Derived factorization categories of non-Thom--Sebastiani-type sums of potentials

We first prove semi-orthogonal decompositions of derived factorization categories arising from sums of potentials of gauged Landau-Ginzburg models, where the sums are not necessarily Thom--Sebastiani type. We then apply the result to the category \HMF^{L_f}(f) of maximally graded matrix factorizations of an invertible polynomial $f$ of chain type, and explicitly construct a full strong exceptional collection E_1,\hdots,E_{\mu} in \HMF^{L_f}(f) whose length $\mu$ is the Milnor number of the Berglund--H\"ubsch transpose $\widetilde{f}$ of $f$. This proves a conjecture, which postulates that for an invertible polynomial $f$ the category \HMF^{L_f}(f) admits a tilting object, in the case when $f$ is a chain polynomial. Moreover, by careful analysis of morphisms between the exceptional objects $E_i$, we explicitly determine the quiver with relations $(Q,I)$ which represents the endomorphism ring of the associated tilting object $\oplus_{i=1}^{\mu}E_i$ in \HMF^{L_f}(f), and in particular we obtain an equivalence \HMF^{L_f}(f)\cong \Db(\fmod kQ/I).Comment: Major improvements. The proof of the existence of a tilting object is added, and we compute the associated quiver with relations. 48 page

Faithful actions from hyperplane arrangements

We show that if X is a smooth quasiprojective 3–fold admitting a flopping contraction, then the fundamental group of an associated simplicial hyperplane arrangement acts faithfully on the derived category of X. The main technical advance is to use torsion pairs as an efficient mechanism to track various objects under iterations of the flop functor (or mutation functor). This allows us to relate compositions of the flop functor (or mutation functor) to the theory of Deligne normal form, and to give a criterion for when a finite composition of 3–fold flops can be understood as a tilt at a single torsion pair. We also use this technique to give a simplified proof of a result of Brav and Thomas (Math. Ann. 351 (2011) 1005–1017) for Kleinian singularities

n-DBI gravity

n-DBI gravity is a gravitational theory introduced in arXiv:1109.1468 [hep-th], motivated by Dirac-Born-Infeld type conformal scalar theory and designed to yield non-eternal inflation spontaneously. It contains a foliation structure provided by an everywhere time-like vector field n, which couples to the gravitational sector of the theory, but decouples in the small curvature limit. We show that any solution of Einstein gravity with a particular curvature property is a solution of n-DBI gravity. Amongst them is a class of geometries isometric to Reissner-Nordstrom-(Anti) de Sitter black hole, which is obtained within the spherically symmetric solutions of n-DBI gravity minimally coupled to the Maxwell field. These solutions have, however, two distinct features from their Einstein gravity counterparts: 1) the cosmological constant appears as an integration constant and can be positive, negative or vanishing, making it a variable quantity of the theory; 2) there is a non-uniqueness of solutions with the same total mass, charge and effective cosmological constant. Such inequivalent solutions cannot be mapped to each other by a foliation preserving diffeomorphism. Physically they are distinguished by the expansion and shear of the congruence tangent to n, which define scalar invariants on each leave of the foliation.Comment: 13 page

On the scalar graviton in n-DBI gravity

n-DBI gravity is a gravitational theory which yields near de Sitter inflation spontaneously at the cost of breaking Lorentz invariance by a preferred choice of foliation. We show that this breakdown endows n-DBI gravity with one extra physical gravitational degree of freedom: a scalar graviton. Its existence is established by Dirac's theory of constrained systems. Firstly, studying scalar perturbations around Minkowski space-time, we show that there exists one scalar degree of freedom and identify it in terms of the metric perturbations. Then, a general analysis is made in the canonical formalism, using ADM variables. It is useful to introduce an auxiliary scalar field, which allows recasting n-DBI gravity in an Einstein-Hilbert form but in a Jordan frame. Identifying the constraints and their classes we confirm the existence of an extra degree of freedom in the full theory, besides the two usual tensorial modes of the graviton. We then argue that, unlike the case of (the original proposal for) Horava-Lifschitz gravity, there is no evidence that the extra degree of freedom originates pathologies, such as vanishing lapse, instabilities and strong self-coupling at low energy scales.Comment: 30 pages, 1 figur

Stability Conditions for 3-fold Flops

Let $f\colon X\to\mathrm{Spec}\, R$ be a 3-fold flopping contraction, where $X$ has at worst Gorenstein terminal singularities and $R$ is complete local. We describe the space of Bridgeland stability conditions on the null subcategory $\mathscr{C}$ of the bounded derived category of $X$, which consists of those complexes that derive pushforward to zero, and also on the affine subcategory $\mathscr{D}$, which consists of complexes supported on the exceptional locus. We show that a connected component of stability conditions on $\mathscr{C}$ is the universal cover of the complexified complement of the real hyperplane arrangement associated to $X$ via the Homological MMP, and more generally that a connected component of normalised stability conditions on $\mathscr{D}$ is a regular covering space of the infinite hyperplane arrangement constructed in Iyama-Wemyss [IW9]. Neither arrangement is Coxeter in general. As a consequence, we give the first description of the Stringy K\"ahler Moduli Space (SKMS) for all smooth irreducible 3-fold flops. The answer is surprising: we prove that the SKMS is always a sphere, minus either 3, 4, 6, 8, 12 or 14 points, depending on the length of the curve.Comment: v2: 39 pages. Section 7 (autoequivalences) now holds in the general settin

Prime thick subcategories on elliptic curves

We classify all prime thick subcategories in the derived category of coherent sheaves on elliptic curves, and determine the Serre invariant locus of Matsui spectrum of derived category of coherent sheaves on any smooth projective curves.Comment: 18 pages, Remark 4.15 is added. To appear in Pacific journal of mathematic

Giant graviton interactions and M2-branes ending on multiple M5-branes

We study splitting and joining interactions of giant gravitons with angular momenta $N^{1/2}\ll J\ll N$ in the type IIB string theory on $AdS_5 \times S^5$ by describing them as instantons in the tiny graviton matrix model introduced by Sheikh-Jabbari. At large $J$ the instanton equation can be mapped to the four-dimensional Laplace equation and the Coulomb potential for $m$ point charges in an $n$-sheeted Riemann space corresponds to the $m$-to-$n$ interaction process of giant gravitons. These instantons provide the holographic dual of correlators of all semi-heavy operators and the instanton amplitudes exactly agree with the pp-wave limit of Schur polynomial correlators in ${\cal N}=4$ SYM computed by Corley, Jevicki and Ramgoolam. By making a slight change of variables the same instanton equation is mathematically transformed into the Basu-Harvey equation which describes the system of M$2$-branes ending on M$5$-branes. As it turns out, the solutions to the sourceless Laplace equation on an $n$-sheeted Riemann space correspond to $n$ M5-branes connected by M2-branes and we find general solutions representing M2-branes ending on multiple M5-branes. Among other solutions, the $n=3$ case describes an M2-branes junction ending on three M5-branes. The effective theory on the moduli space of our solutions might shed light on the low energy effective theory of multiple M5-branes.Comment: 39 pages, 8 figure