D-branes, orbifolds, and Ext groups

3
1
Introduction
One of the predictions of current proposals for the physical significance of derived categories (see [1, 2, 3, 4] for an incomplete list of early references) is that massless boundary Ramond sector states of open strings connecting D-branes wrapped on complex submanifolds of Calabi-Yau’s, with holomorphic gauge bundles, should be counted by mathematical objects known as Ext groups. Specifically, if we have D-branes wrapped on the complex submanifolds i : S ֒→ X and j : T ֒→ X of a Calabi-Yau X , with holomorphic vector bundles E , F , respectively, then massless boundary Ramond sector states should be in one-to-one correspondence with elements of either Extn X (i∗ E , j∗ F ) or Extn X (j∗ F , i∗ E ) (depending upon the open string orientation). In [5] this proposal was checked explicitly for large radius Calabi-Yau’s by using standard well-known BCFT methods to compute the massless Ramond sector spectrum of open strings connecting such D-branes, and then relating that spectrum to Ext groups. Ext groups can be related to ordinary sheaf cohomology groups via spectral sequences. In the absence of background gauge fields, BRST invariance of a vertex operator dictates that the correspond¯-closed, leading to sheaf cohomology groups. When ing bundle-valued differential form is ∂ background gauge fields are turned on, the BRST-invariance condition can be decomposed ¯-closed into tangential and normal components. The tangential components again lead to ∂ differential forms. The normal components can be related to the tangential components by the boundary conditions as modified by the background gauge fields. At least in cases we understand well, these conditions can be reinterpreted as the vanishing of differentials in the spectral sequences we mentioned above. The result is that the massless Ramond sector spectrum is counted directly by Ext groups. There are numerous possible followups to the work in [5]. In this paper we shall extend the methods of [5] to the case of string orbifolds. In particular, we shall demonstrate that there is a one-to-one correspondence between massless boundary Ramond sector states in open string orbifolds and either Extn [X/G] (i∗ E , j∗ F ) or Extn [X/G] (j∗ F , i∗ E ) (depending upon the open string orientation), where [X/G] is known as a “quotient stack.” We do not make any assumptions about the physical relevance of stacks; rather, we compute the massless boundary Ramond sector spectrum directly in BCFT, from first-principles, and 4
5 General intersections 6 Relating orbifolds to large radius, or, the McKay correspondence 6.1 Images of fractional D0 branes . . . . . . . . . . . . . . . . . . . . . . . . . . 2
4 Parallel branes on submanifolds of different dimension 4.1 4.2 Basic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Example: ADHM/ALE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Contents
1 Introduction 2 Warmup: sheaves on stacks 3 Parallel coincident branes on S ֒→ X 3.1 3.2 3.3 Basic analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fractional branes, or, the necessity of stacks . . . . . . . . . . . . . . . . . . Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.3.6 3.3.7 3.3.8 3.3.9 [C3 /Z3 ] [C2 /Zn ] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 7 10 10 13 15 15 16 17 19 20 22 23 26 27 28 28 29 31 33 33
ILL-(TH)-02-10 hep-th/0212218
arXiv:hep-th/0212218v3 18 Dec 2003
ห้องสมุดไป่ตู้
D-branes, Orbifolds, and Ext groups
Sheldon Katz1,2 , Tony Pantev3 , and Eric Sharpe1 1 Department of Mathematics 1409 W. Green St., MC-382 University of Illinois Urbana, IL 61801 2 Department of Physics University of Illinois at Urbana-Champaign Urbana, IL 61801 3 Department of Mathematics University of Pennsylvania David Rittenhouse Lab. 209 South 33rd Street Philadelphia, PA 19104-6395 katz@, tpantev@, ersharpe@
In this note we extend previous work on massless Ramond spectra of open strings connecting D-branes wrapped on complex manifolds, to consider D-branes wrapped on smooth complex orbifolds. Using standard methods, we calculate the massless boundary Ramond sector spectra directly in BCFT, and find that the states in the spectrum are counted by Ext groups on quotient stacks (which provide a notion of homological algebra relevant for orbifolds). Subtleties that cropped up in our previous work also appear here. We also use the McKay correspondence to relate Ext groups on quotient stacks to Ext groups on (large radius) resolutions of the quotients. As stacks are not commonly used in the physics community, we include pedagogical discussions of some basic relevant properties of stacks. December 2002
[C2 /Zn ] revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An example with a nonabelian orbifold group . . . . . . . . . . . . . A non-supersymmetric example: [C/Zn ] . . . . . . . . . . . . . . .
6.2
Images of D0 branes on resolutions . . . . . . . . . . . . . . . . . . . . . . .
36 36 38 38 43 44
7 Conclusions 8 Acknowledgements A Notes on stacks B Proofs of spectral sequences References
Aside: equivariant structures on nontrivial bundles . . . . . . . . . . Z5 action on the Fermat quintic threefold . . . . . . . . . . . . . . .
Z3 5 action on a general quintic threefold . . . . . . . . . . . . . . . . Nontriviality of the spectral sequence . . . . . . . . . . . . . . . . . .