topological A and B models- witten's paper
boltu
This is about Witten's paper on "Mirror Manifolds and Topological String
Theory" , 9112056.
What is the best way to visualise powers, both positive and negative, of
the canonical line bundle? What are the meanings of "K^1/2" and "K^-1"?
What is the relation between twisting the fermions and rotational symmetry
on the world sheet?
Why are the twisting the only ones we could do? If we took the fermons to
be sections of other powers of the canonical line bundle on sigma (of
course tensored with the pullback of the tangent bundle on X) what would
be the effect on the world sheet? Would the sigma-model lagrangian remain
invariant?
What is the relation between twisting with different powers of the line
bundle and the associated U(1) gauge symmetry? Does twisting the fermions
change the spins of the G+- in the superconformal algebra? What exactly is
the relation between changing the relevant bundles of the fermions and the
effect on the superconformal algebra- for example the shift in the
energy-momentum tensor?
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A.J. Tolland
> This is about Witten's paper on "Mirror Manifolds and Topological String
> Theory" , 9112056.
> What is the best way to visualise powers, both positive and negative, of
> the canonical line bundle? What are the meanings of "K^1/2" and "K^-1"?
Remember that for line bundles, dual and inverse are the same
thing.
For one dimensional complex curves, K = T^*X. So K^{-1} = TX.
And K^{-1/2} is a square root of the tangent bundle, one of the
conventional spinor bundles, so K^{1/2} is the dual of
> What is the relation between twisting the fermions and rotational symmetry
> on the world sheet?
Twisting the fermions corresponds to declaring a subgroup of SO(2)
x U(1)_L x U(1)_R other than SO(2) to be the rotational symmetry group.
The twistings correspond to using, instead of J,
J - 1/2 (J_L - J_R) (A-model)
J - 1/2 (J_L + J_R) (B-model)
> Why are the twisting the only ones we could do? If we took the fermons to
> be sections of other powers of the canonical line bundle on sigma (of
> course tensored with the pullback of the tangent bundle on X) what would
> be the effect on the world sheet? Would the sigma-model lagrangian remain
> invariant?
The point of doing this is that we want to treat Q as a BRST
operator, on any curve Sigma. Q has to act globally, so we need the
fermionic parameters of the Lie algebra to be well-defined sections of
some line bundles. This is easy locally, in some small neighborhood.
But it's difficult globally: many line bundles don't have _any_ global
sections. So we twist and put our fermionic parameters in the functions,
where there is always at least the constant sections.
It's a lovely coincidence that this forces us to choose the A or B
twistings. Any twisting would be a symmetry fo the classical Lagrangian.
But only the A and B twistings don't become anomalous when we move to the
quantum theory.
> Does twisting the fermions change the spins of the G+- in the
> superconformal algebra? What exactly is the relation between changing
> the relevant bundles of the fermions and the effect on the
> superconformal algebra- for example the shift in the energy-momentum
> tensor?
Yes, twisting changes the spins. The spins are encoded in the
eigenvalues h and \bar{h} of the grading operators L_0 and \bar{L}_0.
spin = h - \bar{h}. So twisting alters the line bundles by changing how
their fibers transform under rotations; this means that the spins of
fields which arise as sections of these line bundles are altered.
Changing the spins really means changing how we measure them, so we must
alter the components L and \Lbar of the energy momentum tensor.
Hope that helped.
--A.J.
calabisix
I have been studying this paper to get a better understanding of
TQFT's.
I proceed fine until I get to the definition of the fermi field.
what is the significance of defining fermi fields as the product of
K^1/2 and the pullback of the tangent bundle?
Is this a standard Differential Geometry concept?
I have browsed the papers of Aspinwall and Morrison (TFT and Rational
Curves) as well as the papers of Witten (TQFT and Sigma models)...even
his earlier paper on the Susy sigma model. I have yet to find an
intuitive explanation of "why" one creates the product above to proceed
with the theory.
This may be a standard Differential Geometry concept that I just
haven't crossed yet (hence the "Beginner" heading of my post).
Thank you,
calabisix
A.J. Tolland
> I proceed fine until I get to the definition of the fermi field.
> what is the significance of defining fermi fields as the product of
> K^1/2 and the pullback of the tangent bundle?
It's commonly used. The idea is that we want worldsheet fermions
which take values in vector fields on the target space. But you can't
multiply bundles on different spaces (for roughly the same reason that you
can't add vectors which live in different fibers) , so you pull the
tangent bundle back to the worldsheet, using the only map available.
That help?
--A.J.