Loading…
Moonshine
Monstrous moonshine relates distinguished modular functions to the representation theory of the Monster . The celebrated observations that 1 = 1 , 196884 = 1 + 196883 , 21493760 = 1 + 196883 + 21296876 , … … ( * ) illustrate the case of J ( τ )= j ( τ )−744, whose coefficients turn out to be sums of...
Saved in:
| Published in: | Research in the mathematical sciences 2015-12, Vol.2 (1), Article 11 |
|---|---|
| Main Authors: | , , |
| Format: | Article |
| Language: | English |
| Subjects: | |
| Citations: | Items that this one cites Items that cite this one |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| cited_by | cdi_FETCH-LOGICAL-c288t-2bb13bec1873953ffa4bc4b3ccbd494a9121a8c1a1bc68a155fcc10a4d473df73 |
|---|---|
| cites | cdi_FETCH-LOGICAL-c288t-2bb13bec1873953ffa4bc4b3ccbd494a9121a8c1a1bc68a155fcc10a4d473df73 |
| container_end_page | |
| container_issue | 1 |
| container_start_page | |
| container_title | Research in the mathematical sciences |
| container_volume | 2 |
| creator | Duncan, John FR Griffin, Michael J Ono, Ken |
| description | Monstrous moonshine
relates distinguished modular functions to the representation theory of the Monster
. The celebrated observations that
1
=
1
,
196884
=
1
+
196883
,
21493760
=
1
+
196883
+
21296876
,
…
…
(
*
)
illustrate the case of
J
(
τ
)=
j
(
τ
)−744, whose coefficients turn out to be sums of the dimensions of the 194 irreducible representations of
. Such formulas are dictated by the structure of the graded monstrous moonshine modules. Recent works in moonshine suggest deep relations between number theory and physics. Number theoretic Kloosterman sums have reappeared in quantum gravity, and mock modular forms have emerged as candidates for the computation of black hole degeneracies. This paper is a survey of past and present research on moonshine. We also compute the quantum dimensions of the monster orbifold and obtain exact formulas for the multiplicities of the irreducible components of the moonshine modules. These formulas imply that such multiplicities are asymptotically proportional to dimensions. For example, the proportion of 1’s in (*) tends to
dim
(
χ
1
)
∑
i
=
1
194
dim
(
χ
i
)
=
1
5844076785304502808013602136
=
1.711
…
×
1
0
−
28
.
2010 Mathematics Subject Classification:
11F11; 11 F22; 11F37; 11F50; 20C34; 20C35 |
| doi_str_mv | 10.1186/s40687-015-0029-6 |
| format | article |
| fullrecord | <record><control><sourceid>crossref_sprin</sourceid><recordid>TN_cdi_crossref_primary_10_1186_s40687_015_0029_6</recordid><sourceformat>XML</sourceformat><sourcesystem>PC</sourcesystem><sourcerecordid>10_1186_s40687_015_0029_6</sourcerecordid><originalsourceid>FETCH-LOGICAL-c288t-2bb13bec1873953ffa4bc4b3ccbd494a9121a8c1a1bc68a155fcc10a4d473df73</originalsourceid><addsrcrecordid>eNp9j7tOxDAQRS0EEtGyH8BHGGbsiR8lWvGSFtFAbdkTG3YFCbKh4O_JKhRUVHOLOVf3CHGOcIHozGUjMM5KwF4CKC_NkegUeiu9I3v8J5-KdWt7AEBrNGnoRPcwTWN73Y35TJyU-Nby-veuxPPN9dPmTm4fb-83V1vJyrlPqVJCnTKjs9r3upRIiSlp5jSQp-hRYXSMERMbF7HvCzNCpIGsHorVK4FLL9eptZpL-Ki791i_A0I46IRFJ8w64aATzMyohWnz7_iSa9hPX3WcZ_4D_QBxvUvB</addsrcrecordid><sourcetype>Aggregation Database</sourcetype><iscdi>true</iscdi><recordtype>article</recordtype></control><display><type>article</type><title>Moonshine</title><source>Springer Nature</source><creator>Duncan, John FR ; Griffin, Michael J ; Ono, Ken</creator><creatorcontrib>Duncan, John FR ; Griffin, Michael J ; Ono, Ken</creatorcontrib><description>Monstrous moonshine
relates distinguished modular functions to the representation theory of the Monster
. The celebrated observations that
1
=
1
,
196884
=
1
+
196883
,
21493760
=
1
+
196883
+
21296876
,
…
…
(
*
)
illustrate the case of
J
(
τ
)=
j
(
τ
)−744, whose coefficients turn out to be sums of the dimensions of the 194 irreducible representations of
. Such formulas are dictated by the structure of the graded monstrous moonshine modules. Recent works in moonshine suggest deep relations between number theory and physics. Number theoretic Kloosterman sums have reappeared in quantum gravity, and mock modular forms have emerged as candidates for the computation of black hole degeneracies. This paper is a survey of past and present research on moonshine. We also compute the quantum dimensions of the monster orbifold and obtain exact formulas for the multiplicities of the irreducible components of the moonshine modules. These formulas imply that such multiplicities are asymptotically proportional to dimensions. For example, the proportion of 1’s in (*) tends to
dim
(
χ
1
)
∑
i
=
1
194
dim
(
χ
i
)
=
1
5844076785304502808013602136
=
1.711
…
×
1
0
−
28
.
2010 Mathematics Subject Classification:
11F11; 11 F22; 11F37; 11F50; 20C34; 20C35</description><identifier>ISSN: 2197-9847</identifier><identifier>EISSN: 2197-9847</identifier><identifier>DOI: 10.1186/s40687-015-0029-6</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Applications of Mathematics ; Computational Mathematics and Numerical Analysis ; Mathematics ; Mathematics and Statistics ; Review</subject><ispartof>Research in the mathematical sciences, 2015-12, Vol.2 (1), Article 11</ispartof><rights>Duncan et al. 2015. This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c288t-2bb13bec1873953ffa4bc4b3ccbd494a9121a8c1a1bc68a155fcc10a4d473df73</citedby><cites>FETCH-LOGICAL-c288t-2bb13bec1873953ffa4bc4b3ccbd494a9121a8c1a1bc68a155fcc10a4d473df73</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27924,27925</link.rule.ids></links><search><creatorcontrib>Duncan, John FR</creatorcontrib><creatorcontrib>Griffin, Michael J</creatorcontrib><creatorcontrib>Ono, Ken</creatorcontrib><title>Moonshine</title><title>Research in the mathematical sciences</title><addtitle>Mathematical Sciences</addtitle><description>Monstrous moonshine
relates distinguished modular functions to the representation theory of the Monster
. The celebrated observations that
1
=
1
,
196884
=
1
+
196883
,
21493760
=
1
+
196883
+
21296876
,
…
…
(
*
)
illustrate the case of
J
(
τ
)=
j
(
τ
)−744, whose coefficients turn out to be sums of the dimensions of the 194 irreducible representations of
. Such formulas are dictated by the structure of the graded monstrous moonshine modules. Recent works in moonshine suggest deep relations between number theory and physics. Number theoretic Kloosterman sums have reappeared in quantum gravity, and mock modular forms have emerged as candidates for the computation of black hole degeneracies. This paper is a survey of past and present research on moonshine. We also compute the quantum dimensions of the monster orbifold and obtain exact formulas for the multiplicities of the irreducible components of the moonshine modules. These formulas imply that such multiplicities are asymptotically proportional to dimensions. For example, the proportion of 1’s in (*) tends to
dim
(
χ
1
)
∑
i
=
1
194
dim
(
χ
i
)
=
1
5844076785304502808013602136
=
1.711
…
×
1
0
−
28
.
2010 Mathematics Subject Classification:
11F11; 11 F22; 11F37; 11F50; 20C34; 20C35</description><subject>Applications of Mathematics</subject><subject>Computational Mathematics and Numerical Analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Review</subject><issn>2197-9847</issn><issn>2197-9847</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2015</creationdate><recordtype>article</recordtype><recordid>eNp9j7tOxDAQRS0EEtGyH8BHGGbsiR8lWvGSFtFAbdkTG3YFCbKh4O_JKhRUVHOLOVf3CHGOcIHozGUjMM5KwF4CKC_NkegUeiu9I3v8J5-KdWt7AEBrNGnoRPcwTWN73Y35TJyU-Nby-veuxPPN9dPmTm4fb-83V1vJyrlPqVJCnTKjs9r3upRIiSlp5jSQp-hRYXSMERMbF7HvCzNCpIGsHorVK4FLL9eptZpL-Ki791i_A0I46IRFJ8w64aATzMyohWnz7_iSa9hPX3WcZ_4D_QBxvUvB</recordid><startdate>20151201</startdate><enddate>20151201</enddate><creator>Duncan, John FR</creator><creator>Griffin, Michael J</creator><creator>Ono, Ken</creator><general>Springer International Publishing</general><scope>C6C</scope><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20151201</creationdate><title>Moonshine</title><author>Duncan, John FR ; Griffin, Michael J ; Ono, Ken</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c288t-2bb13bec1873953ffa4bc4b3ccbd494a9121a8c1a1bc68a155fcc10a4d473df73</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2015</creationdate><topic>Applications of Mathematics</topic><topic>Computational Mathematics and Numerical Analysis</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Review</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Duncan, John FR</creatorcontrib><creatorcontrib>Griffin, Michael J</creatorcontrib><creatorcontrib>Ono, Ken</creatorcontrib><collection>Springer Nature OA Free Journals</collection><collection>CrossRef</collection><jtitle>Research in the mathematical sciences</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Duncan, John FR</au><au>Griffin, Michael J</au><au>Ono, Ken</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Moonshine</atitle><jtitle>Research in the mathematical sciences</jtitle><stitle>Mathematical Sciences</stitle><date>2015-12-01</date><risdate>2015</risdate><volume>2</volume><issue>1</issue><artnum>11</artnum><issn>2197-9847</issn><eissn>2197-9847</eissn><abstract>Monstrous moonshine
relates distinguished modular functions to the representation theory of the Monster
. The celebrated observations that
1
=
1
,
196884
=
1
+
196883
,
21493760
=
1
+
196883
+
21296876
,
…
…
(
*
)
illustrate the case of
J
(
τ
)=
j
(
τ
)−744, whose coefficients turn out to be sums of the dimensions of the 194 irreducible representations of
. Such formulas are dictated by the structure of the graded monstrous moonshine modules. Recent works in moonshine suggest deep relations between number theory and physics. Number theoretic Kloosterman sums have reappeared in quantum gravity, and mock modular forms have emerged as candidates for the computation of black hole degeneracies. This paper is a survey of past and present research on moonshine. We also compute the quantum dimensions of the monster orbifold and obtain exact formulas for the multiplicities of the irreducible components of the moonshine modules. These formulas imply that such multiplicities are asymptotically proportional to dimensions. For example, the proportion of 1’s in (*) tends to
dim
(
χ
1
)
∑
i
=
1
194
dim
(
χ
i
)
=
1
5844076785304502808013602136
=
1.711
…
×
1
0
−
28
.
2010 Mathematics Subject Classification:
11F11; 11 F22; 11F37; 11F50; 20C34; 20C35</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1186/s40687-015-0029-6</doi><oa>free_for_read</oa></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 2197-9847 |
| ispartof | Research in the mathematical sciences, 2015-12, Vol.2 (1), Article 11 |
| issn | 2197-9847 2197-9847 |
| language | eng |
| recordid | cdi_crossref_primary_10_1186_s40687_015_0029_6 |
| source | Springer Nature |
| subjects | Applications of Mathematics Computational Mathematics and Numerical Analysis Mathematics Mathematics and Statistics Review |
| title | Moonshine |
| url | http://sfxeu10.hosted.exlibrisgroup.com/loughborough?ctx_ver=Z39.88-2004&ctx_enc=info:ofi/enc:UTF-8&ctx_tim=2025-01-07T15%3A37%3A04IST&url_ver=Z39.88-2004&url_ctx_fmt=infofi/fmt:kev:mtx:ctx&rfr_id=info:sid/primo.exlibrisgroup.com:primo3-Article-crossref_sprin&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.genre=article&rft.atitle=Moonshine&rft.jtitle=Research%20in%20the%20mathematical%20sciences&rft.au=Duncan,%20John%20FR&rft.date=2015-12-01&rft.volume=2&rft.issue=1&rft.artnum=11&rft.issn=2197-9847&rft.eissn=2197-9847&rft_id=info:doi/10.1186/s40687-015-0029-6&rft_dat=%3Ccrossref_sprin%3E10_1186_s40687_015_0029_6%3C/crossref_sprin%3E%3Cgrp_id%3Ecdi_FETCH-LOGICAL-c288t-2bb13bec1873953ffa4bc4b3ccbd494a9121a8c1a1bc68a155fcc10a4d473df73%3C/grp_id%3E%3Coa%3E%3C/oa%3E%3Curl%3E%3C/url%3E&rft_id=info:oai/&rft_id=info:pmid/&rfr_iscdi=true |