D-branes, open string vertex operators, and Ext groups

By Sheldon Katz, E. Sharpe. 2002

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In this paper we explicitly work out the precise relationship between Ext groups and massless modes of D-branes wrapped on complex submanifolds of Calabi-Yau manifolds. Specifically, we explicitly compute the boundary vertex operators for massless Ramond sector states, in open string B models describing Calabi-Yau manifolds at large radius, directly in BCFT using standard methods. Naively these vertex operators are in one-to-one correspondence with certain sheaf cohomology groups (as is typical for such vertex operator calculations), which are related to the desired Ext groups via spectral sequences. However, a subtlety in the physics of the open string B model has the effect of physically realizing those spectral sequences in BRST cohomology, so that the vertex operators are actually in one-to-one correspondence with Ext group elements. This gives an extremely concrete physical test of recent proposals regarding the relationship between derived categories and D-branes. We check these results extensively in numerous examples, and comment on several related issues.

Subfields: Geometry and Topology, Nuclear and High Energy Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/24416f9cd46425b0f05664cfa8500b24f253584d

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Citations in the corpus

GUTs in Type IIB Orientifold Compactifications (2008, background): “For two D7-branes wrapping the same divisor Da = Db = D and carrying line bundles La and Lb, matter in the bifundamental representation (Na, Nb) is counted by the extension groups [61]”
D-brane Deconstructions in IIB Orientifolds (2008, background): “2) restricted to S, but rather [120, 59] this bundle tensored by the “line bundle” K −1/2 S , where KS is the canonical line bundle if S (whose first Chern class is −c1(S)).”
D-branes on general Script N = 1 backgrounds: superpotentials and D-terms (2006, background): “phically split into T 1,0 Σ ⊕ N hol Σ some of the h2,0(Σ) inﬁnitesimal embedding deformations may be in fact massive, due to possible obstructions coming from the holomorphic line bundle on the brane [67]. The superpotential (10.20) directly exhibits the possible presence of this kind of obstruction”
Dirichlet Branes and Mirror Symmetry (2009, background): “The way out of this paradox has been pointed out in [156, 298] and involves the possibility of having a nontrivial gauge field”
Lectures on F-theory compactifications and model building (2010, methodology): “This was derived in detail from an analysis of B-model vertex operators in [53].”
Building the standard model on a D3-brane (2005, methodology): “Then the spectrum of massless modes is counted by [40] [33] ExtrX (i∗Fj , i∗Fk) = ∑”
Global F-theory GUTs (2009, background): “In F-theory, it is more appropriate to think in terms of extension groups, which are known to count massless matter at the intersection of two branes [53].”
The effective action of D7-branes in N=1 Calabi-Yau orientifolds (2004, unknown): “Then this spectrum is truncated by keeping only states invariant under O. Mathematically, the massless open string degrees of freedom are given in terms of sections of appropriate sheaf cohomology groups [58] [59] [43].”
M-Strings (2013, background): “The appearance of Extgroups is not unexpected since it has been shown that Ext-groups count the open string states between D-branes wrapped on holomorphic submanifolds [44].”
Codimension-3 Singularities and Yukawa Couplings in F-theory (2009, background): “It is known in Type IIB string theory [9] that open string zero modes from D7–D7 intersection are identified with Ext(1) groups.”
Model building with $F$-theory (2008, unknown): “For a more mathematical perspective see [58, 59].”
Compactifications of heterotic strings on non-Kähler complex manifolds II (2003, background): “always look like bundle-valued differential forms, except in open strings [42], [43] where nontrivial boundary conditions can realize spectral sequences involving bundle-valued differential forms.”