nLab
twisted spin^c structure

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Definition

Topological

For nn \in \mathbb{N} the Lie group Spin c(n)Spin^c(n) 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 .

By the definition at twisted cohomology, for a given class [c]H 3(X,)[c] \in H^3(X, \mathbb{Z}), a cc-twisted spin cspin^c-structure is a choice of homotopy

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

The space/∞-groupoid of all twisted Spin cSpin^c-structures on XX is the homotopy fiber W 3Struc tw(TX)W_3 Struc_{tw}(T X) in the pasting diagram of homotopy pullbacks

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

where the right vertical morphism is the canonical effective epimorphism that picks one point in each connected component.

Smooth

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 Top \simeq ∞Grpd to Smooth∞Grpd.

More precisely, 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.

By the general definition at twisted differential c-structure , the 2-groupoid of smooth twisted spin cspin^c-structures W 3Struc tw(X)\mathbf{W}_3 Struc_{tw}(X) is the joint (∞,1)-pullback

W 3Struc tw(TX) W 3Struc tw(X) tw H smooth 2(X,U(1)) * TX SmoothGrpd(X,BSO(n)) W 3 SmoothGrpd(X,B 2U(1). \array{ \mathbf{W}_3 Struc_{tw}(T X) &\to& \mathbf{W}_3 Struc_{tw}(X) &\stackrel{tw}{\to}& H_{smooth}^2(X, U(1)) \\ \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) } \,.

Applications

Anomaly cancellation in physics

The existence of an ordinary spin structure on a space XX is, as discussed there, the condition for XX to serve as the target space for the spinning particle sigma-model, in that the existence of this structure is precisely the condition that the corresponding fermionic quantum anomaly on the worldline vanishes.

Twisted spin cspin^c-structures appear similarly as the conditions for the analogous quantum anomaly cancellation, but now of the open type II superstring ending on a D-brane. This is also called the Freed-Witten anomaly cancellation.

More precisely, in these applications the class of W 3(TX)HW_3(TX) - H need not vanish, it only needs to be nn-torsion if there is moreover a twisted bundle of rank nn on the DD-brane.

See the references below for details.

References

General

The notion of twisted Spin cSpin^c-structures as such were apparently first discussed in section 5 of

More discussion appears in section 3 of

The refinement to smooth twisted structures is discussed in section 4.1 of

In physics

The need for twisted Spin cSpin^c-structures as Freed-Witten anomaly cancellation condition on the worldvolume of D-branes in string theory was first discussed in

More details are in

A clean formulation and review is provided in

  • Loriano Bonora, Fabio Ferrari Ruffino, Raffaele Savelli, Classifying A-field and B-field configurations in the presence of D-branes (arXiv:0810.4291)

  • Fabio Ferrari Ruffino, Classifying A-field and B-field configurations in the presence of D-branes - Part II: Stacks of D-branes (arXiv:1104.2798)

  • Fabio Ferrari Ruffino, Topics on topology and superstring theory (arXiv:0910.4524)

and

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

In (Laine) the discussion of FW-anomaly cancellation with finite-rank gauge bundles is towards the very end, culminating in equation (3.41).

Revised on August 12, 2012 22:11:26 by Urs Schreiber (82.113.121.69)