox": [70, 462, 507, 478], "spans": [{"bbox": [70, 462, 507, 478], "score": 1.0, "content": "Section 4 of this paper could be used to considerably simplify the proofs of [17,18].", "type": "text"}], "index": 24}], "index": 16.5}, {"type": "text", "bbox": [70, 476, 541, 518], "lines": [{"bbox": [94, 477, 541, 492], "spans": [{"bbox": [94, 477, 541, 492], "score": 1.0, "content": "It is surprising that it is even possible to find all affine fusion automorphisms, and", "type": "text"}], "index": 25}, {"bbox": [70, 491, 541, 506], "spans": [{"bbox": [70, 491, 541, 506], "score": 1.0, "content": "in fact the arguments turn out to be rather short. It is remarkable that the answer is so", "type": "text"}], "index": 26}, {"bbox": [70, 506, 498, 520], "spans": [{"bbox": [70, 506, 498, 520], "score": 1.0, "content": "simple: with few exceptions, they correspond to the Dynkin diagram symmetries.", "type": "text"}], "index": 27}], "index": 26}, {"type": "text", "bbox": [71, 518, 541, 547], "lines": [{"bbox": [95, 521, 540, 535], "spans": [{"bbox": [95, 521, 540, 535], "score": 1.0, "content": "A related task is determining which affine fusion rings are isomorphic. We answer this", "type": "text"}], "index": 28}, {"bbox": [70, 536, 538, 550], "spans": [{"bbox": [70, 536, 538, 550], "score": 1.0, "content": "in section 5 below; as expected most fusion rings with different names are nonisomorphic.", "type": "text"}], "index": 29}], "index": 28.5}, {"type": "text", "bbox": [70, 554, 540, 598], "lines": [{"bbox": [95, 556, 541, 572], "spans": [{"bbox": [95, 556, 541, 572], "score": 1.0, "content": "Acknowledgements. Most of this paper was written during a visit to Max-Planck", "type": "text"}], "index": 30}, {"bbox": [70, 571, 540, 585], "spans": [{"bbox": [70, 571, 540, 585], "score": 1.0, "content": "in Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank", "type": "text"}], "index": 31}, {"bbox": [72, 586, 354, 600], "spans": [{"bbox": [72, 586, 354, 600], "score": 1.0, "content": "Yi-Zhi Huang for clarifying an issue concerning [8,21].", "type": "text"}], "index": 32}], "index": 31}], "layout_bboxes": [], "page_idx": 1, "page_size": [612.0, 792.0], "_layout_tree": [], "images": [], "tables": [], "interline_equations": [{"type": "interline_equation", "bbox": [258, 130, 353, 160], "lines": [{"bbox": [258, 130, 353, 160], "spans": [{"bbox": [258, 130, 353, 160], "score": 0.94, "content": "\\chi_{a}\\chi_{b}=\\sum_{c\\in\\Phi}N_{a b}^{c}\\chi_{c}", "type": "interline_equation"}], "index": 3}], "index": 3}], "discarded_blocks": [], "need_drop": false, "drop_reason": [], "para_blocks": [{"type": "text", "bbox": [70, 70, 541, 115], "lines": [{"bbox": [93, 73, 540, 89], "spans": [{"bbox": [93, 73, 249, 89], "score": 1.0, "content": "The point of introducing the ", "type": "text"}, {"bbox": [249, 75, 268, 88], "score": 0.93, "content": "N_{a b}^{c}", "type": "inline_equation", "height": 13, "width": 19}, {"bbox": [269, 73, 540, 89], "score": 1.0, "content": " in (1.1b) is that they define an algebraic structure,", "type": "text"}], "index": 0}, {"bbox": [70, 87, 541, 104], "spans": [{"bbox": [70, 87, 439, 104], "score": 1.0, "content": "the fusion ring. Consider all formal linear combinations of objects ", "type": "text"}, {"bbox": [440, 93, 453, 101], "score": 0.9, "content": "\\chi_{a}", "type": "inline_equation", "height": 8, "width": 13}, {"bbox": [453, 87, 541, 104], "score": 1.0, "content": " labelled by the", "type": "text"}], "index": 1}, {"bbox": [71, 100, 430, 120], "spans": [{"bbox": [71, 104, 101, 114], "score": 0.91, "content": "a\\in\\Phi", "type": "inline_equation", "height": 10, "width": 30}, {"bbox": [101, 100, 405, 120], "score": 1.0, "content": "; the multiplication is defined to have structure constants ", "type": "text"}, {"bbox": [405, 104, 425, 116], "score": 0.93, "content": "N_{a b}^{c}", "type": "inline_equation", "height": 12, "width": 20}, {"bbox": [425, 100, 430, 120], "score": 1.0, "content": ":", "type": "text"}], "index": 2}], "index": 1, "bbox_fs": [70, 73, 541, 120]}, {"type": "interline_equation", "bbox": [258, 130, 353, 160], "lines": [{"bbox": [258, 130, 353, 160], "spans": [{"bbox": [258, 130, 353, 160], "score": 0.94, "content": "\\chi_{a}\\chi_{b}=\\sum_{c\\in\\Phi}N_{a b}^{c}\\chi_{c}", "type": "interline_equation"}], "index": 3}], "index": 3}, {"type": "text", "bbox": [70, 173, 540, 245], "lines": [{"bbox": [69, 174, 539, 192], "spans": [{"bbox": [69, 174, 412, 192], "score": 1.0, "content": "As an abstract ring, it is not so interesting (the fusion ring over ", "type": "text"}, {"bbox": [412, 178, 421, 187], "score": 0.9, "content": "\\mathbb{C}", "type": "inline_equation", "height": 9, "width": 9}, {"bbox": [422, 174, 513, 192], "score": 1.0, "content": " is isomorphic to ", "type": "text"}, {"bbox": [513, 176, 539, 187], "score": 0.92, "content": "\\mathbb{C}^{||\\Phi||}", "type": "inline_equation", "height": 11, "width": 26}], "index": 4}, {"bbox": [70, 190, 540, 205], "spans": [{"bbox": [70, 190, 313, 205], "score": 1.0, "content": "with operations defined component-wise; over ", "type": "text"}, {"bbox": [314, 192, 324, 203], "score": 0.89, "content": "\\mathbb{Q}", "type": "inline_equation", "height": 11, "width": 10}, {"bbox": [324, 190, 540, 205], "score": 1.0, "content": " it will be a direct sum of number fields).", "type": "text"}], "index": 5}, {"bbox": [69, 204, 542, 221], "spans": [{"bbox": [69, 204, 542, 221], "score": 1.0, "content": "This is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a", "type": "text"}], "index": 6}, {"bbox": [70, 218, 541, 235], "spans": [{"bbox": [70, 218, 541, 235], "score": 1.0, "content": "polynomial ring. Of course it is important in both contexts that we have a preferred basis,", "type": "text"}], "index": 7}, {"bbox": [70, 234, 487, 249], "spans": [{"bbox": [70, 234, 113, 249], "score": 1.0, "content": "namely ", "type": "text"}, {"bbox": [113, 235, 138, 247], "score": 0.93, "content": "\\{\\chi_{a}\\}", "type": "inline_equation", "height": 12, "width": 25}, {"bbox": [138, 234, 487, 249], "score": 1.0, "content": ", and so proper definitions of isomorphisms etc. must respect that.", "type": "text"}], "index": 8}], "index": 6, "bbox_fs": [69, 174, 542, 249]}, {"type": "text", "bbox": [71, 246, 541, 475], "lines": [{"bbox": [94, 247, 541, 263], "spans": [{"bbox": [94, 247, 541, 263], "score": 1.0, "content": "The most important examples of fusion rings are associated to the affine algebras,", "type": "text"}], "index": 9}, {"bbox": [70, 262, 541, 277], "spans": [{"bbox": [70, 262, 541, 277], "score": 1.0, "content": "and it is to these that this paper is devoted. Their automorphisms appear explicitly for", "type": "text"}], "index": 10}, {"bbox": [70, 277, 541, 291], "spans": [{"bbox": [70, 277, 541, 291], "score": 1.0, "content": "instance in the classification of modular invariant (i.e. torus) partition functions [17,18],", "type": "text"}], "index": 11}, {"bbox": [70, 290, 541, 306], "spans": [{"bbox": [70, 290, 541, 306], "score": 1.0, "content": "and also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For", "type": "text"}], "index": 12}, {"bbox": [70, 306, 541, 320], "spans": [{"bbox": [70, 306, 541, 320], "score": 1.0, "content": "instance, fusion-automorphisms (more generally, -homomorphisms) generate large classes", "type": "text"}], "index": 13}, {"bbox": [71, 319, 542, 334], "spans": [{"bbox": [71, 319, 542, 334], "score": 1.0, "content": "of nonnegative integer representations of the fusion-ring, each of which is associated to a", "type": "text"}], "index": 14}, {"bbox": [70, 334, 541, 349], "spans": [{"bbox": [70, 334, 541, 349], "score": 1.0, "content": "boundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the", "type": "text"}], "index": 15}, {"bbox": [71, 349, 541, 363], "spans": [{"bbox": [71, 349, 541, 363], "score": 1.0, "content": "coefficient matrix of the torus partition function is a permutation matrix (in which case the", "type": "text"}], "index": 16}, {"bbox": [70, 363, 540, 377], "spans": [{"bbox": [70, 363, 540, 377], "score": 1.0, "content": "partition function is called an automorphism invariant), we get a fusion ring automorphism.", "type": "text"}], "index": 17}, {"bbox": [70, 377, 540, 392], "spans": [{"bbox": [70, 377, 540, 392], "score": 1.0, "content": "However most torus partition functions are not automorphism invariants (although Moore-", "type": "text"}], "index": 18}, {"bbox": [71, 392, 540, 406], "spans": [{"bbox": [71, 392, 540, 406], "score": 1.0, "content": "Seiberg assert that there is a sense in which any torus partition function can be interpreted", "type": "text"}], "index": 19}, {"bbox": [71, 407, 540, 420], "spans": [{"bbox": [71, 407, 540, 420], "score": 1.0, "content": "as one — see e.g. [3]), and most fusion ring automorphisms do not correspond to partition", "type": "text"}], "index": 20}, {"bbox": [70, 419, 541, 434], "spans": [{"bbox": [70, 419, 541, 434], "score": 1.0, "content": "functions. Nevertheless, the two problems are related. The automorphism invariants for", "type": "text"}], "index": 21}, {"bbox": [71, 435, 541, 448], "spans": [{"bbox": [71, 435, 541, 448], "score": 1.0, "content": "the affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1", "type": "text"}], "index": 22}, {"bbox": [70, 448, 541, 464], "spans": [{"bbox": [70, 448, 541, 464], "score": 1.0, "content": "below) involving q-dimensions will be very useful to us, and conversely the arguments in", "type": "text"}], "index": 23}, {"bbox": [70, 462, 507, 478], "spans": [{"bbox": [70, 462, 507, 478], "score": 1.0, "content": "Section 4 of this paper could be used to considerably simplify the proofs of [17,18].", "type": "text"}], "index": 24}], "index": 16.5, "bbox_fs": [70, 247, 542, 478]}, {"type": "text", "bbox": [70, 476, 541, 518], "lines": [{"bbox": [94, 477, 541, 492], "spans": [{"bbox": [94, 477, 541, 492], "score": 1.0, "content": "It is surprising that it is even possible to find all affine fusion automorphisms, and", "type": "text"}], "index": 25}, {"bbox": [70, 491, 541, 506], "spans": [{"bbox": [70, 491, 541, 506], "score": 1.0, "content": "in fact the arguments turn out to be rather short. It is remarkable that the answer is so", "type": "text"}], "index": 26}, {"bbox": [70, 506, 498, 520], "spans": [{"bbox": [70, 506, 498, 520], "score": 1.0, "content": "simple: with few exceptions, they correspond to the Dynkin diagram symmetries.", "type": "text"}], "index": 27}], "index": 26, "bbox_fs": [70, 477, 541, 520]}, {"type": "text", "bbox": [71, 518, 541, 547], "lines": [{"bbox": [95, 521, 540, 535], "spans": [{"bbox": [95, 521, 540, 535], "score": 1.0, "content": "A related task is determining which affine fusion rings are isomorphic. We answer this", "type": "text"}], "index": 28}, {"bbox": [70, 536, 538, 550], "spans": [{"bbox": [70, 536, 538, 550], "score": 1.0, "content": "in section 5 below; as expected most fusion rings with different names are nonisomorphic.", "type": "text"}], "index": 29}], "index": 28.5, "bbox_fs": [70, 521, 540, 550]}, {"type": "text", "bbox": [70, 554, 540, 598], "lines": [{"bbox": [95, 556, 541, 572], "spans": [{"bbox": [95, 556, 541, 572], "score": 1.0, "content": "Acknowledgements. Most of this paper was written during a visit to Max-Planck", "type": "text"}], "index": 30}, {"bbox": [70, 571, 540, 585], "spans": [{"bbox": [70, 571, 540, 585], "score": 1.0, "content": "in Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank", "type": "text"}], "index": 31}, {"bbox": [72, 586, 354, 600], "spans": [{"bbox": [72, 586, 354, 600], "score": 1.0, "content": "Yi-Zhi Huang for clarifying an issue concerning [8,21].", "type": "text"}], "index": 32}], "index": 31, "bbox_fs": [70, 556, 541, 600]}]}
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[{"type": "text", "bbox": [70, 70, 541, 115], "content": "The point of introducing the in (1.1b) is that they define an algebraic structure, the fusion ring. Consider all formal linear combinations of objects labelled by the ; the multiplication is defined to have structure constants :", "index": 0}, {"type": "interline_equation", "bbox": [258, 130, 353, 160], "content": "", "index": 1}, {"type": "text", "bbox": [70, 173, 540, 245], "content": "As an abstract ring, it is not so interesting (the fusion ring over is isomorphic to with operations defined component-wise; over it will be a direct sum of number fields). This is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a polynomial ring. Of course it is important in both contexts that we have a preferred basis, namely , and so proper definitions of isomorphisms etc. must respect that.", "index": 2}, {"type": "text", "bbox": [71, 246, 541, 475], "content": "The most important examples of fusion rings are associated to the affine algebras, and it is to these that this paper is devoted. Their automorphisms appear explicitly for instance in the classification of modular invariant (i.e. torus) partition functions [17,18], and also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For instance, fusion-automorphisms (more generally, -homomorphisms) generate large classes of nonnegative integer representations of the fusion-ring, each of which is associated to a boundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the coefficient matrix of the torus partition function is a permutation matrix (in which case the partition function is called an automorphism invariant), we get a fusion ring automorphism. However most torus partition functions are not automorphism invariants (although Moore- Seiberg assert that there is a sense in which any torus partition function can be interpreted as one — see e.g. [3]), and most fusion ring automorphisms do not correspond to partition functions. Nevertheless, the two problems are related. The automorphism invariants for the affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1 below) involving q-dimensions will be very useful to us, and conversely the arguments in Section 4 of this paper could be used to considerably simplify the proofs of [17,18].", "index": 3}, {"type": "text", "bbox": [70, 476, 541, 518], "content": "It is surprising that it is even possible to find all affine fusion automorphisms, and in fact the arguments turn out to be rather short. It is remarkable that the answer is so simple: with few exceptions, they correspond to the Dynkin diagram symmetries.", "index": 4}, {"type": "text", "bbox": [71, 518, 541, 547], "content": "A related task is determining which affine fusion rings are isomorphic. We answer this in section 5 below; as expected most fusion rings with different names are nonisomorphic.", "index": 5}, {"type": "text", "bbox": [70, 554, 540, 598], "content": "Acknowledgements. Most of this paper was written during a visit to Max-Planck in Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank Yi-Zhi Huang for clarifying an issue concerning [8,21].", "index": 6}]
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[{"bbox": [93, 73, 540, 89], "content": "The point of introducing the in (1.1b) is that they define an algebraic structure,", "parent_index": 0, "line_index": 0}, {"bbox": [70, 87, 541, 104], "content": "the fusion ring. Consider all formal linear combinations of objects labelled by the", "parent_index": 0, "line_index": 1}, {"bbox": [71, 100, 430, 120], "content": "; the multiplication is defined to have structure constants :", "parent_index": 0, "line_index": 2}, {"bbox": [69, 174, 539, 192], "content": "As an abstract ring, it is not so interesting (the fusion ring over is isomorphic to", "parent_index": 2, "line_index": 0}, {"bbox": [70, 190, 540, 205], "content": "with operations defined component-wise; over it will be a direct sum of number fields).", "parent_index": 2, "line_index": 1}, {"bbox": [69, 204, 542, 221], "content": "This is analogous to the character ring of a Lie algebra, which is isomorphic as a ring to a", "parent_index": 2, "line_index": 2}, {"bbox": [70, 218, 541, 235], "content": "polynomial ring. Of course it is important in both contexts that we have a preferred basis,", "parent_index": 2, "line_index": 3}, {"bbox": [70, 234, 487, 249], "content": "namely , and so proper definitions of isomorphisms etc. must respect that.", "parent_index": 2, "line_index": 4}, {"bbox": [94, 247, 541, 263], "content": "The most important examples of fusion rings are associated to the affine algebras,", "parent_index": 3, "line_index": 0}, {"bbox": [70, 262, 541, 277], "content": "and it is to these that this paper is devoted. Their automorphisms appear explicitly for", "parent_index": 3, "line_index": 1}, {"bbox": [70, 277, 541, 291], "content": "instance in the classification of modular invariant (i.e. torus) partition functions [17,18],", "parent_index": 3, "line_index": 2}, {"bbox": [70, 290, 541, 306], "content": "and also in D-branes and boundary conditions for conformal field theory (see e.g. [1]). For", "parent_index": 3, "line_index": 3}, {"bbox": [70, 306, 541, 320], "content": "instance, fusion-automorphisms (more generally, -homomorphisms) generate large classes", "parent_index": 3, "line_index": 4}, {"bbox": [71, 319, 542, 334], "content": "of nonnegative integer representations of the fusion-ring, each of which is associated to a", "parent_index": 3, "line_index": 5}, {"bbox": [70, 334, 541, 349], "content": "boundary (cylinder) partition function. This will be studied elsewhere. Also, whenever the", "parent_index": 3, "line_index": 6}, {"bbox": [71, 349, 541, 363], "content": "coefficient matrix of the torus partition function is a permutation matrix (in which case the", "parent_index": 3, "line_index": 7}, {"bbox": [70, 363, 540, 377], "content": "partition function is called an automorphism invariant), we get a fusion ring automorphism.", "parent_index": 3, "line_index": 8}, {"bbox": [70, 377, 540, 392], "content": "However most torus partition functions are not automorphism invariants (although Moore-", "parent_index": 3, "line_index": 9}, {"bbox": [71, 392, 540, 406], "content": "Seiberg assert that there is a sense in which any torus partition function can be interpreted", "parent_index": 3, "line_index": 10}, {"bbox": [71, 407, 540, 420], "content": "as one — see e.g. [3]), and most fusion ring automorphisms do not correspond to partition", "parent_index": 3, "line_index": 11}, {"bbox": [70, 419, 541, 434], "content": "functions. Nevertheless, the two problems are related. The automorphism invariants for", "parent_index": 3, "line_index": 12}, {"bbox": [71, 435, 541, 448], "content": "the affine algebras were classified in [17,18]; a Lemma proved there (our Proposition 4.1", "parent_index": 3, "line_index": 13}, {"bbox": [70, 448, 541, 464], "content": "below) involving q-dimensions will be very useful to us, and conversely the arguments in", "parent_index": 3, "line_index": 14}, {"bbox": [70, 462, 507, 478], "content": "Section 4 of this paper could be used to considerably simplify the proofs of [17,18].", "parent_index": 3, "line_index": 15}, {"bbox": [94, 477, 541, 492], "content": "It is surprising that it is even possible to find all affine fusion automorphisms, and", "parent_index": 4, "line_index": 0}, {"bbox": [70, 491, 541, 506], "content": "in fact the arguments turn out to be rather short. It is remarkable that the answer is so", "parent_index": 4, "line_index": 1}, {"bbox": [70, 506, 498, 520], "content": "simple: with few exceptions, they correspond to the Dynkin diagram symmetries.", "parent_index": 4, "line_index": 2}, {"bbox": [95, 521, 540, 535], "content": "A related task is determining which affine fusion rings are isomorphic. We answer this", "parent_index": 5, "line_index": 0}, {"bbox": [70, 536, 538, 550], "content": "in section 5 below; as expected most fusion rings with different names are nonisomorphic.", "parent_index": 5, "line_index": 1}, {"bbox": [95, 556, 541, 572], "content": "Acknowledgements. Most of this paper was written during a visit to Max-Planck", "parent_index": 6, "line_index": 0}, {"bbox": [70, 571, 540, 585], "content": "in Bonn, whose hospitality as always was both stimulating and pleasurable. I also thank", "parent_index": 6, "line_index": 1}, {"bbox": [72, 586, 354, 600], "content": "Yi-Zhi Huang for clarifying an issue concerning [8,21].", "parent_index": 6, "line_index": 2}]
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[]
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[{"bbox": [249, 75, 268, 88], "content": "N_{a b}^{c}", "parent_index": 0, "subtype": "inline"}, {"bbox": [440, 93, 453, 101], "content": "\\chi_{a}", "parent_index": 0, "subtype": "inline"}, {"bbox": [71, 104, 101, 114], "content": "a\\in\\Phi", "parent_index": 0, "subtype": "inline"}, {"bbox": [405, 104, 425, 116], "content": "N_{a b}^{c}", "parent_index": 0, "subtype": "inline"}, {"bbox": [258, 130, 353, 160], "content": "\\chi_{a}\\chi_{b}=\\sum_{c\\in\\Phi}N_{a b}^{c}\\chi_{c}", "parent_index": 1, "subtype": "interline"}, {"bbox": [412, 178, 421, 187], "content": "\\mathbb{C}", "parent_index": 2, "subtype": "inline"}, {"bbox": [513, 176, 539, 187], "content": "\\mathbb{C}^{||\\Phi||}", "parent_index": 2, "subtype": "inline"}, {"bbox": [314, 192, 324, 203], "content": "\\mathbb{Q}", "parent_index": 2, "subtype": "inline"}, {"bbox": [113, 235, 138, 247], "content": "\\{\\chi_{a}\\}", "parent_index": 2, "subtype": "inline"}]
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[]
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# 2.1. The affine fusion ring
The source of some of the most interesting fusion data are the affine nontwisted Kac-Moody algebras $X_{r}^{(1)}$ [23]. Choose any positive integer $k$ . Consider the (finite) set $P_{+}=P_{+}^{k}(X_{r}^{(1)})$ of level $k$ integrable highest weights:
$$
P_{+}\stackrel{\mathrm{def}}{=}\{\sum_{j=0}^{r}\lambda_{j}\Lambda_{j}\mid\lambda_{j}\in\mathbb{Z},\ \lambda_{j}\geq0,\ \sum_{j=0}^{r}a_{j}^{\vee}\lambda_{j}=k\}\ ,
$$
where $\Lambda_{i}$ denote the fundamental weights, and $a_{j}^{\vee}$ are the co-labels, of $X_{r}^{(1)}$ (the $a_{j}^{\vee}$ will be given for each algebra in §3). We will usually drop the (redundant) component $\lambda_{0}\Lambda_{0}$ . KacPeterson [24] found a natural representation of the modular group $\operatorname{SL_{2}}(\mathbb{Z})$ on the complex space spanned by the affine characters $\chi_{\mu}$ , $\mu\in P_{+}$ : most significantly, $\left(\begin{array}{c c}{{0}}&{{-1}}\\ {{1}}&{{0}}\end{array}\right)$ is sent to the Kac-Peterson matrix $S$ with entries
$$
S_{\mu\nu}=c\,\sum_{w\in\overline{{{W}}}}{\operatorname*{det}(w)}\,\exp[-2\pi\mathrm{i}\,\frac{(w(\mu+\rho)|\nu+\rho)}{\kappa}]\ .
$$
An explicit expression for the normalisation constant $c$ is given in e.g. [23, Theorem 13.8(a)]. The inner product in (2.1a) is scaled so that the long roots have norm 2. $\overline{W}$ is the (finite) Weyl group of $X_{r}$ , and acts on $P_{+}$ by fixing $\Lambda_{0}$ . The Weyl vector $\rho$ equals $\sum_{i}\Lambda_{i}$ , and κ d=ef k + i ai∨ . This is the matrix S appearing in (1.1); Φ there is P+ here.
The matrix $S$ is symmetric and unitary. One of the weights, $k\Lambda_{0}$ , is distinguished and will be denoted ‘ $0^{\circ}$ . It is the weight appearing in the denominator of (1.1). A useful fact is that
$$
S_{\lambda0}>0\qquad\mathrm{for~all~}\lambda\in P_{+}\ .
$$
Equation (2.1a) gives us the important
$$
\chi_{\lambda}[\mu]\stackrel{\mathrm{def}}{=}\frac{S_{\lambda\mu}}{S_{0\mu}}=\mathrm{ch}_{\overline{{{\lambda}}}}(-2\pi\mathrm{i}\frac{\overline{{{\mu}}}+\overline{{{\rho}}}}{\kappa})~,
$$
where $\mathrm{ch}_{{\overline{{\lambda}}}}$ is the Weyl character of the $X_{r}$ -module $L(\overline{{\lambda}})$ . Together with the Weyl denominator formula, it provides a useful expression for the $q$ -dimensions:
$$
{\mathcal D}(\lambda)\,\overset{\mathrm{def}}{=}\frac{S_{\lambda0}}{S_{00}}=\prod_{\alpha>0}\frac{\sin(\pi\left(\lambda+\rho\left|\alpha\right)/\kappa\right)}{\sin(\pi\left(\rho\left|\alpha\right)/\kappa\right)}~,
$$
where the product is over the positive roots $\alpha\in\overline{{\Delta}}_{+}$ of $X_{r}$ . Another consequence of (2.1b) is the Kac-Walton formula (2.4).
|
<html><body>
<h1 data-bbox="70 100 214 116">2.1. The affine fusion ring </h1>
<p data-bbox="70 123 541 171">The source of some of the most interesting fusion data are the affine nontwisted Kac-Moody algebras $X_{r}^{(1)}$ [23]. Choose any positive integer $k$ . Consider the (finite) set $P_{+}=P_{+}^{k}(X_{r}^{(1)})$ of level $k$ integrable highest weights: </p>
<div class="equation" data-bbox="174 187 436 226">$$
P_{+}\stackrel{\mathrm{def}}{=}\{\sum_{j=0}^{r}\lambda_{j}\Lambda_{j}\mid\lambda_{j}\in\mathbb{Z},\ \lambda_{j}\geq0,\ \sum_{j=0}^{r}a_{j}^{\vee}\lambda_{j}=k\}\ ,
$$</div>
<p data-bbox="69 239 541 328">where $\Lambda_{i}$ denote the fundamental weights, and $a_{j}^{\vee}$ are the co-labels, of $X_{r}^{(1)}$ (the $a_{j}^{\vee}$ will be given for each algebra in §3). We will usually drop the (redundant) component $\lambda_{0}\Lambda_{0}$ . KacPeterson [24] found a natural representation of the modular group $\operatorname{SL_{2}}(\mathbb{Z})$ on the complex space spanned by the affine characters $\chi_{\mu}$ , $\mu\in P_{+}$ : most significantly, $\left(\begin{array}{c c}{{0}}&{{-1}}\\ {{1}}&{{0}}\end{array}\right)$ is sent to the Kac-Peterson matrix $S$ with entries </p>
<div class="equation" data-bbox="175 338 435 378">$$
S_{\mu\nu}=c\,\sum_{w\in\overline{{{W}}}}{\operatorname*{det}(w)}\,\exp[-2\pi\mathrm{i}\,\frac{(w(\mu+\rho)|\nu+\rho)}{\kappa}]\ .
$$</div>
<p data-bbox="70 389 541 452">An explicit expression for the normalisation constant $c$ is given in e.g. [23, Theorem 13.8(a)]. The inner product in (2.1a) is scaled so that the long roots have norm 2. $\overline{W}$ is the (finite) Weyl group of $X_{r}$ , and acts on $P_{+}$ by fixing $\Lambda_{0}$ . The Weyl vector $\rho$ equals $\sum_{i}\Lambda_{i}$ , and κ d=ef k + i ai∨ . This is the matrix S appearing in (1.1); Φ there is P+ here. </p>
<p data-bbox="70 452 541 494">The matrix $S$ is symmetric and unitary. One of the weights, $k\Lambda_{0}$ , is distinguished and will be denoted ‘ $0^{\circ}$ . It is the weight appearing in the denominator of (1.1). A useful fact is that </p>
<div class="equation" data-bbox="233 497 378 511">$$
S_{\lambda0}>0\qquad\mathrm{for~all~}\lambda\in P_{+}\ .
$$</div>
<p data-bbox="93 519 300 534">Equation (2.1a) gives us the important </p>
<div class="equation" data-bbox="218 549 393 580">$$
\chi_{\lambda}[\mu]\stackrel{\mathrm{def}}{=}\frac{S_{\lambda\mu}}{S_{0\mu}}=\mathrm{ch}_{\overline{{{\lambda}}}}(-2\pi\mathrm{i}\frac{\overline{{{\mu}}}+\overline{{{\rho}}}}{\kappa})~,
$$</div>
<p data-bbox="69 592 540 622">where $\mathrm{ch}_{{\overline{{\lambda}}}}$ is the Weyl character of the $X_{r}$ -module $L(\overline{{\lambda}})$ . Together with the Weyl denominator formula, it provides a useful expression for the $q$ -dimensions: </p>
<div class="equation" data-bbox="200 636 410 672">$$
{\mathcal D}(\lambda)\,\overset{\mathrm{def}}{=}\frac{S_{\lambda0}}{S_{00}}=\prod_{\alpha>0}\frac{\sin(\pi\left(\lambda+\rho\left|\alpha\right)/\kappa\right)}{\sin(\pi\left(\rho\left|\alpha\right)/\kappa\right)}~,
$$</div>
<p data-bbox="69 685 540 715">where the product is over the positive roots $\alpha\in\overline{{\Delta}}_{+}$ of $X_{r}$ . Another consequence of (2.1b) is the Kac-Walton formula (2.4). </p>
</body></html>
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0002044v1
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[{"type": "text", "text": "2.1. The affine fusion ring ", "text_level": 1, "page_idx": 2}, {"type": "text", "text": "The source of some of the most interesting fusion data are the affine nontwisted Kac-Moody algebras $X_{r}^{(1)}$ [23]. Choose any positive integer $k$ . Consider the (finite) set $P_{+}=P_{+}^{k}(X_{r}^{(1)})$ of level $k$ integrable highest weights: ", "page_idx": 2}, {"type": "equation", "text": "$$\nP_{+}\\stackrel{\\mathrm{def}}{=}\\{\\sum_{j=0}^{r}\\lambda_{j}\\Lambda_{j}\\mid\\lambda_{j}\\in\\mathbb{Z},\\ \\lambda_{j}\\geq0,\\ \\sum_{j=0}^{r}a_{j}^{\\vee}\\lambda_{j}=k\\}\\ ,\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "where $\\Lambda_{i}$ denote the fundamental weights, and $a_{j}^{\\vee}$ are the co-labels, of $X_{r}^{(1)}$ (the $a_{j}^{\\vee}$ will be given for each algebra in §3). We will usually drop the (redundant) component $\\lambda_{0}\\Lambda_{0}$ . KacPeterson [24] found a natural representation of the modular group $\\operatorname{SL_{2}}(\\mathbb{Z})$ on the complex space spanned by the affine characters $\\chi_{\\mu}$ , $\\mu\\in P_{+}$ : most significantly, $\\left(\\begin{array}{c c}{{0}}&{{-1}}\\\\ {{1}}&{{0}}\\end{array}\\right)$ is sent to the Kac-Peterson matrix $S$ with entries ", "page_idx": 2}, {"type": "equation", "text": "$$\nS_{\\mu\\nu}=c\\,\\sum_{w\\in\\overline{{{W}}}}{\\operatorname*{det}(w)}\\,\\exp[-2\\pi\\mathrm{i}\\,\\frac{(w(\\mu+\\rho)|\\nu+\\rho)}{\\kappa}]\\ .\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "An explicit expression for the normalisation constant $c$ is given in e.g. [23, Theorem 13.8(a)]. The inner product in (2.1a) is scaled so that the long roots have norm 2. $\\overline{W}$ is the (finite) Weyl group of $X_{r}$ , and acts on $P_{+}$ by fixing $\\Lambda_{0}$ . The Weyl vector $\\rho$ equals $\\sum_{i}\\Lambda_{i}$ , and κ d=ef k + i ai∨ . This is the matrix S appearing in (1.1); Φ there is P+ here. ", "page_idx": 2}, {"type": "text", "text": "The matrix $S$ is symmetric and unitary. One of the weights, $k\\Lambda_{0}$ , is distinguished and will be denoted ‘ $0^{\\circ}$ . It is the weight appearing in the denominator of (1.1). A useful fact is that ", "page_idx": 2}, {"type": "equation", "text": "$$\nS_{\\lambda0}>0\\qquad\\mathrm{for~all~}\\lambda\\in P_{+}\\ .\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "Equation (2.1a) gives us the important ", "page_idx": 2}, {"type": "equation", "text": "$$\n\\chi_{\\lambda}[\\mu]\\stackrel{\\mathrm{def}}{=}\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=\\mathrm{ch}_{\\overline{{{\\lambda}}}}(-2\\pi\\mathrm{i}\\frac{\\overline{{{\\mu}}}+\\overline{{{\\rho}}}}{\\kappa})~,\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "where $\\mathrm{ch}_{{\\overline{{\\lambda}}}}$ is the Weyl character of the $X_{r}$ -module $L(\\overline{{\\lambda}})$ . Together with the Weyl denominator formula, it provides a useful expression for the $q$ -dimensions: ", "page_idx": 2}, {"type": "equation", "text": "$$\n{\\mathcal D}(\\lambda)\\,\\overset{\\mathrm{def}}{=}\\frac{S_{\\lambda0}}{S_{00}}=\\prod_{\\alpha>0}\\frac{\\sin(\\pi\\left(\\lambda+\\rho\\left|\\alpha\\right)/\\kappa\\right)}{\\sin(\\pi\\left(\\rho\\left|\\alpha\\right)/\\kappa\\right)}~,\n$$", "text_format": "latex", "page_idx": 2}, {"type": "text", "text": "where the product is over the positive roots $\\alpha\\in\\overline{{\\Delta}}_{+}$ of $X_{r}$ . Another consequence of (2.1b) is the Kac-Walton formula (2.4). ", "page_idx": 2}]
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Choose any positive integer ", "type": "text"}, {"bbox": [394, 144, 401, 153], "score": 0.89, "content": "k", "type": "inline_equation", "height": 9, "width": 7}, {"bbox": [401, 137, 542, 159], "score": 1.0, "content": ". 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We will usually drop the (redundant) component ", "type": "text"}, {"bbox": [480, 259, 506, 271], "score": 0.9, "content": "\\lambda_{0}\\Lambda_{0}", "type": "inline_equation", "height": 12, "width": 26}, {"bbox": [507, 258, 540, 274], "score": 1.0, "content": ". Kac-", "type": "text"}], "index": 6}, {"bbox": [70, 273, 540, 288], "spans": [{"bbox": [70, 273, 419, 288], "score": 1.0, "content": "Peterson [24] found a natural representation of the modular group ", "type": "text"}, {"bbox": [419, 274, 456, 286], "score": 0.9, "content": "\\operatorname{SL_{2}}(\\mathbb{Z})", "type": "inline_equation", "height": 12, "width": 37}, {"bbox": [457, 273, 540, 288], "score": 1.0, "content": " on the complex", "type": "text"}], "index": 7}, {"bbox": [70, 287, 540, 317], "spans": [{"bbox": [70, 293, 277, 311], "score": 1.0, "content": "space spanned by the affine characters ", "type": "text"}, {"bbox": [278, 299, 292, 308], "score": 0.82, "content": "\\chi_{\\mu}", "type": "inline_equation", "height": 9, "width": 14}, {"bbox": [293, 293, 299, 311], "score": 1.0, "content": ", ", "type": "text"}, {"bbox": [299, 296, 338, 308], "score": 0.91, "content": "\\mu\\in P_{+}", "type": "inline_equation", "height": 12, "width": 39}, {"bbox": [339, 293, 446, 311], "score": 1.0, "content": ": most significantly,", "type": "text"}, {"bbox": [447, 287, 502, 317], "score": 0.95, "content": "\\left(\\begin{array}{c c}{{0}}&{{-1}}\\\\ {{1}}&{{0}}\\end{array}\\right)", "type": "inline_equation", "height": 30, "width": 55}, {"bbox": [505, 294, 540, 308], "score": 1.0, "content": "is sent", "type": "text"}], "index": 8}, {"bbox": [70, 315, 295, 330], "spans": [{"bbox": [70, 315, 220, 330], "score": 1.0, "content": "to the Kac-Peterson matrix ", "type": "text"}, {"bbox": [220, 317, 228, 326], "score": 0.87, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [229, 315, 295, 330], "score": 1.0, "content": " with entries", "type": "text"}], "index": 9}], "index": 7}, {"type": "interline_equation", "bbox": [175, 338, 435, 378], "lines": [{"bbox": [175, 338, 435, 378], "spans": [{"bbox": [175, 338, 435, 378], "score": 0.94, "content": "S_{\\mu\\nu}=c\\,\\sum_{w\\in\\overline{{{W}}}}{\\operatorname*{det}(w)}\\,\\exp[-2\\pi\\mathrm{i}\\,\\frac{(w(\\mu+\\rho)|\\nu+\\rho)}{\\kappa}]\\ .", "type": "interline_equation"}], "index": 10}], "index": 10}, {"type": "text", "bbox": [70, 389, 541, 452], "lines": [{"bbox": [71, 393, 540, 407], "spans": [{"bbox": [71, 393, 343, 407], "score": 1.0, "content": "An explicit expression for the normalisation constant ", "type": "text"}, {"bbox": [344, 397, 349, 403], "score": 0.87, "content": "c", "type": "inline_equation", "height": 6, "width": 5}, {"bbox": [349, 393, 540, 407], "score": 1.0, "content": " is given in e.g. [23, Theorem 13.8(a)].", "type": "text"}], "index": 11}, {"bbox": [70, 406, 540, 421], "spans": [{"bbox": [70, 406, 454, 421], "score": 1.0, "content": "The inner product in (2.1a) is scaled so that the long roots have norm 2. ", "type": "text"}, {"bbox": [455, 407, 468, 417], "score": 0.92, "content": "\\overline{W}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [468, 406, 540, 421], "score": 1.0, "content": " is the (finite)", "type": "text"}], "index": 12}, {"bbox": [70, 420, 541, 436], "spans": [{"bbox": [70, 420, 151, 436], "score": 1.0, "content": "Weyl group of ", "type": "text"}, {"bbox": [151, 423, 167, 433], "score": 0.91, "content": "X_{r}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [167, 420, 242, 436], "score": 1.0, "content": ", and acts on ", "type": "text"}, {"bbox": [242, 423, 258, 434], "score": 0.91, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [258, 420, 313, 436], "score": 1.0, "content": " by fixing ", "type": "text"}, {"bbox": [314, 423, 328, 434], "score": 0.9, "content": "\\Lambda_{0}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [328, 420, 432, 436], "score": 1.0, "content": ". The Weyl vector ", "type": "text"}, {"bbox": [432, 426, 439, 434], "score": 0.87, "content": "\\rho", "type": "inline_equation", "height": 8, "width": 7}, {"bbox": [439, 420, 480, 436], "score": 1.0, "content": " equals", "type": "text"}, {"bbox": [480, 422, 512, 435], "score": 0.93, "content": "\\sum_{i}\\Lambda_{i}", "type": "inline_equation", "height": 13, "width": 32}, {"bbox": [512, 420, 541, 436], "score": 1.0, "content": ", and", "type": "text"}], "index": 13}, {"bbox": [68, 435, 473, 457], "spans": [{"bbox": [68, 435, 473, 457], "score": 1.0, "content": "κ d=ef k + i ai∨ . This is the matrix S appearing in (1.1); Φ there is P+ here.", "type": "text"}], "index": 14}], "index": 12.5}, {"type": "text", "bbox": [70, 452, 541, 494], "lines": [{"bbox": [94, 453, 541, 469], "spans": [{"bbox": [94, 453, 157, 469], "score": 1.0, "content": "The matrix ", "type": "text"}, {"bbox": [158, 456, 166, 465], "score": 0.88, "content": "S", "type": "inline_equation", "height": 9, "width": 8}, {"bbox": [166, 453, 409, 469], "score": 1.0, "content": " is symmetric and unitary. One of the weights, ", "type": "text"}, {"bbox": [409, 456, 430, 467], "score": 0.91, "content": "k\\Lambda_{0}", "type": "inline_equation", "height": 11, "width": 21}, {"bbox": [430, 453, 541, 469], "score": 1.0, "content": ", is distinguished and", "type": "text"}], "index": 15}, {"bbox": [70, 468, 541, 484], "spans": [{"bbox": [70, 468, 159, 484], "score": 1.0, "content": "will be denoted ‘", "type": "text"}, {"bbox": [159, 470, 169, 479], "score": 0.43, "content": "0^{\\circ}", "type": "inline_equation", "height": 9, "width": 10}, {"bbox": [169, 468, 541, 484], "score": 1.0, "content": ". It is the weight appearing in the denominator of (1.1). 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", "type": "text"}, {"bbox": [455, 407, 468, 417], "score": 0.92, "content": "\\overline{W}", "type": "inline_equation", "height": 10, "width": 13}, {"bbox": [468, 406, 540, 421], "score": 1.0, "content": " is the (finite)", "type": "text"}], "index": 12}, {"bbox": [70, 420, 541, 436], "spans": [{"bbox": [70, 420, 151, 436], "score": 1.0, "content": "Weyl group of ", "type": "text"}, {"bbox": [151, 423, 167, 433], "score": 0.91, "content": "X_{r}", "type": "inline_equation", "height": 10, "width": 16}, {"bbox": [167, 420, 242, 436], "score": 1.0, "content": ", and acts on ", "type": "text"}, {"bbox": [242, 423, 258, 434], "score": 0.91, "content": "P_{+}", "type": "inline_equation", "height": 11, "width": 16}, {"bbox": [258, 420, 313, 436], "score": 1.0, "content": " by fixing ", "type": "text"}, {"bbox": [314, 423, 328, 434], "score": 0.9, "content": "\\Lambda_{0}", "type": "inline_equation", "height": 11, "width": 14}, {"bbox": [328, 420, 432, 436], "score": 1.0, "content": ". 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Another consequence of (2.1b)", "type": "text"}], "index": 24}, {"bbox": [69, 702, 243, 718], "spans": [{"bbox": [69, 702, 243, 718], "score": 1.0, "content": "is the Kac-Walton formula (2.4).", "type": "text"}], "index": 25}], "index": 24.5, "bbox_fs": [69, 688, 540, 718]}]}
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[{"type": "title", "bbox": [70, 100, 214, 116], "content": "2.1. The affine fusion ring", "index": 0}, {"type": "text", "bbox": [70, 123, 541, 171], "content": "The source of some of the most interesting fusion data are the affine nontwisted Kac-Moody algebras [23]. Choose any positive integer . Consider the (finite) set of level integrable highest weights:", "index": 1}, {"type": "interline_equation", "bbox": [174, 187, 436, 226], "content": "", "index": 2}, {"type": "text", "bbox": [69, 239, 541, 328], "content": "where denote the fundamental weights, and are the co-labels, of (the will be given for each algebra in §3). We will usually drop the (redundant) component . Kac- Peterson [24] found a natural representation of the modular group on the complex space spanned by the affine characters , : most significantly, is sent to the Kac-Peterson matrix with entries", "index": 3}, {"type": "interline_equation", "bbox": [175, 338, 435, 378], "content": "", "index": 4}, {"type": "text", "bbox": [70, 389, 541, 452], "content": "An explicit expression for the normalisation constant is given in e.g. [23, Theorem 13.8(a)]. The inner product in (2.1a) is scaled so that the long roots have norm 2. is the (finite) Weyl group of , and acts on by fixing . The Weyl vector equals , and κ d=ef k + i ai∨ . This is the matrix S appearing in (1.1); Φ there is P+ here.", "index": 5}, {"type": "text", "bbox": [70, 452, 541, 494], "content": "The matrix is symmetric and unitary. One of the weights, , is distinguished and will be denoted ‘ . It is the weight appearing in the denominator of (1.1). A useful fact is that", "index": 6}, {"type": "interline_equation", "bbox": [233, 497, 378, 511], "content": "", "index": 7}, {"type": "text", "bbox": [93, 519, 300, 534], "content": "Equation (2.1a) gives us the important", "index": 8}, {"type": "interline_equation", "bbox": [218, 549, 393, 580], "content": "", "index": 9}, {"type": "text", "bbox": [69, 592, 540, 622], "content": "where is the Weyl character of the -module . Together with the Weyl denomi- nator formula, it provides a useful expression for the -dimensions:", "index": 10}, {"type": "interline_equation", "bbox": [200, 636, 410, 672], "content": "", "index": 11}, {"type": "text", "bbox": [69, 685, 540, 715], "content": "where the product is over the positive roots of . Another consequence of (2.1b) is the Kac-Walton formula (2.4).", "index": 12}]
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[{"bbox": [72, 102, 213, 118], "content": "2.1. The affine fusion ring", "parent_index": 0, "line_index": 0}, {"bbox": [93, 125, 541, 141], "content": "The source of some of the most interesting fusion data are the affine nontwisted", "parent_index": 1, "line_index": 0}, {"bbox": [69, 137, 542, 159], "content": "Kac-Moody algebras [23]. Choose any positive integer . Consider the (finite) set", "parent_index": 1, "line_index": 1}, {"bbox": [71, 156, 347, 174], "content": "of level integrable highest weights:", "parent_index": 1, "line_index": 2}, {"bbox": [69, 241, 542, 260], "content": "where denote the fundamental weights, and are the co-labels, of (the will be", "parent_index": 3, "line_index": 0}, {"bbox": [70, 258, 540, 274], "content": "given for each algebra in §3). We will usually drop the (redundant) component . Kac-", "parent_index": 3, "line_index": 1}, {"bbox": [70, 273, 540, 288], "content": "Peterson [24] found a natural representation of the modular group on the complex", "parent_index": 3, "line_index": 2}, {"bbox": [70, 287, 540, 317], "content": "space spanned by the affine characters , : most significantly, is sent", "parent_index": 3, "line_index": 3}, {"bbox": [70, 315, 295, 330], "content": "to the Kac-Peterson matrix with entries", "parent_index": 3, "line_index": 4}, {"bbox": [71, 393, 540, 407], "content": "An explicit expression for the normalisation constant is given in e.g. [23, Theorem 13.8(a)].", "parent_index": 5, "line_index": 0}, {"bbox": [70, 406, 540, 421], "content": "The inner product in (2.1a) is scaled so that the long roots have norm 2. is the (finite)", "parent_index": 5, "line_index": 1}, {"bbox": [70, 420, 541, 436], "content": "Weyl group of , and acts on by fixing . The Weyl vector equals , and", "parent_index": 5, "line_index": 2}, {"bbox": [68, 435, 473, 457], "content": "κ d=ef k + i ai∨ . This is the matrix S appearing in (1.1); Φ there is P+ here.", "parent_index": 5, "line_index": 3}, {"bbox": [94, 453, 541, 469], "content": "The matrix is symmetric and unitary. One of the weights, , is distinguished and", "parent_index": 6, "line_index": 0}, {"bbox": [70, 468, 541, 484], "content": "will be denoted ‘ . It is the weight appearing in the denominator of (1.1). A useful fact", "parent_index": 6, "line_index": 1}, {"bbox": [70, 484, 107, 496], "content": "is that", "parent_index": 6, "line_index": 2}, {"bbox": [95, 521, 300, 536], "content": "Equation (2.1a) gives us the important", "parent_index": 8, "line_index": 0}, {"bbox": [70, 595, 540, 611], "content": "where is the Weyl character of the -module . Together with the Weyl denomi-", "parent_index": 10, "line_index": 0}, {"bbox": [70, 610, 424, 625], "content": "nator formula, it provides a useful expression for the -dimensions:", "parent_index": 10, "line_index": 1}, {"bbox": [71, 688, 540, 703], "content": "where the product is over the positive roots of . Another consequence of (2.1b)", "parent_index": 12, "line_index": 0}, {"bbox": [69, 702, 243, 718], "content": "is the Kac-Walton formula (2.4).", "parent_index": 12, "line_index": 1}]
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[{"bbox": [184, 140, 208, 154], "content": "X_{r}^{(1)}", "parent_index": 1, "subtype": "inline"}, {"bbox": [394, 144, 401, 153], "content": "k", "parent_index": 1, "subtype": "inline"}, {"bbox": [71, 156, 152, 173], "content": "P_{+}=P_{+}^{k}(X_{r}^{(1)})", "parent_index": 1, "subtype": "inline"}, {"bbox": [196, 160, 203, 169], "content": "k", "parent_index": 1, "subtype": "inline"}, {"bbox": [174, 187, 436, 226], "content": "P_{+}\\stackrel{\\mathrm{def}}{=}\\{\\sum_{j=0}^{r}\\lambda_{j}\\Lambda_{j}\\mid\\lambda_{j}\\in\\mathbb{Z},\\ \\lambda_{j}\\geq0,\\ \\sum_{j=0}^{r}a_{j}^{\\vee}\\lambda_{j}=k\\}\\ ,", "parent_index": 2, "subtype": "interline"}, {"bbox": [105, 246, 117, 257], "content": "\\Lambda_{i}", "parent_index": 3, "subtype": "inline"}, {"bbox": [316, 245, 329, 259], "content": "a_{j}^{\\vee}", "parent_index": 3, "subtype": "inline"}, {"bbox": [436, 241, 460, 256], "content": "X_{r}^{(1)}", "parent_index": 3, "subtype": "inline"}, {"bbox": [488, 245, 501, 259], "content": "a_{j}^{\\vee}", "parent_index": 3, "subtype": "inline"}, {"bbox": [480, 259, 506, 271], "content": "\\lambda_{0}\\Lambda_{0}", "parent_index": 3, "subtype": "inline"}, {"bbox": [419, 274, 456, 286], "content": "\\operatorname{SL_{2}}(\\mathbb{Z})", "parent_index": 3, "subtype": "inline"}, {"bbox": [278, 299, 292, 308], "content": "\\chi_{\\mu}", "parent_index": 3, "subtype": "inline"}, {"bbox": [299, 296, 338, 308], "content": "\\mu\\in P_{+}", "parent_index": 3, "subtype": "inline"}, {"bbox": [447, 287, 502, 317], "content": "\\left(\\begin{array}{c c}{{0}}&{{-1}}\\\\ {{1}}&{{0}}\\end{array}\\right)", "parent_index": 3, "subtype": "inline"}, {"bbox": [220, 317, 228, 326], "content": "S", "parent_index": 3, "subtype": "inline"}, {"bbox": [175, 338, 435, 378], "content": "S_{\\mu\\nu}=c\\,\\sum_{w\\in\\overline{{{W}}}}{\\operatorname*{det}(w)}\\,\\exp[-2\\pi\\mathrm{i}\\,\\frac{(w(\\mu+\\rho)|\\nu+\\rho)}{\\kappa}]\\ .", "parent_index": 4, "subtype": "interline"}, {"bbox": [344, 397, 349, 403], "content": "c", "parent_index": 5, "subtype": "inline"}, {"bbox": [455, 407, 468, 417], "content": "\\overline{W}", "parent_index": 5, "subtype": "inline"}, {"bbox": [151, 423, 167, 433], "content": "X_{r}", "parent_index": 5, "subtype": "inline"}, {"bbox": [242, 423, 258, 434], "content": "P_{+}", "parent_index": 5, "subtype": "inline"}, {"bbox": [314, 423, 328, 434], "content": "\\Lambda_{0}", "parent_index": 5, "subtype": "inline"}, {"bbox": [432, 426, 439, 434], "content": "\\rho", "parent_index": 5, "subtype": "inline"}, {"bbox": [480, 422, 512, 435], "content": "\\sum_{i}\\Lambda_{i}", "parent_index": 5, "subtype": "inline"}, {"bbox": [158, 456, 166, 465], "content": "S", "parent_index": 6, "subtype": "inline"}, {"bbox": [409, 456, 430, 467], "content": "k\\Lambda_{0}", "parent_index": 6, "subtype": "inline"}, {"bbox": [159, 470, 169, 479], "content": "0^{\\circ}", "parent_index": 6, "subtype": "inline"}, {"bbox": [233, 497, 378, 511], "content": "S_{\\lambda0}>0\\qquad\\mathrm{for~all~}\\lambda\\in P_{+}\\ .", "parent_index": 7, "subtype": "interline"}, {"bbox": [218, 549, 393, 580], "content": "\\chi_{\\lambda}[\\mu]\\stackrel{\\mathrm{def}}{=}\\frac{S_{\\lambda\\mu}}{S_{0\\mu}}=\\mathrm{ch}_{\\overline{{{\\lambda}}}}(-2\\pi\\mathrm{i}\\frac{\\overline{{{\\mu}}}+\\overline{{{\\rho}}}}{\\kappa})~,", "parent_index": 9, "subtype": "interline"}, {"bbox": [106, 597, 125, 610], "content": "\\mathrm{ch}_{{\\overline{{\\lambda}}}}", "parent_index": 10, "subtype": "inline"}, {"bbox": [275, 597, 291, 608], "content": "X_{r}", "parent_index": 10, "subtype": "inline"}, {"bbox": [336, 595, 361, 609], "content": "L(\\overline{{\\lambda}})", "parent_index": 10, "subtype": "inline"}, {"bbox": [351, 615, 357, 623], "content": "q", "parent_index": 10, "subtype": "inline"}, {"bbox": [200, 636, 410, 672], "content": "{\\mathcal D}(\\lambda)\\,\\overset{\\mathrm{def}}{=}\\frac{S_{\\lambda0}}{S_{00}}=\\prod_{\\alpha>0}\\frac{\\sin(\\pi\\left(\\lambda+\\rho\\left|\\alpha\\right)/\\kappa\\right)}{\\sin(\\pi\\left(\\rho\\left|\\alpha\\right)/\\kappa\\right)}~,", "parent_index": 11, "subtype": "interline"}, {"bbox": [301, 688, 341, 702], "content": "\\alpha\\in\\overline{{\\Delta}}_{+}", "parent_index": 12, "subtype": "inline"}, {"bbox": [357, 690, 373, 701], "content": "X_{r}", "parent_index": 12, "subtype": "inline"}]
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"Charge-conjugation is the order 2 permutation of $P_{+}$ given by $C\\lambda\\,=\\,^{t}\\lambda$ , (...TRUNCATED) | "<html><body>\n<p data-bbox=\"70 70 540 100\">Charge-conjugation is the order 2 permutation of $P_{+(...TRUNCATED) |
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"$\\epsilon_{\\ell}(\\lambda)/\\epsilon_{\\ell}^{\\prime}(\\lambda)=\\sigma_{\\ell}(c)/c$ is an unim(...TRUNCATED) | "<html><body>\n<p data-bbox=\"70 70 541 114\">$\\epsilon_{\\ell}(\\lambda)/\\epsilon_{\\ell}^{\\prim(...TRUNCATED) |
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| 4 | 612 | 792 | 1,275 | 1,650 | "[{\"type\": \"text\", \"text\": \"$\\\\epsilon_{\\\\ell}(\\\\lambda)/\\\\epsilon_{\\\\ell}^{\\\\pri(...TRUNCATED) | "{\"layout_dets\": [{\"category_id\": 1, \"poly\": [196, 1736, 1505, 1736, 1505, 1990, 196, 1990], \(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"text\", \"bbox\": [70, 70, 541, 114], \"lines\": [{\"bbox\": [71(...TRUNCATED) | "[{\"type\": \"text\", \"bbox\": [70, 70, 541, 114], \"content\": \"is an unimportant sign independe(...TRUNCATED) | "[{\"bbox\": [71, 73, 540, 89], \"content\": \"is an unimportant sign independent of . This Galois (...TRUNCATED) |
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| "[{\"bbox\": [71, 75, 182, 88], \"content\": \"\\\\epsilon_{\\\\ell}(\\\\lambda)/\\\\epsilon_{\\\\el(...TRUNCATED) |
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"A useful way of identifying weights in affine Weyl orbits involves computing q-dimensions and norms(...TRUNCATED) | "<html><body>\n<p data-bbox=\"71 115 545 200\">A useful way of identifying weights in affine Weyl or(...TRUNCATED) |
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| 5 | 612 | 792 | 1,275 | 1,650 | "[{\"type\": \"text\", \"text\": \"\", \"page_idx\": 5}, {\"type\": \"text\", \"text\": \"A useful w(...TRUNCATED) | "{\"layout_dets\": [{\"category_id\": 1, \"poly\": [195, 559, 1506, 559, 1506, 1014, 195, 1014], \"s(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"text\", \"bbox\": [70, 70, 543, 114], \"lines\": [{\"bbox\": [70(...TRUNCATED) | "[{\"type\": \"text\", \"bbox\": [70, 70, 543, 114], \"content\": \"\", \"index\": 0}, {\"type\": \"(...TRUNCATED) | "[{\"bbox\": [94, 116, 548, 133], \"content\": \"A useful way of identifying weights in affine Weyl (...TRUNCATED) |
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"For instance $\\pi$ must send $J$ -fixed-points to $\\pi(J)$ -fixed-points.\n\nMore generally, a fu(...TRUNCATED) | "<html><body>\n<p data-bbox=\"70 70 392 85\">For instance $\\pi$ must send $J$ -fixed-points to $\\p(...TRUNCATED) |
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| 6 | 612 | 792 | 1,275 | 1,650 | "[{\"type\": \"text\", \"text\": \"For instance $\\\\pi$ must send $J$ -fixed-points to $\\\\pi(J)$ (...TRUNCATED) | "{\"layout_dets\": [{\"category_id\": 1, \"poly\": [196, 241, 1507, 241, 1507, 445, 196, 445], \"sco(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"text\", \"bbox\": [70, 70, 392, 85], \"lines\": [{\"bbox\": [70,(...TRUNCATED) | "[{\"type\": \"text\", \"bbox\": [70, 70, 392, 85], \"content\": \"For instance must send -fixed-(...TRUNCATED) | "[{\"bbox\": [70, 74, 390, 87], \"content\": \"For instance must send -fixed-points to -fixed-po(...TRUNCATED) |
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"2.3. Standard constructions of fusion-symmetries\n\nSimple-currents are a large source of fusion-sy(...TRUNCATED) | "<html><body>\n<p data-bbox=\"72 173 331 188\">2.3. Standard constructions of fusion-symmetries </p>(...TRUNCATED) |
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| 7 | 612 | 792 | 1,275 | 1,650 | "[{\"type\": \"text\", \"text\": \"\", \"page_idx\": 7}, {\"type\": \"text\", \"text\": \"2.3. Stand(...TRUNCATED) | "{\"layout_dets\": [{\"category_id\": 1, \"poly\": [195, 196, 1504, 196, 1504, 447, 195, 447], \"sco(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"text\", \"bbox\": [70, 70, 541, 160], \"lines\": [{\"bbox\": [71(...TRUNCATED) | "[{\"type\": \"text\", \"bbox\": [70, 70, 541, 160], \"content\": \"\", \"index\": 0}, {\"type\": \"(...TRUNCATED) | "[{\"bbox\": [72, 177, 330, 189], \"content\": \"2.3. Standard constructions of fusion-symmetries\",(...TRUNCATED) |
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| "[{\"bbox\": [416, 199, 421, 210], \"content\": \"j\", \"parent_index\": 2, \"subtype\": \"inline\"}(...TRUNCATED) |
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"Our main task in this paper is to find and construct all fusion-symmetries for the affine algebras (...TRUNCATED) | "<html><body>\n<p data-bbox=\"70 99 542 191\">Our main task in this paper is to find and construct a(...TRUNCATED) |
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| 8 | 612 | 792 | 1,275 | 1,650 | "[{\"type\": \"text\", \"text\": \"Our main task in this paper is to find and construct all fusion-s(...TRUNCATED) | "{\"layout_dets\": [{\"category_id\": 1, \"poly\": [196, 276, 1507, 276, 1507, 532, 196, 532], \"sco(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"text\", \"bbox\": [70, 99, 542, 191], \"lines\": [{\"bbox\": [95(...TRUNCATED) | "[{\"type\": \"text\", \"bbox\": [70, 99, 542, 191], \"content\": \"Our main task in this paper is t(...TRUNCATED) | "[{\"bbox\": [95, 102, 541, 117], \"content\": \"Our main task in this paper is to find and construc(...TRUNCATED) |
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"3.2. The algebra $B_{r}^{(1)}$ , $r\\geq3$\n\nA weight $\\lambda$ in $P_{+}$ satisfies $k=\\lambda_(...TRUNCATED) | "<html><body>\n<p data-bbox=\"70 69 221 86\">3.2. The algebra $B_{r}^{(1)}$ , $r\\geq3$ </p>\n<p dat(...TRUNCATED) |
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| 9 | 612 | 792 | 1,275 | 1,650 | "[{\"type\": \"text\", \"text\": \"3.2. The algebra $B_{r}^{(1)}$ , $r\\\\geq3$ \", \"page_idx\": 9}(...TRUNCATED) | "{\"layout_dets\": [{\"category_id\": 1, \"poly\": [196, 1185, 1505, 1185, 1505, 1474, 196, 1474], \(...TRUNCATED) | "{\"preproc_blocks\": [{\"type\": \"text\", \"bbox\": [70, 69, 221, 86], \"lines\": [{\"bbox\": [68,(...TRUNCATED) | "[{\"type\": \"text\", \"bbox\": [70, 69, 221, 86], \"content\": \"3.2. The algebra ,\", \"index\":(...TRUNCATED) | "[{\"bbox\": [68, 67, 219, 92], \"content\": \"3.2. The algebra ,\", \"parent_index\": 0, \"line_in(...TRUNCATED) |
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| "[{\"bbox\": [161, 72, 183, 86], \"content\": \"B_{r}^{(1)}\", \"parent_index\": 0, \"subtype\": \"i(...TRUNCATED) |
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