The emergence of Yangian symmetry and gauge potentials as duals of Kac-Moody currents
Yangian symmetry plays a key role in N=4 super-symmetric gauge theories. What is special in Yangian symmetry is that the algebra contains also multi-local generators. In TGD framework multi-locality would naturally correspond to that with respect to partonic 2-surfaces and string world sheets and the proposal has been that the Super-Kac-Moody algebras assignable to string worlds sheets could generalize to Yangian.
Witten has written a beautiful exposition of Yangian algebras (see this). Yangian is generated by two kinds of generators JA and QA by a repeated formation of commutators. The number of commutations tells the integer characterizing the multi-locality and provides the Yangian algebra with grading by natural numbers. Witten describes a 2-dimensional QFT like situation in which one has 2-D situation and Kac-Moody currents assignable to real axis define the Kac-Moody charges as integrals in the usual manner. It is also assumed that the gauge potentials defined by the 1-form associated with the Kac-Moody current define a flat connection:
∂μjAν- ∂νjAν +[jAμ,jAν]=0 .
This condition guarantees that the generators of Yangian are conserved charges. One can however consider alternative manners to obtain the conservation.
- The generators of first kind - call them JA - are just the conserved Kac-Moody charges. The formula is given by
JA= ∫-∞∞ dxjA0(x,t) .
- The generators of second kind contain bi-local part. They are convolutions of generators of first kind associated with different points of string described as real axis. In the basic formula one has integration over the point of real axis.
QA= fABC∫-∞∞ dx ∫x∞dy jB0(x,t)jC0(y,t)- 2∫-∞∞ jAxdx .
These charges are indeed conserved if the curvature form is vanishing as a little calculation shows.
How to generalize this to the recent context?
- The Kac-Moody charges would be associated with the braid strands connecting two partonic 2-surfaces - Strands would be located either at the space-like 3-surfaces at the ends of the space-time surface or at light-like 3-surfaces connecting the ends. Modified Dirac equation would define Super-Kac-Moody charges as standard Noether charges. Super charges would be obtained by replacing the second quantized spinor field or its conjugate in the fermionic bilinear by particular mode of the spinor field. By replacing both spinor field and its conjugate by its mode one would obtain a conserved c-number charge corresponding to an anti-commutator of two fermionic super-charges. The convolution involving double integral is however not number theoretically attactive whereas single 1-D integrals might make sense.
- An encouraging observation is that the Hodge dual of the Kac-Moody current defines the analog of gauge potential and exponents of the conserved Kac-Moody charges could be identified as analogs for the non-integrable phase factors for the components of this gauge potential. This identification is precise only in the approximation that generators commute since only in this case the ordered integral P(exp(i∫ Adx)) reduces to P(exp(i∫ Adx)).Partonic 2-surfaces connected by braid strand would be analogous to nearby points of space-time in its discretization implying that Abelian approximation works. This conforms with the vision about finite measurement resolution as discretization in terms partonic 2-surfaces and braids.
This would make possible a direct identification of Kac-Moody symmetries in terms of gauge symmetries. For isometries one would obtain color gauge potentials and the analogs of gauge potentials for graviton field (in TGD framework the contraction with M4 vierbein would transform tensor field to 4 vector fields). For Kac-Moody generators corresponding to holonomies one would obtain electroweak gauge potentials. Note that super-charges would give rise to a collection of spartners of gauge potentials automatically. One would obtain a badly broken SUSY with very large value of N defined by the number of spinor modes as indeed speculated earlier (see this).
- The condition that the gauge field defined by 1-forms associated with the Kac-Moody currents are trivial looks unphysical since it would give rise to the analog of topological QFT with gauge potentials defined by the Kac-Moody charges. For the duals of Kac-Moody currents defining gauge potentials only covariant divergence vanishes implying
that curvature form is
Fαβ= εαβ [jμ, jμ] ,
so that the situation does not reduce to topological QFT unless the induced metric is diagonal. This is not the case in general for string world sheets.
- It seems however that there is no need to assume that jμ defines a flat connection. Witten mentions that although the discretization in the definition of JA does not seem to be possible, it makes sense for QA in the case of G=SU(N) for any representation of G. For general G and its general representation there exists no satisfactory definition of Q. For certain representations, such as the fundamental representation of SU(N), the definition of QA is especially simple. One just takes the bi-local part of the previous formula:
QA= fABC∑i<jJBiJCj .
What is remarkable that in this formula the summation need not refer to a discretized point of braid but to braid strands ordered by the label i by requiring that they form a connected polygon. Therefore the definition of JA could be just as above.
- This brings strongly in mind the interpretation in terms of twistor diagrams. Yangian would be identified as the algebra generated by the logarithms of non-integrable phase factors in Abelian approximation assigned with pairs of partonic 2-surfaces defined in terms of Kac-Moody currents assigned with the modified Dirac action. Partonic 2-surfaces connected by braid strand would be analogous to nearby points of space-time in its discretization. This would fit nicely with the vision about finite measurement resolution as discretization in terms partonic 2-surfaces and braids.
The resulting algebra satisfies the basic commutation relations
plus the rather complex Serre relations described in Witten's article).
The connection between Kac-Moody symmetries and gauge symmetries is suggestive and in this case it would be realized in terms of 2-D Hodge duality. Also finite measurement resolution realized in the sense that the points at the ends of given braid strand are regarded to be effectively infinitesimally close so that the gauge algebra is effectively Abelian is essential. Yangian symmetry is crucial for the success of the twistor approach. Zero energy ontology implies that generalized Feynman diagrams contain only massless partonic 2-surfaces with propagators defined by longitudinal momentum components defined in terms of M2⊂ M4 characterizing given causal diamond. There there are excellent hopes that twistor approach applies also in TGD framework. Note that also the conformal transformations of M4 might allow Yangian variants.
For more details see the new chapter The recent vision about preferred extremals and solutions of the modified Dirac equationof "Quantum TGD as Infinite-Dimensional Geometry" or the article with the same title.