数学里面任何精巧的构造都不是accidental的作者： 孔良
It is surprising that the first one and the only one who raised this obvious question in my journal article
http://blog.renren.com/blog/bp/Q7PW8HaVkV
is professor Xiao-Gang Wen. Isn't that an interesting phenomenon? I hope that readers should ponder on this seriously.
既然文老师问了, 我就讲讲我到底从Gelfand的remark: "Kontsevich's deformation quantization is trivial."里面学到了什么? 很抱歉，原文在这里写的很不清楚，其实还有点误导大家。Of course, what I have figured out is not necessarily what Gelfand was trying to tell me.
Before I start, I would like to remind physicists something that they all know. It is generally believed by physicists that a robust phenomenon must have a simple reason (or a simple mechanics) because if the underlining mechanics is not simple, the phenomenon can not be robust. Perhaps at a very foundamental level, only stable or robust phenomena are observable. The same thing is true in mathematics.
If you have seen Kontsevich's construction of deformation quantization, in particular in the R^n case, you will be amazed by the beauty of this construction. It sets your mind free. 我想这也许是我心中一直追求的自由的数学的典范。这个以后再说。I forgot a lot of details of his construction and don't want to waste my time to look at it again. So I will just put it in a simple way. Kontsevich need to construct a sequence of number (structure constants of an algebra), each of which is a coefficient of a basis vector in a "some kind of monomial expansion" (with differential operators, well, polyvector fields if you like), such that all together they satisfy some global algebraic property (associativity). It is just a reformulation of a construction of a solution of deformation quantization. Kontsevich then tried to encode the same data of a basis vector geometrically, by assigning each basis vector a finite graph on the upper half plane with some points on the boundary and some points sitting inside. Now only thing remains is to construct a number from the graph. How to do it? How do we usually do to get a number from a space? You do integration! Kontsevich continued as follows: you can compactify the upper half plane by adding a point so that it becomes a disk. It is well-known in mathematics that there is a hypabolic metric (called Poincare metric) on the disk such that the disk becomes a hypabolic space. Then the graph becomes a graphs in this hypabolic space. We require that edges in the graph follow geodesics. To integrate, we need 1-form that can be constructed from the given angle between each pair of edges ending on the same vertex. Then we integate, we get an number! Kontsevich shows that such constructed a sequence of numbers satisfying all the consistence conditions. Isn't that amazing?
Kontsevich思想的天马行空不是我们要讨论的。我要说的是，为什么Gelfand说它是trivial的。I gradually grasped a simple message from Gelfand only years later (partially influenced by Grothendieck): 数学里面任何精巧的构造都不是accidental的. In other words, there must be a simple or trivial reason behind it. Kontsevich的构造实在是太精巧了，精巧到像是魔术. So it must be trivial. 实在抱歉我并没有在这个问题上继续深入。
In 2006, when I was working in Max-Planck-Institute for Sciences at Lepzig, 其实我当时在读黄易的武侠，somehow got inspired, so I decided to put aside my own research and just to think about this simple message that I just start to understand. It reminded me of the so-called charge-conjugate construction of closed rational CFT in C x C, where C is a modular tensor category obtained from some rational chiral algebra (or vertex operator algebra). I found it by myself a few years ago. This construction is complicated and delicate, and it used all the information of fusion matrices and dualities. I worked until 4:00AM in the morning before I could confirm that it is all correct. Then I told this to my advisor Yi-Zhi Huang the next day. He said that this construction is so natural and delicate, maybe it has already been discovered by physicists. So I checked. Yes, indeed, it was discovered before and was rediscovered by many others. How could physicists missed such a construction? So I just left it as it is. What else can you do? It is well-known to the experts.
但是几年以后，I realized that such a delicate construction must be trivial. 所以我决定放下手边的活，马上思考这个问题。One week later, I found out why it is trivial. What happens is that there is a tensor functor:
C x C --> C, defined by (x, y) --> x \otimes y
which is just the tensor product functor. This is well-known. It turns out that it has a right adjoint functor R. The famous charge-conjugate construction of modular invariant closed CFT is nothing but R(1) where 1 is the unit of C. I also found out that R is not a tensor functor, but it is Frobenius monoidal. I thought that I have invented this concept of Frobenius monoidal functor myself. 呵呵. Such functor preserves Frobenius algebras, i.e. it maps a Frobenius algebra to a Frobenius algebra. Therefore, immediately, we obtain that R(1) is a Frobenius algebra because 1 is. 虽然这些对数学真正的高手来讲都是些trivial的东西，但是我数学基础比较差，我自己发现他们的时候费了不少劲，所以这些小小进展也让我很感动。Actually, this functor R plays an important role in my later works. It simplifies a lot of things. After this, I becomes a believer. How about you? If you try that in your own research, you will like what you find.
我不知道这样讲，大家是不是可以理解。
希望你们也有收获。
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