BPS States Research Articles

REFINED BPS STATE COUNTING FROM NEKRASOV'S FORMULA AND MACDONALD FUNCTIONS

It has been argued that Nekrasov's partition function gives the generating function of refined BPS state counting in the compactification of M theory on local Calabi–Yau spaces. We show that a refined version of the topological vertex we previously proposed (arXiv:hep-th/0502061) is a building block of Nekrasov's partition function with two equivariant parameters. Compared with another refined topological vertex by Iqbal, Kozcaz and Vafa (arXiv:hep-th/0701156), our refined vertex is expressed entirely in terms of the specialization of the Macdonald symmetric functions which is related to the equivariant character of the Hilbert scheme of points on ℂ2. We provide diagrammatic rules for computing the partition function from the web diagrams appearing in geometric engineering of Yang–Mills theory with eight supercharges. Our refined vertex has a simple transformation law under the flop operation of the diagram, which suggests that homological invariants of the Hopf link are related to the Macdonald functions.

Generalized Kac-Moody algebras from CHL dyons

We provide evidence for the existence of a family of generalized Kac-Moody (GKM) superalgebras, N, whose Weyl-Kac-Borcherds denominator formula gives rise to a genus-two modular form at level N, Δk/2(Z), for (N, k) = (1, 10), (2, 6), (3, 4), and possibly (5, 2). The square of the automorphic form is the modular transform of the generating function of the degeneracy of CHL dyons in asymmetric N-orbifolds of the heterotic string compactified on T6. The new generalized Kac-Moody superalgebras all arise as different `automorphic corrections' of the same Lie algebra and are closely related to a generalized Kac-Moody superalgebra constructed by Gritsenko and Nikulin. The automorphic forms, Δk/2(Z), arise as additive lifts of Jacobi forms of (integral) weight k/2 and index 1/2. We note that the orbifolding acts on the imaginary simple roots of the unorbifolded GKM superalgebra, 1, leaving the real simple roots untouched. We anticipate that these superalgebras will play a role in understanding the `algebra of BPS states' in CHL compactifications.

BPS states in (2,0) theory on ℝ ×T5

BPS states in (2,0) theory on ℝ ×<i>T</i>⁵

BPS equations and the stress tensor

We exploit the relationship between the space components Tij of the energy–momentum tensor and the supercurrent to discuss the connection between the BPS equations and the vanishing of the components of the stress tensor in various supersymmetric theories with solitons. Using the fact that certain combination of supercharges annihilate BPS states, we show that Tij=0 for kinks, vortices and dyons, displaying the connection between supersymmetry and non-interacting BPS solitons.

Topological strings on Grassmannian Calabi-Yau manifolds

We present solutions for the higher genus topological string amplitudes on Calabi-Yau-manifolds, which are realized as complete intersections in Grassmannians. We solve the B-model by direct integration of the holomorphic anomaly equations using a finite basis of modular invariant generators, the gap condition at the conifold and other local boundary conditions for the amplitudes. Regularity of the latter at certain points in the moduli space suggests a CFT description. The A-model amplitudes are evaluated using a mirror conjecture for Grassmannian Calabi-Yau by Batyrev, Ciocan-Fontanine, Kim and Van Straten. The integrality of the BPS states gives strong evidence for the conjecture.

Fuzzy ring from M2-brane giant torus

We construct spinning dual M2 giant gravitons in AdS_4 x S^7, which generically become 1/16 BPS states, and show that their world-volumes become torii. By taking an orbifold, we obtain spinning dielectric D2-brane configurations in AdS_4 x CP^3 dual to specific BPS operators in ABJM theory. This reveals a novel mechanism how to give an angular momentum to a dielectric D2-brane. We also find that when its angular momentum in the AdS_4 becomes large, it approaches to a ring-like object. Our result might suggest an existence of supersymmetric black rings in the AdS_4 background. We will also discuss dual giant gravitons in AdS_4 x CP^3.

Cohomological gauge theory, quiver matrix models and Donaldson–Thomas theory

We study the relation between Donaldson–Thomas theory of Calabi–Yau threefolds and a six-dimensional topological Yang–Mills theory. Our main example is the topological U ( N ) gauge theory on flat space in its Coulomb branch. To evaluate its partition function we use equivariant localization techniques on its noncommutative deformation. As a result the gauge theory localizes on noncommutative instantons which can be classified in terms of N-coloured three-dimensional Young diagrams. We give to these noncommutative instantons a geometrical description in terms of certain stable framed coherent sheaves on projective space by using a higher-dimensional generalization of the ADHM formalism. From this formalism we construct a topological matrix quantum mechanics which computes an index of BPS states and provides an alternative approach to the six-dimensional gauge theory.

N = 2 supersymmetric spin foams in three dimensions

We construct the spin foam model for N = 2 supergravity in three dimensions. Classically, it is a BF theory with gauge algebra . This algebra has representations which are not completely reducible. This complicates the procedure when building a state sum. Fortunately, one can and should excise these representations. We show that the restricted subset of representations form a subcategory closed under tensor product. The resulting state sum is once again a topological invariant. Furthermore, within this framework one can identify positively and negatively charged fermions propagating on the spin foam. These results on representations and intertwiners apply more generally to spin network states for N = 2 loop quantum supergravity (in 3 + 1 dimensions) where it allows us to define a notion of BPS states.

𝒩 = 4 SYM onS3with near critical chemical potentials

We study the N = 4 theory at weak coupling, on a three sphere in the grand canonical ensemble with R symmetry chemical potentials. We focus attention on near critical values for the chemical potentials, above which the classical theory has no ground state. By computing a one loop effective potential for the light degrees of freedom in this regime, we show the existence of flat directions of complex dimension N, 2N and 3N for one, two and three critical chemical potentials respectively; these correspond to one half, one quarter and one-eighth BPS states becoming light respectively at the critical values. At small finite temperature we show that the chemical potentials can be continued beyond their classical limiting values to yield a deconfined metastable phase with lifetime diverging in the large N limit. Our low temperaure analysis complements the high temperature metastability found by Yamada and Yaffe. The resulting phase diagram at weak coupling bears a striking resemblance to the strong coupling phase diagram for charged AdS black holes. Our analysis also reveals subtle qualitative differences between the two regimes.

Classification of the BPS states in Bagger-Lambert theory

We classify, in a group theoretical manner, the BPS configurations in the multiple M2-brane theory recently proposed by Bagger and Lambert. We present three types of BPS equations preserving various fractions of supersymmetries: in the first type we have constant fields and the interactions are purely algebraic in nature; in the second type the equations are invariant under spatial rotation SO(2), and the fields can be time-dependent; in the third class the equations are invariant under boost SO(1,1) and provide the eleven-dimensional generalizations of the Nahm equations. The BPS equations for different number of supersymmetries exhibit the division algebra structures: octonion, quarternion or complex.

Giant magnons in the D1-D5 system

We study giant magnons in the the D1-D5 system from both the boundary CFT and as classical solutions of the string sigma model in $AdS_3\times S^3\times T^4$. Re-examining earlier studies of the symmetric product conformal field theory we argue that giant magnons in the symmetric product are BPS states in a centrally extended $SU(1|1)\times SU(1|1)$ superalgebra with two more additional central charges. The magnons carry these additional central charges locally but globally they vanish. Using a spin chain description of these magnons and the extended superalgebra we show that these magnons obey a dispersion relation which is periodic in momentum. We then identify these states on the string theory side and show that here too they are BPS in the same centrally extended algebra and obey the same dispersion relation which is periodic in momentum. This dispersion relation arises as the BPS condition for the extended algebra and is similar to that of magnons in ${\cal N}=4$ Yang-Mills

(Non-)BPS bound states and D-brane instantons

We study non-perturbative effects in four-dimensional N=1 supersymmetric orientifold compactifications due to D-brane instantons which are generically not invariant under the orientifold projection. We show that they can yield superpotential contributions via a multi-instanton process at threshold. Some constituents of this configuration form bound states away from the wall of marginal stability which can decay in other regions of moduli space. A microscopic analysis reveals how contributions to the superpotential are possible when new BPS states compensate for their decay. We study this concretely for D2-brane instantons along decaying special Lagrangians in Type IIA and for D5-branes instantons carrying holomorphic bundles in Type I theory.

BPS states, dualities and determining the mass of elementary particles

BPS states, Montonen and Olive duality which is an extension of ‘tHooft–Polyakov’s symmetry to non-Abelian theories is shown to be consistent with the E-Infinity methodology for calculating the mass spectrum of high energy elementary particles.

Comments on 1/16 BPS quantum states and classical configurations

We formulate the problem of counting 1/16 BPS states of = 4 Yang Mills theory as the enumeration of the local cohomology of an operator acting on holomorphic fields on C2. We study aspects of the enumeration of this cohomology at finite N, especially for operators constructed only out of products of covariant derivatives of scalar fields, and compare our results to the states obtained from the quantization of giant gravitons and dual giants. We physically interpret the holomorphic fields that enter our conditions for supersymmetry semi-classically by deriving a set of Bogomolnyi equations for 1/16-BPS bosonic field configurations in = 4 Yang Mills theory on R4 with reality properties and boundary conditions appropriate to radial quantization. An arbitrary solution to these equations in the free theory is parameterized by holomorphic data on C2 and lifts to a nearby solution of the interacting Bogomolnyi equations only when the constraints equivalent to Q cohomology are obeyed.

New Anomalies in Topological String Theory

We show that the topological string partition function with D-branes on a compact Calabi-Yau manifold has new anomalies that spoil the recursive structure of the holomorphic anomaly equation and introduce dependence on wrong moduli (such as complex structure moduli in the A-model), unless the disk one-point functions vanish. This provides a microscopic explanation for the recent result of Walcher in arXiv:0712.2775 on counting of BPS states in M-theory using the topological string partition function. The relevance of vanishing disk one-point functions to large $N$ duality for compact Calabi-Yau manifolds is noted.

Counting BPS States on the Enriques Calabi-Yau

We study topological string amplitudes for the FHSV model using various techniques. This model has a type II realization involving a Calabi-Yau threefold with Enriques fibres, which we call the Enriques Calabi-Yau. By applying heterotic/type IIA duality, we compute the topological amplitudes in the fibre to all genera. It turns out that there are two different ways to do the computation that lead to topological couplings with different BPS content. One of them gives the standard D0-D2 counting amplitudes, and from the other one we obtain information about bound states of D0-D4-D2 branes on the Enriques fibre. We also study the model using mirror symmetry and the holomorphic anomaly equations. We verify in this way the heterotic results for the D0-D2 generating functional for low genera and find closed expressions for the topological amplitudes on the total space in terms of modular forms, and up to genus three. This model turns out to be much simpler than the generic B-model and might be exactly solvable.

Fusion of conformal interfaces

We study the fusion of conformal interfaces in the c = 1 conformal field theory. We uncover an elegant structure reminiscent of that of black holes in supersymmetric theories. The role of the BPS black holes is played by topological interfaces, which (a) minimize the entropy function, (b) fix through an attractor mechanism one or both of the bulk radii, and (c) are (marginally) stable under splitting. One significant difference is that the conserved charges are logarithms of natural numbers, rather than vectors in a charge lattice, as for BPS states. Besides potential applications to condensed-matter physics and number theory, these results point to the existence of large solution-generating algebras in string theory.

Supersymmetric states of 𝒩 = 4 Yang-Mills from giant gravitons

Mikhailov has constructed an infinite family of (1/8) BPS D3-branes in AdS5 × S5. We regulate Mikhailov's solution space by focussing on finite dimensional submanifolds. Our submanifolds are topologically complex projective spaces with symplectic form cohomologically equal to 2πN times the Fubini-Study Kähler class. Upon quantization and removing the regulator we find the Hilbert Space of N noninteracting Bose particles in a 3d Harmonic oscillator, a result previously conjectured by Beasley. This Hilbert Space is isomorphic to the classical chiral ring of (1/8) BPS states in = 4 Yang-Mills theory. We view our result as evidence that the spectrum of (1/8) BPS states in = 4 Yang Mills theory, which is known to jump discontinuously from zero to infinitesimal coupling, receives no further renormalization at finite values of the `t Hooft coupling.

Supersymmetric states in M5/M2 CFTs

We propose an exact, finite $N$ formula for the partition function over $1/4^{th}$ BPS states in the conformal field theory on the world volume of $N$ coincident M5 branes and $1/8^{th}$ BPS states in the theory of $N$ coincident M2 branes. We obtain our partition function by performing the radial quantization of the Coulomb Branches of these theories, and rederive the same formula from the quantization of supersymmetric giant and dual giant gravitons in $AdS_7 \times S^4$ and $AdS_4 \times S^7$. Our partition function is qualitatively similar to the analogous quantity in ${\cal N}=4$ Yang Mills. It reduces to the sum over supersymmetric multi gravitons at low energies, but deviates from this supergravity formula at energies that scale like a positive power of $N$.

Bubbling AdS and droplet descriptions of BPS geometries in IIB supergravity

This paper focuses on supergravity duals of BPS states in = 4 super Yang-Mills. In order to describe these duals, we begin with a sequence of breathing mode reductions of IIB supergravity: first on S3, then S3 × S1, and finally on S3 × S1 × CP1. We then follow with a complete supersymmetry analysis, yielding 1/8, 1/4 and 1/2 BPS configurations, respectively (where in the last step we take the Hopf fibration of S3). The 1/8 BPS geometries, which have an S3 isometry and are time-fibered over a six-dimensional base, are determined by solving a non-linear equation for the Kahler metric on the base. Similarly, the 1/4 BPS configurations have an S3 × S1 isometry and a four-dimensional base, whose Kahler metric obeys another non-linear, Monge-Ampere type equation. Despite the non-linearity of the problem, we develop a universal bubbling AdS description of these geometries by focusing on the boundary conditions which ensure their regularity. In the 1/8 BPS case, we find that the S3 cycle shrinks to zero size on a five-dimensional locus inside the six-dimensional base. Enforcing regularity of the full solution requires that the interior of a smooth, generally disconnected five-dimensional surface be removed from the base. The AdS5 × S5 ground state corresponds to excising the interior of an S5, while the 1/8 BPS excitations correspond to deformations (including topology change) of the S5 and/or the excision of additional droplets from the base. In the case of 1/4 BPS configurations, by enforcing regularity conditions, we identify three-dimensional surfaces inside the four-dimensional base which separate the regions where the S3 shrinks to zero size from those where the S1 shrinks. We discuss a large class of examples to show the emergence of a universal bubbling AdS picture for all 1/2, 1/4 and 1/8 BPS geometries.