Computation Of Some Zamolodchikov Volumes, With An Application

We compute the Zamolodchikov volumes of some moduli spaces of conformal field theories with target spaces K3, T4, and their symmetric products. As an application we argue that sequences of conformal field theories, built from products of such symmetric products, almost never have a holographic dual with weakly coupled gravity.Comment: 20 pages, 1 figur

A Comment On Berry Connections

When families of quantum systems are equipped with a continuous family of Hamiltonians such that there is a gap in the common spectrum one can define a notion of a Berry connection. In this note we stress that, in general, since the Hilbert bundle defining the family of quantum systems does not come with a canonical trivialization there is in fact not a single Berry connection but rather a family of Berry connections. Two examples illustrate that this remark can have physical consequences.Comment: 17 pages. V2: Some silly misprints fixe

Four-dimensional N=2 Field Theory and Physical Mathematics

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.Comment: 18 pp. Conference Proceeding

Les Houches Lectures on Strings and Arithmetic

These are lecture notes for two lectures delivered at the Les Houches workshop on Number Theory, Physics, and Geometry, March 2003. They review two examples of interesting interactions between number theory and string compactification, and raise some new questions and issues in the context of those examples. The first example concerns the role of the Rademacher expansion of coefficients of modular forms in the AdS/CFT correspondence. The second example concerns the role of the ``attractor mechanism'' of supergravity in selecting certain arithmetic Calabi-Yau's as distinguished compactifications.Comment: 61pp. harvmac b-mode, 3 figures;v2: minor changes;v3: added refs; v4: Important mistake concerning the ``Fareytail transform'' has been fixe

The Partition Function Of Argyres-Douglas Theory On A Four-Manifold

Using the $u$-plane integral as a tool, we derive a formula for the partition function of the simplest nontrivial (topologically twisted) Argyres-Douglas theory on compact, oriented, simply connected, four-manifolds without boundary and with $b_2^+>0$. The result can be expressed in terms of classical cohomological invariants and Seiberg-Witten invariants. Our results hint at the existence of standard four-manifolds that are not of Seiberg-Witten simple type.Comment: 33 page

Conformal blocks for AdS5 singletons

We give a simple derivation of the conformal blocks of the singleton sector of compactifications of IIB string theory on spacetimes of the form X5 x Y5 with Y5 compact, while X5 has as conformal boundary an arbitrary 4-manifold M4. We retain the second-derivative terms in the action for the B,C fields and thus the analysis is not purely topological. The unit-normalized conformal blocks agree exactly with the quantum partition function of the U(1) gauge theory on the conformal boundary. We reproduce the action of the magnetic translation group and the SL(2,Z) S-duality group obtained from the purely topological analysis of Witten. An interesting subtlety in the normalization of the IIB Chern-Simons phase is noted.Comment: 30pp. late

Holographic Action for the Self-Dual Field

We revisit the construction of self-dual field theory in 4l+2 dimensions using Chern-Simons theory in 4l+3 dimensions, building on the work of Witten. Careful quantization of the Chern-Simons theory reveals all the topological subtleties associated with the self-dual partition function, including the generalization of the choice of spin structure needed to define the theory. We write the partition function for arbitrary torsion background charge, and in the presence of sources. We show how this approach leads to the formulation of an action principle for the self-dual field.Comment: 75 pages; 18 figures; LaTe

Flux corrections to anomaly cancellation in M-theory on a manifold with boundary

We show how the coupling of gravitinos and gauginos to fluxes modifies anomaly cancellation in M-theory on a manifold with boundary. Anomaly cancellation continues to hold, after a shift of the definition of the gauge currents by a local gauge invariant expression in the curvatures and E8 fieldstrengths. We compute the first nontrivial correction of this kind. Warning: Ian Moss has called into question several of the numerical coefficients in the extended Dirac operators in this paper. We have not confirmed this but the reader is warned not to trust the precise coefficients in the formulae for the Dirac operators and heat kernel expansions. We believe these possible errors do not change our qualitative conclusions. One of us intends to return to the issue and recheck the formulae. We thank Ian Moss for pointing out these problems.Comment: 35pp. harvmac b-mod

Crossing the Wall: Branes vs. Bundles

We test a recently proposed wall-crossing formula for the change of the Hilbert space of BPS states in d=4,N=2 theories. We study decays of D4D2D0 systems into pairs of D4D2D0 systems and we show how the wall-crossing formula reproduces results of Goettsche and Yoshioka on wall-crossing behavior of the moduli of slope-stable holomorphic bundles over holomorphic surfaces. Our comparison shows very clearly that the moduli space of the D4D2D0 system on a rigid surface in a Calabi-Yau is not the same as the moduli space of torsion free sheaves, even when worldhseet instantons are neglected. Moreover, we argue that the physical formula should make some new mathematical predictions for a future theory of the moduli of stable objects in the derived category.Comment: 23pp. latex, one figur

Comments On The Two-Dimensional Landau-Ginzburg Approach To Link Homology

We describe rules for computing a homology theory of knots and links in $\mathbb{R}^3$. It is derived from the theory of framed BPS states bound to domain walls separating two-dimensional Landau-Ginzburg models with (2,2) supersymmetry. We illustrate the rules with some sample computations, obtaining results consistent with Khovanov homology. We show that of the two Landau-Ginzburg models discussed in this context by Gaiotto and Witten one, (the so-called Yang-Yang-Landau-Ginzburg model) does not lead to topological invariants of links while the other, based on a model with target space equal to the universal cover of the moduli space of $SU(2)$ magnetic monopoles, will indeed produce a topologically invariant theory of knots and links.Comment: 77 pages, 12 figure