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    • CommentRowNumber1.
    • CommentAuthordomenico_fiorenza
    • CommentTimeDec 12th 2010
    • (edited Dec 12th 2010)

    Here is where I’ll blog my redaing of Topological Quantum Field Theories from Compact Lie Groups

    day 1. quickly read the intro and gone directly to section1. main statement there is proposition 1.2, identifying H 2(BG,)H^2(B G,\mathbb{Z}) with the group of abelian characters Hom(G,U(1))Hom(G,U(1)). in nLab we have a nice understanding of this identification as follows: the set of abelian characters is the set of cocycles c:BGBU(1)c:\mathbf{B}G\to \mathbf{B}U(1), and the above identification follows by the long fibration sequence of oo-Lie groupoids

    BBBU(1)B()B 2 \mathbf{B}\mathbb{Z}\to \mathbf{B}\mathbb{R}\to \mathbf{B}U(1)\simeq\mathbf{B}(\mathbb{Z}\hookrightarrow\mathbb{R})\to \mathbf{B}^2\mathbb{Z}\to\cdots

    The description of the 010-1 tqft associated with c:BGBU(1)c:\mathbf{B}G\to \mathbf{B}U(1) is given explicitely. Apparently it can be interpreted as follows: the vector space F(pt)F(pt) associated to a point is the space of sections of the line bundle on the groupoid BG\mathbf{B}G induced by the cocycle cc (this space of sections will be 11- or 00-dimensional depending whether cc is the trivial cocycle or not). The complex number F(S 1)F(S^1) associated to S 1S^1 is the dimension of this vector space. This can be seen as the integral over [S 1,BG][S^1,\mathbf{B}G] of the morphism [S 1,BG][S 1,BU(1)]U(1)[S^1,\mathbf{B}G]\to [S^1,\mathbf{B}U(1)]\to U(1) induced by cc.

    That F(S 1)F(S^1) is related this way to the space of maps from S 1S^1 to BG\mathbf{B}G and to the cocycle cc seems to be what is really crucial here. Yet in section 1 no deep reason for this being “the right answer” is given at this level and it is only remarked that the above described interpretation of dimF(S 1)dim F(S^1) is possible.

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 13th 2010
    • (edited Dec 13th 2010)

    Thanks, for starting this, Domenico.

    The complex number F(S 1)F(S^1) associated to S 1S^1 is the dimension of this vector space. This can be seen as the integral over [S 1,BG][S^1,\mathbf{B}G] of the morphism [S 1,BG][S 1,BU(1)]U(1)[S^1,\mathbf{B}G]\to [S^1,\mathbf{B}U(1)]\to U(1) induced by cc.

    We said the following before elsewhere, but let’s recall it:

    if we think of vector spaces and linear algebra along the lines of integral transforms on sheaves, then the morphism

    [S 1,BG]exp(S())[S 1,BU(1)]U(1)[S^1,\mathbf{B}G] \stackrel{\exp(S(-))}{\to} [S^1,\mathbf{B}U(1)]\to U(1)

    is the path integral over the action, when regarded as an object in

    H/U(1), \mathbf{H}/U(1) \,,

    where H=Grpd\mathbf{H} = \infty Grpd here: we have an \infty-groupoid over U(1)U(1) \hookrightarrow \mathbb{C} and under groupoid-cardinality in \mathbb{C} this is the Haar measure-weighted path integral over exp(S())\exp(S(-)).

    Also just for the record, so that we get set up again for this discussion, this argument corresponds to theorem 6, page 16 of Simon Willerton’s article.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 13th 2010
    • (edited Dec 13th 2010)

    An analogous statement should holds for the space assigned to the point:

    The identification H/BGPSh(BG)\mathbf{H}/\mathbf{B}G \simeq PSh(\mathbf{B}G) goes via the (,1)(\infty,1)-Grothendieck construction. So up to equivalence every element in the over \infty-category is given by a fibration ψ:ΨBG\psi : \Psi \to \mathbf{B}G. This is like a flat bundle of \infty-groupoids over BG\mathbf{B}G: an \infty-groupoid fiber over the point and then autoequivalences for each gGg \in G.

    So the action functional again gives us a composite

    Ψ[pt,BG]exp(S())[pt,BU(1)]BU(1)ρVect\Psi \to [pt, \mathbf{B}G] \stackrel{\exp(S(-))}{\to} [pt, \mathbf{B}U(1)] \simeq \mathbf{B}U(1) \stackrel{\rho}{\to} Vect

    and we ought to be able to interpret this under a cardinality operation as a vector space (by passing to (co)invariants).

    So there should be this picture here: for Σ inΣΣ out\Sigma_{in} \to \Sigma \leftarrow \Sigma_{out} we have

    [Σ in,BG] [Σ,BG] [Σ out,BG] [Σ in,BU(1)] [Σ,BU(1)] [Σ out,BU(1)] [Σ in,Vect] [Σ,Vect] [Σ out,Vect] \array{ [\Sigma_{in}, \mathbf{B}G] &\leftarrow& [\Sigma, \mathbf{B}G] &\to& [\Sigma_{out}, \mathbf{B}G] \\ \downarrow && \downarrow && \downarrow \\ [\Sigma_{in}, \mathbf{B}U(1)] &\leftarrow& [\Sigma, \mathbf{B}U(1)] &\to& [\Sigma_{out}, \mathbf{B}U(1)] \\ \downarrow && \downarrow && \downarrow \\ [\Sigma_{in},Vect] &\leftarrow& [\Sigma, Vect] &\to& [\Sigma_{out}, Vect] }

    and we should be pull-pushing through this int the corresponding over \infty-toposes.

    • CommentRowNumber4.
    • CommentAuthordomenico_fiorenza
    • CommentTimeDec 13th 2010
    • (edited Dec 14th 2010)

    Hi Urs,

    thanks for joining this! :)

    We said the following before elsewhere, but let’s recall it

    yes, recalling that here is a very good idea, thanks.

    and now… day 2 :)

    I’ve skipped to Section 3 to see how the 0-1 tqft case is handled by the general theory. adapting for n=1n=1 the beginning of Section 3, the starting point is the existence of the Bun GBun_G functor Bun G:Bord 1 SOFam 1Bun_G:Bord_1^{SO}\to Fam_1, which maps a manifold MM to the groupoid of principal GG-bundles oner MM (GG is a finite group here). So this is nothing but the functor MH(M,BG)M\mapsto \mathbf{H}(M,\mathbf{B}G) or, in the internal version M[M,BG]M\mapsto [M,\mathbf{B}G].

    Next, fixing VectVect to be the symmetric monoidal category 𝒞\mathcal{C} involved, the classical theory is given by a lift I:Bord 1 SOFam 1(Vect)I: Bord_1^{SO}\to Fam_1(Vect) of Bun GBun_G.Here, objects in Fam 1(Vect)Fam_1(Vect) are objects in Fam 1Fam_1, i.e., finite groupoids, with a morphism to VectVect; morphisms are branes

    Z X Y Vect \array{ && Z&&\\ &\swarrow && \searrow&\\ X && \Rightarrow && Y\\ &\searrow && \swarrow&\\ &&Vect&& }

    In the case of a 0-1 tqft out of a cocycle c:BGBU(1)c:\mathbf{B}G\to \mathbf{B}U(1) the functor II is defined by I(pt +)={BGVect}I(pt_+)=\{\mathbf{B}G\to Vect\} where the morphism to VectVect is induced by cc and by the standard representation of U(1)U(1). This definition of II therefore relies on the cobordism hypotesis in the version given by Theorem 2.1 in the paper: since SO(1)SO(1) is the trivial group, giving I:Bord 1 SOFam 1(Vect)I:Bord_1^SO\to Fam_1(Vect) is the same thing that giving a fully dualizable object in Fam 1(Vect)Fam_1(Vect).

    This leaves it completely implicit what is II on 1-manifolds, which is not completely satisfying. I’ll come back to this later.

  1. Sill on the 0-1 tqft case. So far we have explicitly described only I(pt +)I(pt_+). The definition of I(pt +)I(pt_+) forces I(pt )I(pt_-) to be the representation of GG dual of the one associated with pt +pt_+. So, explicitely, gφ=c(g) 1φg\cdot \varphi=c(g)^{-1}\varphi, for φ:\varphi:\mathbb{C}\to \mathbb{C}. Then I([0,1])I([0,1]) as a cobordism from p +p_+ to p +p_+ is the identity of I(p +)I(p_+) as a representation of GG; similarly, I([0,1])I([0,1]) as a cobordism between p +p p_+\coprod p_- and \emptyset is the canonical pairing I(p )I(p +)I(p_-)\otimes I(p_+)\to \mathbb{C} and I([0,1])I([0,1]) as a cobordism between \emptyset and p +p p_+\coprod p_- is the copairing I(p +)I(p )\mathbb{C}\to I(p_+)\otimes I(p_-). Note that here we are using the fact that I()I(\emptyset) is the trivial 1-dimensional representation of GG, which is forced by the fact that I()I(\emptyset) has to be the unit in the symmetric monoidal category of representations of finite groupoids. Note that II maps a 1-manifold with boundary to a morphism between functors to VectVect rather than to a functor from a groupoid to VectVect. This is obvious, since a 1-manifold is a 1-morphism in Bord 1 SOBord_1^{SO} and so it has to be mapped to a 1-morphism in Fam 1(Vect)Fam_1(Vect).

    Next we come to I(S 1)I(S^1). Since II lifts Bun GBun_G, I(S 1)I(S^1), this has I(S 1)I(S^1) has to be a functor [S 1,BG]0Vect=[S^1,\mathbf{B}G]\to 0Vect=\mathbb{C}. that is I(S 1)I(S^1) has to be a class function on GG. This is is agreement with the brane interpretation of morphisms: from the cospan S 1\emptyset\to S^1\leftarrow \emptyset we get the span of groupoids *[S 1,BG]**\leftarrow [S^1,\mathbf{B}G]\to *, and branes with this roof over VectVect are naturally identified with class functions on GG. and with the given data there is no better candidate for this class function than the cocycle cc itself, seen as a function c:GU(1)c:G\to U(1)\subseteq \mathbb{C}. With this choice,

    F(S 1)=Sum 0(I(S 1))=1G gGc(g) F(S^1)=Sum_0(I(S^1))=\frac{1}{|G|}\sum_{g\in G}c(g)

    in accordance with the formula on page 4 of the paper.

    Note that the fact that II maps S 1S^1 to a functor G//G0VectG//G\to 0Vect has a simple and natural interpretation: since S 1S^1 is a cobordism from the empty manifold to itself, I(S 1)I(S^1) has to be a functor taking values in the space of morphisms of the unit of 1Vect1Vect.

  2. I’m not satisfied with the ad hoc definition of [S 1,BG]0Vect[S^1,\mathbf{B}G]\to 0Vect. I’d like this to be [S 1,BG][S 1,Vect][S^1,\mathbf{B}G]\to [S^1,Vect], induced by BGVect\mathbf{B}G\to Vect, but I’m presently unable to give a rigorous meaning to [S 1,Vect][S^1,Vect]. Yet, I’m convinced precisely this is the key of the whole construction.

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeDec 14th 2010

    but I’m presently unable to give a rigorous meaning to [S 1,Vect][S^1,Vect].

    The canonical interpretation would be as the functor category (or rather its core fr the purposes here) from the groupoid incarnation of S 1S^1 to VectVect. That is the groupoid whose objects are linear endomorphisms on a vector space, and whose morphisms are conjugations by invertible linear maps.

    What is important is, I think, that we regard VectVect as a pointed category, with point the ground field kk regarded as a vector space over itself. The component of core(Func(S 1,Vect))core(Func(S^1,Vect)) on this component is k//k ×k//k^\times (since kEnd k(k)k \simeq End_k(k)). So its 0-truncation is indeed kk.

  3. Hi Urs,

    the functor category interpretation works extremely well! and we do not need to regard VectVect as a pointed category nor to restrict to the core. Indeed, if BGVect\mathbf{B}G\to Vect is the functor induced by cc composed with the standard representation of U(1)U(1), then the induced fnctor [S 1,BG][S 1,Vect][S^1,\mathbf{B}G]\to [S^1,Vect] between functor categories maps the object (*,g)(*,g) to the object (,c(g))(\mathbb{C},c(g)) and the morphism h:(*,g)(*,h 1gh)h: (*,g)\to (*,h^{-1}g h) to the morphism c(h):(,c(g))(,c(h) 1c(g)c(h))c(h):(\mathbb{C},c(g))\to (\mathbb{C},c(h)^{-1} c(g) c(h)) (where I left the conjugation explicit to have a form which is still valid for higher dimensional representations of GG).

    So this part is ok, and to finally land in \mathbb{C} we just need to take the trace, Tr:[S 1,Vect]Tr:[S^1,Vect] \to \mathbb{C}.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeDec 15th 2010

    All right! :-)

    we just need to take the trace

    Hopefully that will be said more intrinsically. At least for the 1-dimensional case, this step is 0-truncation of [S 1,Vect][S^1, Vect].

  4. Hopefully that will be said more intrinsically.

    Yes, having a good intrinsic understanding of what is Tr:[S 1,Vect]Tr:[S^1,Vect] \to \mathbb{C} seems to be a crucial point here. I don’t clearly see this as the truncation morphism [S 1,Vect]τ 0[S 1,Vect][S^1,Vect]\to \tau_{\leq 0}[S^1,Vect] unless “the 1-dimensional case” refers to the dimension of the vector space VV in the object (V,φ)(V,\varphi) of [S 1,Vect][S^1,Vect].

    Apart this, I was thinking that for KK a 1-dimensional manifold (with boundary), the functor category [K,Vect][K,Vect] is a categorical verison of (a particular case of) the labelled diagrams appearing in Joyal-Street-Reshetikhin-Turaev graphical calculus for morphisms. So it seems that on the one hand we have a sort of “decorated cobordism” Bord 1 SO(Vect)Bord_1^{SO}(Vect) equipped with a natural morphism to VectVect given by the graphical calculus, and on the other we have the cocycle cc inducing Bord 1 SO(BG)Bord 1 SO(Vect)Bord_1^{SO}(\mathbf{B}G)\to Bord_1^{SO}(Vect). The composition of these is Bord 1 SO(BG)VectBord_1^{SO}(\mathbf{B}G)\to Vect and pushing forward to the point gives the tqft Bord 1 SOVectBord_1^{SO}\to Vect.

    So the crucial step seems to understand [K,nVect]nVect[K,n{}Vect]\to n{}Vect in arbitrary dimension.

  5. The morphism Bord 1 SO(Vect)VectBord_1^{SO}(Vect)\to Vect should be completely natural and have the following interpretation: Bord 1 SO(Vect)Bord_1^{SO}(Vect) is the free symmetric monoidal category with duals generated by VectVect; hence, for any symmetric monoidal category with duals 𝒞\mathcal{C} and any functor F:Vect𝒞F:Vect\to \mathcal{C}, is induced a duality preserving symmetric monoidal functor Z:Bord 1 SO(Vect)𝒞Z:Bord_1^{SO}(Vect)\to \mathcal{C} extending FF.

    This appears to be a simple repharsing of Joyal-Street-Reshetikhin-Turaev diagrammatics, and suggests the following general statement: Bord n SO()Bord_n^{SO}(-) is the adjoint of the inclusion {\{symmetric monoidal (,n)(\infty,n)-categories with duals}{(,n)\}\hookrightarrow\{(\infty,n)-categories}\}

    Given an arbitrary symmetric monoidal (,n)(\infty,n)-category 𝒞\mathcal{C}, I expect that the full subcategory fd(𝒞)fd(\mathcal{C}) on fully dualizable objects is a symmetric monoidal (,n)(\infty,n)-category with duals, so the conjectured adjuction above would in paticular give an equivalence between duality preserving symmetric monoidal functors Bord n SO(*)fd(𝒞)𝒞Bord_n^{SO}(*)\to fd(\mathcal{C})\hookrightarrow \mathcal{C} and objects in fd(𝒞)fd(\mathcal{C}), i.e., the cobordism hypothesis in its original formulation.

    Coming back to the BGBU(1)\mathbf{B}G\to \mathbf{B}U(1) example, the above seems to say that a symmetric monoidal morphism Bord 1 SO(BG)VectBord_1^{SO}(\mathbf{B}G)\to Vect is completely determined (or better its homotopy class is) by the morphism BGVect\mathbf{B}G\to Vect induced by the cocycle c:BGBU(1)c:\mathbf{B}G\to \mathbf{B}U(1). Next the problem of pushing this forward to a symmetric monoidal morphism Bord 1 SOVectBord_1^{SO}\to Vect seems to be reduced to the problem of pushing forward BGVect\mathbf{B}G\to Vect along BG*\mathbf{B}G\to *. This is a particular case of the problem of pushing forward a morphism XVectX\to Vect along X*X\to *, where XX is any garoupioid, and this can in turn be conveniently expressed in terms of left or right Kan extensions. This precisely transates the limit or colimit formulation of Sum 1Sum_1 in the paper.

  6. according to this MO answer, the left adjoint point of view seems to be correct. I’ll be presently not working on the details of this, but rather will try to read the constructions in Freed-Hopkins-Lurie-Teleman in the light of it. The n=1n=1 case sketched above seems to say that this point of view is promising.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeDec 18th 2010

    I expect that the full subcategory fd(𝒞)fd(\mathcal{C}) on fully dualizable objects is a symmetric monoidal (,n)(\infty,n)-category with duals

    Yes, that’s part of claim 2.3.19 of Classification of TFTs . I have finally started an entry symmetric monoidal (infinity,n)-category and recorded that statement there.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeDec 18th 2010
    • (edited Dec 18th 2010)

    Hi Domenico,

    I had been distracted by some other things and seem to have lost you a bit. Could you say again what you said before about Bord n SO(Vect)Bord_n^{SO}(Vect)? I understand that you say you want to think of this as denoting the free “with dual”ization of VectVect, but I am not sure if I see why and how.

    (I do remember that before we discussed elsewhere that Bord (X)Bord_\infty(X) should be the free symmetric-monoidalization of Π(X)\Pi(X) in some sense.)

    Also you write the cocycle c:BGVectc : \mathbf{B}G \to Vect induces Bord n SO(BG)Bord n SO(Vect)Bord_n^{SO}(\mathbf{B}G) \to Bord^{SO}_n(Vect). But by “Bord n SO(BG)Bord_n^{SO}(\mathbf{B}G)” we used to denote the nn-category of cobordisms equipped with maps to BG\mathbf{B}G. Is that still what you ,mean?

    Sorry for not following, but if you could just restate in a paragraph what you are after now, that might help.

    • CommentRowNumber15.
    • CommentAuthordomenico_fiorenza
    • CommentTimeDec 18th 2010
    • (edited Dec 18th 2010)

    Hi Urs,

    let me use two distinct notations here not to make things foggier than they deserve. On the one side we should have a left adjoint to the inclusion of symmetric monoidal (,n)(\infty,n)-categories with duals into (infy,n)(\infy,n)-categories. Let us denote this by Free n Free^\otimes_n. On the other side we have a notion of 𝒞\mathcal{C}-decorated bordism, that is, precisely, nn-cobordism with maps to 𝒞\mathcal{C}, something we could denote by Bord n SO(𝒞)Bord_n^{SO}(\mathcal{C}) or more explicitly [Bord n SO,𝒞][Bord_n^{SO},\mathcal{C}].

    The arguments in the final part of Classification of TQFTs seem to suggest an equivalence Free n (𝒞)Bord n SO(𝒞)Free^\otimes_n(\mathcal{C})\simeq Bord^{SO}_n(\mathcal{C}). Details of this are still unclear to me, except in the n=1n=1 case where it is a familiar statement from Joyal-Street-Reshetikhin-Turaev. Assuming the statement is true, then by definition of adjointness we have, for any symmetric monoidal (,n)(\infty,n)-category with duals 𝒞\mathcal{C}, a natural morphism Bord n SO(𝒞)𝒞Bord^{SO}_n(\mathcal{C})\to \mathcal{C}. What I’m saying is that (again at least in the n=1n=1 case where I’ve gone through the details) this nicely gives the construction in TQFTs from compact Lie groups.

    Namely, c:BGVectc:\mathbf{B}G\to Vect induces Bord 1 SO(BG)Bord 1 SO(Vect)Bord^{SO}_1(\mathbf{B}G)\to Bord^{SO}_1(Vect). Since VectVect is a symmetric monoidal category with duals (I’m assuming VectVect is denoting the category of finite dimensional vector spaces), then we have the natural symmetric monoidal duality preserving morphism Bord 1 SO(Vect)VectBord^{SO}_1(Vect)\to Vect induced by the universal property of Bord 1 SOBord_1^{SO}, and so we end up with a symmetric monoidal morphism Bord 1 SO(BG)VectBord^{SO}_1(\mathbf{B}G)\to Vect. Now to end up witha tqft we are left with the task of pushing this forward along BG*\mathbf{B}G\to *. Here the idea is: consider the Kan extension of c:BGVectc:\mathbf{B}G\to Vect along BG*\mathbf{B}G\to *, and apply to the homotopy commutative diagram you obtatin the hom-functor [Bord 1 SO,][Bord_1^{SO},-]. This, together with the natural morphism Bord 1 SO(Vect)VectBord^{SO}_1(Vect)\to Vect gives the diagram

    Bord 1 SO(BG) Bord 1 SO(Vect) Vect Bord 1 SO(*) \array{ Bord_1^{SO}(\mathbf{B}G)&\to&Bord_1^{SO}(Vect)&\to&Vect\\ \downarrow&\nearrow&&&\\ Bord_1^{SO}(*) }

    whose “lower profile” should be the seeked tqft.

    edit: is the lower profile the Kan extension

    Bord 1 SO(BG) Vect Bord 1 SO(*) \array{ Bord_1^{SO}(\mathbf{B}G)&\to&Vect\\ \downarrow&\nearrow&&&\\ Bord_1^{SO}(*) }

    ?

  7. the n=2n=2 case seems to go straightforwardly along these lines: a cocylce c:BGB 2U(1)c:\mathbf{B}G\to \mathbf{B}^2 U(1) induces a morphism BG2Vect\mathbf{B}G\to 2 Vect, where 2Vect2 Vect is the 2-category whose objects are \mathbb{C} algebras and whose morphisms are bimodules (this 2-category is denoted AlgAlg in FHLT). The defining representation B 2U(1)2Vect\mathbf{B}^2 U(1)\to 2 Vect maps the unique object of B 2U(1)\mathbf{B}^2 U(1) to the algebra \mathbb{C}, the unique 1-morphism of B 2U(1)\mathbf{B}^2 U(1) to vector space \mathbb{C} (as a (,)(\mathbb{C},\mathbb{C})-bimodule), and the element xU(1)x\in U(1) (as a 2-morphism in B 2U(1)\mathbf{B}^2 U(1)) to the element xU(1)End()x\in U(1)\subseteq End(\mathbb{C}). the Kan extension of BG2Vect\mathbf{B}G\to 2 Vect along BG*\mathbf{B}G \to * is the twisted group algebra of GG (this is example 3.14 in FHLT).

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeDec 19th 2010
    • (edited Dec 19th 2010)

    Hi Domenico,

    sorry to be slowing things down, but I have more questions about what precisely you are thinking of now. Of course I recognize plenty of ideas here, but to make progress I feel I need better grip on where you are thinking of precise statements and where you are sketching how the story ought to run.

    let me use two distinct notations here not to make things foggier than they deserve. On the one side we should have a left adjoint to the inclusion of symmetric monoidal (,n)(\infty,n)-categories with duals into (infy,n)(\infy,n)-categories. Let us denote this by Free n Free^\otimes_n. On the other side we have a notion of 𝒞\mathcal{C}-decorated bordism, that is, precisely, nn-cobordism with maps to 𝒞\mathcal{C},

    What kind of map from a cobordism Σ\Sigma to 𝒞\mathcal{C} are you thinking of here? Do you still map the groupoid Π(Σ)\Pi(\Sigma) to 𝒞\mathcal{C}?

    something we could denote by Bord n SO(𝒞)Bord_n^{SO}(\mathcal{C}) or more explicitly [Bord n SO,𝒞][Bord_n^{SO},\mathcal{C}].

    The latter notation has an evident interpretation as the nn-category of (symmetric monoidal) nn-functors Bord n𝒞Bord_n \to \mathcal{C} (i.e. an extended nnd-TFT with values in 𝒞\mathcal{C}). But I don’t think that that’s what you mean. I’d think it takes a bit more discussion to say what one might mean here. But maybe I am missing something.

    except in the n=1n=1 case where it is a familiar statement from Joyal-Street-Reshetikhin-Turaev

    There is a famous “Reshitikhin-Turaev construction” for n=3n = 3 and 𝒞\mathcal{C} a modular tensor category. That’s not what you mean, though?

    the n=2n=2 case seems to go straightforwardly along these lines: a cocylce c:BGB 2U(1)c:\mathbf{B}G\to \mathbf{B}^2 U(1) induces a morphism BG2Vect\mathbf{B}G\to 2 Vect, where 2Vect2 Vect is the 2-category whose objects are \mathbb{C} algebras and whose morphisms are bimodules (this 2-category is denoted AlgAlg in FHLT). The defining representation B 2U(1)2Vect\mathbf{B}^2 U(1)\to 2 Vect maps the unique object of B 2U(1)\mathbf{B}^2 U(1) to the algebra \mathbb{C}, the unique 1-morphism of B 2U(1)\mathbf{B}^2 U(1) to vector space \mathbb{C} (as a (,)(\mathbb{C},\mathbb{C})-bimodule), and the element xU(1)x\in U(1) (as a 2-morphism in B 2U(1)\mathbf{B}^2 U(1)) to the element xU(1)End()x\in U(1)\subseteq End(\mathbb{C}). the Kan extension of BG2Vect\mathbf{B}G\to 2 Vect along BG*\mathbf{B}G \to * is the twisted group algebra of GG (this is example 3.14 in FHLT).

    Yes, over a point, the quantization construction is a plain colimit. When extending this to all cobordisms, things tend to get trickier.

    Let me see…

    • CommentRowNumber18.
    • CommentAuthordomenico_fiorenza
    • CommentTimeDec 19th 2010
    • (edited Dec 19th 2010)

    Hi Urs,

    no need to apologize: I see what I’m writing here is really foggy. What I’m trying to fix is an essential version of the constructions in FHLT: my impression there is that in many occasions they describe some object coming out of nowhere as the "right" object, and only after they give a simple general recipe which would have produced that specific object. This is particularly evident in the case of the tensor category defined in section 4 of FHLT: the general argument of the paper suggest that should arise very simply as the colimit of the functor BG3VectVect^\tau [G}Vect^\tau [G}\mathbf{B}G\to 3 Vect induced by the cocycle c:BGB 3U(1)c:\mathbf{B}G\to \mathbf{B}^3 U(1), and by the defining representation B 3U(1)3Vect\mathbf{B}^3 U(1)\to 3 Vect mapping the unique object of B 3U(1)\mathbf{B}^3 U(1) to the tensor category of finite dimensional vector spaces over \mathbb{C}. Yet, this seems not to be stated explicitely in the paper.

    So, for how trivial it is, I’m now going to spend some time to convince myself that what I’ve just written is correct :)

    What kind of map from a cobordism Σ\Sigma to 𝒞\mathcal{C} are you thinking of here? Do you still map the groupoid Π(Σ)\Pi(\Sigma) to 𝒞\mathcal{C}?

    Yes, probably that’s the correct way of looking at what I’m at the moment thinking in more naive terms: namely, I’m presently thinking in terms of functors from a triangulation of Σ\Sigma to 𝒞\mathcal{C}. Extremely rough, as you see. I guess you are right and Π(Σ)\Pi(\Sigma) is the right object to be considered.

    something we could denote by Bord n SO(𝒞)Bord_n^{SO}(\mathcal{C}) or more explicitly [Bord n SO,𝒞][Bord_n^{SO},\mathcal{C}].

    You are right, teh second notation is absolutely confusing. What I had in mind is the following: Bord n SO(𝒞)Bord_n^{SO}(\mathcal{C}) maps a triangulated manifold Σ\Sigma to the [Σ,𝒞][\Sigma,\mathcal{C}]. So, adoping the Π(Σ)\Pi(\Sigma) point of view (which I’m conviced should be the correct one), this should be

    Bord n SO(𝒞):Σ[Π(Σ),𝒞] Bord_n^{SO}(\mathcal{C}): \Sigma \to [\Pi(\Sigma),\mathcal{C}]

    at least as a first approximation.

    There is a famous "Reshitikhin-Turaev construction" for n=3n = 3 and 𝒞\mathcal{C} a modular tensor category. That’s not what you mean, though?

    I mean the following: given an arbitrary category 𝒞\mathcal{C}, one can consider the category whose objects are finite sequences of objects of 𝒞\mathcal{C} (each one equipped with a +/+/- sign), and whose morphisms are oriented links between these sequences, with strands decorated by morphisms in 𝒞\mathcal{C}. Juxtaposition of links and the "obvious" braiding makes this a symmetric monoidal category, and if i’m not wrong this should be the free symmetric monoidal category generated by 𝒞\mathcal{C}. I’m not sure of the source for this construction, but I think I’ve learned it from works of Joyal-Street and Reshetikhin-Turaev.

    Yes, over a point, the quantization construction is a plain colimit. When extending this to all cobordisms, things tend to get trickier.

    Absolutely. But I’m confident we can get to the point where tricks disappear and things become completely natural. Having a good understandng of what happens over the point is a good beginning..

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeDec 20th 2010
    • (edited Dec 20th 2010)

    this should be […]

    Yes. This is along the lines indicated in the entry on Topological Quantum Field Theories from Compact Lie Groups:

    we imagine that we have an \infty-topos H\mathbf{H} into which manifolds faithfully embed ManifoldsHManifolds \hookrightarrow \mathbf{H}. Then for every object XHX \in \mathbf{H} there ought to be an (,n)(\infty,n)-functor

    Bord n(*)Fam n(*) Bord_n(*) \to Fam_n(*)

    given by sending a manifold Σ\Sigma with boundaries Σ inΣΣ out\Sigma_{in} \to \Sigma \leftarrow \Sigma_{out} to the span of \infty-groupoids

    H(Σ in,X)H(Σ,X)H(Σ out,X). \mathbf{H}(\Sigma_{in}, X) \leftarrow \mathbf{H}(\Sigma, X) \to \mathbf{H}(\Sigma_{out}, X) \,.

    If H\mathbf{H} is cohesive and XX is of the form LConstALConst A then we have H(Σ,X)Grpd(Π(Σ),A)\mathbf{H}(\Sigma, X) \simeq \infty Grpd(\Pi(\Sigma), A).

    Now, given a morphism XAX \to A (an “action functional”) there ought to be an nn-stack object QC nQC_n which assigns an (,n)(\infty,n)-category of quasicoherent sheaves of nn-modules, and a canonical representation AQC nA \to QC_n. Then the composite XAQC nX \to A \to QC_n ought to be the full action functional. And using this morphism there ought to be a lift of the above

    Z X:Bord nFam n(*) Z_X : Bord_n \to Fam_n(*)

    to Bord nFam n(nVect)Bord_n \to Fam_n(n Vect) or the like.

    And then quantization ought to be postcomposition with a canonical morphism

    :Fam n(nVect)nVect. \int \;\; : \;\; Fam_n(n Vect) \to n Vect \,.
  8. Hi Urs,

    ok, so the (,n)(\infty,n) functor Bord n(*)Fam n(*)Bord_n(*)\to Fam_n(*) associated with an object XX is what I’m calling Bord n(X)Bord_n(X), while the nn-stack object QC nQC_n is what I’ve implicitly been denoting nVectn Vect.

    So the way I figure things is that a morphism XQC nX\to QC_n induces a morphism Bord n(X)Bord n(QC n)Bord_n(X)\to Bord_n(QC_n); next Bord n(QC n)Bord_n(QC_n) should have a natural lift Bord˜ n(QC n)\widetilde{Bord}_n(QC_n) to a functor taking values in Fam n(nVect)Fam_n(n Vect), and this should be the crucial point of the whole construction. Then the composition of these two maps should give

    Bord n(X)Fam n(nVect) Bord_n(X)\to Fam_n(n Vect)

    This is simply restating what you have just written above, stressing attention on Bord n(QC n)Bord_n(QC_n). I mean: it doesn’t really matter how do we get to Bord n(QC n)Bord_n(QC_n) (a way could be, e.g., via a cocycle BGB nU(1)QC n\mathbf{B}G\to \mathbf{B}^n U(1)\to QC_n as we did several times); what really matters is wahat we are able to do once we are on Bord n(QC n)Bord_n(QC_n), i.e., how do we canonically endow the oo-groupoids H(Σ,QC n)\mathbf{H}(\Sigma,QC_n) with morphisms to nVectn Vect.

    Push-forward along Σ*\Sigma\to *?

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeDec 20th 2010
    • (edited Dec 20th 2010)

    ok, so the (,n)(\infty,n) functor Bord n(*)Fam n(*)Bord_n(*)\to Fam_n(*) associated with an object XX is what I’m calling Bord n(X)Bord_n(X),

    Okay, good.

    while the nn-stack object QC nQC_n is what I’ve implicitly been denoting nVectn Vect.

    All right, this is less important, as this is just notation for an object that is kind of obvious. The QCQC-notation is a bit more suggestive for an object in a (,n)(\infty,n)-sheaf topos, but maybe that need not be our primary concern at the moment.

    So the way I figure things is that a morphism XQC nX\to QC_n induces a morphism Bord n(X)Bord n(QC n)Bord_n(X)\to Bord_n(QC_n);

    Hm, okay, I see. That’s a slight variant of saying that the description in terms of that lift. Let me think about it.

    how do we canonically endow the oo-groupoids H(Σ,QC n)\mathbf{H}(\Sigma,QC_n) with morphisms to nVectn Vect.

    I think we should come back to that nice fact that τ nd\tau_{n-d}\mathbf{H}(\Pi(\Sigma_d),\mathbf{B}^n U(1)) \simeq \mathbf{B}^{n-d}U(1)$.

    This means that over Σ d\Sigma_d we get the data suitable for a transformation between the data over Σ d1\Sigma_{d-1}

    H(ΠΣ d,B nU(1)) H(ΠΣ d1,B nU(1)) H(ΠΣ d1,B nU(1)) B n(d1)U(1) (n(d1)Vect). \array{ && \mathbf{H}(\Pi\Sigma_d, \mathbf{B}^n U(1)) \\ & \swarrow && \searrow \\ \mathbf{H}(\Pi\Sigma_{d-1}, \mathbf{B}^n U(1)) &&\swArrow&& \mathbf{H}(\Pi\Sigma'_{d-1}, \mathbf{B}^n U(1)) \\ & \searrow && \swarrow \\ && \mathbf{B}^{n-(d-1)}U(1) \\ && \downarrow \\ && (n-(d-1)Vect) } \,.

    However, I am not yet sure how to organize this properly.

    • CommentRowNumber22.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2010
    • (edited Dec 21st 2010)

    Domenico,

    please remind me: had you worked out the relative version of the truncation formula that is needed for what I said in the above comment?

    For Σ\Sigma an dd-dimensional monifold with boundary Σ inΣ outΣ\Sigma_{in} \coprod \Sigma_{out} \hookrightarrow \Sigma, do we know how to get the natural transformation as above?

    • CommentRowNumber23.
    • CommentAuthordomenico_fiorenza
    • CommentTimeDec 21st 2010
    • (edited Dec 21st 2010)

    Hi Urs,

    yes and no: I had tried to think to that in terms of relative cohomology but was no able to get to a clean result.

    What confuses me about the truncation formula is that it seems to be much more restrictive than one would expect the theory to be: e.g., for n=1n=1 one is delaing with line bundles, whereas things seem to go through for arbitrary vector bundles. That’s why I’d try to think directly in terms of H(,QC n)\mathbf{H}(-,QC_n) rather than in terms of H(,B nU(1))\mathbf{H}(-,\mathbf{B}^n U(1)).

    The basic construction we need to understand, I think, is the following: given a flat vector bundle (or more in general a vector bundle equipped with a connection) on a space XX, taking traces of holonomy gives a section of the End(1 )End(\mathbf{1}_\mathbb{C})-bundle over the loop space X\mathcal{L}X, where 1 \mathbf{1}_\mathbb{C} is the trivial \mathbb{C}-bundle. Here we are hiddenly using the fact that the trace gives a natural brane

    H(ΠS 1,QC 1) * * Vect. \array{ && \mathbf{H}(\Pi S^1, QC_1) \\ & \swarrow && \searrow \\ * &&\Rightarrow&& * \\ & \searrow && \swarrow \\ && Vect } \,.

    where the bottom arrows pick out the algebra object \mathbb{C} in VectVect.

    we have been around this several times, already, but I still think we have not got into the real mening of this; and indeed we are not able to see clearly which is the natural higher dimensional generalization. or at least I am not :)

  9. Hi Urs,

    I’m thinking to what you wrote in #19 again. That’s extremely neat, let me rewrite it in my words to see if I got the point: we have classical theories Bord n(*)Fam n(nVect)Bord_n(*)\to Fam_n( n Vect) and quantum theories Bord(*)nVectBord(*)\to n Vect. The quantization of a classical theory is dealt with in FHLT, and is the natural functor

    :Fam n(nVect)nvect \int: Fam_n( n Vect)\to n vect

    The crucial question, then is: given any object XX in a suitable (,1)(\infty,1)-topos H\mathbf{H}, this gives an obvious funtor Z X:Bord n(*)Fam n(*)Z_X: Bord_n(*)\to Fam_n(*); can this functor be lifted to a classical theory? how?

    these lift I would call classical theories over XX. Each of these gives, in particular a functor XnVectX\to n Vect, corresponding to H)*,X)X\mathbf{H})*,X)\simeq X. The probelm is how much does this functor tell us of the eventual classical theory over XX?

    • CommentRowNumber25.
    • CommentAuthordomenico_fiorenza
    • CommentTimeDec 21st 2010
    • (edited Dec 21st 2010)

    Hey, wait, the answer is: fully dualizable object! Namely, Fam n(nVect)Fam_n(n Vect) is a symmetric monoidal (,n)(\infty,n)-category, so by the cobordism hypotesis giving a classical field theory, i.e., a symmetric monoidal functor I:Bord n(*)Fam n(nVect)I:Bord_n(*)\to Fam_n(n Vect) is the same thing as giving a fully dualizable object in Fam n(nVect)Fam_n(n Vect), i.e. a suitable functor XnVectX\to n Vect with XX an oo-groupoid. More precisely, the datum of II is equivalent to the datum of I(*)={XnVect}I(*)=\{X\to n Vect\}

    Now, composing II with the forgetful morphism Fam n(nVect)Fam n(*)Fam_n(n Vect)\to Fam_n(*) gives a functor F:Bord n(*)Fam n(*)F:Bord_n(*)\to Fam_n(*) which (trivially) is lifted by II. If the functor FF is representable, then it is represented by XX, since F(*)=XF(*)=X. This would mean that II lifts a functor of the form H(,X)\mathbf{H}(-,X) as in the posts above. On the other hand, making the fully dualizable object I(*)I(*) be the cornerstone of the construction makes the issue of the representability of FF an interesting but probably not crucial question.

    So, coming back to the FHLT paper, I think we should ask ourselves the following questions:

    a) does Sum n:Fam n(𝒞)𝒞Sum_n: Fam_n(\mathcal{C})\to \mathcal{C} preserve full dualizable objects?

    b) are the “obvious” functors BGnVect\mathbf{B}G\to n Vect induced by cocycles BGB nU(1)\mathbf{B}G\to \mathbf{B}^n U(1) fully dualizable objects in Fam n(nVect)Fam_n(n Vect)?

    c) is the defining representation B nU(1)nVect\mathbf{B}^n U(1)\to n Vect a fully dualizable objects in Fam n(nVect)Fam_n(n Vect)?

    d) if X𝒞X\to \mathcal{C} is a fully dualizable object in Fam n(𝒞)Fam_n(\mathcal{C}) and YXY\to X is an nn-functor, is the composition Y𝒞Y\to \mathcal{C} a fully dualizable object in Fam n(𝒞)Fam_n(\mathcal{C}) (i.e., does c) implies b)? )

    • CommentRowNumber26.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2010

    re #23, you write:

    What confuses me about the truncation formula is that it seems to be much more restrictive than one would expect the theory to be: e.g., for n=1 one is delaing with line bundles, whereas things seem to go through for arbitrary vector bundles.

    Yes, true, but I would be willing to believe that the case B nU(1)\mathbf{B}^n U(1) plays a special role that justifies it having its tailor-made theorems. After all, the Lagrangians for \infty-Chern-Simons theory that we are trying to figure out the extended quantization of always take values in B nU(1) diff\mathbf{B}^n U(1)_{diff}. So I wouldn’t be shocked if that case has a good general abstract quantization story, while a more general case does not.

    I think the truncation formula we have works too well to easily discard it. After all, it is a general abstract way of saying “the action is the integral over the Lagrangian”. That should not be light-heartedly be discrded! :-)

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeDec 21st 2010

    Concerning your other comments: ’ll think about it and come back to it a little later. Need to get something else done first.

  10. the truncation formula we have works too well to easily discard it. After all, it is a general abstract way of saying “the action is the integral over the Lagrangian”. That should not be light-heartedly be discrded! :-)

    a very good point! absolutely! :)

    concerning #25, the answer to a) is: trivially true. indeed that is an immediate consequence of the cobordism hypothesis and of the fact Sum nSum_n is a symmetric monoidal functor.

  11. concerning question b) in #25, the answer is yes at least for n=1n=1 and n=2n=2. Indeed, by the cobordism hypothesis, the value assigned to the point is a fully dualizable object, and in FHLT, what the functor II does on the point in the 1- and 2- dimensional case is said explicitely: it maps the point to the representation of BG\mathbf{B}G induced by the cocycle c:BGB nU(1)c:\mathbf{B}G\to \mathbf{B}^n U(1). I think one should expect the answer to be yes regardless of the dimension.