Quiver Gauge Theory of Nonabelian Vortices and Noncommutative Instantons in Higher Dimensions

We construct explicit BPS and non-BPS solutions of the Yang-Mills equations on the noncommutative space R^{2n}_\theta x S^2 which have manifest spherical symmetry. Using SU(2)-equivariant dimensional reduction techniques, we show that the solutions imply an equivalence between instantons on R^{2n}_\theta x S^2 and nonabelian vortices on R^{2n}_\theta, which can be interpreted as a blowing-up of a chain of D0-branes on R^{2n}_\theta into a chain of spherical D2-branes on R^{2n} x S^2. The low-energy dynamics of these configurations is described by a quiver gauge theory which can be formulated in terms of new geometrical objects generalizing superconnections. This formalism enables the explicit assignment of D0-brane charges in equivariant K-theory to the instanton solutions.Comment: 45 pages, 4 figures; v2: minor correction

String states, loops and effective actions in noncommutative field theory and matrix models

Refining previous work by Iso, Kawai and Kitazawa, we discuss bi-local string states as a tool for loop computations in noncommutative field theory and matrix models. Defined in terms of coherent states, they exhibit the stringy features of noncommutative field theory. This leads to a closed form for the 1-loop effective action in position space, capturing the long-range non-local UV/IR mixing for scalar fields. The formalism applies to generic fuzzy spaces. The non-locality is tamed in the maximally supersymmetric IKKT or IIB model, where it gives rise to supergravity. The linearized supergravity interactions are obtained directly in position space at one loop using string states on generic noncommutative branes.Comment: 31 pages, 2 figure

Fuzzy hyperspheres via confining potentials and energy cutoffs

We simplify and complete the construction of fully $O(D)$-equivariant fuzzy spheres $S^d_\Lambda$, for all dimensions $d\equiv D-1$, initiated in [G. Fiore, F. Pisacane, J. Geom. Phys. 132 (2018), 423-451]. This is based on imposing a suitable energy cutoff on a quantum particle in $\mathbb{R}^D$ subject to a confining potential well $V(r)$ with a very sharp minimum on the sphere of radius $r=1$; the cutoff and the depth of the well diverge with $\Lambda\in\mathbb{N}$. We show that the irreducible representation ${{\bf\pi}}_\Lambda$ of $O(D\!+\!1)$ on the space of harmonic homogeneous polynomials of degree $\Lambda$ in the Cartesian coordinates of $\mathbb{R}^{D+1}$ (which we express via trace-free completely symmetric projections) is isomorphic to the Hilbert space $\mathcal{H}_{\Lambda}$ of the particle, as a reducible representation of $O(D)\subset O(D\!+\!1)$. Moreover, we show that the algebra of observables is isomorphic to ${{\bf\pi}}_\Lambda\left(Uso(D\!+\!1)\right)$. As $\Lambda$ diverges (commutative limit) so does the dimension of $\mathcal{H}_{\Lambda}$, and we recover ordinary quantum mechanics on the sphere $S^d$; more formally, we have a fuzzy quantization of a coadjoint orbit of $O(D\!+\!1)$ that goes to the classical phase space $T^*S^d$.Comment: Latex file, 40 pages, 3 figure

Membrane matrix models and 3-algebras

In this thesis we study the BPS spectrum and vacuum moduli spaces of membrane matrix models derived from dimensional reduction of the BLG and ABJM M2- brane theories. We explain how these reduced models may be mapped into each other, and describe their relationship with the IKKT matrix model. We construct BPS solutions to the reduced BLG model, and interpret them as quantized Nambu- Poisson manifolds. We study the problem of topologically twisting the reduced ABJM model, and along the way construct a new twist of the IKKT matrix model. We construct a cohomological matrix model whose partition function localizes onto the BPS moduli space of the ABJM matrix model. This partition function computes an equivariant index enumerating framed BPS states with specified R-charges

Boundary State from Ellwood Invariants

Boundary states are given by appropriate linear combinations of Ishibashi states. Starting from any OSFT solution and assuming Ellwood conjecture we show that every coefficient of such a linear combination is given by an Ellwood invariant, computed in a slightly modified theory where it does not trivially vanish by the on-shell condition. Unlike the previous construction of Kiermaier, Okawa and Zwiebach, ours is linear in the string field, it is manifestly gauge invariant and it is also suitable for solutions known only numerically. The correct boundary state is readily reproduced in the case of known analytic solutions and, as an example, we compute the energy momentum tensor of the rolling tachyon from the generalized invariants of the corresponding solution. We also compute the energy density profile of Siegel-gauge multiple lump solutions and show that, as the level increases, it correctly approaches a sum of delta functions. This provides a gauge invariant way of computing the separations between the lower dimensional D-branes.Comment: v2: 63 pages, 14 figures. Major improvements in section 2. Version published in JHE