Instanton strings and hyper-Kähler geometry

We discuss two-dimensional sigma models on moduli spaces of instantons on K3 surfaces. These N = (4, 4) superconformal field theories describe the near-horizon dynamics of the D1-D5-brane system and are dual to string theory on AdS³. We derive a precise map relating the moduli of the K3 type 1113 string compactification to the moduli of these conformal field theories and the corresponding classical hyper-Kahler geometry. We conclude that in the absence of background gauge fields, the metric on the instanton moduli spaces degenerates exactly to the orbifold symmetric product of K3. Turning on a self-dual NS B-field deforms this symmetric product to a manifold that is diffeomorphic to the Hilbert scheme. We also comment on the mathematical applications of string duality to the global issues of deformations of hyper-Kähler manifolds.     

Moduli Space of BPS Walls in Supersymmetric Gauge Theories

Existence and uniqueness of the solution are proved for the ‘master equation’ derived from the BPS equation for the vector multiplet scalar in the U(1) gauge theory with N _F charged matter hypermultiplets with eight supercharges. This proof establishes that the solutions of the BPS equations are completely characterized by the moduli matrices divided by the V-equivalence relation for the gauge theory at finite gauge couplings. Therefore the moduli space at finite gauge couplings is topologically the same manifold as that at infinite gauge coupling, where the gauged linear sigma model reduces to a nonlinear sigma model. The proof is extended to the U(N _C) gauge theory with N _F hypermultiplets in the fundamental representation, provided the moduli matrix of the domain wall solution is U(1)-factorizable. Thus the dimension of the moduli space of U(N _C) gauge theory is bounded from below by the dimension of the U(1)-factorizable part of the moduli space. We also obtain sharp estimates of the asymptotic exponential decay which depend on both the gauge coupling and the hypermultiplet mass differences.     

Exactly solvable string compactifications on manifolds of SU(N) holonomy

A variety of heterotic string compactifications on the K3 surface, manifolds of SU(3) holomony, and higher holomony manifolds, are solved exactly. An example of the quintic hypersurface in CP₄ is worked out in detail. It is conjectured, and demonstrated in part, that any supersymmetric compactification of the heterotic string with an N=2 superconformal theory is equivalent to a compactification on a manifold of SU(N) holonomy, and in particular an arbitrary gluing of the discrete models with c=9 gives a solvable heterotic string compactification on some Calabi-Yau manifold. Calabi-Yau compactifications are seen to be exact vacua of string theory, retaining their topological and geometrical characteristics. Previously unknown enhanced gauge symmetries are found to arise for certain backgrounds.     

Generalizing the O(N)-field theory to N-colored manifolds of arbitrary internal dimension D

We introduce a geometric generalization of the O(N)-field theory that describes N-colored membranes with arbitrary dimension D. As the O(N)-model reduces in the limit N → 0 to self-avoiding polymers, the N-colored manifold model leads to self-avoiding tethered membranes. In the other limit, for inner dimension D → 1, the manifold model reduces to the O(N)-field theory. We analyze the scaling properties of the model at criticality by a one-loop perturbative renormalization group analysis around an upper critical line. The freedom to optimize with respect to the expansion point on this line allows us to obtain the exponent ν of standard field theory to much better precision that the usual 1-loop calculations. Some other field theoretical techniques, such as the large N limit and Hartree approximation, can also be applied to this model. By comparison of low- and high-temperature expansions, we arrive at a conjecture for the nature of droplets dominating the 3d Ising model at criticality, which is satisfied by our numerical results. We can also construct an appropriate generalization that describes cubic anisotropy, by adding an interaction between manifolds of the same color. The two parameter space includes a variety of new phases and fixed points, some with Ising criticality, enabling us to extract a remarkably precise value of 0.6315 for the exponent ν in d = 3. A particular limit of the model with cubic anisotropy corresponds to the random bond Ising problem; unlike the field theory formulation, we find a fixed point describing this system at 1-loop order.     

On the integrability of the SU(N) Hubbard model

We exhibit explicitly the intertwiner operator for the monodromy matrices of the SU(N) Hubbard model recently proposed by Maassarani [Phys. Lett. A 239 (1998) 187]. This produces a new family of non-additive R-matrices and generalizes an earlier result by Shastry [Phys. Rev. Lett. 56 (1986) 2453; J. Stat. Phys. 30 (1988) 57].     

Pseudoholomorphic Curves on Nearly Kähler Manifolds

Let M be an almost complex manifold equipped with a Hermitian form such that its de Rham differential has Hodge type (3,0)+(0,3), for example a nearly Kähler manifold. We prove that any connected component of the moduli space of pseudoholomorphic curves on M is compact. This can be used to study pseudoholomorphic curves on a 6-dimensional sphere with the standard (G ₂-invariant) almost complex structure.     

Reunion Probability of N Vicious Walkers: Typical and Large Fluctuations for Large N

We consider three different models of N non-intersecting Brownian motions on a line segment [0,L] with absorbing (model A), periodic (model B) and reflecting (model C) boundary conditions. In these three cases we study a properly normalized reunion probability, which, in model A, can also be interpreted as the maximal height of N non-intersecting Brownian excursions (called “watermelons” with a wall) on the unit time interval. We provide a detailed derivation of the exact formula for these reunion probabilities for finite N using a Fermionic path integral technique. We then analyze the asymptotic behavior of this reunion probability for large N using two complementary techniques: (i) a saddle point analysis of the underlying Coulomb gas and (ii) orthogonal polynomial method. These two methods are complementary in the sense that they work in two different regimes, respectively for $L\ll O(\sqrt{N})$ and $L\geq O(\sqrt{N})$ . A striking feature of the large N limit of the reunion probability in the three models is that it exhibits a third-order phase transition when the system size L crosses a critical value $L=L_{c}(N)\sim\sqrt{N}$ . This transition is akin to the Douglas-Kazakov transition in two-dimensional continuum Yang-Mills theory. While the central part of the reunion probability, for L～L _c (N), is described in terms of the Tracy-Widom distributions (associated to GOE and GUE depending on the model), the emphasis of the present study is on the large deviations of these reunion probabilities, both in the right [L?L _c (N)] and the left [L?L _c (N)] tails. In particular, for model B, we find that the matching between the different regimes corresponding to typical L～L _c (N) and atypical fluctuations in the right tail L?L _c (N) is rather unconventional, compared to the usual behavior found for the distribution of the largest eigenvalue of GUE random matrices. This paper is an extended version of (Schehr et al. in Phys. Rev. Lett. 101:150601, 2008) and (Forrester et al. in Nucl. Phys. B 844:500–526, 2011).     

F-theory,T-duality on K3 surfaces,and N = 2 supersymmetric gauge theories in four dimensions

We construct T-duality on K3 surfaces. The T-duality exchanges a 4-brane RR charge and a 0-brane RR charge. We study the action of the T-duality on the moduli space of 0-branes located at points of K3 and 4-branes wrapping it. We apply the construction to F-theory compactified on a Calabi-Yau 4-fold and study the duality of N = 2 SU(N_c) gauge theories in four dimensions. We discuss the generalization to the N = 1 duality scenario.     

Monopole and dyon bound states in N = 2 supersymmetric Yang-Mills theories

We study the existence of monopole bound states saturating the BPS bound in N = 2 supersymmetric Yang-Mills theories. We describe how the existence of such bound states relates to the topology of index bundles over the moduli space of BPS solutions. Using an L² index theorem, we prove the existence of certain BPS states predicted by Seiberg and Witten based on their study of the vacuum structure of N = 2 Yang-Mills theories.     

Observations on the moduli space of superconformal field theories

Some aspects of the moduli space of superconformal field theories are discussed. It is helpful to consider the conformal field theory as a background for propagation of strings and to exploit the space-time interpretation. Using this point of view we show that the metric on the moduli space of N = 4 superconformal field theory with c = 6 is locally that of O(20,4)/O(20) × O(4). We also discover some properties of the moduli space of N = 2 superconformal field theories with c = 9. Particular examples of these conformal field theories are sigma models on four- and six-dimensional Calabi-Yau spaces. Therefore, we can use this technique to learn about the moduli space of these spaces. For c = 6 we recover the known moduli space of K₃. Our analysis of the c = 9 system leads to a new coupling in four dimensional supergravity. As a by-product, we prove that gauge couplings cannot depend on the moduli of N = 1 space-time supersymmetric compactifications.     

Geometrical aspects and quantum brachistochrone problem for a collection of N spin-s system with long-range Ising-type interaction

We consider a physical system of N interacting qudits consisting of N spin-s particles coupled via the long-range interaction of Ising-type. We investigate the corresponding dynamics, define the associated quantum state manifold and we give the related Fubini-Study metric. We derive the Gaussian curvature and using the Gauss-Bonnet theorem, we show that the dynamics happen on a two-parametric manifold of spherical topology. We examine the geometrical phase acquired by the system under arbitrary and cyclic evolutions. Further, we study the quantum brachistochrone problem concerning the determination of the smallest possible time to realize a time-optimal evolution. By restricting our study to a two-qubit system under the Ising interaction, a detailed analysis is performed for the Fubini-Study metric, the Gaussian curvature, the geometrical phase and the optimal time in relation with the entanglement of the two qubits.     

O(N) Random Tensor Models

We define in this paper a class of three-index tensor models, endowed with \({O(N)^{\otimes 3}}\) invariance (N being the size of the tensor). This allows to generate, via the usual QFT perturbative expansion, a class of Feynman tensor graphs which is strictly larger than the class of Feynman graphs of both the multi-orientable model (and hence of the colored model) and the U(N) invariant models. We first exhibit the existence of a large N expansion for such a model with general interactions. We then focus on the quartic model and we identify the leading and next-to-leading order (NLO) graphs of the large N expansion. Finally, we prove the existence of a critical regime and we compute the critical exponents, both at leading order and at NLO. This is achieved through the use of various analytic combinatorics techniques.     

E6 breaking in (0,2) string compactifications

The existence of a duality in (0,2) compactifications which is present at the Landau-Ginzburg point allows us to connect in a smooth manner theories with different gauge groups with the same base manifold and same number of effective generations. As we move along the Kahler moduli space of the theories with E₆ gauge group, the VEV's of SO(10) singlets are turned on and break the gauge group to SO(10). We generalize this result and break E₆ down to SU(5).     

Supersymmetric Yang-Mills,octonionic instantons and triholomorphic curves

In four-dimensional gauge theory there exists a well-known correspondence between instantons and holomorphic curves, and a similar correspondence exists between certain octonionic instantons and triholomorphic curves. We prove that this latter correspondence stems from the dynamics of various dimensional reductions of ten-dimensional supersymmetric Yang-Mills theory. More precisely we show that the dimensional reduction of the (5+1)-dimensional supersymmetric sigma model with hyper-Kähler (but otherwise arbitrary) target X to a four-dimensional hyper-Kähler manifold M is a topological sigma model localising on the space of triholomorphic maps M -+ X (or hyperinstantons). When X is the moduli space M_k of instantons on a four-dimensional hyper-Kdhler manifold K, this theory has an interpretation in terms of supersymmetric gauge theory. In this case, the topological sigma model can be understood as an adiabatic limit of the dimensional reduction of ten-dimensional supersymmetric Yang-Mills on the eight-dimensional manifold M × K of holonomy Sp(1) × Sp(1) ⊂ Spin(7), which is a cohomological theory localising on the moduli space of octonionic instantons.     

Extended supergravity on the supergroup manifold: N = 3 and N = 2 theories

In this paper I construct the group-manifold first-order formulation of N = 2 and N = 3 supergravity based on the $\text{Osp}\text{(}\text{4}\text{2}\text{)}\text{and Osp}\text{(}\text{4}\text{3}\text{)}$ supergroups, respectively. In the case N = 2, a group manifold version of the theory was already presented in a previous paper. Here a simpler formulation is given which shows exact factorization in the SO(2) subgroup absent in the previous one. Particular attention is paid to the algebraic role played by the $\text{spin}\text{-}\text{1}\text{2}$ field which is the novel feature of the N = 3 case with respect to N = 2. It is shown how the “non-geometrical” term in the gravitino transformation law in the N = 2 theory arises from the rheonomic symmetry mechanism.     

Moduli in N = 1 heterotic/F-theory duality

The moduli in a 4D N = 1 heterotic compactification on an elliptic CY, as well as in the dual F-theoretic compactification, break into “base” parameters which are even (under the natural involution of the elliptic curves), and “fiber” or twisting parameters; the latter include a continuous part which is odd, as well as a discrete part. We interpret all the heterotic moduli in terms of cohomology groups of the spectral covers, and identify them with the corresponding F-theoretic moduli in a certain stable degeneration. The argument is based on the comparison of three geometric objects: the spectral and cameral covers and the ADE del Pezzo fibrations. For the continuous part of the twisting moduli, this amounts to an isomorphism between certain abelian varieties: the connected component of the heterotic Prym variety (a modified Jacobian) and the F-theoretic intermediate Jacobian. The comparison of the discrete part generalizes the matching of heterotic 5-brane/F-theoretic 3-brane impurities.     

The complete N = 3 matter coupled supergravity

The complete N = 3 matter coupling to supergravity is obtained in a geometrical framework. This coupling always exists if the 3n complex scalars of the n vector multiplets are co-ordinates of the Kähler-grassmannian manifold SU(3, n)/SU(3) × SU(n) × U(1). Subgroups of SO(3, n) ⊂ SU(3, n) of dimension 3 + n can be gauged and give rise to a non-trivial scalar potential. The techniques used in this paper allow for the calculation of scalar potentials of extended supergravities in any dimension without explicit construction of the lagrangian. This opens the possibility of discussing patterns of partial supersymmetry and gauge symmetry breaking on a purely group-theoretical ground.     

Geometric phase in SU(N) interferometry

An interferometric scheme to study Abelian geometric phase shift over the manifold SU(N)/U(N − 1) is presented. Presented by H. de Guise at the DI-CRM Workshop held in Prague, 18–21 June 2000. This work has been supported by two Macquarie University Research Grants and by an Australian Research Council Large Grant.