Geometry, Topology, and Physics Seminar, Winter 2011

Organizer: Dave Morrison.
Meets 4:00 - 5:30 p.m. Fridays in South Hall 6635.

Jan. 14

### K3 surfaces, modular forms, and non-geometric heterotic compactifications

Abstract: Type IIB string theory has an SL(2,Z) symmetry and a complex scalar field tau valued in the upper half plane, on which SL(2,Z) acts by fractional linear transformations; this naturally suggests building models in which tau is allowed to vary. Although the SL(2,Z)-invariant function j(tau) can reveal some of the structures of these models, for their full construction and study we need SL(2,Z) modular forms, particularly the Eisenstein series E_4(tau) and E_6(tau) and the corresponding Weierstrass equations. The Weierstrass equations can also be analyzed in algebraic geometry via the theory of elliptic curves. This approach leads to the "F-theory" compactifications of type IIB theory. Similarly, the heterotic string compactified on T^2 has a large discrete symmetry group SO(2,18;Z), which acts on the scalars in the theory in a natural way; there have been a number of attempts to construct models in which these scalars are allowed to vary by using SO(2,18;Z)-invariant functions. In our new work, we give (in principle) a more complete construction of these models, using SO(2,18;Z)-modular forms analogous to the Eisenstein series. In practice, we restrict to special cases in which either there are no Wilson lines -- and SO(2,2;Z) symmetry -- or there is a single Wilson line -- and SO(2,3;Z) symmetry. In those cases, the modular forms can be analyzed in detail and there turns out to be a precise theory of K3 surfaces with prescribed singularities which corresponds to the structure of the modular forms. Using these two approaches -- modular forms on the one hand, and the algebraic geometry of the K3 surfaces on the other hand -- we can construct non-geometric compactifications of the heterotic theory. This is a report on two joint projects: one with McOrist and Sethi and the other with Malmendier.

Audio [ mp3, wma ]; Lecture notes.

Jan. 21

Jan. 28

### Gauged linear sigma models and non-complete-intersection Calabi-Yau varieties

Abstract: A large class of examples of Calabi-Yau varieties is provided by "complete intersections" in toric Fano varieties: that is, Calabi-Yau varieties whose homogeneous ideal has the same number of generators as the codimension of the variety. This class has been well-studied both in mathematics and in physics, in part because of Witten's "gauged linear sigma model" (GLSM) construction which provides a two-dimensional superconformal field theory corresponding to the Calabi-Yau variety. The GLSM corresponding to a complete intersection always has an abelian gauge group.

From the viewpoint of projective geometry, it is well-understood that "most" varieties are not complete intersections. A classical result of Serre shows that a codimension two variety can be expected to be a complete intersection, but in codimension three, Buchsbaum and Eisenbud showed around 1970 that a different construction is needed: the variety can be expected to be a "Pfaffian variety," determined by a rank condition on a skew-symmetric matrix of homogeneous polynomials.

In 1998, Rødland studied Calabi-Yau threefolds built from a $7\times 7$ skew-symmetric matrix, and in 2006, Hori and Tong showed how to describe these Calabi-Yau threefolds with a GLSM using a non-abelian gauge group.

There have been other constructions of Pfaffian Calabi-Yau threefolds in recent years, notably by Tonoli and Kanazawa. This report on work in progress will describe how to extend the Hori-Tong construction to give GLSM descriptions of these Calabi-Yau threefolds. In principle, the construction should work for any Calabi-Yau variety of codimension three in a toric Fano ambient space.

Audio [ mp3, wma ]; Lecture notes v1, Lecture notes v2.

Feb. 4

### Pfaffian Calabi-Yau varieties

Abstract: We will continue the discussion of Pfaffian Calabi-Yau varieties and gauged linear sigma models.

Audio [ mp3, wma ];

Feb. 11

Feb. 18

Feb. 25

March 4

### Ricci flow through singularities

Abstract: We construct and describe smooth forward Ricci flow evolutions of singular initial metrics resulting from rotationally symmetric neckpinches, without performing an intervening surgery. In the restrictive context of rotational symmetry, our construction gives evidence in favor of Perelman's hope for a "canonically defined Ricci flow through singularities." This is joint work with Sigurd Angenent and Cristina Caputo.