p****o
发帖数: 13 | |
w****1
发帖数: 4931 | 2 what do you want to talk about?
【在 p****o 的大作中提到】
: 看这里人气不错，问一句，哈哈。
|
p****o
发帖数: 13 | 3 Let's see. How about some simple stuff, like a galois deformation proof of
abelian reciprocity? I wonder where I can find the complete argument
involving iwasawa algebra etc. |
b*******i
发帖数: 548 | 4 中国人里
做Langlands的只有江迪化一个
做几何Langlands的只有两个 博士快毕业了
Mirrow Symmetry是上个世纪末的事情了
现在不知道还有谁做 |
b*******i
发帖数: 548 | 5 Galois deformation没办法证明互反律
实际情况都是反过来用class field theory计算universal deformation ring of
Galois representation
Iwasawa理论最好看Iwasawa本人的文章
比所有后来人做的都干净漂亮
【在 p****o 的大作中提到】
: Let's see. How about some simple stuff, like a galois deformation proof of
: abelian reciprocity? I wonder where I can find the complete argument
: involving iwasawa algebra etc.
|
e*******y
发帖数: 73 | 6 geometric L 是不是和数论没有什么关系啊 |
w****1
发帖数: 4931 | 7 don't you know that geometric Langlands IS mirror symmetry? :D
Seriously though, can someone give an in-depth explanation of why geometric
Langlands is important?
【在 b*******i 的大作中提到】
: 中国人里
: 做Langlands的只有江迪化一个
: 做几何Langlands的只有两个 博士快毕业了
: Mirrow Symmetry是上个世纪末的事情了
: 现在不知道还有谁做
|
r**a
发帖数: 536 | 8
Do you mean witten's paper? That mirror symmetry is different with the "
normal" mirror symmetry. Normal mirror means for the CY3 compactification of
typeII strings there will be a pair of CY3s corresponding to the same low
energy physics. And mathematicians want to know the rigorous statements and
proof about it. It will relate with Gopakumar-Vafa, Donaldson-thomas, blabla
...
"Mirror symmetry" in GL is different. This mirror borrowed the initial
physical meaning of mirror symmetry. But the co
【在 w****1 的大作中提到】
: don't you know that geometric Langlands IS mirror symmetry? :D
: Seriously though, can someone give an in-depth explanation of why geometric
: Langlands is important?
|
w****1
发帖数: 4931 | 9 It is not different. Mirror symmetry is a statement about 1+1 dimensional
superconformal field theories. The geometric version has to do with SCFTs
arising from sigma models on Calabi-Yau manifolds. The Calabi-Yau manifold
in the GL context is the Hitchin moduli space, which is hyperkahler and is a
special case of the more general story. N=4 SYM motivates this story but is
not a necessarily ingredient.
of
and
blabla
different.
【在 r**a 的大作中提到】
:
: Do you mean witten's paper? That mirror symmetry is different with the "
: normal" mirror symmetry. Normal mirror means for the CY3 compactification of
: typeII strings there will be a pair of CY3s corresponding to the same low
: energy physics. And mathematicians want to know the rigorous statements and
: proof about it. It will relate with Gopakumar-Vafa, Donaldson-thomas, blabla
: ...
: "Mirror symmetry" in GL is different. This mirror borrowed the initial
: physical meaning of mirror symmetry. But the co
|
r**a
发帖数: 536 | 10 可能我没有表达清楚。我和你的侧重点不同。正如你所说Mirror in GL就是借用了最一
开始Mirror symmetry 在SCFT中的意思。但是"normal” mirror symmetry有很多方面
侧重于spacetime的意义，而不是worldsheet上面的。"normal” mirror symmetry对于
理解弦论如何能得到低能物理起着很重要的作用。他解释了typeIIA and typeII B之间
的关系。这也就是为啥向gromov-witten, donaldson-thomas, gupakumar-vafa都能联
系在一起的原因。我的意思是说从spacetime的这个角度来看，normal mirror 和
mirror in GL是很不一样的。我个人感觉，如果说某个人是做mirror symmetry的话，
似乎意味着此人的工作应着重于gromov-witten, donaldson-thomas, gupakumar-vafa
等等，而不是说GL or N=4 SYM。至少这句话在物理圈子里面适用。
你的话更强调的是从worldsheet角度理解。另
【在 w****1 的大作中提到】
: It is not different. Mirror symmetry is a statement about 1+1 dimensional
: superconformal field theories. The geometric version has to do with SCFTs
: arising from sigma models on Calabi-Yau manifolds. The Calabi-Yau manifold
: in the GL context is the Hitchin moduli space, which is hyperkahler and is a
: special case of the more general story. N=4 SYM motivates this story but is
: not a necessarily ingredient.
:
: of
: and
: blabla
|
|
|
r**a
发帖数: 536 | 11 另外，hitchin moduli space上面的那个mirror是不是已经被证实是对的了？我忘了是
否有这个结论了。印象里这里的mirror好像是两种hitchin fibration之间的一个对应
，具体的时间太长了想不起来了。
a
is
【在 w****1 的大作中提到】
: It is not different. Mirror symmetry is a statement about 1+1 dimensional
: superconformal field theories. The geometric version has to do with SCFTs
: arising from sigma models on Calabi-Yau manifolds. The Calabi-Yau manifold
: in the GL context is the Hitchin moduli space, which is hyperkahler and is a
: special case of the more general story. N=4 SYM motivates this story but is
: not a necessarily ingredient.
:
: of
: and
: blabla
|
w****1
发帖数: 4931 | 12 The mirror symmetry of GL is what you call "normal" mirror symmetry. The
mirror symmetry acts on the Hitchin moduli space as fiber-wise T-duality, as
described by Strominger-Yau-Zaslow years ago. Both physicists and
mathematicians are interested in the mapping between branes on the two sides
of the mirror. A special feature of the Hitchin moduli space is that there
are a lot of singular fibers, which must be carefully treated. I'm no expert
on this -- perhaps someone else can elaborate. Another
【在 r**a 的大作中提到】
: 可能我没有表达清楚。我和你的侧重点不同。正如你所说Mirror in GL就是借用了最一
: 开始Mirror symmetry 在SCFT中的意思。但是"normal” mirror symmetry有很多方面
: 侧重于spacetime的意义，而不是worldsheet上面的。"normal” mirror symmetry对于
: 理解弦论如何能得到低能物理起着很重要的作用。他解释了typeIIA and typeII B之间
: 的关系。这也就是为啥向gromov-witten, donaldson-thomas, gupakumar-vafa都能联
: 系在一起的原因。我的意思是说从spacetime的这个角度来看，normal mirror 和
: mirror in GL是很不一样的。我个人感觉，如果说某个人是做mirror symmetry的话，
: 似乎意味着此人的工作应着重于gromov-witten, donaldson-thomas, gupakumar-vafa
: 等等，而不是说GL or N=4 SYM。至少这句话在物理圈子里面适用。
: 你的话更强调的是从worldsheet角度理解。另
|
r**a
发帖数: 536 | 13
as
这个我已经说过，SYZ的东西可以被称为mirror的定义了。但是不是每一种mirror
symmetry都能像normal mirror那样对物理产生那么大的影响的。我之所以说mirror in
GL和normal mirror不同，就是因为对于物理的影响不同，所处的地位也不同。还是那
句话，同与不同，要看你站在哪个角度说话。如果你站在SYZ上，所有的mirror都是一
样的。不就是t对偶嘛。
Both physicists and
sides
A special feature of the Hitchin moduli space is that there
expert
which
ordinary
【在 w****1 的大作中提到】
: The mirror symmetry of GL is what you call "normal" mirror symmetry. The
: mirror symmetry acts on the Hitchin moduli space as fiber-wise T-duality, as
: described by Strominger-Yau-Zaslow years ago. Both physicists and
: mathematicians are interested in the mapping between branes on the two sides
: of the mirror. A special feature of the Hitchin moduli space is that there
: are a lot of singular fibers, which must be carefully treated. I'm no expert
: on this -- perhaps someone else can elaborate. Another
|
w****1
发帖数: 4931 | 14 As I said, there are two new features of the mirror symmetry for GL that we
are not used to, namely very singular fibers and cc branes, which is
presumably what's interesting about it.
There is a lot more to mirror symmetry than enumerative invariants, even for
Calabi-Yau 3-folds. We are interested in the category of branes, which has
a variety of presumably equivalent descriptions, in terms of bounded derived
category of coherent sheaves, in terms of special Lagrangians, in terms of
matrix fact
【在 r**a 的大作中提到】
:
: as
: 这个我已经说过，SYZ的东西可以被称为mirror的定义了。但是不是每一种mirror
: symmetry都能像normal mirror那样对物理产生那么大的影响的。我之所以说mirror in
: GL和normal mirror不同，就是因为对于物理的影响不同，所处的地位也不同。还是那
: 句话，同与不同，要看你站在哪个角度说话。如果你站在SYZ上，所有的mirror都是一
: 样的。不就是t对偶嘛。
: Both physicists and
: sides
: A special feature of the Hitchin moduli space is that there
|
w****1
发帖数: 4931 | 15
That can't be true. I have met two already.:)
【在 b*******i 的大作中提到】
: 中国人里
: 做Langlands的只有江迪化一个
: 做几何Langlands的只有两个 博士快毕业了
: Mirrow Symmetry是上个世纪末的事情了
: 现在不知道还有谁做
|
p****o
发帖数: 13 | 16 Well, firstly there are many methods to prove this 1-d case which is
kronecker-weber. And it can be proved by the similar R=T argument using
taylor-wiles patching. But I dont know the details as I just heard it from
some experts.
I wonder if it will look like the usual argument in class field theory
starting from the local case and do some group cohomologies.
【在 b*******i 的大作中提到】
: Galois deformation没办法证明互反律
: 实际情况都是反过来用class field theory计算universal deformation ring of
: Galois representation
: Iwasawa理论最好看Iwasawa本人的文章
: 比所有后来人做的都干净漂亮
|
