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Oct
17
comment Is ghost-number a physical reality/observable?
... maps from that space to a certain classifying space. The point now is that this concept makes sense more generally than just for topological spaces. For instance also morphisms between cochain complexes (graded vector spaces equipped with a nilpotent linear endomorphism d of degree + 1) have a notion of gauge transformations between them, called "cochain homotopies" in this context. Therefore there is also a notion of cohomology on these. Indeed the ordinary definition of the degree-n cohomology of a cochain complex (ker d / im d) is equivalently the space of cochain homomorphisms from...
Oct
17
comment Is ghost-number a physical reality/observable?
Cohomology is a concept that generally applies as soon as we are in a homotopical situation, meaning as soon as there is a notion of gauge transformations, gauge-of-gauge-transformations, etc. The notion of cohomology on a topological space rests on the fact that continuous maps between topological spaces have "gauge transformations" between them -- called homotopies in this context -- and gauge-of-gauge transformations -- called homotopies of homotopies, etc. For instance the degree-n integral cohomology of a topological space is the set of homotopy-classes (gauge equivalence classes) of...
Oct
17
comment Physical interpretation to the category of CFTs
Generally, if you pass from just the category of VOAs to that of full CFTs, you see more exmaples of the direct physical role of homomorphisms. For instance morphisms from a trivial (n+1)-dimensional theory to a nontrivial one define "twisted" n-dimensional theories, as in arxiv.org/abs/1108.0189 . In a similar fashion the very definition of rational 2d CFT is holographically given, as discussed in arxiv.org/abs/hep-th/0612306 .
Oct
17
comment Physical interpretation to the category of CFTs
Moreover, without the homomorphisms defined, no universal construction of the objects exists. For instance the adjoint construction of open-closed TCFTs from open (T)CFTs in arxiv.org/abs/math/0412149 rests on the definition of morphisms between (T)CFTs.
Oct
17
comment Physical interpretation to the category of CFTs
Any kind of object only has an interpretation with a given definition of homomorphism. The very fact that the definition of a VOA has something to do with CFT rests in the morphisms. For instance, if we redefine the morphisms between VOAs to be linear maps of the underlying vector spaces, then VOAs would become equivalent to just vector spaces, and all the CFT structure were lost. Or, even more drastically, if you declared that there is precisely one morphism from any VOA to any other, then that would make the theory of VOAs become equivalent to the theory of the contractible space.
Oct
16
comment Is ghost-number a physical reality/observable?
I can walk you through it. What's your first question?
Oct
15
comment Chern-Simons theory
quick reply on the last bit: when r=0 then the tensor product or r representations is -- essentially by definition of tensor product -- the trivial 1-dimensional representation, because that is the tensor unit in the category of representations. Since every element in the trivial representation is invariant, the passage to G-invariants does not change this statement, and hence for r = 0 that formula yields the 1-dimensional vector space.
Oct
15
answered Physical interpretation to the category of CFTs
Oct
14
comment Geometric Langlands as a partially defined topological field theory
Concerning to what extent this applies to KW theory: as I tried to indicate, at least we know that it has compactifications to 2d that in certain parts of the parameter space reproduce the A-model and the B-model. For these 2d TCFTs we know excactly what's going on (via Lurie's section 4.2). Since these are simple special cases induced from KW theory, it seems to follow that KW theory is "at least as non-fully defined" as this. Not sure if this helps, but this is the statement that I can see so far.
Oct
14
answered Geometric Langlands as a partially defined topological field theory
Oct
14
answered which letter to use for a CFT?
Oct
13
comment Sympletic structure of General Relativity
Thanks, Igor, for taking the time to look at the article. You could or should maybe collect these comments and re-post them as an answer to the question.
Oct
13
comment Is ghost-number a physical reality/observable?
For me, coming from the other end, it is curious to see where the jargon is located here. Whether and which jargon is "necessary" may depend on what one wants to achieve. I can imagine students who take the statement of the above form "...you should never take them too seriously..." as a satisfactory explanation for what's going on. And maybe even most of the students reading here. But I am hoping once in a while a student comes by who looks for more genuine understanding of what's going on. But of course its good to offer both versions.
Oct
12
answered Is ghost-number a physical reality/observable?
Oct
12
answered Models of higher Chern-Simons type
Oct
12
answered Sympletic structure of General Relativity
Oct
11
comment Is string theory local?
@KellyDavis: Pavel is quite right, cobordism representations do not just serve to axiomatize TQFT, but also QFT with metric structure, notably CFT, as first emphasized by Graeme Segal. This is quite standard by now. References are here: ncatlab.org/nlab/show/conformal+field+theory#FQFTReferences . The morphism associated by such a metric QFT to a cobordism is indeed effectively the S-matrix. The question whether string field theory falls into this pattern is perfectly reasonable (even though the answer might be: no, it does not).
Oct
10
comment Quantum Field Theory from a mathematical point of view
Thanks. I have added a comment to the bottom of my answer above in an attempt to clarify this. I think it is important to realize that much of physics, even the most established theories, is "vague and speculative" from the point of view of actual mathematics, of actual precision of argument and certainty of truth. This is not at all to say that this physics is bad. But realizing this gap to the non-vagueness and non-speculation of maths is the necessary first step for appreciating what it means -- or would mean -- to genuinely have "QFT from a mathematical point of view".
Oct
10
comment Quantum Field Theory from a mathematical point of view
I didn't mean to be polemical at all. Where do you sense polemics?
Oct
9
answered Quantum Field Theory from a mathematical point of view