Types of quantum field thories
then the Hodge star operator squares to (Lorentian signature) or (Euclidean signature) on . Therefore it makes sense in these dimensions to impose the self-duality or chirality constraint
With this duality constraint imposed, one speaks of self-dual higher gauge fields or chiral higher gauge fields or higher gauge fields with self-dual curvature. (These are a higher degree/dimensional generalization of what in Yang-Mills theory are called Yang-Mills instanton field configruations.)
Their quantum field theory is more subtle than usual: first of all the above standard action functional now vanishes constantly.
But sense can be made of the theory of self-dual gauge fields by other means. Notably – by a version of the holographic principle the – partition function of the self-dual theory on an of dimension is given by the state (wave function) of an abelian higher Chern-Simons theory in dimension .
: 7-dimensional Chern-Simons theory is related to a fivebrane model on its boundary;
provides the polarization by splitting into self-dual and anti-self-dual forms:
notice that (by the formulas at Hodge star operator) we have on mid-dimensional forms
Therefore it provides a complex structure on .
We see that the symplectic structure on the space of forms can equivalently be rewritten as
Here on the right now the Hodge inner product of with appears, which is invariant under applying the Hodge star to both arguments.
We then decompose into the -eigenspaces of : say is imaginary self-dual if
and imaginary anti-self-dual if
Then for imaginary self-dual and we find that the symplectic pairing is
Therefore indeed the symplectic pairing vanishes on the self-dual and on the anti-selfdual forms. Evidently these provide a decomposition into Lagrangian subspaces.
Therefore a state of higher Chern-Simons theory on may locally be thought of as a function of the self-dual forms on . Under holography this is (therefore) identified with the correlator of a self-dual higher gauge theory on .
By the above discussion (…) the partition function of self-dual higher gauge theory is given by (a multiple of) the unique holomorphic section of a square root of the line bundle classified by the secondary intersection pairing. (Witten I, Hopkins-Singer).
The worldvolume theory of the M5-brane, the 6d (2,0)-superconformal QFT, contains a self-dual 2-form field. Its AdS7-CFT6 holographic description by 7-dimensional Chern-Simons theory is due to (Witten I).
|line bundle||square root||choice corresponds to|
|canonical bundle||Theta characteristic||over Riemann surface and Hermitian manifold (e.g.Kähler manifold): spin structure|
|density bundle||half-density bundle|
|canonical bundle of Lagrangian submanifold||metalinear structure||metaplectic correction|
|determinant line bundle||Pfaffian line bundle|
|quadratic secondary intersection pairing||partition function of self-dual higher gauge theory||integral Wu structure|
An survey and introduction is in
Original reference on self-dual/chiral fields include
Mans Henningson, Bengt E.W. Nilsson, Per Salomonson, Holomorphic factorization of correlation functions in (4k+2)-dimensional (2k)-form gauge theory (arXiv:hep-th/9908107)
M. Henningson, The quantum Hilbert space of a chiral two-form in dimensions (arxiv:hep-th/0111150)
The chiral boson in 2d is discussed in detail in
A quick exposition of the basic idea is in
A precise formulation of the phenomenon in terms of ordinary differential cohomology is given in
The Uncertainty of Fluxes Commun.Math.Phys.271:247-274 (2007) (arXiv:hep-th/0605198)
Heisenberg Groups and Noncommutative Fluxes , AnnalsPhys.322:236-285 (2007) (arXiv:hep-th/0605200)
Conceptual aspects of this are also discussed in section 6.2 of
Motivated by this the ordinary differential cohomology of self-dual fields had been discussed in
The generalization of this to generalized differential cohomology is discussed from p. 26 on in
More discussion of the general principle is in
Discussion of the quantum anomaly of self-dual theories is in
Samuel Monnier, The anomaly line bundle of the self-dual field theory (arXiv:1109.2904)
Samuel Monnier, Geometric quantization and the metric dependence of the self-dual field theory (arXiv:1011.5890)
For the case of nonabelian self-dual 1-form gauge fields see the references at Yang-Mills instanton.