Ktheory (physics)
String theory 

Fundamental objects 
Perturbative theory 
Nonperturbative results 
Phenomenology 
Mathematics 
Theorists

In string theory, the Ktheory classification refers to a conjectured application of Ktheory (in abstract algebra and algebraic topology) to superstrings, to classify the allowed RamondRamond field strengths as well as the charges of stable Dbranes.
In condensed matter physics Ktheory has also found important applications, specially in the topological classification of topological insulators, superconductors and stable Fermi Surfaces (Kitaev (2009), Horava (2005)).
Contents
History
This conjecture, applied to Dbrane charges, was first proposed by Minasian & Moore (1997). It was popularized by Witten (1998) who demonstrated that in type IIB string theory arises naturally from Ashoke Sen's realization of arbitrary Dbrane configurations as stacks of D9 and antiD9branes after tachyon condensation.
Such stacks of branes are inconsistent in a nontorsion NeveuSchwarz (NS) 3form background, which, as was highlighted by Kapustin (2000), complicates the extension of the Ktheory classification to such cases. Bouwknegt & Varghese (2000) suggested a solution to this problem: Dbranes are in general classified by a twisted Ktheory, that had earlier been defined by Rosenberg (1989).
Applications
String theory 

Fundamental objects 
Perturbative theory 
Nonperturbative results 
Phenomenology 
Mathematics 
Theorists

The Ktheory classification of Dbranes has had numerous applications. For example, Hanany & Kol (2000) used it to argue that there are eight species of orientifold oneplane. Uranga (2001) applied the Ktheory classification to derive new consistency conditions for flux compactifications. Ktheory has also been used to conjecture a formula for the topologies of Tdual manifolds by Bouwknegt, Evslin & Varghese (2004). Recently Ktheory has been conjectured to classify the spinors in compactifications on generalized complex manifolds.
Open problems
Despite these successes, RR fluxes are not quite classified by Ktheory. Diaconescu, Moore & Witten (2003) argued that the Ktheory classification is incompatible with Sduality in IIB string theory.
In addition, if one attempts to classify fluxes on a compact tendimensional spacetime, then a complication arises due to the selfduality of the RR fluxes. The duality uses the Hodge star, which depends on the metric and so is continuously valued and in particular is generically irrational. Thus not all of the RR fluxes, which are interpreted as the Chern characters in Ktheory, can be rational. However Chern characters are always rational, and so the Ktheory classification must be replaced. One needs to choose a half of the fluxes to quantize, or a polarization in the geometric quantizationinspired language of Diaconescu, Moore, and Witten and later of Varghese & Sati (2004). Alternately one may use the Ktheory of a 9dimensional time slice as has been done by Maldacena, Moore & Seiberg (2001).
Ktheory classification of RR fluxes
In the classical limit of type II string theory, which is type II supergravity, the RamondRamond field strengths are differential forms. In the quantum theory the welldefinedness of the partition functions of Dbranes implies that the RR field strengths obey Dirac quantization conditions when spacetime is compact, or when a spatial slice is compact and one considers only the (magnetic) components of the field strength which lie along the spatial directions. This led twentieth century physicists to classify RR field strengths using cohomology with integral coefficients.
However some authors have argued that the cohomology of spacetime with integral coefficients is too big. For example, in the presence of NeveuSchwarz Hflux or nonspin cycles some RR fluxes dictate the presence of Dbranes. In the former case this is a consequence of the supergravity equation of motion which states that the product of a RR flux with the NS 3form is a Dbrane charge density. Thus the set of topologically distinct RR field strengths that can exist in branefree configurations is only a subset of the cohomology with integral coefficients.
This subset is still too big, because some of these classes are related by large gauge transformations. In QED there are large gauge transformations which add integral multiples of two pi to Wilson loops. The pform potentials in type II supergravity theories also enjoy these large gauge transformations, but due to the presence of ChernSimons terms in the supergravity actions these large gauge transformations transform not only the pform potentials but also simultaneously the (p+3)form field strengths. Thus to obtain the space of inequivalent field strengths from the forementioned subset of integral cohomology we must quotient by these large gauge transformations.
The AtiyahHirzebruch spectral sequence constructs twisted Ktheory, with a twist given by the NS 3form field strength, as a quotient of a subset of the cohomology with integral coefficients. In the classical limit, which corresponds to working with rational coefficients, this is precisely the quotient of a subset described above in supergravity. The quantum corrections come from torsion classes and contain mod 2 torsion corrections due to the FreedWitten anomaly.
Thus twisted Ktheory classifies the subset of RR field strengths that can exist in the absence of Dbranes quotiented by large gauge transformations. Daniel Freed has attempted to extend this classification to include also the RR potentials using differential Ktheory.
Ktheory classification of Dbranes
Ktheory classifies Dbranes in noncompact spacetimes, intuitively in spacetimes in which we are not concerned about the flux sourced by the brane having nowhere to go. While the Ktheory of a 10d spacetime classifies Dbranes as subsets of that spacetime, if the spacetime is the product of time and a fixed 9manifold then Ktheory also classifies the conserved Dbrane charges on each 9dimensional spatial slice. While we were required to forget about RR potentials to obtain the Ktheory classification of RR field strengths, we are required to forget about RR field strengths to obtain the Ktheory classification of Dbranes.
Ktheory charge versus BPS charge
As has been stressed by Petr Hořava, the Ktheory classification of Dbranes is independent of, and in some ways stronger than, the classification of BPS states. Ktheory appears to classify stable Dbranes missed by supersymmetry based classifications.
For example, Dbranes with torsion charges, that is with charges in the order N cyclic group \mathbf Z_N, attract each other and so can never be BPS. In fact, N such branes can decay, whereas no superposition of branes that satisfy a Bogomolny bound may ever decay. However the charge of such branes is conserved modulo N, and this is captured by the Ktheory classification but not by a BPS classification. Such torsion branes have been applied, for example, to model DouglasShenker strings in supersymmetric U(N) gauge theories.
Ktheory from tachyon condensation
Ashoke Sen has conjectured that, in the absence of a topologically nontrivial NS 3form flux, all IIB brane configurations can be obtained from stacks of spacefilling D9 and anti D9 branes via tachyon condensation. The topology of the resulting branes is encoded in the topology of the gauge bundle on the stack of the spacefilling branes. The topology of the gauge bundle of a stack of D9s and anti D9s can be decomposed into a gauge bundle on the D9's and another bundle on the anti D9's. Tachyon condensation transforms such a pair of bundles to another pair in which the same bundle is direct summed with each component in the pair. Thus the tachyon condensation invariant quantity, that is, the charge which is conserved by the tachyon condensation process, is not a pair of bundles but rather the equivalence class of a pair of bundles under direct sums of the same bundle on both sides of the pair. This is precisely the usual construction of topological Ktheory. Thus the gauge bundles on stacks of D9's and antiD9's are classified by topological Ktheory. If Sen's conjecture is right, all Dbrane configurations in type IIB are then classified by Ktheory. Petr Horava has extended this conjecture to type IIA using D8branes.
Twisted Ktheory from MMS instantons
While the tachyon condensation picture of the Ktheory classification classifies Dbranes as subsets of a 10dimensional spacetime with no NS 3form flux, the Maldacena, Moore, Seiberg picture classifies stable Dbranes with finite mass as subsets of a 9dimensional spatial slice of spacetime.
The central observation is that Dbranes are not classified by integral homology because Dpbranes wrapping certain cycles suffer from a FreedWitten anomaly, which is cancelled by the insertion of D(p2)branes and sometimes D(p4)branes that end on the afflicted Dpbrane. These inserted branes may either continue to infinity, in which case the composite object has an infinite mass, or else they may end on an antiDpbrane, in which case the total Dpbrane charge is zero. In either case, one may wish to remove the anomalous Dpbranes from the spectrum, leaving only a subset of the original integral cohomology.
The inserted branes are unstable. To see this, imagine that they extend in time away (into the past) from the anomalous brane. This corresponds to a process in which the inserted branes decay via a Dpbrane that forms, wraps the forementioned cycle and then disappears. MMS^{[1]} refer to this process as an instanton, although really it need not be instantonic.
The conserved charges are thus the nonanomolous subset quotiented by the unstable insertions. This is precisely the AtiyahHirzebruch spectral sequence construction of twisted Ktheory as a set.
Reconciling twisted Ktheory and Sduality
Diaconescu, Moore, and Witten have pointed out that the twisted Ktheory classification is not compatible with the Sduality covariance of type IIB string theory. For example, consider the constraint on the RamondRamond 3form field strength G_{3} in the AtiyahHirzebruch spectral sequence (AHSS):
 d_3G_3=Sq^3G_3+H\cup G_3=G_3\cup G_3+H\cup G_3=0
where d_{3}=Sq^{3}+H is the first nontrivial differential in the AHSS, Sq^{3} is the third Steenrod square and the last equality follows from the fact that the nth Steenrod square acting on any nform x is x\cupx.
The above equation is not invariant under Sduality, which exchanges G_{3} and H. Instead Diaconescu, Moore, and Witten have proposed the following Sduality covariant extension
 G_3\cup G_3+H\cup G_3+H\cup H=P
where P is an unknown characteristic class that depends only on the topology, and in particular not on the fluxes. Diaconescu, Freed & Moore (2007) have found a constraint on P using the E_{8} gauge theory approach to Mtheory pioneered by Diaconescu, Moore, and Witten.
Thus Dbranes in IIB are not classified by twisted Ktheory after all, but some unknown Sdualitycovariant object that inevitably also classifies both fundamental strings and NS5branes.
However the MMS prescription for calculating twisted Ktheory is easily Scovariantized, as the FreedWitten anomalies respect Sduality. Thus the Scovariantized form of the MMS construction may be applied to construct the Scovariantized twisted Ktheory, as a set, without knowing having any geometric description for just what this strange covariant object is. This program has been carried out in a number of papers, such as Evslin & Varadarajan (2003) and Evslin (2003a), and was also applied to the classification of fluxes by Evslin (2003b). Bouwknegt et al. (2006) use this approach to prove Diaconescu, Moore, and Witten's conjectured constraint on the 3fluxes, and they show that there is an additional term equal to the D3brane charge. Evslin (2006) shows that the KlebanovStrassler cascade of Seiberg dualities consists of a series of Sdual MMS instantons, one for each Seiberg duality. The group, \mathbf Z_N of universality classes of the SU(M+N)\times SU(M) supersymmetric gauge theory is then shown to agree with the Sdual twisted Ktheory and not with the original twisted Ktheory.
Some authors have proposed radically different solutions to this puzzle. For example, Kriz & Sati (2005) propose that instead of twisted Ktheory, II string theory configurations should be classified by elliptic cohomology.
Researchers
Prominent researchers in this area include Ed Witten, Peter Bouwknegt, Angel Uranga, Emanuel Diaconescu, Gregory Moore, Anton Kapustin, Jonathan Rosenberg, Ruben Minasian, Amihay Hanany, Hisham Sati, Nathan Seiberg, Juan Maldacena, Daniel Freed, and Igor Kriz.
See also
Notes
 ^ Juan Maldacena, Gregory Moore and Nathan Seiberg. DBrane Instantons and KTheory Charges. http://arxiv.org/abs/hepth/0108100
References
 Bouwknegt, Peter; Evslin, Jarah; Jurco, Branislav; Varghese, Mathai; Sati, Hisham (2006), "Flux Compactifications on Projective Spaces and The SDuality Puzzle", Advances in Theoretical and Mathematical Physics 10: 345–394, Bibcode:2005hep.th....1110B, arXiv:hepth/0501110, doi:10.4310/atmp.2006.v10.n3.a3.
 Bouwknegt, Peter; Evslin, Jarah; Varghese, Mathai (2004), "TDuality: Topology Change from Hflux", Communications in Mathematical Physics 249 (2): 383–415, Bibcode:2004CMaPh.249..383B, arXiv:hepth/0306062, doi:10.1007/s0022000411156.
 Bouwknegt, Peter; Varghese, Mathai (2000), "Dbranes, Bfields and twisted Ktheory", Journal of High Energy Physics 0003 (007): 007, Bibcode:2000JHEP...03..007B, arXiv:hepth/0002023, doi:10.1088/11266708/2000/03/007.
 Diaconescu, Emanuel; Freed, Daniel S.; Moore, Gregory (2007), "The Mtheory 3form and E_{8} gauge theory", in Miller, Haynes R.; Ravenel, Douglas C., Elliptic Cohomology: Geometry, Applications, and Higher Chromatic Analogues, Cambridge University Press, pp. 44–88, Bibcode:2003hep.th...12069D, arXiv:hepth/0312069.
 Diaconescu, Emanuel; Moore, Gregory; Witten, Edward (2003), "E_{8} Gauge Theory, and a Derivation of KTheory from MTheory", Advances in Theoretical and Mathematical Physics 6: 1031–1134, Bibcode:2000hep.th....5090D, arXiv:hepth/0005090.
 Evslin, Jarah (2003a), "IIB Soliton Spectra with All Fluxes Activated", Nuclear Physics B 657: 139–168, Bibcode:2003NuPhB.657..139E, arXiv:hepth/0211172, doi:10.1016/S05503213(03)001548.
 Evslin, Jarah (2003b), "Twisted KTheory from Monodromies", Journal of High Energy Physics 0305 (030): 030, Bibcode:2003JHEP...05..030E, arXiv:hepth/0302081, doi:10.1088/11266708/2003/05/030.
 Evslin, Jarah (2006), "The Cascade is a MMS Instanton", Advances in Soliton Research, Nova Science Publishers, pp. 153–187, Bibcode:2004hep.th....5210E, arXiv:hepth/0405210.
 Evslin, Jarah; Varadarajan, Uday (2003), "KTheory and SDuality: Starting Over from Square 3", Journal of High Energy Physics 0303 (026): 026, Bibcode:2003JHEP...03..026E, arXiv:hepth/0112084, doi:10.1088/11266708/2003/03/026.
 Hanany, Amihay; Kol, Barak (2000), "On Orientifolds, Discrete Torsion, Branes and M Theory", Journal of High Energy Physics 0006 (013): 013, Bibcode:2000JHEP...06..013H, arXiv:hepth/0003025, doi:10.1088/11266708/2000/06/013.
 Kapustin, Anton (2000), "Dbranes in a topologically nontrivial Bfield", Advances in Theoretical and Mathematical Physics 4: 127–154, Bibcode:1999hep.th....9089K, arXiv:hepth/9909089.
 Kriz, Igor; Sati, Hisham (2005), "Type IIB String Theory, SDuality, and Generalized Cohomology", Nuclear Physics B 715 (3): 639–664, Bibcode:2005NuPhB.715..639K, arXiv:hepth/0410293, doi:10.1016/j.nuclphysb.2005.02.016.
 Maldacena, Juan; Moore, Gregory; Seiberg, Nathan (2001), "DBrane Instantons and KTheory Charges", Journal of High Energy Physics 0111 (062): 062, Bibcode:2001JHEP...11..062M, arXiv:hepth/0108100, doi:10.1088/11266708/2001/11/062.
 Minasian, Ruben; Moore, Gregory (1997), "Ktheory and RamondRamond charge", Journal of High Energy Physics 9711 (002): 002, Bibcode:1997JHEP...11..002M, arXiv:hepth/9710230, doi:10.1088/11266708/1997/11/002.
 Olsen, Kasper; Szabo, Richard J. (1999), "Constructing DBranes from KTheory", Advances in Theoretical and Mathematical Physics 3: 889–1025, Bibcode:1999hep.th....7140O, arXiv:hepth/9907140.
 Rosenberg, Jonathan (1989), "ContinuousTrace Algebras from the Bundle Theoretic Point of View", Journal of the Australian Mathematical Society, Series A 47 (03): 368–381, doi:10.1017/S1446788700033097.
 Uranga, Angel M. (2001), "Dbrane probes, RR tadpole cancellation and Ktheory charge", Nuclear Physics B 598: 225–246, Bibcode:2001NuPhB.598..225U, arXiv:hepth/0011048, doi:10.1016/S05503213(00)007872.
 Varghese, Mathai; Sati, Hisham (2004), "Some Relations between Twisted Ktheory and E_{8} Gauge Theory", Journal of High Energy Physics 0403 (016): 016, Bibcode:2004JHEP...03..016M, arXiv:hepth/0312033, doi:10.1088/11266708/2004/03/016.
 Witten, Edward (1998), "DBranes and KTheory", Journal of High Energy Physics 9812 (019): 019, Bibcode:1998JHEP...12..019W, arXiv:hepth/9810188, doi:10.1088/11266708/1998/12/019.
References (Condensed Matter Physics)
 Kitaev, Alexei (2009), Periodic table for topological insulators and superconductors, Bibcode:2009AIPC.1134...22K, arXiv:0901.2686, doi:10.1063/1.3149495.
 Horava, Petr (2005), "Stability of Fermi Surfaces and K Theory", Physical Review Letters 95 (016405).
Further reading
An excellent introduction to the Ktheory classification of Dbranes in 10 dimensions via Ashoke Sen's conjecture is the original paper "Dbranes and Ktheory" by Edward Witten; there is also an extensive review by Olsen & Szabo (1999).
A very comprehensible introduction to the twisted Ktheory classification of conserved Dbrane charges on a 9dimensional timeslice in the presence of NeveuSchwarz flux is Maldacena, Moore & Seiberg (2001).
External links
 Ktheory on arxiv.org