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We explore the zero-temperature behavior of an assembly of bosons interacting through a zero-range, attractive potential. Because the twobody interaction admits a bound state, the many-body model is best described by a Hamiltonian that includes the coupling between atomic and molecular components. Due to the positive scattering length, the low-density collection is expected to remain stable against collapse despite the attraction between particles. Although a variational many-body analysis indicates a collapsing solution with only a molecular component to its condensate at low density, the expected atomic condensate solution can be obtained if the chemical potential is allowed to be complex valued. In addition to revealing two discrete eigenfrequencies associated with the molecular case, an expansion in small oscillations quantifies the imaginary part of the chemical potential as proportional to a coherent decay rate of the atomic condensate into a continuum of collective phonon excitations about the collapsing lower state.
Surface operators in gauge theory are analogous to Wilson and ’t Hooft line operators except that they are supported on a two-dimensional surface rather than a one-dimensional curve. In a previous paper, we constructed a certain class of half-BPS surface operators in N = 4 super Yang–Mills theory, and determined how they transform under S-duality. Those surface operators depend on a relatively large number of freely adjustable parameters. In the present paper, we consider the opposite case of half-BPS surface operators that are “rigid” in the sense that they do not depend on any parameters at all. We present some simple constructions of rigid half-BPS surface operators and attempt to determine how they transform under duality. This attempt is only partially successful, suggesting that our constructions are not the whole story. The partial match suggests interesting connections with quantization. We discuss some possible refinements and some string theory constructions which might lead to a more complete picture.
A recent attempt to extend the geometric Langlands duality to affine Kac–Moody groups has led Braverman and Finkelberg to conjecture a mathematical relation between the intersection cohomology of the moduli space of G-bundles on certain singular complex surfaces, and the integrable representations of the Langlands dual of an associated affine G-algebra, where G is any simply-connected semisimple group. For the AN−1 groups, where the conjecture has been mathematically verified to a large extent, we show that the relation has a natural physical interpretation in terms of six-dimensional compactifications of M-theory with coincident five-branes wrapping certain hyperkähler four-manifolds; in particular, it can be understood as an expected invariance in the resulting spacetime BPS spectrum under string dualities. By replacing the singular complex surface with a smooth multi-Taub-NUT manifold, we find agreement with a closely related result demonstrated earlier via purely field-theoretic considerations by Witten. By adding OM five-planes to the original analysis, we argue that an analogous relation involving the non-simply-connected DN groups ought to hold as well. This is the first example of a string-theoretic interpretation of such a two-dimensional extension to complex surfaces of the geometric Langlands duality for the A–D groups.
We discuss B-type tensor product branes in mirrors of two-parameter Calabi–Yau hypersurfaces, using the language of matrix factorizations. We determine the open string moduli of the branes at the Gepner point. By turning on both bulk and boundary moduli we then deform the brane away from the Gepner point. Using the deformation theory of matrix factorizations we compute Massey products. These contain the information about higher order deformations and obstructions. The obstructions are encoded in the F-term equations, which we obtain from the Massey product algorithm. We show that the F-terms can be integrated to an effective superpotential. Our results provide an ingredient to open/closed mirror symmetry for these hypersurfaces.
We use our recently developed algebraic methods for the calculation of the heat kernel on homogeneous bundles over symmetric spaces to evaluate the non-perturbative low-energy effective action in quantum general relativity and Yang–Mills gauge theory in curved space. We obtain an exact integral repesentation for the effective action that generates all terms in the standard asymptotic epxansion of the effective action without derivatives of the curvatures effectively summing up the whole infinite subseries of all quantum corrections with low momenta.