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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2010

    created stubs for

    This is not supposed to be satisfactory content. I just wanted these pages to exist right now, so that links to them work.

  2. what about defining a genus as a functor from an ∞\infty-category of cobordism to an associative ring spectrum and considering the ring homomorphism as a particular case?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2010
    • (edited Jun 21st 2010)

    what about defining a genus as a functor from an ∞-category of cobordism to an associative ring spectrum and considering the ring homomorphism as a particular case?

    I was about to post something about that here:

    it ought to be true, up to technical details that “the” cobordism ring is the decategorificaiton of “the” (∞,∞)(\infty,\infty)-category Bord ∞Bord_{\infty}, which is in fact an ∞\infty-groupoid (being an ∞\infty-category with all duals), and is in fact a ring spectrum.

    Then for any coefficient ring spectrum AA, and every “∞\infty-TFT”

    Bord ∞→A Bord_\infty \to A

    there is the corresponding decategorification

    Ω→R \Omega \to R

    which is just a morphism of rings.

    Something like that. See also John Baez’s comment here.

    But I am hazy on telling this story in more detail. Can anyone?

    Meanwhile I have created a stub for cobordism ring.

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2010

    This should relate to our other discussion about the free symmetric monoidal (oo,1)-category with duals on paths:

    It would seem that Bord ∞(X)Bord_\infty(X) should be closely related to the free symmetric etc. on the path ∞\infty-groupoid Π(x)\Pi(x).

    We said this kind of thing before, my point here being: the path n-groupoid is most natural in its untruncated ∞\infty-version. There it looks like it ought to be related to Bord ∞(X)Bord_\infty(X).

  5. But I am hazy on telling this story in more detail. Can anyone?

    I’m staying away two days. If none has been expanding this till friday, I’ll give a try as I’m back

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 21st 2010

    If none has been expanding this till friday, I’ll give a try as I’m back

    I probably won’t look into this any more right now. I need to concentrate on something else. Am looking forward to whatever you can come up with!

  7. the naive picture I’ve in mind is the following: consider the (∞,0)(\infty,0)-category Ω n,k\Omega_{n,k} of nn-manifolds with a stratified boundary, down to kk-dimensional boundary strata, with diffeomorphisms, homotopies between diffeomorphisms and so on. then consider a two-dimensional page Ω *,*\Omega_{*,*}, where in position (p,−q)(p,-q) one has Ω p,q\Omega_{p,q}. Then forming the graded object associated to this bigraded object, the “degree zero” part consists of closed nn-dimensional manifolds, for any nn. the “degree 1” part consists of cobordisms between closed manifolds (again, in any dimension).

    this way (I mean, if one is able to correctly formalize this naive idea) one should be able to collect all extended cobordisms, with all “levels of extension” into a single object, which should be the “true” cobordism ring Ω\Omega. in particular, any representation of Ω\Omega restricted to Ω n,0\Omega_{n,0} would be an extended TQFT. so one could consider a representation of Ω\Omega as a compatible collection of extended TQFTs.

    very vague and foggy by now, I’ll try to think better to all this as I’ll be back.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 22nd 2010
    • (edited Jun 22nd 2010)

    Hm, but aren’t you secretly “just” rephrasing that the cobordism ring should be the decategorification of a subcategory of the (∞,n)(\infty,n)-category Bord nBord_n as n→∞n \to \infty?

    That would make the relation how to get genera as decategorifications of TQFTs pretty immediate.

  9. Hm, but aren’t you secretly “just” rephrasing…

    probably. and I actually hope that it can be seen in such a neat way. what I wrote was very vague and foggy, but I preferred writing it there so that anyone could eventually think on it during this couple of days I’ve been away. anyway, just came back and ready to work on today’s question: “is the cobordism ring the decategorification of a subcategory of the (∞,n)(\infty,n)-category Bord nBord_n as n→∞n\to \infty? :)

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJun 24th 2010

    Great. It seems that there are some people out there to whom this answer is obvious. I spoke to at least one expert on these matters once casually who said it’s all clear. But we ^had to interrupt before he could say more.

    In any case, even if this is well know to some small select circle, it certainly deserves to be made explicit.

  11. looking at the (∞,n)(\infty,n)-category Bord nBord_n here could be a bit confusing at first. namely that would refer to a sort of extended cobordism ring, which is probably more than we presently want. the naive categorification of the cobordism ring should work with the (∞,1)(\infty,1)-categories whose objects are closed (n−1)(n-1)-dimensional manifolds, and whose 1-morphisms are nn-dimensional cobordisms between these. then one has diffeomorphisms between cobordisms, homotopies between diffeomorphisms, and so on. so, rather than a subcategory of the (∞,n)(\infty,n)-category Bord nBord_n it seems we are dealing with a shift (unfolding?) of it. we could denote this by Bord n[n−1]Bord_n[n-1], or maybe Ω n−1Bord n\Omega^{n-1}Bord_n. here we could distinguish between a closed sector Ω n−1Bord n cl\Omega^{n-1}Bord_n^{cl} where only (n−1)(n-1)-dimensional closed manifolds are considered, and the open/closed sector where (n−1)(n-1)-dimensional manifods with boundary are admitted. but I won’t go into this right now.

    rather, I’d like to write Ω n−1Bord n≃Ω nBBord n\Omega^{n-1}Bord_n\simeq \Omega^n\mathbf{B}Bord_n, and then to think of the union of these over all dimensions nn as hocolim nΩ nBBord nhocolim_n\Omega^n\mathbf{B}Bord_n, which has a nice Galatius-Madsen-Tillmann-Weiss sound.

    • CommentRowNumber12.
    • CommentAuthordomenico_fiorenza
    • CommentTimeJun 26th 2010
    • (edited Jun 26th 2010)

    apparently, there is a well established notion of cobordism ring spectra in existing literature. for instance theorem 1.5 in Topological modular forms (after Hopkins, Miller, and Lurie) by Paul Goerss reads:

    The Witten genus can be realized as a morphism of commutative ring spectra σ:MString→tmf\sigma: MString \to tmf. The map σ\sigma is surjective on homotopy in non-negative degrees.

    Here MStringMString is the string cobordism ring spectrum: a commutative ring spectrum whose homotopy groups are the string cobordism ring. Also, Goerss paper has a nice section on ring spectra with cobordism ring spectra as an example; it contains no proofs, but it’s worth giving a look at.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2010
    • (edited Jun 26th 2010)

    But does he have any indication on how the cobordism spectrum would be Bord (∞,n)Bord_{(\infty,n)} for n→∞n \to \infty?

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2010

    I added brief remarks and references to genus and Witten genus. But needs to be polished and expanded (drastically).

    • CommentRowNumber15.
    • CommentAuthordomenico_fiorenza
    • CommentTimeJun 26th 2010
    • (edited Jun 26th 2010)

    But does he have any indication on how the cobordism spectrum would be Bord(∞,n)Bord(\infty,n) for n→∞n\to\infty?

    unfortunately not. but if I’m not wrong in my considerations in #11, one should be able to found this in Galatius-Madsen-Tillmann-Weiss, or in Lurie’s paper on the classification of TQFT. let me have a look.. yes, here’s a reference (short and undetailed, but it very clearly describes the geometric idea):

    http://math.northwestern.edu/~jnkf/classes/mflds/2cobordism.pdf

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2010

    Ah, thanks!!

    Darn, so I see this just shows that I didn’t read the cob hypothesis proof in enough detail.

    Anyway:

    Bord (∞,∞)≃Ω ∞MO. Bord_{(\infty,\infty)} \simeq \Omega^\infty M O \,.

    Cool, so I’ll put this into some entry now. Hm, where should this go…

  17. I was also thinking that we have Thom spectra MOMO, MSOMSO, MSpinMSpin and MStringMString, so there should be also a Thom spectrum MFivebraneMFivebrane and correspondingly a naturally defined Schreiber genus.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2010

    I have now added remarks along the lines Bord (∞,∞)≃Ω ∞MOBord_{(\infty,\infty)} \simeq \Omega^\infty M O to

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2010
    • (edited Jun 26th 2010)

    I was also thinking that we have Thom spectra MO, MSO, MSpin and MString,

    Sure. Feel like starting Thom spectrum? If so, please include the floating TOC that you can find now at cobordism ring.

    so there should be also a Thom spectrum MFivebrane

    Yes, certainly. And the next one should be MNinebrane, though to date there remain some things to be understood as to its relation to physics.

    and correspondingly a naturally defined Schreiber genus.

    Well, first of all if anything here, it should be named after Hisham Sati. He was the one to identify the relation of the 7-connected cover of O(n)O(n) to anomaly cancellation in the the dual heterotic string with its fundamerntal NS-fivebrane (all I did was observe that if so, I have a theory for how to form differential refinements of these structures and deduce the dual Green-Schwarz mechanism formula from that). But maybe more importantly, wouldn’t we first need to conjure a morphism MFivebrane→xyzMFivebrane \to xyz to be justified of speaking of a genus?

    What might xyz be? I have no clue at the moment.

  20. edited a bit [cobordism ring] (minor edits). I will start Thom spectrum in a minute.

    What might xyz be? I have no clue at the moment.

    neither do I :(

    but I’m confident there will be a natural xyzxyz, we will eventually be able to work out some day :)

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2010
    • (edited Jun 26th 2010)

    Domenico,

    so it would be nice now that this is out of the way to continue our discussion from elsewhere (forget where):

    one thing I am wondering about now is this:

    there should now be a nice way to see functorially how the Witten genus relates to a 2dCFT. The 2dCFT is an (∞,2)(\infty,2)-functor

    Z X:Bord (∞,2) conf→abc Z_X : Bord_{(\infty,2)}^{conf} \to abc

    and the Witten genus is an (∞,0)(\infty,0)-functor

    Bord (∞,∞) String→tmf. Bord_{(\infty,\infty)}^{String} \to tmf \,.

    We know that the 2d SCFT that we want to be looking at comes frrom one single such string manifold XX (the target space for the heterotic string).

    So somehow the second ∞\infty-functor above goes out of parts of the moduli space of the type of (∞,2)(\infty,2)-functors Bord (∞,2)→abcBord_{(\infty,2)} \to abc.

    What might be a systematic way to relate these?

  22. created stub for Thom spectrum.

    What might be a systematic way to relate these?

    an interesting question deserving an elegant answer :)

    I’ll think to this

    • CommentRowNumber23.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2010

    thanks.

    for completeness, I cretaed stub for connective spectrum essentially by copy-and-pasting material from spectrum

    • CommentRowNumber24.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2010

    oh, notice that we do have already an entry complex cobordism spectrum. I once wrote this (stubby, though) but forgot about it. Should be merged/harmonized with Thom spectrum

  25. from what I found reading here and there trying to fix my vague ideas on genera, it seems that the spectra version of genera is called orientation in the literature on the subject. for instance, one has

    • K-orientation: a map of spectra MU→KMU\to K, giving rise to the Todd genus (here KK is the complex K-theory spectrum);

    • the Atiyah-Bott-Shapiro orientations MSpin→KOMSpin \to KO, giving rise to the A^\hat{A}-genus (here KOKO is the real K-theory spectrum)

    so one could talk of tmf-orientation for the map MString→tmfMString \to tmf inducing the Witten genus.

    • CommentRowNumber26.
    • CommentAuthordomenico_fiorenza
    • CommentTimeJun 27th 2010
    • (edited Jun 27th 2010)

    a very nice interpretation of the maps of spectra above can be found reading through the lines in Stolza and Teichner What is an elliptic object?. the basic idea is, roughly, that the real K-theory spectrum KOKO is the spectrum of 1-dimensional susy real euclidean field theories, and tmf is the spectrum of 2-dimensional susy conformal field theories (and let me add that this picture should be completed by saying that the Eilenberg-MacLane spectrum of ℝ\mathbb{R} should be the spectrum of 0-dimensional susy field theories, a statement I’m quite sure having also found somewhere on the Lab).

    then, let us adopt the naive (and brilliant, in my opinion) point of view on Thom spectra given in the notes on cobordism by Francis and Gwilliam cited in the references to cobordism ring: MOMO is the moduli space of manifolds up to cobordism (and similarly for MSOMSO, MUMU, MSpinMSpin, MStringMString,…). so we see that giving a map of spectra σMSpin→KO\sigma MSpin \to KO or σ:MString→tmf\sigma:MString \to tmf are “just” the neat formalization of the naive idea that the path integral sigma model associates with any spin manifold a 1-dimensional susy real euclidean field theory and to a string manifold a 2-dimensional susy conformal field theory (the cobordism invariance of this is still obscure to me, yet).

    this suggests what should be the hypothetical xyzxyz in σ:MFivebrane→xyz\sigma:MFivebrane\to xyz: which kind of field theory do we associate via path-integral sigma model to a fivebrane manifold?

    • CommentRowNumber27.
    • CommentAuthorUrs
    • CommentTimeJun 27th 2010

    Right, that sounds good.

    In fact what Stolz-Teichner show in low degree is that it is concordance classes of those QFTs that give real cohomology, K-theory, and conjecturally tmf. It should be true that without dividing out concordance one gets the differential refinements of these.

    With that it should be true that the genera we are looking at are morphisms

    σ^:H^(−,MSpin)→H^(−,KO) \hat \sigma : \hat H(-, M Spin) \to \hat H(-, K O)

    of differential cohomology theories, to be interpreted as σ\sigma-model quantization that takes a target space geometry to the corresponding worldvolume QFT.

    Hmm..

  28. but by Yoneda such a collection of morphisms is the same thing as a single morphism MSpin→KOMSpin \to KO, so by looking at differential refinements what one is doing is setting up the right categorical setting. I agree with this.

    • CommentRowNumber29.
    • CommentAuthorUrs
    • CommentTimeJun 27th 2010

    but by Yoneda such a collection of morphisms is the same thing as a single morphism MSpin→KO

    Without the hat in H^\hat H, yes.

  30. so I see I’m a bit lost here: what does the hat stand for?

  31. ah, ok, now I’ve looked back at differential cohomology. I see..

    so, in particular the relation between the map of spectra σ:MSpin→KO\sigma:MSpin \to KO and field theory should better be seen at a differential cohomology level, as you write above. and it is precisely at this refined level that the relation between KOKO and 1-dimensional euclidean susy field theory becomes clear: it is differential cohomology with coefficients in KOKO (i.e., roughly, vector bundles with spin connections) to be (at least conjecturally) realted to 1-dimensional euclidean susy field theory.

    actually, a non susy version of this statement can be found in http://arxiv.org/abs/0903.0121, I just come across.

  32. and let me add that this picture should be completed by saying that the Eilenberg-MacLane spectrum of ℝ\mathbb{R} should be the spectrum of 0-dimensional susy field theories..

    this can indeed be found in http://math.ucr.edu/home/baez/twf_ascii/week255, as part of a wider and nice discussion.

    • CommentRowNumber33.
    • CommentAuthorUrs
    • CommentTimeJun 28th 2010

    Yes, yes. The 0-dimensional case has been very much clarified by various students of Stolz and Teichner. Depending on the amount of supersymmetry on the point one gets either real cohomology/de Rham cohomology or its 2-periodic version^.

  34. I’ve just come across Lazard ring which is another instance of how cobordism rings seems to be relevant to classical cohomology theory. I’m now looking for a detailed account on the formal group law/cohomology theory dictionary, where can I found it? in particular I’m interested in the details of the following statements I learned from Lurie’s survey of elliptic cohomology:

    a) the additive group law gives (periodic) ordinary (e.g. singular) cohomology

    b) the multiplicative group law gives K-theory

    c) the exponential homomorphism {additive} →\to {multiplicative} gives the Chern character

    • CommentRowNumber35.
    • CommentAuthorJohn Baez
    • CommentTimeJun 30th 2010

    I explained how any complex oriented cohomology theory gives a formal group law back in week150. The harder part is going back, or showing that complex oriented cobordism theory gives the universal formal group law. For the last, you should probably read Quillen’s original paper on the subject! And repay me by typing this reference into the nnLab article!

  36. Thanks. I edited Lazard ring adding the references.

    I’ve not been able to find the statement “the exponential homomorphism {additive} →\to {multiplicative} gives the Chern character”, which however should not be hard to prove. but since it is interesting (at least to me), I find it would be worth having a detailed accunt on this. so, before trying to think seriously to the details of this, I’d like to see whether I can find a ready-made reference :)

    • CommentRowNumber37.
    • CommentAuthorUrs
    • CommentTimeJul 1st 2010
    • (edited Jul 1st 2010)

    By the way, Uli Bunke and Thomas Schick have an explicit multiplicative model for the differential cobordism cohomology theory, the thing that I denoted H^(−,MG)\hat H(-, M G) above (for some GG).

    • CommentRowNumber38.
    • CommentAuthorUrs
    • CommentTimeJul 1st 2010

    … and this induces a model for Landweber-exact differential elliptic cohomology.

  39. there’s something completely trivial here I need to focus..

    Let L=MP(*)L=MP(*) be the Lazard ring, and let RR be any ring. by the universal property of the Lazard ring, formal group laws on RR are the same thing as ring homomorphisms L→RL\to R. but such a morphism is nothing but an RR-valued genus. so it seems we are saying that each Landweber-exact cohomology theory has its own distinctive genus (the one which determines the whole cohomology theory). a statement of this kind a bit surprises me, since I can’t remember having seen this clearly stated anywhere. on the other hand this seems to fit what happened with the elliptic genus, where the genus pre-dates the cohomology (if I’m not wrong here, defining elliptic cohomology out of the elliptic genus was Landweber motivating example).

    • CommentRowNumber40.
    • CommentAuthordomenico_fiorenza
    • CommentTimeJul 3rd 2010
    • (edited Jul 3rd 2010)

    on second thought, the way from cohomology to genus is more direct: a (good) cohomology theory gives a formal group law using the ℬU(1)\mathcal{B}U(1) trick. so associated to it there is a genus.

    the remarkable fact here seems to be that the whole cohomology and the genus carry the same amunt of information!

  41. another naive comment on genera: if I’m not wrong in saying that every formal group law defines a genus, then a family of formal group laws defines a family of genera, which can be seen as a function on (or rather a section of a bundle over) the base of the family. for instance, the elliptic genus (when suitably formalized as, e.g. il Lurie’s survey) is an example of this.

    I’d like to write up something to sum up all we have on elliptic cohomology and elliptic genus so far on the Lab, but I’m almost unable to work on the Lab these days.. :(

  42. still unable to work on the Lab.. :(

    just a trivial remak: I was wondering about lifting genera (classical genera, i.e, with values in some subring of ℂ\mathbb{C}, I mean) to differential forms. the idea (which does not pretend to be original at all) should be the following: assume we can associate with any manifold MM (with boundary) a non-homogenous closed differential form ω M\omega_M, such that:

    • ω M×N=π M *ω M∧π N *ω N\omega_{M\times N}=\pi_M^*\omega_M\wedge \pi_N^*\omega_N;

    • ω M∐N=ω M⊕ω N\omega_{M\coprod N}=\omega_M\oplus \omega_N;

    • ω ∂M=i ∂M *ω M\omega_{\partial M}=i^*_{\partial M}\omega_M.

    Then g(M)=∫ Mω Mg(M)=\int_M\omega_M would be a genus (cobordism invariance is just Stokes formula).

  43. in the IAS notes Witten describes a path-integral derivation of the A^\hat{A}-genus. in those notes cobordism invariance of A^\hat{A}-genus is not discussed, but I feel it should be immediate from path-integral nonsense (what I can imagine is a sort of Stokes formula for path integrals). any reference for this?

    • CommentRowNumber44.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2010
    • (edited Jul 14th 2010)

    Domenico,

    I was thinking about the following, with Thomas Nikolaus:

    As we said, the ∞\infty-category Bord ∞Bord_\infty is the Thom spectrum. But the cobordism hypothesis in its vanilla form is actually stated not for Bord nBord_n, but for Bord n frBord^{fr}_n – framed cobordisms. I gather that Bord ∞ frBord_\infty^{fr} should be the sphere spectrum.

    Now, here is something possibly noteworthy: there is an embedding i *:sSet→dSeti_* : sSet \to dSet of simplicial sets into dendroidal sets, which regards a simplicial set as an ∞\infty-operad with no non-unary operations. Based on earlier discussion we had, Thomas thinks he has now a proof that the model structure on dSetdSet localized at all horn fillers is equivalent to connective spectra: a dendroidal set that fills all horns is the (∞,1)(\infty,1)-operad coming from a groupal symmetric monoidal ∞\infty-groupoid (the nn-ary operations are the nn-fold tensor product).

    It is then pretty clear that the fibrant replacement P(i *dSet)P (i_* dSet) of the point in dSetdSet under this structure is the sphere spectrum. Which in turn should be Bord ∞ frBord^{fr}_\infty.

    You remember that we were discussing what the free symmetric monoidal ∞\infty-groupoid with all duals on the path ∞\infty-groupoid Π(X)\Pi(X) should be, namely Bord ∞ fr(X)Bord_\infty^{fr}(X). In view of the above, there is a suggestive guess for an answer: consider

    [C op,sSet] ←LConst→Π sSet ↓ ↓ [C op,dSet] ←LConst→Π d dSet \array{ [C^{op}, sSet] &\stackrel{\overset{\Pi}{\to}}{\underset{LConst}{\leftarrow}}& sSet \\ \downarrow && \downarrow \\ [C^{op}, dSet] &\stackrel{\overset{\Pi_d}{\to}}{\underset{LConst}{\leftarrow}}& dSet }

    So for XX a manifold, regarded as a 0-truncated ∞\infty-stack on C=CartSpC = CartSp, to form Π d(X)\Pi_d(X) which might be Bord ∞ fr(X)Bord^{fr}_\infty(X) we first form Π(X)∈sSet\Pi(X) \in sSet, which is its fundamental ∞\infty-groupoid, then regard this under the embedding i *Π(X)i_* \Pi(X) in dSetdSet and there fibrantly replace.

    Just an idea.

  45. Let me see if I’m following: what we are aiming at is defining Bord ∞ frBord_\infty^{fr} in a (∞,1)(\infty,1)-topos. so we want to take the familiar Bord ∞ fr(X)Bord_\infty^{fr}(X) of usual cobordism theory and express it in a way involving only operations available in every (∞,1)(\infty,1)-topos. and these should be forming the fundamental ∞\infty-groupoid, going from simplicial sets to dendroidal ones, and taking fibrant replacements.

    I should study more dendroidal sets before being able to see if this actually reproduces Bord ∞ frBord^{fr}_\infty or not. but I trust you on this. and if it does, then the presentation you give seems to me general and clean enough to be a very good candidate for a definition of Bord ∞ frBord_\infty^{fr} in a (∞,1)(\infty,1)-topos. sorry to be unable to check details at the moment :(

    • CommentRowNumber46.
    • CommentAuthorUrs
    • CommentTimeJul 14th 2010

    Actually, what I just said would (if it makes any sense) give Bord ∞ fr(X)Bord_\infty^{fr}(X) not in the ∞\infty-topos, but as a bare (symmetric monoidal etc.) ∞\infty-groupoid.

    I’ll try to get more details on this worked out…

    • CommentRowNumber47.
    • CommentAuthorzskoda
    • CommentTimeJul 15th 2010

    Possibly related MPI seminar, few days ago (anybody can get notes ?)

    12.07.2010, Topics in Topology Pokman Cheung (MIT Cambridge/MPI): Chiral differential operators and topology

    In this talk, I will first describe a reformulation of a construction by Gorbounov, Malikov and Schechtman in terms of global geometric data. This will then be applied to define vertex algebraic versions of Dolbeault complexes and obtain a geometric description of the Witten and elliptic genera of complex manifolds

    • CommentRowNumber48.
    • CommentAuthorUrs
    • CommentTimeMay 22nd 2012

    added references to Witten genus

1 to 48 of 48

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