Article

# 2d Wall-Crossing, R-twisting, and a Supersymmetric Index

• ##### Cumrun Vafa
02/2010;
Source: arXiv

ABSTRACT Starting from N=2 supersymmetric theories in 2 dimensions, we formulate a novel time-dependent supersymmetric quantum theory where the R-charge is twisted along the time. The invariance of the supersymmetric index under variations of the action for these theories leads to two predictions: In the IR limit it predicts how the degeneracy of BPS states change as we cross the walls of marginal stability. On the other hand, its equivalence with the UV limit relates this to the spectrum of the U(1) R-charges of the Ramond ground states at the conformal point. This leads to a conceptually simple derivation of results previously derived using tt* geometry, now based on time-dependent supersymmetric quantum mechanics. Comment: 28 pages

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##### Article: Four-dimensional N=2 Field Theory and Physical Mathematics
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ABSTRACT: We give a summary of a talk delivered at the 2012 International Congress on Mathematical Physics. We review d=4, N=2 quantum field theory and some of the exact statements which can be made about it. We discuss the wall-crossing phenomenon. An interesting application is a new construction of hyperkahler metrics on certain manifolds. Then we discuss geometric constructions which lead to exact results on the BPS spectra for some d=4, N=2 field theories and on expectation values of -- for example -- Wilson line operators. These new constructions have interesting relations to a number of other areas of physical mathematics.
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