|
|
| Line 1: |
Line 1: |
| In algebraic geometry, the '''Witten conjecture''' is a conjecture about intersection numbers of stable classes on the [[moduli space of curves]], introduced by {{harvs|txt|last=Witten|authorlink=Edward Witten|year=1991}}, and generalized in {{harv|Witten|1993}}.
| | Futures Trader Pete Grupe from Ridgetown, usually spends time with hobbies and interests which includes airbrushing, free email service and building. Is a travel enthusiast and these days made vacation to Archaeological Site of Sabratha.<br><br>My blog: [http://maiz.ca/pricing/ canadian email provider] |
| Witten's original conjecture was proved by {{harvtxt|Kontsevich|1992}}.
| |
| | |
| Witten's motivation for the conjecture was that two different models of 2-dimensional quantum gravity should have the same partition function. The partition function for one of these models can be described in terms of intersection numbers on the moduli stack of algebraic curves, and the partition function for the other is the logarithm of the τ-function of the [[KdV hierarchy]]. Identifying these partition functions gives Witten's conjecture that a certain generating function formed from intersection numbers should satisfy the differential equations of the KdV hierarchy.
| |
| | |
| ==Statement==
| |
| | |
| Suppose that ''M''<sub>''g'',''n''</sub> is the moduli stack of compact Riemann surfaces of genus ''g'' with ''n'' distinct marked points ''x''<sub>1</sub>,...,''x''<sub>''n''</sub>,
| |
| and {{overline|''M''}}<sub>''g'',''n''</sub> is its Deligne–Mumford compactification. There are ''n'' line bundles ''L''<sub>''i''</sub> on | |
| {{overline|''M''}}<sub>''g'',''n''</sub>, whose fiber at a point of the moduli stack is given by the cotangent space of a Riemann surface at the marked point ''x''<sub>''i''</sub>. The intersection index 〈τ<sub>''d''<sub>1</sub></sub>, ..., τ<sub>''d''<sub>''n''</sub></sub>〉 is the intersection index of Π ''c''<sub>1</sub>(''L''<sub>''i''</sub>)<sup>''d''<sub>''i''</sub></sup> on {{overline|''M''}}<sub>''g'',''n''</sub> where Σ''d''<sub>''i''</sub> = dim{{overline|''M''}}<sub>''g'',''n''</sub> = 3''g'' – 3 + ''n'', and 0 if no such ''g'' exists, where ''c''<sub>1</sub> is the first [[Chern class]] of a line bundle. Witten's generating function
| |
| :<math>F(t_0,t_1,\ldots)
| |
| = \sum\langle\tau_0^{k_0}\tau_1^{k_1}\cdots\rangle\prod_{i\ge 0} \frac{t_i^{k_i}}{k_i!}
| |
| =\frac{t_0^3}{6}+ \frac{t_1}{24} + \frac{t_0t_2}{24} + \frac{t_1^2}{24}+ \frac{t_0^2t_3}{48} + \cdots
| |
| </math>
| |
| encodes all the intersection indices as its coefficients.
| |
| | |
| Witten's conjecture states that the partition function ''Z'' = exp ''F'' is a τ-function for the [[KdV hierarchy]], in other words it satisfies a certain series of partial differential equations corresponding to elements ''L''<sub>''i''</sub> for ''i''≥–1 of the [[Virasoro algebra]].
| |
| | |
| ==Proof==
| |
| | |
| Kontsevich used a combinatorial description of the moduli spaces in terms of ribbon graphs to show that
| |
| <math>\sum_{d_1+\cdots+d_n=3g-3+n}\langle \tau_{d_1},\ldots,\tau_{d_n}\rangle \prod_{1\le i\le n} \frac{(2d_i-1)!!}{\lambda_i^{2d_i+1}}
| |
| =\sum_{\Gamma\in G_{g,n}}\frac{2^{-|X_0|}}{|\text{Aut} \Gamma|}\prod_{e\in X_1}\frac{2}{\lambda(e)}
| |
| </math>
| |
| | |
| Here the sum on the right is over the set ''G''<sub>''g'',''n''</sub> of ribbon graphs ''X'' of compact Riemann surfaces of genus ''g'' with ''n'' marked points. The set of edges ''e'' and points of ''X'' are denoted by ''X''<sub> 0</sub> and ''X''<sub>1</sub>. The function λ is thought of as a function from the marked points to the reals, and extended to edges of the ribbon graph by setting λ of an edge equal to the sum of λ at the two marked points corresponding to each side of the edge.
| |
| | |
| By Feynman diagram techniques, this implies that
| |
| ''F''(''t''<sub>0</sub>,...) is an asymptotic expansion of
| |
| :<math> \log\int \exp(i \text{tr} X^3/6)d\mu</math>
| |
| as Λ lends to infinity, where Λ and Χ are positive definite ''N'' by ''N'' hermitian matrices, and ''t''<sub>''i''</sub> is given by
| |
| :<math> t_i = \frac{- \text{tr} \Lambda^{-1-2i}}{1\times3\times5\times\cdots\times (2i-1)}</math>
| |
| and the probability measure μ on the positive definite hermitian matrices is given by
| |
| :<math> d\mu =c_\Lambda\exp(-\text{tr} X^2\Lambda/2)dX</math> | |
| where ''c''<sub>Λ</sub> is a normalizing constant. This measure has the property that
| |
| :<math>\int X_{ij}X_{kl}d\mu = \delta_{il}\delta_{jk}\frac{2}{\Lambda_i+\Lambda_j}</math>
| |
| which implies that its expansion in terms of Feynman diagrams is the expression for ''F'' in terms of ribbon graphs.
| |
| | |
| From this he deduced that exp F is a τ-function for the KdV hierarchy, thus proving Witten's conjecture.
| |
| | |
| ==See also==
| |
| | |
| The [[Virasoro conjecture]] is a generalization of the Witten conjecture.
| |
| | |
| ==References==
| |
| | |
| *{{Citation | last1=Cornalba | first1=Maurizio | last2=Arbarello | first2=Enrico | last3=Griffiths | first3=Phillip A. | title=Geometry of algebraic curves. Volume II | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] | isbn=978-3-540-42688-2 | doi=10.1007/978-3-540-69392-5 | id={{MR|2807457}} | year=2011 | volume=268}}
| |
| *{{Citation | last1=Kazarian | first1=M. E. | last2=Lando | first2=Sergei K. | title=An algebro-geometric proof of Witten's conjecture | url=http://dx.doi.org/10.1090/S0894-0347-07-00566-8 | doi=10.1090/S0894-0347-07-00566-8 | id={{MR|2328716}} | year=2007 | journal=[[Journal of the American Mathematical Society]] | issn=0894-0347 | volume=20 | issue=4 | pages=1079–1089}}
| |
| *{{Citation | last1=Kontsevich | first1=Maxim | title=Intersection theory on the moduli space of curves and the matrix Airy function | url=http://projecteuclid.org/getRecord?id=euclid.cmp/1104250524 | id={{MR|1171758}} | year=1992 | journal=Communications in Mathematical Physics | issn=0010-3616 | volume=147 | issue=1 | pages=1–23|doi=10.1007/BF02099526}}
| |
| *{{Citation | last1=Lando | first1=Sergei K. | last2=Zvonkin | first2=Alexander K. | title=Graphs on surfaces and their applications | publisher=[[Springer-Verlag]] | location=Berlin, New York | series=Encyclopaedia of Mathematical Sciences | isbn=978-3-540-00203-1 | id={{MR|2036721}} | year=2004 | volume=141 |url=http://www.springer.com/cda/content/document/cda_downloaddocument/9783540002031-c1.pdf?SGWID=0-0-45-100940-p13863104}}
| |
| *{{Citation | last1=Witten | first1=Edward | author1-link=Edward Witten | title=Surveys in differential geometry (Cambridge, MA, 1990) | publisher=Lehigh Univ. | location=Bethlehem, PA | isbn=978-0-8218-0168-0 | id={{MR|1144529}} | year=1991 | volume=1 | chapter=Two-dimensional gravity and intersection theory on moduli space | pages=243–310}}
| |
| *{{Citation | last1=Witten | first1=Edward | author1-link=Edward Witten | editor1-last=Goldberg | editor1-first=Lisa R. | editor2-last=Phillips | editor2-first=Anthony V. | title=Topological methods in modern mathematics (Stony Brook, NY, 1991) | publisher=Publish or Perish | location=Houston, TX | series=Proceedings of the symposium in honor of John Milnor's sixtieth birthday held at the State University of New York, Stony Brook, New York, June 14–21, 1991. | isbn=978-0-914098-26-3 | id={{MR|1215968}} | year=1993 | chapter=Algebraic geometry associated with matrix models of two-dimensional gravity | pages=235–269}}
| |
| | |
| {{Algebraic curves navbox}}
| |
| | |
| [[Category:Moduli theory]]
| |
| [[Category:Algebraic geometry]]
| |
Futures Trader Pete Grupe from Ridgetown, usually spends time with hobbies and interests which includes airbrushing, free email service and building. Is a travel enthusiast and these days made vacation to Archaeological Site of Sabratha.
My blog: canadian email provider