14.4 KB · ⇊ 10 parts · ⤓ Micron 14.4 KB · ⤓ HTML 63.8 KB

top

In topology, a branch of mathematics, a string group is an infinite-dimensional group String ⁡ ⁡ ( n ) {\displaystyle \operatorname {String} (n)} introduced by Stolz (1996) as a 3 {\displaystyle 3} -connected cover of a spin group. A string manifold is a manifold with a lifting of its frame bundle to a string group bundle. This means that in addition to being able to define holonomy along paths, one can also define holonomies for surfaces going between strings. There is a short exact sequence of topological groups

0 → → K ( Z , 2 ) → → String ⁡ ⁡ ( n ) → → Spin ⁡ ⁡ ( n ) → → 0 {\displaystyle 0\rightarrow {\displaystyle K(\mathbb {Z} ,2)}\rightarrow \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow 0}

where K ( Z , 2 ) {\displaystyle K(\mathbb {Z} ,2)} is an Eilenberg–MacLane space and Spin ⁡ ⁡ ( n ) {\displaystyle \operatorname {Spin} (n)} is a spin group. The string group is an entry in the Whitehead tower (dual to the notion of Postnikov tower) for the orthogonal group:

⋯ ⋯ → → Fivebrane ⁡ ⁡ ( n ) → → String ⁡ ⁡ ( n ) → → Spin ⁡ ⁡ ( n ) → → SO ⁡ ⁡ ( n ) → → O ⁡ ⁡ ( n ) {\displaystyle \cdots \rightarrow \operatorname {Fivebrane} (n)\to \operatorname {String} (n)\rightarrow \operatorname {Spin} (n)\rightarrow \operatorname {SO} (n)\rightarrow \operatorname {O} (n)}

It is obtained by killing the π π 3 {\displaystyle \pi _{3}} homotopy group for Spin ⁡ ⁡ ( n ) {\displaystyle \operatorname {Spin} (n)} , in the same way that Spin ⁡ ⁡ ( n ) {\displaystyle \operatorname {Spin} (n)} is obtained from SO ⁡ ⁡ ( n ) {\displaystyle \operatorname {SO} (n)} by killing π π 1 {\displaystyle \pi _{1}} . The resulting manifold cannot be any finite-dimensional Lie group, since all finite-dimensional compact Lie groups have a non-vanishing π π 3 {\displaystyle \pi _{3}} . The fivebrane group follows, by killing π π 7 {\displaystyle \pi _{7}} .

More generally, the construction of the Postnikov tower via short exact sequences starting with Eilenberg–MacLane spaces can be applied to any Lie group G, giving the string group String(G).

• Intuition for the string group

• Understanding the geometry

• Fivebrane group and higher groups

• See also

• References

• External links

The relevance of the Eilenberg-Maclane space K ( Z , 2 ) {\displaystyle K(\mathbb {Z} ,2)} lies in the fact that there are the homotopy equivalences

K ( Z , 1 ) ≃ ≃ U ( 1 ) ≃ ≃ B Z {\displaystyle K(\mathbb {Z} ,1)\simeq U(1)\simeq B\mathbb {Z} }

for the classifying space B Z {\displaystyle B\mathbb {Z} } , and the fact K ( Z , 2 ) ≃ ≃ B U ( 1 ) {\displaystyle K(\mathbb {Z} ,2)\simeq BU(1)} . Notice that because the complex spin group is a group extension

0 → → K ( Z , 1 ) → → Spin C ⁡ ⁡ ( n ) → → Spin ⁡ ⁡ ( n ) → → 0 {\displaystyle 0\to K(\mathbb {Z} ,1)\to \operatorname {Spin} ^{\mathbb {C} }(n)\to \operatorname {Spin} (n)\to 0}

the String group can be thought of as a "higher" complex spin group extension, in the sense of higher group theory since the space K ( Z , 2 ) {\displaystyle K(\mathbb {Z} ,2)} is an example of a higher group. It can be thought of the topological realization of the groupoid B U ( 1 ) {\displaystyle \mathbf {B} U(1)} whose object is a single point and whose morphisms are the group U ( 1 ) {\displaystyle U(1)} . Note that the homotopical degree of K ( Z , 2 ) {\displaystyle K(\mathbb {Z} ,2)} is 2 {\displaystyle 2} , meaning its homotopy is concentrated in degree 2 {\displaystyle 2} , because it comes from the homotopy fiber of the map

String ⁡ ⁡ ( n ) → → Spin ⁡ ⁡ ( n ) {\displaystyle \operatorname {String} (n)\to \operatorname {Spin} (n)}

from the Whitehead tower whose homotopy cokernel is K ( Z , 3 ) {\displaystyle K(\mathbb {Z} ,3)} . This is because the homotopy fiber lowers the degree by 1 {\displaystyle 1} .

The geometry of String bundles requires the understanding of multiple constructions in homotopy theory,cite-ref-1[1] but they essentially boil down to understanding what K ( Z , 2 ) {\displaystyle K(\mathbb {Z} ,2)} -bundles are, and how these higher group extensions behave. Namely, K ( Z , 2 ) {\displaystyle K(\mathbb {Z} ,2)} -bundles on a space M {\displaystyle M} are represented geometrically as bundle gerbes since any K ( Z , 2 ) {\displaystyle K(\mathbb {Z} ,2)} -bundle can be realized as the homotopy fiber of a map giving a homotopy square

P → → ∗ ∗ ↓ ↓ ↓ ↓ M → K ( Z , 3 ) {\displaystyle {\begin{matrix}P&\to &*\\\downarrow &&\downarrow \\M&\xrightarrow {} &K(\mathbb {Z} ,3)\end{matrix}}}

where K ( Z , 3 ) = B ( K ( Z , 2 ) ) {\displaystyle K(\mathbb {Z} ,3)=B(K(\mathbb {Z} ,2))} . Then, a string bundle S → → M {\displaystyle S\to M} must map to a spin bundle S → → M {\displaystyle \mathbb {S} \to M} which is K ( Z , 2 ) {\displaystyle K(\mathbb {Z} ,2)} -equivariant, analogously to how spin bundles map equivariantly to the frame bundle.

The fivebrane group can similarly be understoodcite-ref-2[2] by killing the π π 7 ( Spin ⁡ ⁡ ( n ) ) ≅ ≅ π π 7 ( O ⁡ ⁡ ( n ) ) {\displaystyle \pi _{7}(\operatorname {Spin} (n))\cong \pi _{7}(\operatorname {O} (n))} group of the string group String ⁡ ⁡ ( n ) {\displaystyle \operatorname {String} (n)} using the Whitehead tower. It can then be understood again using an exact sequence of higher groups

0 → → K ( Z , 6 ) → → Fivebrane ⁡ ⁡ ( n ) → → String ⁡ ⁡ ( n ) → → 0 {\displaystyle 0\to K(\mathbb {Z} ,6)\to \operatorname {Fivebrane} (n)\to \operatorname {String} (n)\to 0}

giving a presentation of Fivebrane ⁡ ⁡ ( n ) {\displaystyle \operatorname {Fivebrane} (n)} it terms of an iterated extension, i.e. an extension by K ( Z , 6 ) {\displaystyle K(\mathbb {Z} ,6)} by String ⁡ ⁡ ( n ) {\displaystyle \operatorname {String} (n)} . Note map on the right is from the Whitehead tower, and the map on the left is the homotopy fiber.

• Gerbe

• N-group (category theory)

• Elliptic cohomology

• String bordism

cite-note-11. ↑ citerefjurco2011Jurco, Branislav (August 2011). "Crossed Module Bundle Gerbes; Classification, String Group and Differential Geometry". International Journal of Geometric Methods in Modern Physics. 08 (5): 1079–1095. arXiv:math/0510078. Bibcode:2011IJGMM..08.1079J. doi:10.1142/S0219887811005555. ISSN 0219-8878. S2CID 1347840.

cite-note-22. ↑ citerefsatischreiberstasheff2009Sati, Hisham; Schreiber, Urs; Stasheff, Jim (November 2009). "Fivebrane Structures". Reviews in Mathematical Physics. 21 (10): 1197–1240. arXiv:0805.0564. Bibcode:2009RvMaP..21.1197S. doi:10.1142/S0129055X09003840. ISSN 0129-055X. S2CID 13307997.

• citerefhenriquesdouglashill2011Henriques, André G.; Douglas, Christopher L.; Hill, Michael A. (2011), "Homological obstructions to string orientations", Int. Math. Res. Notices, 18: 4074–4088, arXiv:0810.2131, Bibcode:2008arXiv0810.2131D

• citerefwockelsachsenikolaus2013Wockel, Christoph; Sachse, Christoph; Nikolaus, Thomas (2013), "A Smooth Model for the String Group", International Mathematics Research Notices, 2013 (16): 3678–3721, arXiv:1104.4288, Bibcode:2011arXiv1104.4288N, doi:10.1093/imrn/rns154

• citerefstolz1996Stolz, Stephan (1996), "A conjecture concerning positive Ricci curvature and the Witten genus", Mathematische Annalen, 304 (4): 785–800, doi:10.1007/BF01446319, ISSN 0025-5831, MR 1380455, S2CID 123359573

• citerefstolzteichner2004Stolz, Stephan; Teichner, Peter (2004), "What is an elliptic object?" (PDF), Topology, geometry and quantum field theory, London Math. Soc. Lecture Note Ser., vol. 308, Cambridge University Press, pp. 247–343, doi:10.1017/CBO9780511526398.013, ISBN 9780521540490, MR 2079378

• citerefbaez2007Baez, J. (2007), Higher Gauge Theory and the String Group

• From Loop Groups to 2-groups - gives a characterization of String(n) as a 2-group

• string group at the nLab

• Whitehead tower at the nLab

• What is an elliptic object?