Minimal geodescis
Montag, 22.4.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
A geodesic \(c:\bmathbb{R}\bto M\) is called minimal if a lift to the universal covering globally minimizes distance. On the \(2\)-dimensional torus with an arbitrary Riemannian metric there are uncountably many minimal geodesic. In dimension at least \(3\), there may be very few minimal geodesics. Let us assume that \(M\) is closed. In 1990 Victor Bangert has shown that the number of geometrically distinct minimal geodesics is bounded below by the first Betti number \(b_1\).\n\nIn joint work with Clara Löh, we improve Bangert's lower bound and we show that this number is at least \(b_1^2+2b_1\).\n\nThe talk will have many ties to previous research done in Freiburg many years ago: to the research of Victor Bangert, to the Diploma thesis I have written in Freiburg in 1994 in Bangert's group, to the research of the younger Burago, when he was\na long term guest in Freiburg and other aspects.\n
Multiplication of BPS states in heterotic torus theories
Montag, 29.4.24, 16:15-17:15, Raum 125, Ernst-Zermelo-Str. 1
The space of states of an N = 2 superconformal field theory contains an infinite-dimensional subspace of Bogomol'nyi–Prasad–Sommerfield (BPS) states, defined as states with minimal energy given their charge. In particular, they arise in worldsheet theories of strings. In this setting, Harvey and Moore introduced a bilinear map on BPS states.\n\nThis talk presents a mathematically rigorous approach to this construction, which has been considered promising but not properly understood for almost 30 years now. The example used throughout is that of a heterotic string with all but four dimensions compactified on a torus. For this case, the BPS states were claimed to form a Borcherds–Kac–Moody algebra, as introduced in Borcherds' proof of the monstrous moonshine conjectures.\n\nThe first half of the talk, unfortunately, consists in pointing out problems with the proposed construction. The second half will provide more details on selected aspects, such as the existence of a finite-dimensional Lie algebra of massless BPS states.