aa r X i v : . [ h e p - t h ] J un June 27, 2011 YITP-SB-11-20
Yang-Mills by dimensionallyreducing Chern-Simons
W. Siegel*
C. N. Yang Institute for Theoretical PhysicsState University of New York, Stony Brook, NY 11794-3840
ABSTRACT
We derive the usual first-order form of the Yang-Mills action in arbitrarydimensions by dimensional reduction from a Chern-Simons-like action. Theantisymmetric tensor auxiliary field of the first-order action appears as agauge field for the extra dimensions. The higher-dimensional geometry wasintroduced in our previous paper by adding dimensions “dual” to spin, assuggested by the superstring’s affine Lie algebra.* mailto:[email protected]://insti.physics.sunysb.edu/˜siegel/plan.html
Chern-Simons analogs
We consider a class of theories related to Chern-Simons, but defined in more thanthree dimensions, and dependent on the geometry. The theories are defined as usualin terms of Yang-Mills covariant derivatives ∇ A = d A + iA A , but the free derivatives d A are nonabelian. Dividing up the derivatives as A = ( α, i ),[ d A , d B } = T ABC d C : [ d α , d β } = T αβ i d i , [ d α , d i } = [ d i , d j } = 0The field strengths are F AB = d [ A A B ) − T ABC A C + iA [ A A B ) Defining as usual the Chern-Simons form (but taking into account the torsion) X ABC = A [ A d B A C ) − A [ A T BC ) D A D + iA [ A A B A C ) the Lagrangian takes the form (appearing as its trace in the action) L = a αβi X αβi for some constant tensor a invariant under the desired symmetries.All such actions have an important difference from standard Chern-Simons terms:The torsion introduces a contribution − η ij A i A j , η ij ≡ a αβ ( i T αβ j ) We assume η is invertible, which allows A i to be removed from the action as anauxiliary field by its field equation. The action thus resembles the first-order form ofa standard Yang-Mills action: L = ( − η ij A i A j + A i ˆ F i ) − a αβi A α d i A β , ˆ F i ≡ a αβi ( d [ α A β ) + iA [ α A β ) )But ˆ F is not covariant; in fact the auxiliary field equation is η ij A j = ˆ F i .There are several nontrivial examples of such actions in the literature. One isthe minimal supersymmetrization of the usual 3D Chern-Simons form [1]: In thatcase d α are the usual 2 supersymmetry derivatives and d i the 3 translations, while T and a are both γ matrices. The auxiliary field equation is just the usual con-ventional constraint determining the vector gauge field in terms of the spinor. Asimilar expression yields the action for 4D N=1 super Yang-Mills (with α a 4-spinorindex), if the representation-preserving constraints are imposed by hand [2]. An-other example is the action for maximally supersymmetric Yang-Mills in 4D N=3harmonic superspace [3]: In that case the 2 d α and 1 d i are the 3 lowering operatorsof SU(3)/U(1) . Unlike the previous cases, not all the derivatives of the space appearin that Lagrangian. Dimensional reduction
In previous papers [4] we derived the affine Lie algebra of the superstring, contain-ing the generalization of the superparticle’s spinor and vector derivatives, but also aspinor “dual” to that spinor derivative. The spinor field strength of super Yang-Millsappeared as a “gauge field” to the new spinor derivative, in the same way as the usualspinor and vector gauge superfields did for the other two derivatives.In a recent paper [5] we considered the natural extension of this affine Lie algebrato include a derivative for which the antisymmetric tensor field strength is the gaugefield. It appeared as a necessary consequence of including spin operators in thealgebra, as their dual. (“Prepotentials” appeared as gauge fields for the spin.)We then applied this algebra directly to super Yang-Mills and supergravity, with-out direct reference to strings. Superspace was derived by a combination of isotropyconstraints, imposed in terms of gauge covariant derivatives (for the spin derivatives),and dimensional reduction, imposed in terms of dual symmetry generators. The for-mer eliminated also the corresponding gauge fields, in covariant gauges, while thelatter kept the corresponding gauge fields, but as field strengths (as in conventionalforms of dimensional reduction).In this paper we consider only bosonic Yang-Mills, so we limit our algebra ofderivatives to only the usual translations and the derivatives dual to spin. The algebrais a special case of that considered in the previous section:[ d a , d b ] = d ab , [ d ab , d c ] = [ d ab , d cd ] = 0with α = a , i = ab . We now have T a,bcd = δ c [ a δ db ] , a a,b,cd = η a [ c η d ] b , η ab,cd = η a [ c η d ] b ˆ F ab = d [ a A b ] + iA [ a A b ] In our previous paper we did not consider an action for the theory before dimen-sional reduction, although this is generally how this reduction is applied, especially insupersymmetric theories. We now see that the Chern-Simons-like action above is theone suited for this purpose: (1) It directly identifies the gauge field A ab for the dualspin coordinates as the usual field strength ˆ F , which is covariant upon reduction be-cause the noncovariant d ab term in its transformation law dies. (2) It gives the usualfirst-order Yang-Mills action upon reduction, since the extra A a d ab A b term drops out.In D=4, the extra dimensions (and thus the extra field) can be restricted to beself-dual. Conclusions
We have shown a natural way of introducing first-order formalisms, where theauxiliary field is originally nontrivial, appearing as a gauge field. This field (andthe corresponding extra dimensions) is also suggested by an algebraic analysis of thesuperstring. This approach might have advantages similar to those of the manifestlyT-dual formulation of the actions for low energy states of strings, which is treated asdimensional reduction from twice the coordinates [6].Generalization to higher spins (e.g., gravity) and supersymmetry should be con-sidered. For example, the usual auxiliary fields of 4D N=1 and 2 super Yang-Mills(dimension-2 scalars in the adjoint of U(1) and SU(2) R-symmetry, respectively)would appear as the gauge fields for extra dimensions dual to R-symmetry (insteadof spin), since they appear in generalized d’Alembertians as the R-symmetry ana-log of the relativistic Pauli term. Another possible application would be to find afirst-order formalism for string field theory, with on-shell field strengths built into theformalism. A first-quantized formulation of the approach would be a first step.
Acknowledgment
This work is supported in part by National Science Foundation Grant No. PHY-0969739.
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