spin^c structure


Higher spin geometry



Special and general types

Special notions


Extra structure






For nn \in \mathbb{N} the Lie group spin^c is a central extension

U(1)Spin c(n)SO(n) U(1) \to Spin^c(n) \to SO(n)

of the special orthogonal group by the circle group. This comes with a long fiber sequence

BU(1)BSpin c(n)BSO(n)W 3B 2U(1), \cdots \to B U(1) \to B Spin^c(n) \to B SO(n) \stackrel{W_3}{\to} B^2 U(1) \,,

where W 3W_3 is the third integral Stiefel-Whitney class .

An oriented manifold XX has Spin cSpin^c-structure if the characteristic class [W 3(X)]H 3(X,)[W_3(X)] \in H^3(X, \mathbb{Z})

W 3(X)W 3(TX):XTXBSO(n)W 3B 2U(1)K(,3) W_3(X) \coloneqq W_3(T X) \;\colon\; X \stackrel{T X}{\to} B SO(n) \stackrel{W_3}{\to} B^2 U(1) \simeq K(\mathbb{Z},3)

is trivial. This is the Dixmier-Douady class of the circle 2-bundle/bundle gerbe that obstructs the existence of a Spin cSpin^c-principal bundle lifting the given tangent bundle.

A manifold XX is equipped with Spin cSpin^c-structure η\eta if it is equipped with a choice of trivializaton

η:1W 3(TX). \eta : 1 \stackrel{\simeq}{\to} W_3(T X) \,.

The homotopy type/∞-groupoid of Spin cSpin^c-structures on XX is the homotopy fiber W 3Struc(TX)W_3 Struc(T X) in the pasting diagram of homotopy pullbacks

W 3Struc(TX) W 3Struc(X) * * TX Top(X,BSO(n)) W 3 Top(X,B 2U(1)). \array{ W_3 Struc(T X) &\to& W_3 Struc(X) &\to& * \\ \downarrow && \downarrow && \downarrow \\ * &\stackrel{T X}{\to}& Top(X, B SO(n)) &\stackrel{W_3}{\to}& Top(X,B^2 U(1)) } \,.

If the class does not vanish and if hence there is no Spin cSpin^c-structure, it still makes sense to discuss the structure that remains as twisted spin^c structure .


Since U(1)Spin cSOU(1) \to Spin^c \to SO is a sequence of Lie groups, the above may be lifted from the (∞,1)-topos L wheL_{whe} Top \simeq ∞Grpd of discrete ∞-groupoids to that of smooth ∞-groupoids, Smooth∞Grpd.

More in detail, by the discussion at Lie group cohomology (and smooth ∞-groupoid -- structures) the characteristic map W 3:BSOB 2U(1)W_3 : B SO \to B^2 U(1) in Grpd\infty Grpd has, up to equivalence, a unique lift

W 3:BSOB 2U(1) \mathbf{W}_3 : \mathbf{B} SO \to \mathbf{B}^2 U(1)

to Smooth∞Grpd, where on the right we have the delooping of the smooth circle 2-group.

Accordingly, the 2-groupoid of smooth spin cspin^c-structures W 3Struc(X)\mathbf{W}_3 Struc(X) is the joint (∞,1)-pullback

W 3Struc(TX) W 3Struc(X) * * TX SmoothGrpd(X,BSO(n)) W 3 SmoothGrpd(X,B 2U(1)). \array{ \mathbf{W}_3 Struc(T X) &\to& \mathbf{W}_3 Struc(X) &\to& * \\ \downarrow && \downarrow && \downarrow \\ * &\stackrel{T X}{\to}& Smooth \infty Grpd(X, \mathbf{B} SO(n)) &\stackrel{\mathbf{W}_3}{\to}& Smooth \infty Grpd(X,\mathbf{B}^2 U(1)) } \,.

Higher spin cspin^c-structures

In parallel to the existence of higher spin structures there are higher analogs of Spin cSpin^c-structures, related to quantum anomaly cancellation of theories of higher dimensional branes.


Of Spin cSpin^c


The group Spin cSpin^c is the fiber product

Spin c :=Spin× 2U(1) =(Spin×U(1))/ 2, \begin{aligned} Spin^c & := Spin \times_{\mathbb{Z}_2} U(1) \\ & = (Spin \times U(1))/{\mathbb{Z}_2} \,, \end{aligned}

where in the second line the action is the diagonal action induced from the two canonical embeddings of subgroups 2\mathbb{Z}_2 \hookrightarrow \mathbb{Z} and 2U(1)\mathbb{Z}_2 \hookrightarrow U(1).


We have a homotopy pullback diagram

BSpin c BU(1) c 1mod2 BSO w 2 B 2 2. \array{ \mathbf{B} Spin^c &\to& \mathbf{B}U(1) \\ \downarrow && \downarrow^{\mathrlap{\mathbf{c}_1 mod 2}} \\ \mathbf{B} SO &\stackrel{w_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 } \,.

We present this as usual by simplicial presheaves and ∞-anafunctors.

The first Chern class is given by the ∞-anafunctor

B() c 1 B(1)=B 2 BU(1). \array{ \mathbf{B}(\mathbb{Z} \to \mathbb{R}) &\stackrel{c_1}{\to}& \mathbf{B}(\mathbb{Z} \to 1) = \mathbf{B}^2 \mathbb{Z} \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B} U(1) } \,.

The second Stiefel-Whitney class is given by

B( 2Spin) w 2 B( 21)=B 2 2 BSO. \array{ \mathbf{B}(\mathbb{Z}_2 \to Spin) &\stackrel{w_2}{\to}& \mathbf{B}(\mathbb{Z}_2 \to 1) = \mathbf{B}^2 \mathbb{Z}_2 \\ \downarrow^{\mathrlap{\simeq}} \\ \mathbf{B} SO } \,.

Notice that the top horizontal morphism here is a fibration.

Therefore the homotopy pullback in question is given by the ordinary pullback

Q B() B( 2Spin) B 2 2. \array{ Q &\to& \mathbf{B}(\mathbb{Z} \to \mathbb{R}) \\ \downarrow && \downarrow \\ \mathbf{B}(\mathbb{Z}_2 \to Spin) &\to& \mathbf{B}^2 \mathbb{Z}_2 } \,.

This pullback is B(Spin×)\mathbf{B}(\mathbb{Z} \stackrel{\partial}{\to} Spin \times \mathbb{R}), where

:n(nmod2,n). \partial : n \mapsto ( n mod 2 , n) \,.

This is equivalent to

B( 2Spin×U(1)) \mathbf{B}(\mathbb{Z}_2 \stackrel{\partial'}{\to} Spin \times U(1))

where now

:σ(σ,σ). \partial' : \sigma \mapsto (\sigma, \sigma) \,.

This in turn is equivalent to

B(Spin× 2U(1)), \mathbf{B} (Spin \times_{\mathbb{Z}_2} U(1)) \,,

which is the original definition.

This factors the above characterization of BSpin c\mathbf{B}Spin^c as the homotopy fiber of W 3\mathbf{W}_3:


We have a pasting diagram of homotopy pullbacks of smooth infinity-groupoids of the form

BSpin c BU(1) * c 1mod2 BSO w 2 B 2 2β 2B 2U(1). \array{ \mathbf{B} Spin^c &\to& \mathbf{B}U(1) &\to& \ast \\ \downarrow && \downarrow^{\mathrlap{c_1 \, mod\, 2}} && \downarrow \\ \mathbf{B}SO &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 \stackrel{\mathbf{\beta}_2}{\to} \mathbf{B}^2 U(1) } \,.

This is discussed at Spin^c – Properties – As the homotopy fiber of smooth w3.

As KUKU-orientation

For XX an oriented manifold, the map X*X \to \ast is generalized oriented in periodic complex K-theory precisely if XX has a Spin cSpin^c-structure.

Relation to metaplectic structures

Let (X,ω)(X,\omega) be a compact symplectic manifold equipped with a Kähler polarization 𝒫\mathcal{P} hence a Kähler manifold structure JJ. A metaplectic structure of this data is a choice of square root Ω 0,n\sqrt{\Omega^{0,n}} of the canonical line bundle. This is equivalently a spin structure on XX (see the discussion at Theta characteristic).

Now given a prequantum line bundle L ωL_\omega, in this case the Dolbault quantization of L ωL_\omega coincides with the spin^c quantization of the spin^c structure induced by JJ and L ωΩ 0,nL_\omega \otimes \sqrt{\Omega^{0,n}}.

This appears as (Paradan 09, prop. 2.2).


From almost complex structures

An almost complex structure canonically induces a Spin cSpin^c-structure:


For all nn \in \mathbb{N} we have a homotopy-commuting diagram

BU(n) BU(1) c 1mod2 BSO(2n) w 2 B 2 2, \array{ \mathbf{B}U(n) &\to& \mathbf{B}U(1) \\ \downarrow &\swArrow& \downarrow^{\mathrlap{\mathbf{c_1} mod 2}} \\ \mathbf{B}SO(2n) &\stackrel{\mathbf{w}_2}{\to}& \mathbf{B}^2 \mathbb{Z}_2 } \,,

where the vertical morphism is the canonical morphism induced from the identification of real vector spaces 2\mathbb{C} \to \mathbb{R}^2, and where the top morphism is the canonical projection BU(n)BU(1)\mathbf{B}U(n) \to \mathbf{B}U(1) (induced from U(n)U(n) being the semidirect product group U(n)SU(n)U(1)U(n) \simeq SU(n) \rtimes U(1)).


By the general relation between c 1c_1 of an almost complex structure and w 2w_2 of the underlying orthogonal structure, discussed at Stiefel-Whitney class – Relation to Chern classes.


By prop. 1 and the universal property of the homotopy pullback this induces a canonical morphism

k:BU(n)BSpin c. k \colon \mathbf{B}U(n) \to \mathbf{B}Spin^c \,.

and this is the universal morphism from almost complex structures:


For c:XBU(n)c \colon X \to \mathbf{B}U(n) modulating an almost complex structure/complex vector bundle over XX, the composite

kc:XcBU(n)kBSpin c k c \colon X \stackrel{c}{\to} \mathbf{B}U(n) \stackrel{k}{\to} \mathbf{B}Spin^c

is the corresponding Spin cSpin^c-structure.



A canonical textbook reference is

  • H.B. Lawson and M.-L. Michelson, Spin Geometry , Princeton University Press, Princeton, NJ, (1989)

Other accounts include

  • Blake Mellor, Spin cSpin^c-manifolds (pdf)

  • Stable complex and Spin cSpin^c-structures (pdf)

  • Peter Teichner, Elmar Vogt, All 4-manifolds have Spin cSpin^c-structures (pdf)

As KUKU-orientation/anomaly cancellation in type II string theory

That the U(1)U(1)-gauge field on a D-brane in type II string theory in the absense of a B-field is rather to be regarded as part of a spin cspin^c-structure was maybe first observed in

The twisted spin^c structure (see there for more details) on the worldvolume of D-branes in the presence of a nontrivial B-field was discussed in

See at Freed-Witten-Kapustin anomaly cancellation.

A more recent review is provided in

  • Kim Laine, Geometric and topological aspects of Type IIB D-branes (arXiv:0912.0460)

See also

The relation to metaplectic corrections is discussed in

See also

  • O. Hijazi, S. Montiel, F. Urbano, Spin cSpin^c-geometry of Kähler manifolds and the Hodge Laplacian on minimal Lagrangian submanifolds (pdf)

Revised on August 29, 2013 17:21:44 by Urs Schreiber (