Extended TQFT and CFT have been puzzling me for while. While I understand the mathematical motivation behind them, I don't quite understand the physical meaning. In particular, it's not clear to me to which extent these constructions produce more information (i.e. allow for several "extended" versions of the same "non-extended" theory) vs. restring the "allowed" theories (by ruling out theories that cannot be "extended).

The most bugging question, however, is in the context of string theory. In string theory we are supposed to consider the moduli space of all SCFTs, more precisely for type II we need all SCFTs with central charge 10, for type I we need BSCFTs with certain properties etc. The question is,

What is the signficance of extended SCFTs in this context?

Is there a reason that what we really need is the moduli space of extended theories? Is it rather that "extendable" theories play some special role within the larger moduli space?

  • $\begingroup$ I am not sure what you mean by extended in this context, can you elaborate or give a reference? $\endgroup$ – user566 Feb 5 '12 at 19:24
  • $\begingroup$ @Moshe extended TQFT is explained in ncatlab.org/nlab/show/extended+topological+quantum+field+theory and extended CFT is supposed to be analogous $\endgroup$ – Squark Feb 5 '12 at 19:53

One should notice that extended CFT hasn't been fully formalized yet. ("TCFT" is only superficially conformal, in fact it axiomatizes topological strings. Also the Moore-Segal discussion is about 2d TFT, as far as I am aware.)

While not fully formalized yet, there are a few results that already come very close. For rational 2d CFT the FRS formalims and its related constructions probably goes the furthest: it gives constructions of open-closed rational CFTs in terms of "state sum constructions" that are entirely analogous to those for 2d extended TFTs, notably entirely analogous to the Fukuma-Hosono-Kawai construction. In both cases there is an algebra of open string states assigned to the interval, and from it the rest of the structure is induced. The crucial difference is that for TFT this algebra is an ordinary algebra (albeit $A_\infty$), while for CFT in the FRS formulation it is an algebra object internal to the modular tensor category of representations of the vertex operator algebra, that describes the CFT to be described locally.

From this one reads off the following use of extended CFT for string theory:

  1. the refinement of the VOA to a "full CFT" in the sense of a full representation of the conformal cobordism 1-category is precisely a solution to the sewing constraints. This involves modular invariance, but also all its higher genus analogs. This is already a step not always correctly done in the physics literature. In the FRS articles you find examples of "modular invariant CFTs" to which corresponds one, none or several full CFTs.

  2. the further refinement to a extended conformal cobordisms representation (to the extent that it has been formalized) takes care of taking also all possble boundary conditions hence all possible D-brane configurations into account.

A modern (re)view of the state of the art of such "extended" 2d CFT is in

Liang Kong, Conformal field theory and a new geometry.

Related aspects are in

Stephan Stolz, Peter Teichner, Supersymmetric field theories and generalized cohomology

There the non-topological QFT is regarded explicitly as a cobordism representation, albeit not yet fully extended.

Structures that are expected to eventually serve as ingredients for fully extended CFT have been discussed in

Chris Douglas, André Henriques, Topological modular forms and conformal nets .

They discuss a 3-category of conformal data such that the 3d extended TQFT induced by fully dualizabe objects in there is holographic dual to the given 2d CFT.

André has recently been giving talks on more of the story of how to more explicitly obtain the extnded CFT from this data. See the video of his talk at the recent Mathematical foundations of quantum field theory workshop.

  • $\begingroup$ Thx Urs! I'm still confused though. Your point 1 above suggests that extended CFT is relevant regardless the introduction of open strings / D-branes. Is this the case? Are you suggesting that "extendedness" is necessary for the full formalization of "regular" i.e. closed CFT? Also your remark about modular invariance for higher genus is very interesting. I believe it contradicts Motl's answer to my question $\endgroup$ – Squark Feb 6 '12 at 9:22
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    $\begingroup$ Hi Squark, yes, in these extended pictures the closed sector typically arises from the open sector, which is more fundamental. You will see this amplified in the article by Kong that I mentioned. It is worked out in much detail in work by Fuchs-Runkel-Schweigert (FRS). See notably their article "Uniqueness of open/closed rational CFT with given algebra of open states" projecteuclid.org/… $\endgroup$ – Urs Schreiber Feb 6 '12 at 11:25
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    $\begingroup$ Hi Squark, try out this crisp summary (3 pages) of FRS: mth.kcl.ac.uk/staff/i_runkel/PDF/ost.pdf I suppose this addresses several of the points raised here, keeping in mind that the results that they refer to are obtained with a state-sum construction as for extedned TFT but internal to the given modular tensor category of VOA representations. $\endgroup$ – Urs Schreiber Feb 6 '12 at 11:40
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    $\begingroup$ Examples of modular invariants that do not correspond to a full CFT are listed on page 3 of arxiv.org/abs/hep-th/0204148. $\endgroup$ – Urs Schreiber Feb 6 '12 at 11:49

First, there are two kinds of extended TFTs: open-closed TFTs (as in Moore-Segal and Costello's papers) and higher-categorical TFTs (as in Lurie's paper). These are somewhat related, but they are not the same. In fact, open-closed formulation is the one originating in string theory.

Let $Z_{CFT}$ be a CFT with the right central charge. Then the closed string partition function is roughly $$Z_{closed}=\sum_{g=0}^\infty g_s^{2g-2}\int_{\Sigma\in\mathcal{M}_g} Z_{CFT}(\Sigma),$$ where $\mathcal{M}_g$ is the moduli space of closed Riemann surfaces of genus $g$.

Open-closed CFTs come in when you want to compute the open string partition function, where strings end on a D-brane $\Lambda$ (possibly multiple D-branes). The open string partition function is then $$Z_{open}=\sum_{g,h}g_s^{2g-2+h}\int_{\Sigma\in\mathcal{N}_{g,h}}Z_{CFT}(\Sigma, \Lambda),$$ where $\mathcal{N}_{g,h}$ is the moduli space of Riemann surfaces with $h$ boundary components and $Z_{CFT}$ is the CFT partition function on $\Sigma$ with labels $\Lambda$ attached to each boundary component.

This of course extends to the case of insertions on the boundary and in the interior.

Regarding extensions oF TFTs, you can always think about a closed TFT (CFT) as an open-closed TFT (CFT) with an empty set of D-branes. The question is of course what are the possible nontrivial categories of D-branes. If the closed algebra of a TFT is semisimple, Moore and Segal classified possible categories of boundary conditions (the maximal category of boundary conditions is the category of finitely generated modules over the closed algebra).

  • $\begingroup$ So, is it a good translation to my language to say that "extended" CFT is an attempt to formalize and classify boundary CFTs as used in the context of string theory? $\endgroup$ – user566 Feb 6 '12 at 0:03
  • $\begingroup$ Yes, open-closed CFTs are precisely that. There is also something called "boundary CFT" (arxiv.org/abs/math-ph/0405067), which is a non-equivalent axiomatization of the same object using conformal nets (i.e. within the Haag-Kastler framework). $\endgroup$ – Pavel Safronov Feb 6 '12 at 1:05
  • $\begingroup$ Thx Pavel, this helps. However now the question that's troubling me is what is the precise relation between open-closed (boundary) TFT and higher-categorical TFT? And what is the physical meaning of the later, if any? $\endgroup$ – Squark Feb 6 '12 at 7:13
  • $\begingroup$ Given an extended 2d TFT $Z$ valued in the 2-category of linear categories, one has an open-closed TFT whose category of boundary conditions is $Z(pt)$. Physically, extended TFTs (perhaps, with some structure on cobordisms) appear when one studies various defects: e.g., in $d$ dimensions point defects (local operators) are classified by $Z(S^{d-1})$, loop operators are given by $Z(S^{d-2}\times S^1)$ (in which case you might be interested in $Z(S^{d-2})$) etc. Kapustin has a nice discussion in his ICM talk: arxiv.org/abs/1004.2307. $\endgroup$ – Pavel Safronov Feb 8 '12 at 2:25
  • $\begingroup$ Some good observations on a possible systematic formalization of the relation between extended QFT and open-closed QFT/ QFT with defects are towards the end of the slides "Topological Defects and Classifying Local Topological Field Theories in Low Dimensions" ncatlab.org/nlab/files/SchommerPriesDefects.pdf by Chris Schommer-Pries. There from slide 65 on it is shown that specifiying a ("almost natural") transformation between two 2d extended TQFTs (regarded as 2-functors on extended cobordisms) amounts to giving the data of a defect junction between the two QFTs. ... $\endgroup$ – Urs Schreiber Feb 8 '12 at 12:38

Not entirely sure what's being asked, but maybe some random remarks about the topological string might help.

It's actually easier to think about specifying the D-branes and the open string tree amplitudes. In Costello's formulation, the TCFT axioms mean that this data is a type of Calabi-Yau A_oo category. Lurie generalized this and showed that this is really an (oo,1) category. (I don't remember if the category needs to be stable -- certainly most of the categories one runs across in string theory are stable [which implies that the homotopy category is triangulated]).

What's interesting is that you can get the space of closed string states from this data. (You can see the diagrams for this in my paper Deformations and D-branes, but the idea is an old one.) In mathy terms, the cyclic cohomology of the D-brane category is the space of closed string states. It's conjectured (and proven in some cases) that the analog of the Hodge-de Rham spectral sequence degenerates, which almost gives you a Hodge structure. Specifying a splitting should be enough to give you all the perturbative open and closed string amplitudes. So, in a sense, the "extended" TCFT is the perturbative topological string theory.

  • $\begingroup$ Thx Aaron. It seems you are assuming extended TFT/CFT is the same as boundary TFT/CFT however see Pavel's answer and my comment on it $\endgroup$ – Squark Feb 6 '12 at 7:15
  • $\begingroup$ I think it would be really helpful if, following Moshe's suggestion, you spelled out exactly what you meant by the question. I suspect the answer will be that what you're looking for is the same as a perturbative string background. $\endgroup$ – Aaron Feb 7 '12 at 1:05

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