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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 1st 2013

    since the story of the various duals, compactifications and twists of gauge field theories which constitute “Witten’s grand story” (or whatever it should be called in total) gets a bit long, I thought it would be good to have a birds-eye view digest of it – and so I created a survey table

    gauge theory from AdS-CFT – table

    and included it into some relevant entries.

    (There is clearly still room for expansion and further details, but maybe it’s a start).

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeJul 1st 2013

    Are there other dimensions possible for AdS-CFT duality, other than 5/4 and 7/6?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 1st 2013
    • (edited Jul 1st 2013)

    Yes. The following horizon limits of branes yield conformal worldvolume field theories, and hence AdS/CFT correspondences:

    • D3 brane: AdS5/CFT5

    • M5-brane: AdS7/CFT6

    • M2-brane: AdS4/CFT3

    • D1-D5 brane bound state: AdS3/CFT2

    But also non-conformal field theories, such as those on Dp-branes for p4p \neq 4, can to some extent be captured holographically, at least that is the idea. In particular plain QCD is supposed to have in some limit and in some approximation a hologrpahic description.

    That leads to the currently very active field of “holographic solid state physics” and the like, where for instance from the temperature of a black hole in a high dimensional AdS spacetime people try to compute something as mundane as the shear viscosity of a quark-gluon plasma (or the like, I am not an expert on this). See the references listed here.

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 24th 2013

    I thought the geometric Langlands correspondence derives from S-duality and beyond that from 6d (2,0)-supersymmetric QFT. So why does it appear in the second table of gauge theory from AdS-CFT – table?

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 24th 2013

    According to this

    And this process continues further down in dimension: N=2 D=4 super Yang-Mills theory itself has further KK-reductions to 2-dimensional field theories. After passing to the topologically twisted theory, these are an A-model and a B-model topological string TCFT, respectively. Now what used to be S-duality/Montonen-Olive duality in 4d and conformal invariance in 6d and topological invariant in 7d and 11d here becomes geometric Langlands duality in 2d (see there for more on this) and produces Donaldson theory,

    it would seem ’geometric Langlands duality’ should be in the top table.

    While we’re on that page, where it says

    For instance various deep but rather mysterious properties of 4-dimensional Yang-Mills theory (the central theory of the standard model of particle physics), or at least of its supersymmetric versions N=2 D=4 super Yang-Mills theory/N=2 D=4 super Yang-Mills theory, find their natural geometric interpretation by understanding this 4d theory as the KK-compactification of the 6d (2,0)-superconformal QFT on a torus fiber,

    did you mean

    N=2 D=4 super Yang-Mills theory/N=4 D=4 super Yang-Mills theory

    rather than

    N=2 D=4 super Yang-Mills theory/N=2 D=4 super Yang-Mills theory?

    • CommentRowNumber6.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 24th 2013

    So at N=4 D=4 super Yang-Mills theory

    Among all gauge theory Lagrangians that of N=4, D=4 SYM is special in several ways, in particular of course in that it is conformally invariant and in that it has maximal supersymmetry; and ultimately by the fact that it is the KK-reduction of the very special 6d (2,0)-superconformal QFT and related by AdS7-CFT6 duality to the very special theory of 11-dimensional supergravity/M-theory,

    do you mean N=4, D=4 SYM should appear in both gauge theory from AdS-CFT – tables?

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 29th 2013

    Urs, as mentioned in the last two comments, I think there’s some obscurities (possibly mistakes) at gauge theory from AdS-CFT – table, but I don’t know how to straighten it out.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeSep 29th 2013

    Sorry, I had missed that. Thanks for alerting me. I’ll look into it.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeSep 29th 2013

    Okay, so I have fixed one N=2N = 2 to N=4N=4 at KK-reduction – Cascades of KK-reductions .

    The Kapustin-Witten construction of geometric Langlands starts indeed with N=4N=4 sYM. The slip came in from the fact that this also of course exhibits S-duality, while elsewhere it was mentioned that the S-duality of N=2N=2 sYM is explained by the compactification from the 6d theory.

    Now the numbering in the entry should be right, but of course this issue well deserves to be further expanded on, eventually.

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeSep 29th 2013

    Thanks again for catching this, David! Good eye.

    • CommentRowNumber11.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 7th 2014

    As N=4 D=4 super Yang-Mills theory goes through its transformations:

    ↓ topological twist

    topologically twisted N=4 D=4 super Yang-Mills theory

    ↓ KK-compactification on Riemann surface

    A-model on BunG and B-model on LocG, geometric Langlands correspondence,

    is there a parallel path for its holographical dual 5d Chern-Simons theory?

    A bit of idle Sunday morning speculating about where to find something 3d and Langlands-ish to fit with Ben-Zvi’s remark:

    This explains the “categorification” (need for a function-sheaf dictionary, which is the weak part of the analogy) that takes place in passing from classical to geometric Langlands — if you study the corresponding QFT on such three-manifolds, you get structures much closer to those of the classical Langlands correspondence.

    • CommentRowNumber12.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 7th 2014

    I had an inklng I’d asked something similar before. Very nearly 3 years ago in fact.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2014
    • (edited Sep 7th 2014)

    Just one remark on the content of the question: the full dual of 4dSYM is more than just a 5d CS theory, it is (supposedly) the full type II string theory compactified on a 5-sphere and with asymptotic AdS boundary conditions. But inside that sits a CS term, and in a suitable limit that CS term dominates. This is the content of Witten 98.

    In general, there ought to be parallels for everything done on one side of the holographic duality also on the other side. In fact the folklore is that both sides are “equivalent”. If true, this would guarantee an answer to your question. However, I find that statement dubious, because in that limit where the CS term dominates and the whole duality actually becomes precisey mathematics, then the two sides are known to be closely related but are not really “equivalent”. In any case, whatever happens on one side should have incarnation on the other side.

    Quite likely there is literature addressing your question more concretely, but I’d have to dig around to find it.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeSep 7th 2014
    • (edited Sep 7th 2014)

    Another comment on that idea of “improving the analogy” by realizing that one side has a secret categorification:

    this refers to the traditional idea that to make an analogy, then one needs to promote automorphic representations, which are functions on the arithmetic analog of Bun G(Σ)Bun_G(\Sigma), to sheaves on that stack.

    I trust that something along these lines may be done, but I do notice that when I read Langlands’ number theoretic conjecture, then I see a somewhat different story. The automorphic representations are not the end of the “Langlands correspondence” but an intermediate step. The end is “conjecture 3” which is about assigning L-functions to Galois representations (with automorphic forms being a kind of intermediate technical step to make that happen).

    Here, L-functions, as we discussed at length elsewhere, are transformations of or otherwise are some information extracted from theta-functions. Thought of this way, geometric quantization of Chern-Simons theory involves a map that takes (certain) 𝒪\mathcal{O}-modules on Loc G(Σ)Loc_G(\Sigma) (namely prequantum line bundles on the phase space of CS theory over Sigma) to (certain, twisted) 𝒟\mathcal{D}-modules on Σ\mathcal{M}_\Sigma – namely the bundle of conformal blocks equipped with the Hitchin connection – whose fibers are essentially the theta functions (z,τ)θ(z,τ)(\mathbf{z},\mathbf{\tau})\mapsto \theta(\mathbf{z}, \mathbf{\tau}), with z\mathbf{z} being a local coordinate on Loc G(Σ)Loc_G(\Sigma) and τ\tau a local coordinate on Σ\mathcal{M}_\Sigma.

    Hence, very schematically, traditional geometric quantization of Chern-Simons theory involves a theta-function construction roughly of the form

    𝒪(Loc G(Σ))𝒟( Σ) \mathcal{O}(Loc_G(\Sigma)) \longrightarrow \mathcal{D}(\mathcal{M}_\Sigma).

    If we remember that theta-functions induce L-functions (theta functions are a bit more refined than their L-functions), then this story sounds a lot like the number theoretic Langlands correspondence all the way through to “conjecture 3”. And there’d be no funny extra categorification for which one would have to bend over backwards.

    I may be wrong. But this is presently my idea, and this is what I am trying to make more precise.

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2014

    This here should bring automorphic forms in more explicitly:

    start with a Chern-Simons 3-bundle BGB 3U(1)\mathbf{B}G\to \mathbf{B}^3 U(1). Regard it as a fully dualizable object in Corr n(H /B 3U(1))Corr_n(\mathbf{H}_{/\mathbf{B}^3 U(1)}). The corresponding TQFT sends a closed surface Σ\Sigma to the theta bundle

    Loc G(Σ)=[Π(Σ),BG]BU(1). Loc_G(\Sigma) = [\Pi(\Sigma), \mathbf{B}G] \to \mathbf{B}U(1) \,.

    But in fact the TQFT is defined on framed manifolds. At least if GG is a torus group (abelian CS) then a choice of framing of Σ\Sigma induces a choice of framing of Loc G(Σ)Loc_G(\Sigma). Since a framing orients every cohomology theory, this allows to produce the space of theta functions/quantum states by pushforward. This way we get theta functions on the moduli space of framed surfaces. That should be the automorphicity in question.

    • CommentRowNumber16.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 8th 2014

    Re #14, how dependent is that story on the dimension of Σ\Sigma? Surely there’s something attractive in the idea that the mystery of Weil’s Rosetta Stone is at last revealed when we come to see that it’s all about the QFT you can do when dealing with 3d entities which are fibred over a circle. David Ben-Zvi commented to me:

    it seems to me that at least as far as the Langlands correspondence is concerned, things fall in line much better if we think of the analog of a function field over a finite field as a three manifold fibered over the circle - ie think of the finite field itself as the analog of a circle. This is precisely the reasoning in the arithmetic topology picture you mention, but it also works really nicely with the 4d QFT perspective on geometric Langlands - the structures that 4d QFT attaches to surface bundles over the circle (or to circle bundles over the circle, however boring those are) are much closer in line with what the global Langlands correspondence attaches to a function field (or the local Langlands correspondence, to a nonarchimedean local field)…

    Oh, why does a 4d QFT operate on a 3d (surface bundles over the circle) or even 2d (circle bundles over the circle) kind of thing? Am I missing something important? As you may have guessed that was why I was asking about holographic duals in #11, trying to get an odd dimension going.

    • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 8th 2014

    If number field are 3d, and QFTs fit in some cascade of various dimensions, there ought to be higher dimensions around in arithmetic. What’s the etale cohomological dimension of a higher local field?

    • CommentRowNumber18.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 8th 2014

    Drauoil seems to have worked on this and on the cohomology of curves over higher fields.

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeSep 8th 2014
    • (edited Sep 8th 2014)

    Hi David,

    re #16, yes, I understand that this is the idea, that’s what I was referring to in #14 for instance.

    This fix nonwithstanding, at this point I feel a bit disillusioned and demotivated about the idea of seeing S-duality as analogous to number theoretic Langlands. I may not have much more to add to this for the time being (not having anymore the unbounded time supplies that I used to have). I am currently interested in seeing number theoretic Langlands as analogous to geometric quantization of Chern-Simons, much more like it may be seen back in Frenkel’s 05 lecture notes, in the last section, but with some different emphasis.

    It may all turn out to be related, of course. After all, from quantization via the A-model we sort of know that geometric quantization is reflected in aspects of mirror symmetry. But presently I find the picture via genuine geometric quantization looks more compelling. I may be wrong, but this will be what I’ll be concentrating on for the moment.

    • CommentRowNumber20.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 9th 2014

    Interesting. David B.-Z. also commented that he thinks there is far more structure on the QFT side.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2014
    • (edited Sep 9th 2014)

    Not sure what you mean here by “the QFT side”, probably you mean the QFT side as in the Witten-Kapustin-style proposal to capture geometric Langlands?

    What I mean is this: for GL 2GL_2 then adelic automorphic forms are essentially the classical automorphic forms, hence are modular functions, hence functions on the moduli space of complex curves. So in this case at least “𝒟(Bun GL 2)\mathcal{D}(Bun_{GL_2})” is like 𝒟( Σ)\mathcal{D}(\mathcal{M}_{\Sigma}).

    What I have in mind is that geometric quantization of Chern-Simons theory, namely the construction of Hitchin connectios, is a map

    𝒪(Loc G(Σ))𝒟( Σ) \mathcal{O}(Loc_G(\Sigma)) \longrightarrow \mathcal{D}(\mathcal{M}_\Sigma)

    at least if on the left we restrict to line bundles and if on the the right we understand projectively flat connections, namely is the map that takes a prequantum line bundle on the phase space of Chern-Simons theory over Σ\Sigma and for each complex structure on Σ\Sigma produces the space of holomorphic sections of the induced holomorphic structure of that prequantum line bundle.

    This process has a very general abstract formulation in terms of “Local prequantum field theory (schreiber)”: start with the Chern-Simons 3-bundle BGB 3U(1)\mathbf{B}G\to \mathbf{B}^3 U(1) regarded as a fully dualizable object in Corr n(H /B 3U(1))Corr_n(\mathbf{H}_{/\mathbf{B}^3 U(1)}), and hence as defining a framed extended TFT. This “prequantum TFT” sends a surface Σ\Sigma to the transgression of the 3-bundle which is the prequantum line bundle Loc G(Σ)BU(1)Loc_G(\Sigma)\to \mathbf{B}U(1). At least for abelian GG then each framing of Σ\Sigma induces a framing of Loc G(Σ)Loc_G(\Sigma) and hence a “motivic quantization” by forming sections in any cohomology theory EE which receives twists from BU(1)B U(1), such as notably E=KUE = KU. This way weget a flat KUKU-line bundle on the moduli space of framings, whose KU-classes in each fiber are the tradtional Spin cSpin^c-quantization. Similar story holds for complex structure instead of framing.

    Either way, this construction is general abstract and holds for all kinds of geoemtries and for all choices of (higher) Chern-Simons bundles.

    I am thinking: number theoretic Langlands really looks like an arithmetic geometry version of THIS construction.

    Of course I’d need to give detailed proof, still. But the same is true for the proposal that number theoretic Langlands is an arithmetic incarnation of S-duality.

    • CommentRowNumber22.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 9th 2014

    Not sure what you mean here by “the QFT side”

    I took it to mean that most likely way to extract the inter-geometric core of all this is through getting to the essence of the physics, since that’s where the best developed structure is, and expressing it in sufficiently abstract general way, i.e., what you’re doing.

    But maybe I haven’t realised how abstract general your account is, if it

    holds for all kinds of geometries.

    So, every single term in that paragraph beginning

    This process has a very general abstract formulation…

    is expressible in any situation where we have an \infty-topos which is (differentially) cohesive of a base \infty-topos? And we have such a case in E-∞ geometry?

    • CommentRowNumber23.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 9th 2014

    Why in that first table at differential cohesion and idelic structure in the ’arithmetic geometry’ row of the column ’moduli of fields’ is it geometric Langlands, and not arithmetic Langlands?

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2014
    • (edited Sep 9th 2014)

    Regarding that table: true, that didn’t make sense what I had there. The analog of “moduli of CS-fields” in the arithmetic context is moduli of Galois representations. I have edited that table entry a bit, but I should get back to this and fine-tune more. Thanks for pointing this out.

    Regarding generality: right, so what I sketched above works for H\mathbf{H} any \infty-topos, FieldsB n𝔾\mathbf{Fields} \to \mathbf{B}^n \mathbb{G} any morphism in there, for EE any spectrum object and B 2𝔾BGL 1(E)\mathbf{B}^2 \mathbb{G}\to \mathbf{B}GL_1(E) any map (what I called a “choice of superposition principle” in dcct) – if Fields\mathbf{Fields} is such that the moduli stack [Π(Σ),Fields][\Pi(\Sigma),\mathbf{Fields}] naturally inherits EE-orientation from the (n+1)(n+1) framings (or otherwise the given GG-structure) on Σ\Sigma.

    I am not claiming yet that I have realized such a setup in arithmetic geometry. I am just thinking that trying to do so is what is suggested by number theoretic Langlands.

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2014
    • (edited Sep 9th 2014)

    Maybe to highlight, the following two basic facts seem important to keep in mind:

    First, the cobordism theorem says that the cobordisms do not “really consist of smooth manifolds”. Manifolds are just one way to neatly represent the homotopy types in Bord nBord_n. And second, the Hitchin connection, being flat, tells us that whatever geoemtric structure we may happen to put on our cobordisms, it is only the homotopy type of the geometric realization of the moduli space of these which will matter.

    So given a surface Σ\Sigma, then the cobordism hypothesis says that to quantize CS we need to produce some flat connection on a bundle of quantum states over the moduli space of 3-framings on Σ\Sigma.

    That moduli space however may have different incarnations. For instance it is a bunch of connected components in the geometric realization | Σ|\vert \mathcal{M}_\Sigma\vert of the usual Riemann moduli space on which the usual Hitchin connection is defined. Similarly, there may be arithmetic moduli spaces that are similar. The close relation between automorphic forms and modular forms suggests this. I still need to organize my thoughts on this, though.

    • CommentRowNumber26.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 9th 2014
    • (edited Sep 9th 2014)

    So maybe that’s the proper way to finish a conversation I had with John 8 years ago:

    Me: But I’m left wondering why this special kind of nn-category, i.e., ones with duals, relate to rather special kinds of spaces, i.e., smooth real manifolds. Why smooth? And why real?

    John: Well, this is the mystery that makes the Tangle Hypothesis so interesting! You start with an algebraically natural sort of nn-category, namely kk-tuply monoidal nn-categories with duals, and you discover that the free such gadget on one object describes nn-dimensional surfaces in R n+kR^{n+k}!

    Now

    You: The cobordism theorem says that the cobordisms do not “really consist of smooth manifolds”. Manifolds are just one way to neatly represent the homotopy types in Bord nBord_n.

    • CommentRowNumber27.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 9th 2014

    There is at least an idea of arithmetic bordism. We have algebraic cobordism but not étale cobordism. Probably not very useful fishing about on my part.

    • CommentRowNumber28.
    • CommentAuthorUrs
    • CommentTimeSep 9th 2014

    Just to clarify, I am not looking for a full category of cobordisms with geometric structure. What I am saying instead is that when forming the bundle of spaces of quantum states in codimension 1, then this is a flat bundle (“Hitchin connection”) which hence only depends on the homotopy type of the moduli stack of geometric structures which one uses to “polarize” and hence quantize. In the standard story and for instance over the pointed torus, then this is the moduli stack of elliptic curves and its homotopy type is just the delooping of the mapping class group of the torus. But plenty of other geometric structures also have moduli stacks with geometric realization of roughly this form. For instance 2-framings of surfaces have as moduli a disjoint union of stabilizer subgroups of the mapping class group. (We’d really need to look at 3-framings here…) So when one is just looking at the in-to-out black-box of geometric quantization, then it need not necessarily be complex-analytic structure that is used for polarization.

    Sorry, maybe I am not making myself clear. I will be trying to produce a more concrete statement.

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeSep 10th 2014

    Here is the issue now phrased as an MO question.

    • CommentRowNumber30.
    • CommentAuthorUrs
    • CommentTimeSep 11th 2014

    Turns out the answer to the last question is actually positive, it seems: the projective Hitchin connection does lift to a flat connection on the moduli of 3-framings. Chris Schommer-Pries very kindly pointed me to the relavant facts (in particular section 5 of Segal’s CFT notes).

    • CommentRowNumber31.
    • CommentAuthorDavidRoberts
    • CommentTimeSep 11th 2014

    Someone, not necessarily you, Urs, and ideally Chris S-P, should add an answer to the MO question.

    • CommentRowNumber32.
    • CommentAuthorUrs
    • CommentTimeSep 12th 2014

    Yes, I will be taking care of this story being exposed. Busy right now and the week to come (travelling to Italy), but I’ll be looking into this.

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