Abstract
We analyze the de Sitter construction of [1] using ten-dimensional supergravity, finding exact agreement with the four-dimensional effective theory. Starting from the fermionic couplings in the D7-brane action, we derive the ten-dimensional stress-energy due to gaugino condensation on D7-branes. We demonstrate that upon including this stress-energy, as well as that due to anti-D3-branes, the ten-dimensional equations of motion require the four-dimensional curvature to take precisely the value determined by the four-dimensional effective theory of [1].
Highlights
A foundational problem in cosmology is to characterize de Sitter solutions of string theory
Provided that the generalized complex geometry superpotential continues to equal the full superpotential in off-shell configurations — which we find very plausible but do not prove here — our ten-dimensional computation of the scalar potential for the Kähler modulus continues to precisely match the four-dimensional theory, in the presence of anti-D3-branes as well as off-shell
We have shown that the ten-dimensional equation of motion (2.22), incorporating the stress-energy Tμλνλ in (3.33), requires that the Einstein-frame scalar curvature R4[g] takes exactly the value demanded by the four-dimensional Einstein equation (2.24) with the scalar potential (3.12), i.e. the value computed in the four-dimensional effective theory in [1]
Summary
A foundational problem in cosmology is to characterize de Sitter solutions of string theory. Beginning with the ten-dimensional action of type I string theory, we derive the two-gaugino and four-gaugino couplings on D7-branes, and compute the ten-dimensional stress-energy sourced by a gaugino bilinear expectation value λλ. We show that couplings of the D7-brane gauginos, including the couplings to flux derived by Dymarsky and Martucci in [23] following [24], source a contribution Tμλνλ to the stress-energy tensor Including this stress-energy in the ten-dimensional equations of motion, we compute the four-dimensional scalar curvature, and find perfect agreement with that determined by the F-term potential in the four-dimensional N = 1 supersymmetric effective theory of [1]. We repeat this computation for a compactification containing a D3-brane, with analogous results
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