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The '''Nahm equations''' are a system of [[ordinary differential equation]]s introduced by [[Werner Nahm]] in the context of the [[Nahm transform]] – an alternative to [[Richard S. Ward|Ward]]'s [[twistor]] construction of [[monopole (mathematics)|monopoles]]. The Nahm equations are formally analogous to the algebraic equations in the [[ADHM construction]] of [[instanton]]s, where finite order matrices are replaced by differential operators. | |||
Deep study of the Nahm equations was carried out by [[Nigel Hitchin]] and [[Simon Donaldson]]. Conceptually, the equations arise in the process of infinite-dimensional [[hyperkähler reduction]]. Among their many applications we can mention: Hitchin's construction of monopoles, where this approach is critical for establishing nonsingularity of monopole solutions; Donaldson's description of the [[moduli space]] of monopoles; and the existence of [[hyperkähler manifold|hyperkähler structure]] on [[coadjoint orbit]]s of complex semisimple Lie groups, proved by [[Peter Kronheimer]], Olivier Biquard, and A.G. Kovalev. | |||
== | == Equations == | ||
Let ''T''<sub>1</sub>(''z''),''T''<sub>2</sub>(''z''), ''T''<sub>3</sub>(''z'') be three matrix-valued meromorphic functions of a complex variable ''z''. The Nahm equations are a system of matrix differential equations | |||
== | :<math> | ||
\begin{align} | |||
\frac{dT_1}{dz}&=[T_2,T_3]\\[3pt] | |||
\frac{dT_2}{dz}&=[T_3,T_1]\\[3pt] | |||
\frac{dT_3}{dz}&=[T_1,T_2], | |||
\end{align} | |||
</math> | |||
together with certain analyticity properties, reality conditions, and boundary conditions. The three equations can be written concisely using the [[Levi-Civita symbol]], in the form | |||
== | :<math>\frac{dT_i}{dz}=\frac{1}{2}\sum_{j,k}\epsilon_{ijk}[T_j,T_k]=\sum_{j,k}\epsilon_{ijk}T_j T_k. </math> | ||
More generally, instead of considering ''N'' by ''N'' matrices, one can consider Nahm's equations with values in a Lie algebra '''g'''. | |||
== | === Additional conditions === | ||
The variable ''z'' is restricted to the open interval (0,2), and the following conditions are imposed: | |||
# <math>T^*_i = -T_i;</math> | |||
# <math>T_i(2-z)=T_i(z)^{T};\,</math> | |||
# ''T''<sub>''i''</sub> can be continued to a meromorphic function of ''z'' in a neighborhood of the closed interval [0,2], analytic outside of 0 and 2, and with simple poles at ''z'' = 0 and ''z'' = 2; and | |||
# At the poles, the residues of (''T''<sub>1</sub>,''T''<sub>2</sub>, ''T''<sub>3</sub>) form an irreducible representation of the group [[SU(2)]]. | |||
== Nahm–Hitchin description of monopoles == | |||
There is a natural equivalence between | |||
# the monopoles of charge ''k'' for the group SU(2), modulo gauge transformations, and | |||
# the solutions of Nahm equations satisfying the additional conditions above, modulo the simultaneous conjugation of ''T''<sub>1</sub>,''T''<sub>2</sub>, ''T''<sub>3</sub> by the group O(k,'''R'''). | |||
== Lax representation == | |||
The Nahm equations can be written in the [[Lax form]] as follows. Set | |||
:<math> | |||
\begin{align} | |||
& A_0=T_1+iT_2, \quad A_1=-2i T_3, \quad A_2=T_1-iT_2 \\[3 pt] | |||
& A(\zeta)=A_0+\zeta A_1+\zeta^2 A_2, \quad B(\zeta)=\frac{1}{2}\frac{dA}{d\zeta}=\frac{1}{2}A_1+\zeta A_2, | |||
\end{align} | |||
</math> | |||
then the system of Nahm equations is equivalent to the Lax equation | |||
:<math> \frac{dA}{dz}=[A,B]. </math> | |||
As an immediate corollary, we obtain that the spectrum of the matrix ''A'' does not depend on ''z''. Therefore, the characteristic equation | |||
:<math> \det(\lambda I+A(\zeta,z))=0, </math> | |||
which determines the so-called '''spectral curve''' in the twistor space ''TP''<sup>1</sup>, is invariant under the flow in ''z''. | |||
== See also == | |||
*[[Bogomolny equation]] | |||
*[[Yang–Mills–Higgs equations]] | |||
== | == References == | ||
*{{cite paper |last=Nahm |first=W. |title=All self-dual multimonopoles for arbitrary gauge groups |work=CERN, preprint TH. 3172 |year=1981 |url=http://cdsweb.cern.ch/record/131817 }} | |||
*{{cite journal |authorlink=Nigel Hitchin |first=Nigel |last=Hitchin |title=On the construction of monopoles |journal=Communications in Mathematical Physics |volume=89 |issue=2 |year=1983 |pages=145–190 |doi=10.1007/BF01211826 }} | |||
*{{cite journal |authorlink=Simon Donaldson |first=Simon |last=Donaldson |title=Nahm's equations and the classification of monopoles |journal=Communications in Mathematical Physics |volume=96 |issue=3 |year=1984 |pages=387–407 |doi=10.1007/BF01214583 }} | |||
*{{cite book |authorlink=Michael Atiyah |first=Michael |last=Atiyah |last2=Hitchin |first2=N. J. |title=The geometry and dynamics of magnetic monopoles |series=M. B. Porter Lectures |publisher=Princeton University Press |location=Princeton, NJ |year=1988 |isbn=0-691-08480-7 }} | |||
*{{cite journal |last=Kovalev |first=A. G. |title=Nahm's equations and complex adjoint orbits |journal=[[Quarterly Journal of Mathematics|Quart. J. Math. Oxford]] |volume=47 |year=1996 |issue=185 |pages=41–58 |doi=10.1093/qmath/47.1.41 }} | |||
*{{cite journal |last=Biquard |first=Olivier |title={{lang|fr|Sur les équations de Nahm et la structure de Poisson des algèbres de Lie semi-simples complexes}} |trans_title=Nahm equations and Poisson structure of complex semisimple Lie algebras |journal=[[Mathematische Annalen|Math. Ann.]] |volume=304 |year=1996 |issue=2 |pages=253–276 |doi=10.1007/BF01446293 }} | |||
== External links == | |||
*[http://www.maths.tcd.ie/~islands/index.php?title=Main_Page Islands project] – a wiki about the Nahm equations and related topics | |||
[[Category:Differential equations]] | |||
[[Category:Mathematical physics]] | |||
[[Category:Integrable systems]] | |||
[[Category:Equations of physics]] | |||
Revision as of 19:22, 16 January 2014
The Nahm equations are a system of ordinary differential equations introduced by Werner Nahm in the context of the Nahm transform – an alternative to Ward's twistor construction of monopoles. The Nahm equations are formally analogous to the algebraic equations in the ADHM construction of instantons, where finite order matrices are replaced by differential operators.
Deep study of the Nahm equations was carried out by Nigel Hitchin and Simon Donaldson. Conceptually, the equations arise in the process of infinite-dimensional hyperkähler reduction. Among their many applications we can mention: Hitchin's construction of monopoles, where this approach is critical for establishing nonsingularity of monopole solutions; Donaldson's description of the moduli space of monopoles; and the existence of hyperkähler structure on coadjoint orbits of complex semisimple Lie groups, proved by Peter Kronheimer, Olivier Biquard, and A.G. Kovalev.
Equations
Let T1(z),T2(z), T3(z) be three matrix-valued meromorphic functions of a complex variable z. The Nahm equations are a system of matrix differential equations
![{\displaystyle {\begin{aligned}{\frac {dT_{1}}{dz}}&=[T_{2},T_{3}]\\[3pt]{\frac {dT_{2}}{dz}}&=[T_{3},T_{1}]\\[3pt]{\frac {dT_{3}}{dz}}&=[T_{1},T_{2}],\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/38e366d0117b19f42738ef776613da9bdc6276b3)
together with certain analyticity properties, reality conditions, and boundary conditions. The three equations can be written concisely using the Levi-Civita symbol, in the form
![{\displaystyle {\frac {dT_{i}}{dz}}={\frac {1}{2}}\sum _{j,k}\epsilon _{ijk}[T_{j},T_{k}]=\sum _{j,k}\epsilon _{ijk}T_{j}T_{k}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/92c5841757b9f2c36b03f4c2d8563dd6710da3e1)
More generally, instead of considering N by N matrices, one can consider Nahm's equations with values in a Lie algebra g.
Additional conditions
The variable z is restricted to the open interval (0,2), and the following conditions are imposed:
- Ti can be continued to a meromorphic function of z in a neighborhood of the closed interval [0,2], analytic outside of 0 and 2, and with simple poles at z = 0 and z = 2; and
- At the poles, the residues of (T1,T2, T3) form an irreducible representation of the group SU(2).
Nahm–Hitchin description of monopoles
There is a natural equivalence between
- the monopoles of charge k for the group SU(2), modulo gauge transformations, and
- the solutions of Nahm equations satisfying the additional conditions above, modulo the simultaneous conjugation of T1,T2, T3 by the group O(k,R).
Lax representation
The Nahm equations can be written in the Lax form as follows. Set
![{\begin{aligned}&A_{0}=T_{1}+iT_{2},\quad A_{1}=-2iT_{3},\quad A_{2}=T_{1}-iT_{2}\\[3pt]&A(\zeta )=A_{0}+\zeta A_{1}+\zeta ^{2}A_{2},\quad B(\zeta )={\frac {1}{2}}{\frac {dA}{d\zeta }}={\frac {1}{2}}A_{1}+\zeta A_{2},\end{aligned}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4188e336b8f963a60333438b8a080d602d8cc663)
then the system of Nahm equations is equivalent to the Lax equation
![{\displaystyle {\frac {dA}{dz}}=[A,B].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6286482c241788cc501c2383fc5540c26a2ad2a4)
As an immediate corollary, we obtain that the spectrum of the matrix A does not depend on z. Therefore, the characteristic equation
which determines the so-called spectral curve in the twistor space TP1, is invariant under the flow in z.
See also
References
- Template:Cite paper
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The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
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Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang
External links
- Islands project – a wiki about the Nahm equations and related topics