User urs schreiber - MathOverflow

KK-theory by abelianized correspondences of smooth stacks?

2013-05-18T00:28:49Z

Whith (Kasparov, bivariant) KK-theory I am left with the nagging feeling that the theory is "more fundamental", than has been made explicit, that there is a "more profound" universal characterization still to be uncovered.

This feeling is particularly driven by the characterization of certain KK-groups as abelianizations of correspondences of spaces, as recalled here. Since "most" $C^\ast$-algebras arise as topological/smooth groupoid convolution algebras an evident open question here seems to be the following:

Shouldn't KK-theory have a neat characterization in terms of an abelianization/stabilization of correspondences of differentiable stacks? In particular if we allow at least the correspondence spaces themselves to be more general smooth groupoids, maybe?

Put this way, this seems to suggest another question:

Should KK-theory be thought of as an incarnation in topology/differential geometry of the same general principle which in algebraic geometry produces motivic cohomology?

Because in both cases one builds abelianizations of correspondences of the relevant "spaces".

Looking around, I see that Grigory Garkusha recently seems to talking about something at least very similar sounding, here, though I still need to really absorb this.

Maybe here is another way to look at what I am after:

for $\mathbf{H}$ a cohesive infinity-topos and $n \in \mathbb{N}$, the (infinity,n)-category of spans $Span_n(\mathbf{H}, \flat \mathbf{B}^n U(1))$ in $\mathbf{H}$ over the coefficient object for $n$-localized action functionals is -- as recalled and discussed at nLab:prequantum field theory -- the codomain for (topological) local prequantum field theories

$$ \exp(i S) : Bord_n \to Span_n(\mathbf{H}, \flat \mathbf{B}^n U(1)) \,. $$

For $n=2$ we have some results (indicated/announced briefly at the end in the examples-section of Higher geometric prequantum field theory ) that show that the quantization of such a prequantum field theory wants to land in KK-theory, as a "geometric" improvement of the 2-category 2Mod of bare algebras and bare bimodules. In view of the partial characterization of KK-theory in terms of just equivalence classes of precisely such spans above, this makes me wonder:

might the quantization of $\exp(i S)$ be just the postcomposition with a kind of stabilization functor that sends spans/correspondences in $\mathbf{H}$ to their motivic/KK-theoretic abelianization?

For the case of discrete geometry, hence $\mathbf{H} = \infty Grpd \simeq L_{whe} sSet$, this idea or something close is appears in Baez, Hoffnung Walker (for 1-groupoids) and at least roughly also in Freed-Hopkins-Lurie-Teleman (for general $\infty$-groupoids). But I am after the geometric case here:

Doesn't it look like KK-theory wants to be the answer to "What is the abelianization of spans of smooth groupoids?" ?

What is known? What can one say? What seems likely?

Answer by Urs Schreiber for Braided Monoidal 2-categories with duals

2013-05-14T20:27:00Z

One simple way of producing symmetric monoidal $(\infty,n)$-categories with all duals is to form $n$-fold spans/correspondences, hence an (∞,n)-category of spans $Span_n(\mathbf{H})$ in some ambient $\infty$-topos $\mathbf{H}$.

This is discussed around section 3.2 in Jacob Lurie's "On the classification of TFTs".

In fact in $Span_n(\mathbf{H})$ every object is fully self-dual even. For low $n$ this is spelled out a bit at the beginning of these notes here

Domenico Fiorenza, Urs Schreiber, et al, Higher Chern-Simons local prequantum field theory .

For $X \in \mathbf{H} \hookrightarrow Span_n(\mathbf{H})$ any object, the corresponding invariant assigned to a closed framed $n$-manifold $\Sigma$ is $X^{\Pi(\Sigma)}$, where $\Pi(\Sigma) \in \infty Grpd \simeq L_{whe} sSet$ is the homotopy type of $\Sigma$ and the exponential notation denotes the powering of $\mathbf{H}$ over $\infty Grpd$.

While these are not the quantum invariants that you are looking for, this are in some precise sense the PREquantum invariants of a local field with moduli sstack $X$, before quantization. An exposition of this is in the lecture notes geometry of physics in the section on prequantum field theory

A slight variant of this (also discussed there in more detail) works as follows: for $G \in Grp(\mathbf{H})$ an abelian $\infty$-group object, also the $(\infty,n)$-category $Span_n(\mathbf{H}_{/G})$ of $n$-fold spans in the slice $\infty$-topos over $G$ is symmetric monoidal with all duals. Objects are now maps $\exp(i S) : X \to G$ and their duals are now

$$ \exp(-i S) : X \to G $$

(using the inversion operation in $G$). As the notation suggests, the manifold invariant induced by that now are prequantum fields equipped with a local action functional.

These are still not the interesting quantum invariant that you are looking for, but this is now that data which upon "quantization" should give rise to them.

For discrete higher gauge theories (Dijkgraaf-Witten-type theories) this is indicated in sections 3 and 8 of Freed-Hopkins-Lurie-Teleman.

Answer by Urs Schreiber for Resolutions of Lie algebras

2013-05-12T04:26:09Z

Going back to Quillen in 1969 there is a resolution adjunction between dg-coalgebras and dg-Lie algebras, which restricts to a rectification resolution adjunction between $L_\infty$-algebras and dg-Lie algebras. This is an equivalence of homotopy theories due to theorem 3.2 in

Vladimir Hinich, DG coalgebras as formal stacks (arXiv:math/9812034)

Here the resolution functor sends an $L_\infty$-algebra (and hence in particular a Lie algebra) to a dg-Lie algebra whose underlying graded Lie algebra is free on the underlying chain complex.

More details and more pointers are at

http://www.ncatlab.org/nlab/show/model+structure+on+dg-Lie+algebras#RectificationResolution .

Answer by Urs Schreiber for Second nonabelian group cohomology: cocycles vs. gerbes

2013-05-10T15:44:21Z

The cocycle data which you review together is a map of 2-groupoids $B \Gamma \to B Aut(B G)$ to the delooping of the automorphism 2-group "of $G$" (really: of $BG$). As for any cocycle with coefficients in an automorphism group, there is the corresponding associatived 2-bundle, hence a $BG$-fiber bundle. That's the corresponding Giraud G-gerbe and that's essentially the group extension.

There are some details on this spelled out in the nLab entry nonabelian group cohomology, though maybe that needs another touch.

The general abstract story of nonabelian cocycle, infinity-gerbes and associated infinity-bundles is in section 4 of

Thomas Nikolaus, Urs Schreiber, Danny Stevenson, Principal infinity-bundles - General theory (arXiv:1207.0248, web)

Answer by Urs Schreiber for Gerstenhaber bracket out of $L_\infty$ algebras

2013-04-29T12:40:49Z

There is a notion of universal enveloping algebra for $L_\infty$-algebras, see the references listed at http://ncatlab.org/nlab/show/universal+enveloping+E-n+algebra .

Answer by Urs Schreiber for Simplicial presheaves that are colimits of themselves?

2013-04-27T11:35:30Z

Yes, that's right. I once wrote out some details behind this (of course it's standard) in the nLab at

Reedy model structure -- Examples -- Over the simplex category

and at

Homotopy limit -- Examples -- Homotopy colimits over simplicial diagrams .

Answer by Urs Schreiber for What is the homotopy colimit of the Cech nerve as a bi-simplical set?

2013-04-25T12:36:09Z

So, as you almost say yourself, the statement holds precisely for effective epimorphisms, which are characterized (or even defined, depending on where you start) by the fact that they are the hocolims over their own Cech nerve.

So the remaining question is how to identify effective epimorphisms. The crucially useful statement here is this:

In an infinity-topos, the effective epimorphisms are precisely those maps that induce epimoprhisms on connected components. (See here for statement and references.)

Here "epimorphism on connetced components" means in general "epimorphism in the underlying sheaf topos after 0-truncation". But for the special case of the $\infty$-topos of $\infty$-groupoids, which is the one presented by simplicial sets with weak equivalences the weak homotopy equivalences, it means simply this:

*A map of simplicial sets $f : X \to Y$ is an effective epi precisely if it induces an epi on connected components, hence precise if $\pi_0(f) : \pi_0(X) \to \pi_0(Y)$ is a surjection of sets.*

So you just need to check that all connected components are hit.

Now, you were talking about "epimorphisms of simplicial sets". If by that you mean epis in the 1-category of simplicial sets, hence degreewise epis, then these are of course in particular epis on $\pi_0$, hence are effective epis even in the $\infty$-topos. But the condition to be effective epi on a simplicial set is much weaker than a degreewise epi.

Answer by Urs Schreiber for Intuitionistic logic as quantization of classical logic?

2013-04-24T18:04:20Z

No, intuitionistic logic is not the quantization of classical logic in a useful sense. (It is obtained from classical logic by removing artifical constraints, which is a step rather different from quantization.)

But there is a way to see the point of intuitionistic logic from a perspective motivated from quantization.

To see this, the main point to notice is that intuitionistic logic is the logic that allows to carry out reasoning in ambient toposes other than that of plain sets, hence in geometric contexts. That's the whole point of it.

And this is something that all of quantization theory starts with (usually secretly, of course): the "classical" (really: pre-quantum) data that quantization is applied to lives in generalized differential geometry where path spaces and spaces of differential forms etc. exist as smooth spaces. And this means that all of quantization starts (usually secretly, of course) in an ambient intuitionistic context. Of course there are plenty of ways to fight noticing this, which is why it's still an exotic perspective sociologically, even though it is the natural perspective fundamentally. The lecture notes geometry of physics try to give the natural perspective.

Urs Schreiber, Mike Shulman, Quantum gauge field theory in Cohesive homotopy type theory

an intuitionistic axiomatization of (pre-)quantum physics is laid out, and in

Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher geometric prequantum theory

it is is discussed how Heisenberg groups, Poisson Lie algebras, Heisenberg Lie algebras, (pre-)quantum operators etc. drop out naturally from these intuitionistic axioms.

Indeed, once axiomatized in intuitionistic logic this way, all these concept become much simpler than they are usually thought of. They are really sitting just a tad above the very foundations (univalent foundations, that is). See the above articles for why and how.

Answer by Urs Schreiber for How to understand Chern-Simons action

2013-04-18T19:59:14Z

Often in the literature by "Chern-Simons theory" is meant by default $G$-Chern-Simons theory whose gauge group is a connected and simply connected semisimple compact group $G$, such as $G = SU$. In this case it so happens that all $G$-principal bundles on a 3-manifold $\Sigma_3$ are trivializable, and hence one can identify the space of G-principal connections on $\Sigma_3$ just with that of $\mathfrak{g}$-valued differential forms. So one gets away with the naive formula that you recall above.

In stark contrast to this is what may seem to be a simpler example, namely $U(1)$-Chern-Simons theory. Since $U(1)$ is not simply connected, clearly, there are of course non-trivial $U(1)$-principal bundles on $\Sigma_3$, in general, and hence the above naive approach fails, as you notice.

In this case the correct Chern-Simons action is instead obtained this way: given a field configuration $\nabla$ which is a circle-principal connection, we can form its differential cup-product square in ordinary differential cohomology. This yields a $\mathbf{B}^2 U(1)$-principal 3-connection $\nabla \cup \nabla$, often known as a bundle 2-gerbe with connection or else as a degree-4 cocycle in Deligne cohomology. This now has a connection 3-form and hence has a volume holonomy over $\Sigma_3$. And this now is the correct action functional for Chern-Simons theory. For more on this see at nLab:higher dimensional Chern-Simons theory.

Secretly this higher principal connection structure also governs the first, seemingly simpler case. The action functional of Chern-Simons theory is always the volume holonomy of a 3-connection, the Chern-Simons circle 3-connection.

This is in fact the general abstract characterization of Chern-Simons theories and all its higher (and lower) dimensional variants. A Chern-Simons-type action functional is always the volume holonomy of a refinement of a universal characteristic class to ordinary differential cohomology. Further remarks along these lines are for instance in

Domenico Fiorenza, Hisham Sati, Urs Schreiber, A higher stacky perspective on Chern-Simons theory.

Answer by Urs Schreiber for Reconciling two notions of geometric quantization.

2013-04-12T05:38:13Z

Klaas Landsman has been proposing that geometric quantization is best understood as taking values in KK-cocycles. (A quick review of this idea is for instance around p. 134 of his student's thesis available here.) If one remembers the KK-cocycle before passing it through the Baum-Connes assembly map to compute its index (which is supposedly an iso anyway...), then this means remembering the module structure that you are asking for.

From this and various other perspectives it looks like regarding geometric quantization as a map into KK-theory is a rather attractive idea.

A collection of relevant references by Landsman, his students, and related articles is here: http://ncatlab.org/nlab/show/geometric+quantization+by+push-forward#References

Answer by Urs Schreiber for What is the DGLA controlling the deformation theory of a complex submanifold?

2013-02-14T12:55:54Z

This here might be helpful:

Domenico Fiorenza, Elena Martinengo, A short note on ∞-groupoids and the period map for projective manifolds, Publications of the nLab vol. 2 no. 1 (2012) (arXiv:0911.3845)

Answer by Urs Schreiber for "Philosophical" meaning of the Yoneda Lemma

2009-10-29T01:45:38Z

One way to look at it is this:

for $C$ a category, one wants to look at presheaves on $C$ as being "generalized objects modeled on $C$" in the sense that these are objects that have a sensible rule for how to map objects of $C$ into them. You can "probe" them by test objects in $C$.

For that interpretation to be consistent, it must be true that some $X$ in $C$ regarded as just an object of $C$ or regarded as a generalized object is the same thing. Otherwise it is inconsistent to say that presheaves on $C$ are generalized objects on $C$.

The Yoneda lemma ensures precisely that this is the case.

I wrote up a more detailed expository version of this story at motvation for sheaves, cohomology and higher stacks.

Answer by Urs Schreiber for What is an $(\infty,1)$-topos, and why is this a good setting for doing differential geometry?

2013-02-11T08:23:51Z

If the question is not really about principal bundle theory but just about: why do we need higher differential geometry at all, then of course there are plenty of further answers:

Classical differential geometry includes orbifolds

http://ncatlab.org/nlab/show/orbifold

as objects that handle non-free quotients of smooth manifolds. These are really the first kinds of examples of Lie groupoids

http://ncatlab.org/nlab/show/Lie+groupoid

hence of stacks on the category of smooth manifolds. All of classical foliation theory

http://ncatlab.org/nlab/show/foliation

is secretly Lie groupoid theory. And the only sane way of understanding the collection of Lie groupoids, with their correct notion of Morita equivalence and of homotopy, is as understanding them as the objects of the $(2,1)$-topos of stacks over smooth manifolds.

Also Lie theory itself breaks out of the category of smooth manifolds. For instance where Lie's three theorems fail: not every infinite-dimensional Lie algebra integrates to a Lie group, but it instead integrates to a certain Lie 2-group, a group object in Lie groupoids/smooth stacks. This is all the more true as soon as you admit that Lie algebroids are part of differential geometry

http://ncatlab.org/nlab/show/Lie+algebroid .

Lie algebroids directly encode PDEs

http://ncatlab.org/nlab/show/exterior+differential+system

Most Lie algebroids integrate to Lie groupoids, but some want to integrate to Lie 2-groupoids, which liven in the $(3,1)$-topos over smooth manifolds. And so on.

Doing differential geometry and not stopping when classical constructions fail invariably leads one to higher differential geometry, hence to working in the $(\infty,1)$-topos over smooth manifolds.

Maybe that's what the question was really asking. If in addition one feels like refining the site of smooth manifolds itself such as to include "derived smooth manifolds", then one gets something even richer, as mentioned in another reply here. Such derived and higher differential geometry is notably the home of BV-BRST formalism

http://ncatlab.org/nlab/show/BV-BRST+formalism

hence of the modern form of variational calculus, symplectic reduction and homological integration theory.

Answer by Urs Schreiber for What is an $(\infty,1)$-topos, and why is this a good setting for doing differential geometry?

2013-02-10T19:43:34Z

Here is one way to say it, which makes the relation to principal bundle theory most manifest ( http://ncatlab.org/schreiber/show/Principal+%E2%88%9E-bundles+--+theory,+presentations+and+applications ):

An $\infty$-topos is a context for homotopy theory that satisfies three extra axioms, the "Giraud-Rezk-Lurie"-axioms (for all keywords see the pointers behind the above link). In

Thomas Nikolaus, Urs Schreiber, Danny Stevenson Principal infinity-bundles - General theory http://arxiv.org/abs/1207.0248

it is shown that precisely two of these axioms make the theory of principal bundles work well, with their classification by nonabelian cohomology. This is purely axiomatic, hence completely general. It can for instance be implemented in homotopy type theory.

In the second part

Thomas Nikolaus, Urs Schreiber, Danny Stevenson Principal infinity-bundles - Presentations http://arxiv.org/abs/1207.0249

are discussed convenient ways to implement this general theory in geometric contexts such as topology, differential geometry, algebraic geometry, etc. This way the axiomatic theory recovers tradional theory, inclduding theory of gerbes, higher gerbes, twisted cohomology, twisted bundles, simplicial bundles etc.

There is announced a thrid part "Principal infinity-bundles - Examples and applications", which is not out yet. But loads of examples and applications are discussed in the text

Differential cohomology in a cohesive topos http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos

There are corresponding lecture notes

The geometry of physics http://ncatlab.org/nlab/show/geometry+of+physics

A few weeks back in January I was giving the first three lectures of the second part of these notes in Singapore at the TFT meeting. If you have any familiary with physics, check out for instance the session

geometry fof physics - Fields http://ncatlab.org/nlab/show/geometry%20of%20physics#Fields

which includes a discussion of how the traditional notion of "field bundles" in quantum field theory is not completely correct, and how its correction leads one to studying bundle theory in an infinity-topos. Lots of examples follow.

Answer by Urs Schreiber for How are infinite-dimensional manifolds most commonly treated?

2013-02-09T18:32:38Z

Different notions of manifolds may be useful in different approaches. Maybe more important than finding "universal consensus" on which one is suppoosed to be used where is to have a language to treat the various notions uniformly such as to be able pass between them in a useful way.

One such more general category is that of "diffeological spaces". In

http://ncatlab.org/nlab/show/diffeological+space

is discussed how for instance Frechet manifolds faithfully embed into these.

Diffeological spaces form a "quasi-topos". Following Grothendieck's lead, it is better to go one step further to an actual topos for differential geometry. The topos generalization of diffeological spaces is that of "smooth spaces" (smooth sets/smooth 0-types)

http://ncatlab.org/nlab/show/smooth+spaces

which is the sheaf topos over the category of smooth manifolds (or equivalently just over that of Euclidean spaces with smooth maps between them). Variants of this with a bit more information about the differential aspect of differential geometry include for instance the "Cahier topos"

http://ncatlab.org/nlab/show/Cahiers+topos

See there for pointers for how "convenient vector spaces" and hence the infinite-dimensional manifolds modeled on them ("convenient manifolds") are faithfully embedded into that topos.

In these toposes for instance all mapping spaces exist and can be usefully treated, while they agree with the infinite-dimensional manifold structures on mapping spaces whenever those actually exist. Similar statements hold for all other universal constructions.

Thereby topos theory transforms the question of finding "universal consensus" on which definition is best to a more relevant technical question: which concrete definition happens to constitute a presentation of a universaly existing construction in the topos. Presentations are useful, but are man-made. They may apply or not, may be useful here or there. But the smooth spaces which they present exist universally, robustly and meaniningfully irrespective of such choices.

Answer by Urs Schreiber for Forcing in Homotopy Type Theory

2013-01-14T16:11:03Z

In as far as we regard forcing as forming internal sheaves, the question is asking how to say "internal category of sheaves" in homotopy type theory.

It is expected that this works in directed analogy with the situation in ordinary type theory by considering internal sites, except that there is a technical problem currently not fully solved: in homotoy type theory an internal category is necessarily an internal (infinity,1)-category and in order to say this one needs to be able to say "(semi-)simplicial object in the homotopy type theory". This might seem immediate, but is a little subtle, due to the infinite tower of higher coherences involved. For truncated internal categories one might proceed "by hand" as indicated here. A general formalization has recently been proposed by Voevodsky -- see the webpage UF-IAS-2012 -- Semi-simplicial types -- but this definition does not in fact at the moment work in homotopy type theory. (Last I heard was that Voevodsky had been thinking about changing homotopy type theory itself to make this work. But this is second-hand information only, we need to wait for one of the IAS-HoTT fellows to see this here and give us first-order news on this.)

However, that all said, there is something else which one can do and which does work: if an $\infty$-topos is equipped with a notion of (formally) étale morphisms, then one can speak about internal sheaves over the canonical internal site of any object without explicitly considering the internal site itself: the "petit (infintiy,1)-topos" of sheaves on a given object $X \in \mathbf{H}$ is the full sub-(oo,1)-category of the slice over $X$ on the (formally) étale maps

It may seem surprising on first sight, but this can be saiD in homotopy type theory and it can be said naturally and elegantly:

For this we simply add the axioms of differential cohesive homotopy type theory to plain homotopy type theory. This means that we declare there to be two adjoint triples of idempotent (co)monadic modalities called

shape modality $\dashv$ flat modality $\dashv$ sharp modality

and

reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality .

Using this we say: a function is formally étale if its naturality square of the unit of the "infinitesimal shape modality" is a homotopy pullback square. Then we have available in the homotopy type theory the sub-slice over any $X$ on those maps that are formally étale. This is internally the $\infty$-topos of $\infty$-stacks over $X$, hence the "forcing of $X$" in terms of the standard interpretation of forcing as passing to sheaves.

What I just indicated is discussed in detail in section 3.10.4 and 3.10.7 of the notes

Differential cohomology in a cohesive $\infty$-topos

( http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos )

For more details on the above axioms of cohesive homotopy type theory see the first section of my article with Mike Shulman:

Quantum gauge field theory in Cohesive homotopy type theory

( http://ncatlab.org/schreiber/show/Quantum+gauge+field+theory+in+Cohesive+homotopy+type+theory )

Answer by Urs Schreiber for What do whitehead towers have to do with physics?

2012-12-13T15:37:53Z

Hi Dave,

I only just saw this question now. Maybe I can still react anyway.

The anomalies that we are talking about here mean the following: the action functional of a given QFT may turn out to be not quite a function on the configuration space, but instead a section of some line bundle with connection over configuration space. Hence for the theory to make sense, that line bundle with connection must be trivialized, and hence first of all must be trivializable. The Chern class of that line bundle is the the global anomaly, the obstruction to there existing a trivialization of the underlying bundle. It's curvature is the local anomaly, a measure for the obstruction for it to trivialize also as a bundle with connection.

That such anomalous contributions to the action functional appear for heterotic-type super-branes ("spinning branes") comes from the fact that for these the fermionic path integral is not a function on the bosonic configurations, but is a section of the Pfaffian line bundle of the given Dirac operator.

So anomaly cancellation is the process where we identify those constraints on the field content which make these anomaly line bundles become trivialized. This, and the standard references on it, is reviewed here:

http://ncatlab.org/nlab/show/quantum+anomaly

That the anomaly of the spinning particle vaishes if the target space has Spin structure is classical. That the anomaly of the heterotic string vanishes if the target has String structure is due to Killingback and Witten, originally, by an argument that was recently made rigorous by Bunke. See the references at

http://ncatlab.org/nlab/show/differential+string+structure

That the cancellation of the anomaly of the super 5-brane is similarly related to higher topological structures, and the introduction of the term "Fivebrane group" and "Fivebrane structure" is originally due to

Hisham Sati, Urs Schreiber, Jim Stasheff, Fivebrane structures Rev.Math.Phys.21:1197-1240 (2009) (arXiv:0805.0564)

That article also includes a review of the whole story of anomaly cancellation in its section 3.

In the successor

Hisham Sati, Urs Schreiber, Jim Stasheff, Twisted differential string and fivebrane structures Communications in Mathematical Physics (2012) (arXiv:0910.4001)

we develop the full differential geometry corresponding to this. Various related articles with further developments are listed here

http://ncatlab.org/nlab/show/Geometric+and+topological+structures+related+to+M-branes http://ncatlab.org/schreiber/show/differential+cohomology+in+a+cohesive+topos#Subprojects

Answer by Urs Schreiber for The Chern-Simons/Wess-Zumino-Witten correspondence

2012-07-06T12:10:12Z

There is yet one more perspective on the relation between $G$-Chern-Simons theory and the WZW-model on $G$: the background B-field of the latter can be regarded as being the prequantum circle 2-bundle in codimension 2 for a "higher/extended geometric quantization" of Chern-Simons theory.

This is spelled out a bit at

nLab:Chern-Simons theory -- Geometirc quantization -- In higher codimension.

In brief the story is this:

We have constructed in Cech cocycles for differential characteristic classes a refinement of the generator of $H^4(B G, \mathbb{Z})$ to a morphism of smooth moduli $\infty$-stacks $\mathbf{c}_c : \mathbf{B}G_c \to \mathbf{B}^3 U(1)_c$ from that of $G$-principal bundles with connection to that of circle 3-bundles (bundle 2-gerbes) with connection

(for $G$ a simple, simply connected Lie group).

This is such that when transgressed to the mapping $\infty$-stack from a closed compact oriented 3d manifold $\Sigma_3$ it yields the Chern-Simons action functional

$$ \exp(2 \pi i \int_{\Sigma_3} [\Sigma_3, \mathbf{c}_{conn}]) : CSFields(\Sigma_3) = [\Sigma_3, \mathbf{B}G_{conn}] \to U(1) \,. $$

But one can similarly transgress to mapping stacks out of a $0 \leq k \leq 3$-dimensional manifold $\Sigma_k$. For $k = 1$ with $\Sigma_1 = S^1$ one obtains a canonical circle 2-bundle (circle bundle gerbe) with connection on the smooth moduli stack of $G$-principal connections on the circle

$$ \exp(2 \pi i \int_{S^1} [S^1, \mathbf{c}_{conn}]) : [\Sigma_1, \mathbf{B}G_{conn}] \to \mathbf{B}^2 U(1) \,. $$

Now since $\mathbf{B}$ is "categorical delooping" while $[S^1, -]$ is "geometric looping", the mapping stack on the left if not quite equivalent to $G$ itself, but it receives a canonical map from it

$$ \bar \nabla_{can} : G \to [S^1, \mathbf{B}G_{conn}] \,. $$

In fact, the internal hom adjunct of this map is a canonical $G$-principal connection $\nabla_{can}$ on $S^1 \times G$, and this is precisely that from def. 3.3 of the article by Carey et al that Konrad mentions in his reply.

So the composite

$G \to [S^1, \mathbf{B}G_{conn}] \stackrel{transgression}{\to} \mathbf{B}^2 U(1)_{conn}$

is thw WZW circle 2-bundle on $G$, or equivalently the Chern-Simons prequantum circle 2-bundle in codimension 2.

(The math parser here gets confused when I type in the full formulas. But you can find them at the above link).

Answer by Urs Schreiber for The Chern-Simons/Wess-Zumino-Witten correspondence

2012-07-03T09:13:52Z

Quantum field theories are understood/formalized at various levels of detail (e.g. action functional only, space of states/partition function only, full functorial QFT, full extended QFT). Accordingly there are such different levels at which people will say "It is well-known that...".

For the general holographic principle there are still lots of gaps, but for the special case of 3dChern-Simons-TQFT/2dWZW-CFT things are pretty well understood.

The nLab entry

holographic principle -- Ordinary Chern-Simons theory / WZW-model

gives a list of pointers, some of which coincide with what is being said in other replies here.

First of all there is a direct relation between the action functionals: the CS action functional on a manifold with boundary is not gauge invariant. The boundary term that appears is the action functional of the WZW model (the topological term, at least, and the kinetic term with due fine-tuning).

More abstractly, the Chern-Simons action for $G$ simply connected arises by transgression of a differential universal characteristic map on higher smooth moduli stacks $\mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn}$. The WZW action (the topolological term) similarly arises simply by the (differentially twisted) looping (as smooth $\infty$-stacks) of this map.

Then the famous original observation: geometric quantization of this action functional yields a space of states for Chern-Simons theory that may be naturally identified with the partition function of the WZW model.

To promote this further to a relation between ful QFTs, one needs to know what the full QFT corresponding to Chern-Simons theory is. This hasn't as yet been fully established via quantization, but the expectation is that it is what the Reshetikhin-Turaev construction gives when fed the modular tensor category of loop group representations of the gauge group. Assuming this, there is a very detailed construction by Fuchs-Runkel-Schweigert and others that effectively construct the rational WZW CFT (as a full Segal-style CFT) from the TQFT.

Recently the holographic aspect of this construction has been further amplified by Kapustin-Saulina and then by Fuchs-Schweigert-Valentino.

See at the above link for references to all these items.

Answer by Urs Schreiber for Analog of "Spin" Chern-Simons Theory

2012-06-29T12:34:15Z

As David Roberts mentions in the comments above, indeed as one climbs up the Whitehead tower of the orthogonal group the higher Pontraygin classes become divisible by higher factors.

Notably in the next step, if you demand that also $\tfrac{1}{2}p_1$ vanishes, hence that you have not only a Spin-structure but even a String structure, then the second Pontryagian class becomes divisble by 6.

In this case one can also divide the Lagrangian for 7-dimensional Chern-Simons theory by 6. This is discussed in

Fiorenza, Sati, Schreiber, Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory.

transfinite composition of weak equivalences in sSet

2010-01-23T15:25:36Z

Weak equivalences in the standard model structure on simplicial sets are allegedly closed under transfinite composition.

What's a reference for that?

Compact space in site -> compact object in topos

2012-04-30T08:32:15Z

Given a site $C$, there are various standard notions for an object $X \in C$ being compact. For instance:

Every covering family $\lbrace U_i \to X \rbrace$ has a finite subfamily that is still covering.
The functor $C(X,-)$ commutes with filtered colimits.
After Yoneda-embedding, the functor $Sh_C(X, -)$ commutes with filtered colimits.
After $\infty$-Yoneda-embedding, the functor $\infty Sh_C(X, -)$ commutes with filtered $\infty$-colimits.

These notions are closely related but subtly different. For instance for $C = Top$ it is well known that the first two are not equivalent without further fine-tuning.

What can one say about the relation of 1. to 3. and 4. ?

It seems to me that one can say for instance: compactness in the first sense implies that $Sh_C(X,-)$ commutes with mono-filtered colimits, and this should generalize to the $\infty$-case in the suitable sense.

What else can one say?

Answer by Urs Schreiber for How are the classifying space of $E_8$ and $K(\mathbb{Z},4)$ related?

2012-02-04T13:40:08Z

re 1): André's answer is superb, but just for the record, original references determining the homotopy groups of E8 are here: http://ncatlab.org/nlab/show/E8#HomotopyGroupsReferences

re 2):

One way to think about the phenomenon $B E_8 \simeq_{15} B^3 U(1) \simeq K(\mathbb{Z},4)$ from the point of view of string theory is to compare it to

a) the equivalence $B \mathrm{PU}(\mathcal{H}) \simeq B^2 U(1) \simeq K(\mathbb{Z},3)$ that controls Freed-Witten anomaly cancellation over D-branes

b) generalized complex geometry and exceptional generalized geometry that controls various other geometric structures in string theory.

In all these cases, one is looking at geometry which arises from reduction of structure groups along maps $G \to K$ of groups with the property that they are weak homotopy equivalences.

This is true for the inclusions of maximal compact subgroups that control generalized complex and exceptional generalized geometry, hence the U-duality symmetry of supergravity theories in various dimensions.

Notice that these inclusions are far from being equivalences as morphisms of Lie groups. But they are equivalences of the underlying topological spaces.

This situation now has a good analog in higher smooth geometry, which "explains" the role of $E_8$.

Namely, there is a smooth 3-group $\mathbf{B}^2 U(1)$ (a smooth group 2-stack) and the universal degree 4-class on $B E_8$ has a smooth refinement to a morphism of smooth 3-groups (group 2-stacks)

$$ \Omega \mathbf{a} : E_8 \to \mathbf{B}^2 U(1) . $$

There is a higher analog of the notion of "reduction of structure groups" along such higher maps, and this controls the geometry of the supergravity C-field. For comparison, there is similarly a morphism of smooth 2-groups (smooth group stacks)

$$ \Omega \mathbf{dd} : \mathrm{PU}(\mathcal{H}) \to \mathbf{B}U(1) $$

and its induced "generalized geometry" by "reduction of higher structure groups" controls precisely the Chan-Paton bundles on D-branes twisted by the $B$-field.

Both of these morphisms of smoth higher stacks become equivalences of topological spaces under geometric realization (the first on 15-coskeleta, hence over the relevant spacetimes). So we may think of this as saying that:

"The Lie group $E_8$ is a generalized maximal compact subgroup of the smooth 3-group $\mathbf{B}^2 U(1)$. The geometry of the $C$-field is the 'generalized geometry' controled by this 'inclusion'."

For more details on all this, see around section 4.3 of

http://arxiv.org/abs/1201.5277

and the big overview tables in section 4.4.1 of

http://ncatlab.org/schreiber/files/cohesivedocumentv032.pdf

Cohesive ∞-toposes for analytic geometry

2012-01-05T13:26:04Z

There is a class of big ∞-toposes that come with a good supply of intrinsic notions of differential geometry and differential cohomology: called cohesive ∞-toposes (after Lawvere's cohesive toposes).

One way to get hold of cohesive ∞-toposes is to construct them over suitable sites of definition, such as ∞-cohesive sites. For instance, by using variants of the site of smooth manifolds, one obtains this way ∞-toposes for things like smooth geometry, synthetic differential geometry, supergeometry and the like. An account is here. One key technical point is that smooth manifolds are, of course, locally contractible, and that the large site of all of them has a small dense subsite of contractible spaces, which is an ∞-cohesive site.

I would like to construct cohesive ∞-toposes for further kinds of geometry. Currently I am focusing on analytic geometry in the Berkovich style (as referenced for instance here). Because Berkovich has the nice result that every k-analytic space locally embeddable into a smooth space is locally contractible, with the contractible patches being directed colimits of analytic domains.

This seems to suggest that we can faithfully embed this analytic geometry (and its higher analogs) in the ∞-sheaf ∞-topos over a site that consists, maybe, of contractible ind-objects of k-analytic spaces, or something similar.

But I am just learning some basics of Berkovich theory, and this is where my question starts: can anyone help me see if there is an ∞-cohesive site (or some variant, for instance we can use hypercovers instead of the covers mentioned there) suitable for (Berkovich style) analytic geometry?

Answer by Urs Schreiber for What are $n$-poset?

2011-10-12T20:29:03Z

It seems to me that you are effectively asking why a $(0,1)$-category is a poset. Because if that is so, it makes sense to define an $n$-poset to be an $(n-1,n)$-category.

To see why a $(0,1)$-category is a poset, just unwind the definition: it contains possibly non-invertible 1-morphisms, but any two of them that have the same source and target are equivalent (and the space of choices of equivalences between them is contractible). By the characterization of posets as categories that means it is a poset.

derived critical locus

2011-03-02T00:10:12Z

I am looking for discussion in the literature that properly formalizes the heuristic idea that a BV-BRST complex is a model for the "derived critical locus of a function on an $\infty$-Lie algebroid".

The kind of statement that I am after would be in the following style:

Pass to the ambient $\infty$-topos of $\infty$-sheaves on the $\infty$-site of formal duals to commutative cochain dg-algebras in non-positive degree over a field of characteristic 0 (for some topology, which I think won't matter much for the following question): the context of dg-geometry. There is then a derived functor $dgAlg^{op} \to Sh_\infty(dgAlg_-^{op})$ that interprets unbounded dg-algebras as objects in this $\infty$-topos, and this I shall make use of in the following.

In there we should have a canonical morphism

$$ \theta : \mathbb{A}^1 \to \mathbb{L}\Omega^1_K(-) $$

from the line to the $\infty$-sheaf of cotangent complexes, that sends over $A \in dgAlg_-$ an element $a \in Q A \simeq \mathbb{A}^1(A)$ to $d a$, for $Q A$ a cofibrant replacement.

Now consider an $\infty$-Lie algebroid, for instance as a simple standard example the homotopy quotient of a Lie algebra action on an ordinary affine, for which sugestive notation would be $X//\mathfrak{g}$. The dg-algebra corresponding to this dually is the corresponding Chevalley-Eilenberg algebra / BRST complex (in non-negative degree). Then a morphism

$$ S : X//\mathfrak{g} \to \mathbb{A}^1 $$

is a $\mathfrak{g}$-invariant "action functional". The composite

$$ d S : X//\mathfrak{g} \stackrel{S}{\to} \mathbb{A}^1 \stackrel{\theta}{\to} \mathbb{L} \Omega^1_K(-) $$

would be its differential. The derived critical locus of $S$ ought to be the homotopy fiber $hofib (d S)$ (over the global point given by the 0-forms).

Is the BV-BRST complex in $dgAlg$ of the data $(X, \mathfrak{g}, S)$ a model for $hofib (d S)$ ?

Or do you know writeups of details about statements of a similar flavor?

Answer by Urs Schreiber for $\infty$-forms and $\infty$-plectic geometry

2011-09-22T12:25:51Z

The way you phrase your question it sounds to me as if you are mixing up the $n$ in "$n$-category" with the $n$ in "$n$-dimensional manifold". While there are relations (for instance the n-catgeory of cobordisms is "built from" $n$-dimensional manifolds) in other contexts these are two entirely unrelated numbers.

So when you say "$\infty$-plectic" I am guessing that you are thinking on "n-plectic geometry" possibly as absorbed into some context of higher symplectic geometry that knows about symplectic ∞-groupoids?

But, as far as I can see, all the actual forms appearing here, representing certain de Rham classes on smooth ∞-groupoids, are of finite degree.

I wouldn't readily know what it means to have a form of infinite degree. Except maybe this: one can argue that the degree-0 elements of the BV-complex for a $dim > 1 $-dimensional field theory represent "mid $\infty$-dimensional" forms. But I don't think that's what you mean here.

Answer by Urs Schreiber for Is there a relation between 4-dimentional general relativity and exotic smooth structures on $\mathbb{R}^4$?

2011-09-20T15:18:31Z

There have been many proposals over the time for why four dimensions (or four large dimensions!) might be singled out by theory or by some dynamics. Just recently one could see for instance the following arXiv preprint

Sang-Woo Kim, Jun Nishimura, Asato Tsuchiya, Expanding (3+1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9+1)-dimensions (arXiv:1108.1540)

claiming that computer simulations of a certain description of nonperturbative string theory show that exactly 3+1 dimensions dynamically become macroscopic in this theory. Similar statements have been made every now and then. One needs to be a bit careful.

Notice that your statement about the role of Calabi-Yau compactification in string theory is not correct. There is nothing in the theory itself that singles out spacetimes that contain a 6-dimensional Calabi-Yau space as a factor (locally). Rather, a little computation shows that IF one assumes the background geometry to be of this form, with the Riemannian size of the CY factor very small, then it follows that the effective QFT after the Kaluza-Klein compactification in the remaining four dimensions has precisely one global supersymmetry at intermediate energy scales. Until very recently, it was widely expected that this is a property that corresponds to our observed world, and that was the only reason for considering these backgrounds. This may be changing as we speak: new experimental results from the LHC these days increasingly disfavor this prejedudice. You may find this related blog discussion here useful: Local and global supersymmetry

Answer by Urs Schreiber for Lie $2$-groups and differential equations

2011-09-19T22:47:26Z

A well-studied special case of higher symmetries of differential equations is that of differential equations that arise as Euler-Lagrange equations of local action functionals. The symmetries and symmetries-of-symmetries and symmetries-of-symmetries-of-symmetries of such a system of equations form an $\infty$-groupoid whose infinitesimal version is encoded by the corresponding BRST complex -- which is the Chevalley-Eilenberg algebra of the corresponding L-∞ algebroid. In simple cases (or else locally) this is the global quotient by a smooth ∞-group: the "ghosts" in the BRST complex are the cotangents to the local symmetries, the "ghosts-of-ghosts" are the cotangents to the local symmetries-of-symmetries, and so on.

For instance

for the action functional of the Yang-Mills field the symmetries form an ordinary Lie group;
for the action functional of the Kalb-Ramond field the symmetries form the circle 2-group $\mathbf{B}U(1) = (U(1) \to 1)$, (or rather the 2-group of functions with values in the circle 2-group);
for the action functional of the supergravity C-field the symmetries are governed by the circle 3-group $\mathbf{B}^2 U(1) = (U(1) \to 1 \to 1)$;
the higher abelian Chern-Simons theory in dimension $4k+3$ has the circle (2k+1)-group $\mathbf{B}^{2k} U(1)$ as its gauge group;
the symmetries of full string field theory form a general $\infty$-group (not an $n$-group for any finite $n$) the structure of which nobody really understands, I think.
every ∞-Chern-Simons theory (or equivalently its Euler-Lagrange equations) has a higher group of symmetries. In general, this is not just a higher gauge group, but even a higher gauge groupoid .
- the gauge groupoid of the Poisson sigma-model is controled by the Lie integration of a [Poisson Lie algebroid](http://ncatlab.org/nlab/show/Poisson+Lie algebroid), which is a [symplectic+groupoid](http://ncatlab.org/nlab/show/symplectic groupoid);
- the gauge 2-groupoid of the Courant sigma-model is controled by the Lie integration of a Courant Lie 2-algebroid, which is a symplectic Lie 2-algebroid;
- the gauge $n$-groupoid of a grade $n$ AKSZ sigma-model is similarly controled by a symplectic Lie n-groupoid.
- the 7-dimensional "fivebrane Chern-Simons theory" has string 2-group-symmetries

Answer by Urs Schreiber for Group cohomology of compact Lie group with integer coeffient

2011-09-15T07:05:54Z

A collection of relevant references (general, standard as well as specific ones) is here:

http://ncatlab.org/nlab/show/group+cohomology#OnTopologicalGroups

Comment by Urs Schreiber

2013-05-14T20:08:52Z

@B.Bischop, could you point to the page that you are thinking of? Thanks.

Comment by Urs Schreiber

2013-05-05T17:25:29Z

ncatlab.org/nlab/show/…

Comment by Urs Schreiber

2013-04-25T17:04:50Z

An survey of and introduction to the Heunen-Landsman-Spitters (and others') idea of "Bohr toposes" is here: ncatlab.org/nlab/show/Bohr+topos . That's indeed a good point to mention.

Comment by Urs Schreiber

2013-04-24T19:58:15Z

@Margaret, right, so Heyting algebras are to toposes as logic is to type theory (ncatlab.org/nlab/show/type%20theory): the latter includes the former. So what I said above involves not just intuitionistic logic, but also intuitionistic type theory (ncatlab.org/nlab/show/intuitionistic%20type%20theory). But that's only natural.

Comment by Urs Schreiber

2013-04-24T17:09:16Z

@Margaret, yes, and even apart from this non-analogy it should be made clear that the answer to the original question is definitely NO.

Comment by Urs Schreiber

2013-04-23T19:03:04Z

The relevant chapter is on the arXiv: Deligne, Freed, Supersolutions ([arXiv:hep-th/9901094](arxiv.org/abs/hep-th/9901094))

Comment by Urs Schreiber

2013-04-23T11:06:58Z

A decent discussion of this comment of Feynman's happens to be in item 7 here: wired.co.uk/magazine/archive/2012/05/start/… (By the way, I am not implying anything about the above discussion. I was just in the kind of mood to recall this "joke". In particular Domenico's reply is excellent. Also, I'd dare say that Feynman's point is to be taken with grains of salt. For instance, that most equations of motion in physics are equivalent to delta S = 0 for suitable functional S, once you reorganize them a bit, is of genuine importance.)

Comment by Urs Schreiber

2013-04-22T19:01:26Z

Take one of the connections to be trivial and classified by the constant map, but the other nontrivial.

Comment by Urs Schreiber

2013-04-22T18:50:39Z

This comes from stating Hamilton's equations in the standard form for which rate-of-change-of-position-and-momentum is on one side of the equation and derivatives-of-the-Hamiltonian is on the other. Signs are inserted to make this come out right. Every since Newton physicists usually state their equations of motion this way (rate of change = source of rate of change) -- unless they are Richard Feynman and are joking around, then they may bring each and everything to the left and have a zero on the right.

Comment by Urs Schreiber

2013-04-19T19:04:10Z

By the way, strictly speaking this is not "another point of view", but is part of what it means to refine a universal characteristic class to differential cohomology! And it holds more generally than for traditional Chern-Weil theory, too, notably it holds also for invariant polynomials not just on matrix Lie algebras, but generally on Lie algebroids and higher Lie algebroids. This then identifies "AKSZ sigma-models" as Chern-Simons type theories (ncatlab.org/schreiber/show/…) and in fact generalizes them to globalized field data.

Comment by Urs Schreiber

2013-04-18T21:50:31Z

Using the differential cup product that I mentioned, one can build higher dimensional theories by coupling lower dimensional ones. For instance one gets a 5d CS-type theory on a U(1)-field by cup-cubing the differential first Chern class, and one on pairs consisting of an SU- and of a U(1)-gauge field by forming the cup product of the differential second and first Chern-class, respectively. This is described here: ncatlab.org/schreiber/show/…

Comment by Urs Schreiber

2013-04-14T13:12:14Z

A commented list of references is here: ncatlab.org/nlab/show/…

Comment by Urs Schreiber

2013-04-12T05:27:12Z

Discussion of how Dwyer-Kan style homotopy theory in Kan-complex enriched categories relates to the notions in quasi-category theory is in appendix A.3.3 of Lurie "Higher Topos Theory". A quick survey is on the nLab here: ncatlab.org/nlab/show/homotopy+Kan+extension .

Comment by Urs Schreiber

2013-04-11T19:51:47Z

This is a good question. I think has a good answer if one thinks of the Spin^c-quantization as taking values in KK-cocycles, as promoted particularly in articles by Landsman. But before we get to that, a remark: it is not true that all functions on phase space act on the geometric quantum states, but only those whose flow fixes the polarization leafs do. This is in general a rather small subspace of functions/observables. But still, it is a good question to ask how push-forward quantization can remember this.

Comment by Urs Schreiber

2013-04-03T12:19:25Z

Symplectic 3-form?? 2-plectic form? ncatlab.org/nlab/show/2-plectic+geometry