Physics:Pure spinor

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Short description: Class of spinors constructed using Clifford algebras

In the domain of mathematics known as representation theory, pure spinors (or simple spinors) are spinors that are annihilated, under the Clifford algebra representation, by a maximal isotropic subspace of a vector space V with respect to a scalar product Q. They were introduced by Élie Cartan[1] in the 1930s and further developed by Claude Chevalley.[2]

They are a key ingredient in the study of spin structures and higher dimensional generalizations of twistor theory,[3] introduced by Roger Penrose in the 1960s. They have been applied to the study of supersymmetric Yang-Mills theory in 10D,[4][5] superstrings,[6] generalized complex structures[7] [8] and parametrizing solutions of integrable hierarchies.[9][10][11]

Clifford algebra and pure spinors

Consider a complex vector space V, with either even dimension 2n or odd dimension 2n+1, and a nondegenerate complex scalar product Q, with values Q(u,v) on pairs of vectors (u,v). The Clifford algebra Cl(V,Q) is the quotient of the full tensor algebra on V by the ideal generated by the relations

uv+vu=2Q(u,v), u,vV.

Spinors are modules of the Clifford algebra, and so in particular there is an action of the elements of V on the space of spinors. The complex subspace Vψ0V that annihilates a given nonzero spinor ψ has dimension mn. If m=n then ψ is said to be a pure spinor. In terms of stratification of spinor modules by orbits of the spin group Spin(V,Q), pure spinors correspond to the smallest orbits, which are the Shilov boundary of the stratification by the orbit types of the spinor representation on the irreducible spinor (or half-spinor) modules.

Pure spinors, defined up to projectivization, are called projective pure spinors. For V of even dimension 2n, the space of projective pure spinors is the homogeneous space SO(2n)/U(n); for V of odd dimension 2n+1, it is SO(2n+1)/U(n).

Irreducible Clifford module, spinors, pure spinors and the Cartan map

The irreducible Clifford/spinor module

Following Cartan[1] and Chevalley,[2] we may view V as a direct sum

V=VnVn*  or  V=VnVn*𝐂,

where VnV is a totally isotropic subspace of dimension n, and Vn* is its dual space, with scalar product defined as

Q(v1+w1,v2+w2):=w2(v1)+w1(v2),v1,v2Vn, w1,w2Vn*,

or

Q(v1+w1+a1,v2+w2+a2):=w2(v1)+w1(v2)+a1a2,a1,a2𝐂,

respectively.

The Clifford algebra representation ΓXEnd(Λ(Vn)) as endomorphisms of the irreducible Clifford/spinor module Λ(Vn), is generated by the linear elements XV, which act as

Γv(ψ)=vψ  (wedge product)  for vVn  and Γw(ψ)=ι(w)ψ  (inner product) for wVn*,

for either V=VnVn* or V=VnVn*𝐂, and

Γaψ=(1)pa ψ,a𝐂, ψΛp(Vn),

for V=VnVn*𝐂, when ψ is homogeneous of degree p.

Pure spinors and the Cartan map

A pure spinor ψ is defined to be any element ψΛ(Vn) that is annihilated by a maximal isotropic subspace wV with respect to the scalar product Q. Conversely, given a maximal isotropic subspace it is possible to determine the pure spinor that annihilates it, up to multiplication by a complex number, as follows.

Denote the Grassmannian of maximal isotropic (n-dimensional) subspaces of V as 𝐆𝐫n0(V,Q). The Cartan map [1][12][13]

𝐂𝐚:𝐆𝐫n0(V,Q)𝐏(Λ(Vn))

is defined, for any element w𝐆𝐫n0(V,Q), with basis (X1,,Xn), to have value

𝐂𝐚(w):=Im(ΓX1ΓXn);

i.e. the image of Λ(Vn) under the endomorphism formed from taking the product of the Clifford representation endomorphisms {ΓXiEnd(Λ(Vn))}i=1,,n, which is independent of the choice of basis (X1,,Xn). This is a 1-dimensional subspace, due to the isotropy conditions,

Q(Xi,Xj)=0,1i,jn,

which imply

ΓXiΓXj+ΓXjΓXi=0,1i,jn,

and hence 𝐂𝐚(w) defines an element of the projectivization 𝐏(Λ(Vn)) of the irreducible Clifford module Λ(Vn). It follows from the isotropy conditions that, if the projective class [ψ] of a spinor ψΛ(Vn) is in the image 𝐂𝐚(w) and Xw, then

ΓX(ψ)=0.

So any spinor ψ with [ψ]𝐂𝐚(w) is annihilated, under the Clifford representation, by all elements of w. Conversely, if ψ is annihilated by ΓX for all Xw, then [ψ]𝐂𝐚(w).

If V=VnVn* is even dimensional, there are two connected components in the isotropic Grassmannian 𝐆𝐫n0(V,Q), which get mapped, under 𝐂𝐚, into the two half-spinor subspaces Λ+(Vn),Λ(Vn) in the direct sum decomposition

Λ(Vn)=Λ+(Vn)Λ(Vn),

where Λ+(Vn) and Λ(Vn) consist, respectively, of the even and odd degree elements of Λ(Vn) .

The Cartan relations

Define a set of bilinear forms {βm}m=0,2n on the spinor module Λ(Vn), with values in Λm(V*)Λm(V) (which are isomorphic via the scalar product Q), by

βm(ψ,ϕ)(X1,,Xm):=β0(ψ,ΓX1ΓXmϕ),for ψ,ϕΛ(Vn), X1,,XmV,

where, for homogeneous elements ψΛp(Vn), ϕΛq(Vn) and volume form Ω on Λ(Vn),

β0(ψ,ϕ)Ω={ψϕif p+q=n0otherwise. 

As shown by Cartan,[1] pure spinors ψΛ(Vn) are uniquely determined by the fact that they satisfy the following set of homogeneous quadratic equations, known as the Cartan relations:[1][12][13]

βm(ψ,ψ)=0 mnmod(4),0m<n

on the standard irreducible spinor module.

These determine the image of the submanifold of maximal isotropic subspaces of the vector space V, with respect to the scalar product Q, under the Cartan map, which defines an embedding of the Grassmannian of isotropic subspaces of V in the projectivization of the spinor module (or half-spinor module, in the even dimensional case), realizing these as projective varieties.

There are therefore, in total,

0mn1mn, mod 4(dim(V)m)

Cartan relations, signifying the vanishing of the bilinear forms βm with values in the exterior spaces Λm(V) for mn, mod 4, corresponding to these skew symmetric elements of the Clifford algebra. However, since the dimension of the Grassmannian of maximal isotropic subspaces of V is 12n(n1) when V is of even dimension 2n and 12n(n+1) when V has odd dimension 2n+1, and the Cartan map is an embedding of the connected components of this in the projectivization of the half-spinor modules when V is of even dimension and in the irreducible spinor module if it is of odd dimension, the number of independent quadratic constraints is only

2n112n(n1)1

in the 2n dimensional case, and

2n12n(n+1)1

in the 2n+1 dimensional case.

In 6 dimensions or fewer, all spinors are pure. In 7 or 8  dimensions, there is a single pure spinor constraint. In 10 dimensions, there are 10 constraints

ψΓμψ=0,μ=1,,10,

where Γμ are the Gamma matrices that represent the vectors in 10 that generate the Clifford algebra. However, only 5 of these are independent, so the variety of projectivized pure spinors for V=10 is 10 (complex) dimensional.

Applications of pure spinors

Supersymmetric Yang Mills theory

For d=10 dimensional, N=1 supersymmetric Yang-Mills theory, the super-ambitwistor correspondence,[4][5] consists of an equivalence between the supersymmetric field equations and the vanishing of supercurvature along super null lines, which are of dimension (1|16), where the 16 Grassmannian dimensions correspond to a pure spinor. Dimensional reduction gives the corresponding results for d=6, N=2 and d=4, N=3 or 4.

String theory and generalized Calabi-Yau manifolds

Pure spinors were introduced in string quantization by Nathan Berkovits.[6] introduced generalized Calabi–Yau manifolds, where the generalized complex structure is defined by a pure spinor. These spaces describe the geometry of flux compactifications in string theory.

Integrable systems

In the approach to integrable hierarchies developed by Sato,[14] and his students,[15][16] equations of the hierarchy are viewed as compatibility conditions for commuting flows on an infinite dimensional Grassmannian. Under the (infinite dimensional) Cartan map, projective pure spinors are equivalent to elements of the infinite dimensional Grassmannian consisting of maximal isotropic subspaces of a Hilbert space under a suitably defined complex scalar product. They therefore serve as moduli for solutions of the BKP integrable hierarchy,[9][10][11] parametrizing the associated BKP τ-functions, which are generating functions for the flows. Under the Cartan map correspondence, these may be expressed as infinite dimensional Fredholm Pfaffians.[11]

References

  1. 1.0 1.1 1.2 1.3 1.4 Cartan, Élie (1981) [1938]. The theory of spinors. New York: Dover Publications. ISBN 978-0-486-64070-9. https://books.google.com/books?isbn=0486640701. 
  2. 2.0 2.1 Chevalley, Claude (1996). The Algebraic Theory of Spinors and Clifford Algebras (reprint ed.). Columbia University Press (1954); Springer (1996). ISBN 978-3-540-57063-9. 
  3. Penrose, Roger; Rindler, Wolfgang (1986) (in en). Spinors and Space-Time. Cambridge University Press. pp. Appendix. doi:10.1017/cbo9780511524486. ISBN 9780521252676. 
  4. 4.0 4.1 Witten, E. (1986). "Twistor-like transform in ten dimensions". Nuclear Physics B266 (2): 245–264. doi:10.1016/0550-3213(86)90090-8. Bibcode1986NuPhB.266..245W. 
  5. 5.0 5.1 Harnad, J.; Shnider, S. (1986). "Constraints and Field Equations for Ten Dimensional Super Yang-Mills Theory". Commun. Math. Phys. 106 (2): 183–199. doi:10.1007/BF01454971. Bibcode1986CMaPh.106..183H. http://projecteuclid.org/euclid.cmp/1104115696. 
  6. 6.0 6.1 {{cite journal |last1=Berkovits |first1=Nathan |year=2000 |title=Super-Poincare Covariant Quantization of the Superstring
  7. Hitchin, Nigel (2003). "Generalized Calabi-Yau manifolds". Quarterly Journal of Mathematics 54 (3): 281–308. doi:10.1093/qmath/hag025. 
  8. Gualtieri, Marco (2011). "Generalized complex geometry". Annals of Mathematics. (2) 174 (1): 75–123. doi:10.4007/annals.2011.174.1.3. 
  9. 9.0 9.1 Date, Etsuro; Jimbo, Michio; Kashiwara, Masaki; Miwa, Tetsuji (1982). "Transformation groups for soliton equations IV. A new hierarchy of soliton equations of KP type". Physica 4D (11): 343–365. 
  10. 10.0 10.1 Date, Etsuro; Jimbo, Michio; Kashiwara, Masaki; Miwa, Tetsuji (1983). M. Jimbo and T. Miwa. ed. "Transformation groups for soliton equations". In: Nonlinear Integrable Systems - Classical Theory and Quantum Theory (World Scientific (Singapore)): 943–1001. 
  11. 11.0 11.1 11.2 Balogh, F.; Harnad, J.; Hurtubise, J. (2021). "Isotropic Grassmannians, Plücker and Cartan maps". Journal of Mathematical Physics 62 (2): 121701. doi:10.1063/5.0021269. 
  12. 12.0 12.1 Harnad, J.; Shnider, S. (1992). "Isotropic geometry and twistors in higher dimensions. I. The generalized Klein correspondence and spinor flags in even dimensions". Journal of Mathematical Physics 33 (9): 3197–3208. doi:10.1063/1.529538. 
  13. 13.0 13.1 Harnad, J.; Shnider, S. (1995). "Isotropic geometry and twistors in higher dimensions. II. Odd dimensions, reality conditions, and twistor superspaces". Journal of Mathematical Physics 36 (9): 1945–1970. doi:10.1063/1.531096. 
  14. Sato, Mikio (1981). "Soliton equations as dynamical systems on infinite dimensional Grassmann manifolds". Kokyuroku, RIMS, Kyoto Univ.: 30–46. 
  15. Date, Etsuro; Jimbo, Michio; Kashiwara, Masaki; Miwa, Tetsuji (1981). "Operator Approach to the Kadomtsev-Petviashvili Equation–Transformation Groups for Soliton Equations III–". Journal of the Physical Society of Japan (Physical Society of Japan) 50 (11): 3806–3812. doi:10.1143/jpsj.50.3806. ISSN 0031-9015. Bibcode1981JPSJ...50.3806D. 
  16. Jimbo, Michio; Miwa, Tetsuji (1983). "Solitons and infinite-dimensional Lie algebras". Publications of the Research Institute for Mathematical Sciences (European Mathematical Society Publishing House) 19 (3): 943–1001. doi:10.2977/prims/1195182017. ISSN 0034-5318. 

Bibliography