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 configurations.)
Since imposing the self-duality constraint on the fields that enter the above action functional makes that functional vanish identically, self-dual higher gauge theory is notorious for being subtle in that either it does not have a Lagrangian field theory description, or else a somewhat intricate indirect one (e.g. Witten 96, p. 7, Belov-Moore 06a). But instead one may regard the self-duality condition rather as part of the quantum theory (Witten 96, p.8, Witten 99, section 3 DMW 00b, page 3), namely as a choice of polarization of the phase space of an unconstrained theory in one dimension higher. By such as “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 dimensional Chern-Simons theory in dimension .
The way this works is understood in much mathematical detail for , see at AdS3-CFT2 and CS-WZW correspondence. Motivated by this it has been proposed and studied in a fair bit of mathematical detail for (Witten 96, Hopkins-Singer 02), see at M5-brane and 6d (2,0)-superconformal QFT and AdS7/CFT6. In both these cases the higher gauge fields are cocycles in ordinary differential cohomology. In (DMW 00, Belov-Moore 06b) it is suggested that similarly taking the self-dual fields to be cocycles in (differential) complex K-theory produces the RR-fields of type II superstring theory in dimension 10.
Imagine the self-dual -form field on to be coupled to sources given by -form gauge fields, hence by cocycles in ordinary differential cohomology of degree . One may try to write down a Lagrangian for this coupling (and that is what is discussed in (Belov-Moore 06I)) and the resulting partition function is then locally a function of the source fields, hence a function on the intermediate Jacobian and globally (since the action functional will not be strictly gauge invariant) a holomorphic section of a certain holomorphic line bundle on that space.
However, no choice of Lagrangian will enforce a strictly self-dual gauge field (the anti-self dual component will remain in the space of fields, hence the partition function will depend on it). Therefore the idea is to turn this around, reject the idea that self-dual higher gauge theory is directly itself a Lagrangian quantum field theory, and instead define the partition function of the self-dual higher gauge theory (and thereby, indirectly, the self-dual higher gauge theory itself!) as being a certain section of a certain line bundle on the intermediate Jacobian.
The question then is: which line bundle? Some plausibility arguments show that it must be a Theta characteristic bundle, hence a square root of the canonical bundle on the intermediate Jacobian. These have 1-dimensional spaces of holomorphic sections (theta functions) and hence any choice of that already uniquely fixes the would-be partition function, too, up to a choice of global factor. (For that reason later authors often regard the partition function as a line bundle, somewhat abusing the terminology.)
Such quadratic refinement turns out to be given for by a Spin structure and for by a Wu structure (and the latter case is the actual example of interest in (Witten 96)). This is what the central theorem of (Hopkins-Singer 02) establishes rigorously.
One notices now that while the self-dual -dimensional gauge field theory thus itself is not a Lagrangian quantum field theory, it arises holographically form a field theory in dimension that is, namely the intersection pairing above is the Lagrangian for higher dimensional Chern-Simons theory and the holomorphic line bundle which it gives rise to on the intermediate Jacobian, is the prequantum line bundle of Chern-Simons theory, whose holomorphic sections are its quantum states (see at quantization of Chern-Simons theory). From this perspective the square root quadratic refinement is the metaplectic correction in the geometric quantization of Chern-Simobs theory.
This kind of relation
|self-dual d gauge theory||d Chern-Simons theory|
|partition function/correlator||wavefunction/quantum state|
|conformal blocks||space of quantum states|
is the hallmark of the holographic principle.
: 11-dimensional Chern-Simons theory (of fields which are cocycles in (twisted differential) complex K-theory) is related to a parts of a type II string theory on its boundary (or that of the space-filling D9-brane, if one wishes) (Belov-Moore 06b).
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.
(See also at Serre duality.)
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 96, Hopkins-Singer).
|Calabi-Cau n-fold||line n-bundle||moduli of line n-bundles||moduli of flat/degree-0 n-bundles||Artin-Mazur formal group of deformation moduli of line n-bundles||complex oriented cohomology theory||modular functor/self-dual higher gauge theory of higher dimensional Chern-Simons theory|
|unit in structure sheaf||multiplicative group/group of units||formal multiplicative group||complex K-theory|
|elliptic curve||line bundle||Picard group/Picard scheme||Jacobian||formal Picard group||elliptic cohomology||3d Chern-Simons theory/WZW model|
|K3 surface||line 2-bundle||Brauer group||intermediate Jacobian||formal Brauer group||K3 cohomology|
|Calabi-Yau 3-fold||line 3-bundle||intermediate Jacobian||CY3 cohomology||7d Chern-Simons theory/M5-brane|
For the self-dual theory abelian is that of a scalar field on a real-2-dimensional surface such that is self-dual. For any complex structure on , making it a complex torus, this makes a “chiral” boson, in the sense of a chiral half of the -WZW model.
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 96).
The RR-field in type II string theory are self-dual. Since the RR-fields are cocycles in (differential) K-theory, the proper discussion of this now involves generalizing from the above story ordinary cohomology and hence generalizing the concept of principally polarized intermediate Jacobians from ordinary cohomology to K-theory.
(Of course one may, as a warmup, “approximate” K-theory classes on a 10-manifold by integral cohomology lifts of the degree-5 component of their Chern character and hence discuss that as abelian self-dual higher gauge theory in dimension 10. This is discussed in (Witten 99, section 4)).
The relevant analog of the intermediate Jacobian for K-tehory on a 10-manifold is naturally taken to be
(where the action is via the Chern character/realification ).
where for ordinary cohomology the (quadratic refinement of the) intersection product (= Deligne-Beilinson cup product followed by fiber integration in ordinary differential cohomology) provided the symplectic structure/polarization of the intermediate Jacobian, here it is tensoring of Dirac operators followed by fiber integration in differential K-theory, hence the index map in K-theory.
|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|
Original articles on the general issue include
Self-duality for higher abelian gauge fields (in ordinary differential cohomology):
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
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.
D. Diaconescu, Gregory Moore, Edward Witten, Gauge Theory, and a Derivation of K-Theory from M-Theory, Adv.Theor.Math.Phys.6:1031-1134,2003 (arXiv:hep-th/0005090), summarised in A Derivation of K-Theory from M-Theory (arXiv:hep-th/0005091)
Stefan Müller-Stach, Chris Peters, Vasudevan Srinivas, Abelian varieties and theta functions associated to compact Riemannian manifolds; constructions inspired by superstring theory,Journal de mathématiques pures et appliquées 98.5 (2012): 542-573 (arXiv:1105.4108, pdf slides)
Review is in (Szabo 12, section 3.6 and 4.6).