# TGD: Physics as Infinite-Dimensional Geometry

Note: Newest contributions are at the top!

 Year 2014

### Higgs and p-adic mass calculations

In the earlier blog posting I told that the boundary condition for the Kähler-Dirac equation is massless Dirac equation with the analog of Higgs term added. The interpretation in terms of Higgs mechanism however fails since the term can be also tachyonic. Intriguingly, p-adic mass calculations require the ground state conformal weight to be negative half odd integer. This raises the question whether the the boundary condition for Kähler-Dirac equation could be equivalent for the mass formulation given by the condition that the scaling generator L0 annihilates the physical states for Super Virasoro representations. This equivalence is suggested by quantum classical correspondence.

If this is the case, the two mass shell conditions would be equivalent. This possibility is discussed more precisely below.

1. The boundary condition for K-D equation reads as

(pkγk+ Γn)Ψ=0 .

(pkγk is algebraic Dirac operator in Minkowski space and Γn is Kähler Dirac gamma matrix defined as contraction of the canonical energy momentum current of Kähler action with imbedding space gamma matrices.

Mass shell condition corresponds to the vanishing of the square of the algebraic Dirac operator and should be equivalent with the mass shell condition given by the vanishing of the action of L0:

pkpk== p2= m02× (hgr +n) ==mn2 .

m0 is CP2 mass scale dictated by CP2 size scale and analogous to that given by string tension. m0 is evaluated for the standard value of Planck constant. hgr is ground state conformal weight and n is the conformal weight assignable to the Super-Virasoro generators creating the state.

p-Adic mass calculations require that hgr is negative and half odd integer valued so that ground state would be tachyonic. The first principle explanation for this could be the presence of Minkowskian time-like contribution in Γn coming from the canonical momentum density for Kähler action. One cannot exclude a p-adically small deviation of hgr from the negative half odd integer value proportional to at least second power of prime p perhaps assignable to Higgs like contribution or contribution of string like objects assignable to elementary particle.

2. One can decompose Γn to M4 and CP2 parts corresponding to the contractions of the canonical momentum density with M4 and CP2 gamma matrices respectively:

Γn = Tn(M4)+ Tn(CP2) .

Tn(M4) involves M4 gamma matrices is determined by the energy momentum tensor TK of Kähler action determined by its imbedding space variation coming from the induced metric. Tn(CP2) involves CP2 gamma matrices and is sum coming from the imbedding space variations coming from a variation with respect to the induced metric and induced Kähler form. M4 and CP2 contributions are orthogonal to each other as imbedding space vectors.

3. The square of the mass boundary condition gives

(p+Tn(M4))k(p+Tn(M4))k +Tnk(CP2)Tnk(CP2)=0 .

This condition can be simplified if one assumes that the direction of classical energy momentum density vector Tnk(M4) is same as four momentum pk. This assumption is motivated by quantum classical correspondence. This would give

Tnk(M4)= α (x) pk .

The coeffcient α can depend on the position along string.

4. With these assumptions the condition reads

(1+α)2 p2 +Tnk(CP2)Tnk(CP2)=0 .

and gives

Tnk(CP2)Tnk(CP2)/(1+α)2=- mn2 .

where mn2 is the mass squared associated with the state as given by the vanishing of L0 action on the state.

In coordinate changes the left hand changes in position dependent manner but the change of the factor α compensates the change of T2(CP2) term so that the condition is general coordinate invariant statement.

5. Combiging this with the mass shell condition coming from the vanishing of the action of L0 gives

Tnk(CP2)Tnk(CP2)= -(1+α)2m02(hgr+n) .

One can solve α from this condition:

α=+/- S/Mn -1 , S2k== - Tnk(CP2)Tnk(CP2) (≥ 0) .

6. The interpretation of the effective metric defined by the Kähler-Dirac gamma matrices has been a longstanding problem. It seems that the geffnn of this metric appears naturally if one assumes that Super-Virasoro conditions for L0 is equivalent with that given by the boundary condition for Kähler-Dirac equation.
The conclusion is that the Higgs like term could provide classical space correlate for the basic stringy mass formulate. p-Adic mass calculations apply thermodynamics with mass squared replacing the energy in the usual thermodynamics. In Zero Energy Ontology p-adic thermodynamics is replaced with its square root and one would have quantum superposition of space-time surfaces with mass squared values mn2 with probabilities given by p-adic thermodynamics. The 3-momenta could be identical for these contributions but energies would differ. This does not break Lorenz invariance but would mean extremely small breaking of time translation invariance characterized by the inverse of p-adic prime. The breaking is of the order of of 10-38 for electron characterized by Mersenne prime M127. For the evolution of TGD view about the relationship of Higgs mechanism and p-adic mass calculations see the chapter "Recent view about Kähler geometry and spin structure of WCW".

### String world sheets, partonic 2-surfaces and vanishing of induced (classical) weak fields

Well-definedness of the em charge is the fundamental on spinor modes. Physical intuition suggests that also classical Z0 field should vanish - at least in scales longer than weak scale. Above the condition guaranteeing vanishing of em charge has been discussed at very general level. It has however turned out that one can understand situation by simply posing the simplest condition that one can imagine: the vanishing of classical W and possibly also Z0 fields inducing mixing of different charge states.

1. Induced W fields mean that the modes of Kähler-Dirac equation do not in general have well-defined em charge. The problem disappears if the induced W gauge fields vanish. This does not yet guarantee that couplings to classical gauge fields are physical in long scales. Also classical Z0 field should vanish so that the couplings would be purely vectorial. Vectoriality might be true in long enough scales only. If W and Z0 fields vanish in all scales then electroweak forces are due to the exchanges of corresponding gauge bosons described as string like objects in TGD and represent non-trivial space-time geometry and topology at microscopic scale.
2. The conditions solve also another long-standing interpretational problem. Color rotations induce rotations in electroweak-holonomy group so that the vanishing of all induced weak fields also guarantees that color rotations do not spoil the property of spinor modes to be eigenstates of em charge.
One can study the conditions quite concretely by using the formulas for the components of spinor curvature .
1. The representation of the covariantly constant curvature tensor is given by

R01= e0 ∧ e1-e2∧ e3 , R23= e0∧ e1- e2∧ e3 ,

R02=e0∧ e2-e3 ∧ e1 , R31 = -e0∧ e2+e3∧ e1 ,

R03 = 4e0∧ e3+2e1∧ e2 , R12 = 2e0∧ e3+4e1∧ e2 .

R01=R23 and R03= R31 combine to form purely left handed classical W boson fields and Z0 field corresponds to Z0=2R03.

Kähler form is given by

J= 2(e0∧e3+e1∧ e2) .

2. The vanishing of classical weak fields is guaranteed by the conditions

e0∧ e1-e2∧e3 =0 ,

e0∧ e2-e3 ∧e1 ,

4e0∧ e3+2e1∧e2 .

3. There are many manners to satisfy these conditions. For instance, the condition e1= a× e0 and e2=-a×e3 with arbitrary a which can depend on position guarantees the vanishing of classical W fields. The CP2 projection of the tangent space of the region carrying the spinor mode must be 2-D.

Also classical Z0 vanishes if a2= 2 holds true. This guarantees that the couplings of induced gauge potential are purely vectorial. One can consider other alternaties. For instance, one could require that only classical Z0 field or induced Kähler form is non-vanishing and deduce similar condition.

4. The vanishing of the weak part of induced gauge field implies that the CP2 projection of the region carrying spinor mode is 2-D. Therefore the condition that the modes of induced spinor field are restricted to 2-surfaces carrying no weak fields sheets guarantees well-definedness of em charge and vanishing of classical weak couplings. This condition does not imply string world sheets in the general case since the CP2 projection of the space-time sheet can be 2-D.
How string world sheets could emerge?
1. Additional consistency condition to neutrality of string world sheets is that Kähler-Dirac gamma matrices have no components orthogonal to the 2-surface in question. Hence various fermionic would flow along string world sheet.
2. If the Kähler-Dirac gamma matrices at string world sheet are expressible in terms of two non-vanishing gamma matrices parallel to string world sheet and sheet and thus define an integrable distribution of tangent vectors, this is achieved. What is important that modified gamma matrices can indeed span lower than 4-D space and often do so (massless extremals and vacuum extremals representative examples). Induced gamma matrices defined always 4-D space so that the restriction of the modes to string world sheets is not possible.
3. String models suggest that string world sheets are minimal surfaces of space-time surface or of imbedding space but it might not be necessary to pose this condition separately.
In the proposed scenario string world sheets emerge rather than being postulated from beginning.
1. The vanishing conditions for induced weak fields allow also 4-D spinor modes if they are true for entire spatime surface. This is true if the space-time surface has 2-D projection. One can expect that the space-time surface has foliation by string world sheets and the general solution of K-D equation is continuous superposition of the 2-D modes in this case and discrete one in the generic case.
2. If the CP2 projection of space-time surface is homologically non-trivial geodesic sphere S2, the field equations reduce to those in M4× S2 since the second fundamental form for S2 is vanishing. It is possible to have geodesic sphere for which induced gauge field has only em component?
3. If the CP2 projection is complex manifold as it is for string like objects, the vanishing of weak fields might be also achieved.
4. Does the phase of cosmic strings assumed to dominate primordial cosmology correspond to this phase with no classical weak fields? During radiation dominated phase 4-D string like objects would transform to string world sheets.Kind of dimensional transmutation would occur.
Right-handed neutrino has exceptional role in K-D action.
1. Electroweak gauge potentials do not couple to νR at all. Therefore em neutrality condition is un-necessary if the induced gamma matrices do not mix right handed neutrino with left-handed one. This is guaranteed if M4 and CP2 parts of Kähler-Dirac operator annihilate separately right-handed neutrino spinor mode. Also νR modes can be interpreted as continuous superpositions of 2-D modes and this allows to define overlap integrals for them and induced spinor fields needed to define WCW gamma matrices and super-generators.
2. For covariantly constant right-handed neutrino mode defining a generator of super-symmetries is certainly a solution of K-D. Whether more general solutions of K-D exist remains to be checked out.

### Class field theory and TGD: does TGD reduce to number theory?

The intriguing general result of class field theory) -something extremely abstract for physicist's brain - is that the the maximal Abelian extension for rationals is homomorphic with the multiplicative group of ideles. This correspondence plays a key role in Langlands correspondence (see this,this, this, and this).

Does this mean that it is not absolutely necessary to introduce p-adic numbers? This is actually not so. The Galois group of the maximal abelian extension is rather complex objects (absolute Galois group, AGG, defines as the Galois group of algebraic numbers is even more complex!). The ring Z of adeles defining the group of ideles as its invertible elements homeomorphic to the Galois group of maximal Abelian extension is profinite group. This means that it is totally disconnected space as also p-adic integers and numbers are. What is intriguing that p-dic integers are however a continuous structure in the sense that differential calculus is possible. A concrete example is provided by 2-adic units consisting of bit sequences which can have literally infinite non-vanishing bits. This space is formally discrete but one can construct differential calculus since the situation is not democratic. The higher the pinary digit in the expansion is, the less significant it is, and p-adic norm approaching to zero expresses the reduction of the insignificance.

1. Could TGD based physics reduce to a representation theory for the Galois groups of quaternions and octonions?

Number theoretical vision about TGD raises questions about whether adeles and ideles could be helpful in the formulation of TGD. I have already earlier considered the idea that quantum TGD could reduce to a representation theory of appropriate Galois groups. I proceed to make questions.

1. Could real physics and various p-adic physics on one hand, and number theoretic physics based on maximal Abelian extension of rational octonions and quaternions on one hand, define equivalent formulations of physics?
2. Besides various p-adic physics all classical number fields (reals, complex numbers, quaternions, and octonions) are central in the number theoretical vision about TGD. The technical problem is that p-adic quaternions and octonions exist only as a ring unless one poses some additional conditions. Is it possible to pose such conditions so that one could define what might be called quaternionic and octonionic adeles and ideles?

It will be found that this is the case: p-adic quaternions/octonions would be products of rational quaternions/octonions with a p-adic unit. This definition applies also to algebraic extensions of rationals and makes it possible to define the notion of derivative for corresponding adeles. Furthermore, the rational quaternions define non-commutative automorphisms of quaternions and rational octonions at least formally define a non-associative analog of group of octonionic automorphisms (see this).

3. I have already earlier considered the idea about Galois group as the ultimate symmetry group of physics. The representations of Galois group of maximal Abelian extension (or even that for algebraic numbers) would define the quantum states. The representation space could be group algebra of the Galois group and in Abelian case equivalently the group algebra of ideles or adeles. One would have wave functions in the space of ideles.

The Galois group of maximal Abelian extension would be the Cartan subgroup of the absolute Galois group of algebraic numbers associated with given extension of rationals and it would be natural to classify the quantum states by the corresponding quantum numbers (number theoretic observables).

If octonionic and quaternionic (associative) adeles make sense, the associativity condition would reduce the analogs of wave functions to those at 4-dimensional associative sub-manifolds of octonionic adeles identifable as space-time surfaces so that also space-time physics in various number fields would result as representations of Galois group in the maximal Abelian Galois group of rational octonions/quaternions. TGD would reduce to classical number theory!

4. Absolute Galois group is the Galois group of the maximal algebraic extension and as such a poorly defined concept. One can however consider the hierarchy of all finite-dimensional algebraic extensions (including non-Abelian ones) and maximal Abelian extensions associated with these and obtain in this manner a hierarchy of physics defined as representations of these Galois groups homomorphic with the corresponding idele groups.
5. In this approach the symmetries of the theory would have automatically adelic representations and one might hope about connection with Langlands program.

2. Adelic variant of space-time dynamics and spinorial dynamics?

As an innocent novice I can continue to pose stupid questions. Now about adelic variant of the space-time dynamics based on the generalization of Kähler action discussed already earlier but without mentioning adeles (see this).

1. Could one think that adeles or ideles could extend reals in the formulation of the theory: note that reals are included as Cartesian factor to adeles. Could one speak about adelic or even idelic space-time surfaces endowed with adelic or idelic coordinates? Could one formulate variational principle in terms of adeles so that exponent of action would be product of actions exponents associated with various factors with Neper number replaced by p for Zp. The minimal interpretation would be that in adelic picture one collects under the same umbrella real physics and various p-adic physics.
2. Number theoretic vision suggests that 4:th/8:th Cartesian powers of adeles have interpretation as adelic variants of quaternions/ octonions. If so, one can ask whether adelic quaternions and octonions could have some number theretical meaning. Note that adelic quaternions and octonions are not number fields without additional assumptions since the moduli squared for a p-adic analog of quaternion and octonion can vanish so that the inverse fails to exist.

If one can pose a condition guaranteing the existence of inverse, one could define the multiplicative group of ideles for quaternions. For octonions one would obtain non-associative analog of the multiplicative group. If this kind of structures exist then four-dimensional associative/co-associative sub-manifolds in the space of non-associative ideles define associative/co-associative ideles and one would end up with ideles formed by associative and co-associative space-time surfaces.

3. What about equations for space-time surfaces. Do field equations reduce to separate field equations for each factor? Can one pose as an additional condition the constraint that p-adic surfaces provide in some sense cognitive representations of real space-time surfaces: this idea is formulated more precisely in terms of p-adic manifold concept (see this). Or is this correspondence an outcome of evolution?

Physical intuition would suggest that in most p-adic factors space-time surface corresponds to a point, or at least to a vacuum extremal. One can consider also the possibility that same algebraic equation describes the surface in various factors of the adele. Could this hold true in the intersection of real and p-adic worlds for which rationals appear in the polynomials defining the preferred extremals.

4. To define field equations one must have the notion of derivative. Derivative is an operation involving division and can be tricky since adeles are not number field. If one can guarantee that the p-adic variants of octonions and quaternions are number fields, there are good hopes about well-defined derivative. Derivative as limiting value df/dx= lim ( f(x+dx)-f(x))/dx for a function decomposing to Cartesian product of real function f(x) and p-adic valued functions fp(xp) would require that fp(x) is non-constant only for a finite number of primes: this is in accordance with the physical picture that only finite number of p-adic primes are active and define "cognitive representations" of real space-time surface. The second condition is that dx is proportional to product dx × ∏ dxp of differentials dx and dxp, which are rational numbers. dx goes to xero as a real number but not p-adically for any of the primes involved. dxp in turn goes to zero p-adically only for Qp.
5. The idea about rationals as points commont to all number fields is central in number theoretical vision. This vision is realized for adeles in the minimal sense that the action of rationals is well-defined in all Cartesian factors of the adeles. Number theoretical vision allows also to talk about common rational points of real and various p-adic space-time surfaces in preferred coordinate choices made possible by symmetries of the imbedding space, and one ends up to the vision about life as something residing in the intersection of real and p-adic number fields. It is not clear whether and how adeles could allow to formulate this idea.
6. For adelic variants of imbedding space spinors Cartesian product of real and p-adc variants of imbedding spaces is mapped to their tensor product. This gives justification for the physical vision that various p-adic physics appear as tensor factors. Does this mean that the generalized induced spinors are infinite tensor products of real and various p-adic spinors and Clifford algebra generated by induced gamma matrices is obtained by tensor product construction? Does the generalization of massless Dirac equation reduce to a sum of d'Alembertians for the factors? Does each of them annihilate the appropriate spinor? If only finite number of Cartesian factors corresponds to a space-time surface which is not vacuum extremal vanishing induced Kähler form, Kähler Dirac equation is non-trivial only in finite number of adelic factors.

3. Objections

The basic idea is that appropriately defined invertible quaternionic/octonionic adeles can be regarded as elements of Galois group assignable to quaternions/octonions. The best manner to proceed is to invent objections against this idea.

1. The first objection is that p-adic quaternions and octonions do not make sense since p-adic variants of quaternions and octonions do not exist in general. The reason is that the p-adic norm squared ∑ xi2 for p-adic variant of quaternion, octonion, or even complex number can vanish so that its inverse does not exist.
2. Second objection is that automorphisms of the ring of quaternions (octonions) in the maximal Abelian extension are products of transformations of the subgroup of SO(3) (G2) represented by matrices with elements in the extension and in the Galois group of the extension itself. Ideles separate out as 1-dimensional Cartesian factor from this group so that one does not obtain 4-field (8-fold) Cartesian power of this Galois group.
If the p-adic variants of quaternions/octonions are be rational quaternions/octonions multiplied by p-adic number, these objections can be circumvented.
1. This condition indeed allows to construct the inverse of p-adic quaternion/octonion as a product of inverses for rational quaternion/octonion and p-adic number! The reason is that the solutions to ∑ xi2=0 involve always p-adic numbers with an infinite number of pinary digits - at least one and the identification excludes this possibility.
2. This restriction would give a rather precise content for the idea of rational physics since all p-adic space-time surfaces would have a rational backbone in well-defined sense.
3. One can interpret also the quaternionicity/octonionicity in terms of Galois group. The 7-dimensional non-associative counterparts for octonionic automorphisms act as transformations x→ gxg-1. Therefore octonions represent this group like structure and the p-adic octonions would have interpretation as combination of octonionic automorphisms with those of rationals.

Adelic variants of of octonions would represent a generalization of these transformations so that they would act in all number fields. Quaternionic 4-surfaces would define associative local sub-groups of this group-like structure. Thus a generalization of symmetry concept reducing for solutions of field equations to the standard one would allow to realize the vision about the reduction of physics to number theory.

See the new chapter Unified Number Theoretical Vision or the article with the same title.

### General ideas about octonions, quaternions, and twistors

Octonions, quaternions, quaternionic space-time surfaces, octonionic spinors and twistors and twistor spaces are highly relevant for quantum TGD. In the following some general observations distilled during years are summarized.

There is a beautiful pattern present suggesting that H=M4× CP2 is completely unique on number theoretical grounds. Consider only the following facts. M4 and CP2 are the unique 4-D spaces allowing twistor space with Kähler structure. M8-H duality allows to deduce M4× CP2 via number theoretical compactification. Octonionic projective space OP2 appears as octonionic twistor space (there are no higher-dimensional octonionic projective spaces). Octotwistors generalise the twistorial construction from M4 to M8 and octonionic gamma matrices make sense also for H with quaternionicity condition reducing OP2 to to the twistor space of H.

A further fascinating structure related to octo-twistors is the non-associated analog of Lie group defined by automorphisms by octonionic imaginary units: this group is topologically six-sphere. Also the analogy of quaternionicity of preferred extremals in TGD with the Majorana condition central in super string models is very thought provoking. All this suggests that associativity indeed could define basic dynamical principle of TGD.

See the new chapter Unified Number Theoretical Vision or the article.

### Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds"?

The construction of Kähler geometry of WCW ("world of classical worlds") is fundamental to TGD program. I ended up with the idea about physics as WCW geometry around 1985 and made a breakthrough around 1990, when I realized that Kähler function for WCW could correspond to Kähler action for its preferred extremals defining the analogs of Bohr orbits so that classical theory with Bohr rules would become an exact part of quantum theory and path integral would be replaced with genuine integral over WCW. The motivating construction was that for loop spaces leading to a unique Kähler geometry. The geometry for the space of 3-D objects is even more complex than that for loops and the vision still is that the geometry of WCW is unique from the mere existence of Riemann connection.

The basic idea is that WCW is union of symmetric spaces G/H labelled by zero modes which do not contribute to the WCW metric. There have been many open questions and it seems the details of the earlier approach must be modified at the level of detailed identifications and interpretations. What is satisfying that the overall coherence of the picture has increased dramatically and connections with string model and applications of TGD as WCW geometry to particle physics are now very concrete.

1. A longstanding question has been whether one could assign Equivalence Principle (EP) with the coset representation formed by the super-Virasoro representation assigned to G and H in such a manner that the four- momenta associated with the representations and identified as inertial and gravitational four-momenta would be identical. This does not seem to be the case. The recent view will be that EP reduces to the view that the classical four- momentum associated with Kähler action is equivalent with that assignable to modified Dirac action supersymmetrically related to Kähler action: quantum classical correspondence (QCC) would be in question. Also strong form of general coordinate invariance implying strong form of holography in turn implying that the super-symplectic representations assignable to space-like and light-like 3-surfaces are equivalent could imply EP with gravitational and inertial four-momenta assigned to these two representations.
2. The detailed identification of groups G and H and corresponding algebras has been a longstanding problem. Symplectic algebra associated with δM4+/-× CP2 (δM4+/- is light-cone boundary - or more precisely, with the boundary of causal diamond (CD) defined as Cartesian product of CP2 with intersection of future and past direct light cones of M4 has Kac-Moody type structure with light-like radial coordinate replacing complex coordinate z. Virasoro algebra would correspond to radial diffeomorphisms.

I have also introduced Kac-Moody algebra assigned to the isometries and localized with respect to internal coordinates of the light-like 3-surfaces at which the signature of the induced metric changes from Minkowskian to Euclidian and which serve as natural correlates for elementary particles (in very general sense!). This kind of localization by force could be however argued to be rather ad hoc as opposed to the inherent localization of the symplectic algebra containing the symplectic algebra of isometries as sub-algebra. It turns out that one obtains direct sum of representations of symplectic algebra and Kac-Moody algebra of isometries naturally as required by the success of p-adic mass calculations.

3. The dynamics of Kähler action is not visible in the earlier construction. The construction also expressed WCW Hamiltonians as 2-D integrals over partonic 2-surfaces. Although strong form of general coordinate invariance (GCI) implies strong form of holography meaning that partonic 2-surfaces and their 4-D tangent space data should code for quantum physics, this kind of outcome seems too strong. The progress in the understanding of the solutions of modified Dirac equation led however to the conclusion that spinor modes other than right-handed neutrino are localized at string world sheets with strings connecting different partonic 2-surfaces.

This leads to a modification of earlier construction in which WCW super-Hamiltonians were essentially 2-D flux integrals. Now they are 2-D flux integrals with super-Hamiltonian replaced Noether super charged for the deformations in G and obtained by integrating over string at each point of partonic 2-surface. Each spinor mode gives rise to super current and the modes of right-handed neutrino and other fermions differ in an essential manner. Right-handed neutrino would correspond to symplectic algebra and other modes to the Kac-Moody algebra and one obtains the crucial 5 tensor factors of Super Virasoro required by p-adic mass calculations.

The matrix elements of WCW metric between Killing vectors are expressible as anticommutators of super-Hamiltonians identifiable as contractions of WCW gamma matrices with these vectors and give Poisson brackets of corresponding Hamiltonians. The anti-commutation relates of induced spinor fields are dictated by this condition. Everything is 3-dimensional although one expects that symplectic transformations localized within interior of X3 act as gauge symmetries so that in this sense effective 2-dimensionality is achieved. The components of WCW metric are labelled by standard model quantum numbers so that the connection with physics is extremely intimate.

4. An open question in the earlier visions was whether finite measurement resolution is realized as discretization at the level of fundamental dynamics. This would mean that only certain string world sheets from the slicing by string world sheets and partonic 2-surfaces are possible. The requirement that anti-commutations are consistent suggests that string world sheets correspond to surfaces for which Kähler magnetic field is constant along string in well defined sense (Jμνεμνg1/2 remains constant along string). It however turns that by a suitable choice of coordinates of 3-surface one can guarantee that this quantity is constant so that no additional constraint results.
See the new chapter Recent View about Kähler Geometry and Spin Structure of "World of Classical Worlds or the article with the same title.