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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 16th 2010

    started a stub for the B-model

    • CommentRowNumber2.
    • CommentAuthorzskoda
    • CommentTimeJun 16th 2010

    I made few corrections.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJun 16th 2010


    the B-model is not a supersymmetric theory. It is obtained from a supersymmetric theory by “twisting” that, such that what used to be supercharges become BRST-like operators. The supersymmetry disappears and gives rise to topological invariance.

    • CommentRowNumber4.
    • CommentAuthorzskoda
    • CommentTimeJun 16th 2010
    • (edited Jun 16th 2010)

    Take for exmaple Cox-Katz standard textbook, page 421:

    These (topologically twisted, Z.Š.) models are still supersymmetric (but now have only N = 1 supersymmetry). Furthermore, if V and V are a mirror pair, then the A-model derived from a Calabi-Yau manifold V is mirror symmetric to the B-model derived from its mirror manifold V (for corresponding choices of the complex structure of V and Kahler structure on V).

    Also, 421-422

    the remaning terms can be compactly written in terms of a certain fermionic operator, the BRST operator Q. This operator is the supersymmetry transformation which survives the reduction in supersymmetry from the original N=2 symmetry to the twisted N=1 theory.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJun 16th 2010

    Hm, I am not sure I buy into this. The N=1 susy sigma model is something else, namely the heterotic string on the Calabi-Yau. Not every graded symmetry should be called a “supersymmetry”.

    But in any case it seems we agree that the A- and B-model are not N=2 supersymmetric, as you write. So can we at least say N=1 in the entry, instead of N=2?

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeJun 16th 2010

    Right, I changed it to N=1 (though I still remember that Vafa claimed it is N=2 still but the second one is not physical, because of nonobservable U(1) operator).

    Not every graded symmetry should be called a “supersymmetry”.

    The question is if there is a supersymmetry algebra combined with CFT operators. Do we have just Virasoro or super Virasoro. It does not matter if it is at BRST level or anything as long as formal algebra relations are satisfied (“survive”).

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeJun 16th 2010

    Do we have just Virasoro or super Virasoro.

    Well, we have neither. The odd generator now squares to 0, instead of to a nontrivial translation generator. This is of course the hallmark of the fact that the twisted theory is “topological” and not “conformal” anymore.

    • CommentRowNumber8.
    • CommentAuthorzskoda
    • CommentTimeJun 16th 2010

    The N=1 susy sigma model is something else, namely the heterotic string on the Calabi-Yau.

    Did I quote anything contradicting this assertion ? By twisting N=2 supersymmetric nonlinear sigma model you get N=1 supersymmetric theory, not a N=1 supersymmetric sigma model.

    • CommentRowNumber9.
    • CommentAuthorzskoda
    • CommentTimeJun 16th 2010

    This is of course the hallmark of the fact that the twisted theory is “topological” and not “conformal” anymore.

    Independence from conformal structure does not break necessary the conformal invariance ins’t it ?; it just associates no dynamics to that part. First examples of TQFTs are in Riemannian setting. A common way is to vary over all such structures to get independence.

    • CommentRowNumber10.
    • CommentAuthorKevin Lin
    • CommentTimeJun 25th 2010
    • (edited Jun 25th 2010)
    I added a few words regarding the B-model of a Landau-Ginzburg model.

    I hope that someone can also add/say something about BCOV.
    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJun 13th 2013

    added to B-model pointers to its “second quantization” to “Kodeira-Spencer gravity”, “BCOV theory”.

    • CommentRowNumber12.
    • CommentAuthorzskoda
    • CommentTimeJun 17th 2013

    Kodaira – corrected at B-model.