Theory Fivebrane Research Articles

$${\widehat{Z}}$$ at Large N: From Curve Counts to Quantum Modularity

Reducing a 6d fivebrane theory on a 3-manifold $Y$ gives a $q$-series 3-manifold invariant $\widehat{Z}(Y)$. We analyse the large-$N$ behaviour of $F_K=\widehat{Z}(M_K)$, where $M_K$ is the complement of a knot $K$ in the 3-sphere, and explore the relationship between an $a$-deformed ($a=q^N$) version of $F_{K}$ and HOMFLY-PT polynomials. On the one hand, in combination with counts of holomorphic annuli on knot complements, this gives an enumerative interpretation of $F_K$ in terms of counts of open holomorphic curves. On the other, it leads to closed form expressions for $a$-deformed $F_K$ for $(2,2p+1)$-torus knots. They suggest a further $t$-deformation based on superpolynomials, which can be used to obtain a $t$-deformation of ADO polynomials, expected to be related to categorification. Moreover, studying how $F_K$ transforms under natural geometric operations on $K$ indicates relations to quantum modularity in a new setting.

BPS spectra and 3-manifold invariants

We provide a physical definition of new homological invariants [Formula: see text] of 3-manifolds (possibly, with knots) labeled by abelian flat connections. The physical system in question involves a 6d fivebrane theory on [Formula: see text] times a 2-disk, [Formula: see text], whose Hilbert space of BPS states plays the role of a basic building block in categorification of various partition functions of 3d [Formula: see text] theory [Formula: see text]: [Formula: see text] half-index, [Formula: see text] superconformal index, and [Formula: see text] topologically twisted index. The first partition function is labeled by a choice of boundary condition and provides a refinement of Chern–Simons (WRT) invariant. A linear combination of them in the unrefined limit gives the analytically continued WRT invariant of [Formula: see text]. The last two can be factorized into the product of half-indices. We show how this works explicitly for many examples, including Lens spaces, circle fibrations over Riemann surfaces, and plumbed 3-manifolds.

String theory of supertubes

String theory on AdS3 backgrounds arises as an IR limit of Little String Theory on NS5-branes. A wide variety of holographic RG flows from the fivebrane theory in the UV to (orbifolds of) AdS3 in the IR is amenable to exact treatment in worldsheet string theory as a class of null-gauged WZW models. The condensate of stringy winding operators which resolves the near-source structure of fivebranes on the Coulomb branch plays a crucial role in AdS3, revealing stringy structure invisible to the supergravity approximation. The D-brane sector contains precursors of the long strings which dominate black hole entropy in the dual spacetime CFT.

Chern-Simons theory and S-duality

We study S-dualities in analytically continued SL(2) Chern-Simons theory on a 3-manifold M. By realizing Chern-Simons theory via a compactification of a 6d five-brane theory on M, various objects and symmetries in Chern-Simons theory become related to objects and operations in dual 2d, 3d, and 4d theories. For example, the space of flat SL(2, C ) connections on M is identified with the space of supersymmetric vacua in a dual 3d gauge theory. The hidden symmetry h → -4π^2/h of SL(2) Chern-Simons theory can be identified as the S-duality transformation of N=4 super-Yang-Mills theory (obtained by compactifying the five-brane theory on a torus); whereas the mapping class group action in Chern-Simons theory on a three-manifold M with boundary C is realized as S-duality in 4d N=2 super-Yang-Mills theory associated with the Riemann surface C. We illustrate these symmetries by considering simple examples of 3-manifolds that include knot complements and punctured torus bundles, on the one hand, and mapping cylinders associated with mapping class group transformations, on the other. A generalization of mapping class group actions further allows us to study the transformations between several distinguished coordinate systems on the phase space of Chern-Simons theory, the SL(2) Hitchin moduli space.

An associative star-three-product and applications to M two/M five-brane theory

The star-product between functions enable us to take the large $N$ limit in a controlled way. At finite $N$ it serves as a substitute for matrix multiplications. Non-abelian gauge theory can be deconstructed from lower dimensional gauge theories using star-products. In this paper we extend the star-product to a star-three-product. We then apply the star-three-product to realize hermitian three-algebra of ABJM theory. We define a fuzzy three-torus. We deconstruct Abelian M five-brane in a constant background three-form potential on a fuzzy three-torus. We deconstruct non-Abelian extensions which might be related with multiple M five branes. We also mention the fuzzy three-sphere case.

Supersymmetry breaking vacua from M theory fivebranes

We consider intersecting brane configurations realizing = 2 supersymmetric gauge theories broken to = 1 by multitrace superpotentials, and softly to = 0. We analyze, in the framework of M5-brane wrapping a curve, the supersymmetric vacua and the analogs of spontaneous supersymmetry breaking and soft supersymmetry breaking in gauge theories. We show that the M5-brane does not exhibit the analog of metastable spontaneous supersymmetry breaking, and does not have non-holomorphic minimal volume curves with holomorphic boundary conditions. However, we find that any point in the = 2 moduli space can be rotated to a non-holomorphic minimal volume curve, whose boundary conditions break supersymmetry. We interpret these as the analogs of soft supersymmetry breaking vacua in the gauge theory.

Dynamical topology change in M theory

We study the topology change in M theory compactifications on Calabi–Yau three-folds in the presence of G flux (the four form field strength). In particular, we discuss vacuum solutions in strongly coupled heterotic string theory in which the topology change is inevitable within a single space–time background. For rather generic choices of initial conditions, the field equations drive the Kähler moduli outside the classical moduli space of a Calabi–Yau manifold. Consistency of the solution suggests that degenerate flop curves—just as wrapped M theory fivebranes—carry magnetic charges under the four form field strength.

Flowing from a noncommmutative (OM) five brane via its supergravity dual

The dual supergravity description of the flow between (2,0) five-brane theory and the noncommutative five-brane (OM) theory is examined at critical five-brane field strength. The self-duality of the field strength is shown to arise as a consequence of the supergravity solution. Open membrane solutions are examined in the background of the five-brane giving rise to an M analogue of the noncommutative open string (NCOS) solution.

M theory fivebrane and confining phase of N = 1 SO ( Nc) gauge theories

The moduli space of vacua for the confining phase of N = 1 SO ( N c ) supersymmetric gauge theories in four dimensions is analyzed by studying the M theory fivebrane. The type IIA brane configuration consists of a single NS5 brane, multiple copies of NS'5 branes, D4 branes between them, and D6 branes intersecting D4 branes. We construct M theory fivebrane configuration corresponding to the superpotential perturbation where the power of adjoint field is connected to the number of NS'5 branes. At a singular point in the moduli space where mutually local dyons become massless, the quadratic degeneracy of the N = 2 SO ( N c ) hyperelliptic curve determines whether this singular point gives a N = 1 vacua or not. The comparison of the meson vevs in M theory fivebrane configuration with field theory results turns out that the effective superpotential by the “integrating in” method with enhanced gauge group SU (2) is exact.

M theory fivebrane and SQCD

Talk presented at Strings `97 in Amsterdam (June 16 - 21, 1997)

A modular invariant partition function for the fivebrane

We compute an SL(6, Z) invariant partition function for the chiral two-form of the M theory fivebrane compactified on the six-torus T 6. From a manifestly SL(5, Z) invariant formalism, we prove that the partition function has an additional SL(2, Z) symmetry. The combination of these two symmetries ensures SL(6, Z) invariance. Thus, whether or not a fully covariant Lagrangian is available, the fivebrane on the six-torus has a consistent quantum theory.

Sp(Nc)gauge theories and the M theory fivebrane

We analyze an M theory fivebrane in order to study the moduli space of vacua of $N=1$ supersymmetric $\mathrm{Sp}{(N}_{c})$ gauge theories with ${N}_{f}$ flavors in four dimensions. We show how the $N=2$ Higgs branch can be encoded in M theory by studying the orientifold which plays a crucial role in our work. When all the quark masses are the same, the surface of the M theory spacetime representing a nontrivial ${\mathbf{S}}^{1}$ bundle over ${\mathbf{R}}^{3}$ develops ${A}_{{N}_{f}\ensuremath{-}1}$ type singularities at two points where D6 branes are located. Furthermore, by turning off the masses, two singular points on the surface collide and produce an ${A}_{{2N}_{f}\ensuremath{-}1}$ type singularity. The sum of the multiplicities of rational curves on the resolved surface gives the dimension of an $N=2$ Higgs branch which agrees with the counting from the brane configuration picture of type IIA string theory. By rotating M theory fivebranes we get the strongly coupled dynamics of $N=1$ theory and describe the vacuum expectation values of the meson field parametrizing the Higgs branch which are in complete agreement with the field theory results. Finally, we take the limit where the mass of an adjoint chiral multiplet goes to infinity and compare it with field theory results. For the massive case, we comment on some relations with recent work which deals with the $N=1$ duality in the context of M theory.

Supersymmetric gauge theories with classical groups via M theory fivebrane

We study the moduli space of vacua of four-dimensional N = 1and N = 2 supersymmetric gauge theories with the gauge groups Sp(2 N c ), SO(2 N c ) and SO(2 N c + 1) using the M theory fivebrane. Higgs branches of the N = 2 supersymmetric gauge theories are interpreted in terms of the M theory fivebrane and the type IIA s-rule is realized in it. In particular, we construct the fivebrane configuration which corresponds to a special Higgs branch root. This root is analogous to the baryonic branch root in the SU( N c ) theory which remains as a vacuum after the adjoint mass perturbation to break N = 2 to N = 1. Furthermore, we obtain the monopole condensations and the meson vacuum expectation values in the confining phase of N = 1 supersymmetric gauge theories using the fivebrane technique. These are in complete agreement with the field theory results for the vacua in the phase with a single confined photon.

M theory and the “integrating in” method with an antisymmetric tensor

Recently, a non-hyperelliptic curve describing the Coulomb branch of N=2 SUSY $SU(N_c)$ Yang-Mills theory with an antisymmetric tensor matter was proposed using a configuration of a single M theory five-brane. We study the singular surface in the moduli space of the curve to compare it with results from the ``integrating in'' method in field theory. In order to achieve the consistency, we find it necessary to take account of an additional superpotential $W_{\Delta}$ which has been neglected so far. The explicit form of $W_{\Delta}$ is worked out.

The M theory five-brane and the heterotic string

Brane actions with chiral bosons present special challenges. Recent progress in the description of the two main examples — the M theory five-brane and the heterotic string — is described. Also, double dimensional reduction of the M theory five-brane on K3 is shown to give the heterotic string.

Branes and dynamical supersymmetry breaking

We study dynamical supersymmetry breaking in four dimensions using the fivebrane of M theory, in particular for the Izawa-Yanagida-Intriligator-Thomas (IYIT) model, which we realize as the worldvolume theory of a certain M theory fivebrane configuration. From the brane point of view, supersymmetry is broken when a holomorphic configuration with the proper boundary conditions does not exist. We discuss the difference between explicit and spontaneous supersymmetry breaking and between runaway behavior and having a stable vacuum. As a preparation for the study of the IYIT model, we examine a realization of the orientifold four-plane in M theory. We derive known as well as new results on the moduli spaces of N = 2 and N = 1 theories with symplectic gauge groups. These results are based on a hypothesis that a certain intersection of the fivebrane and the Z 2 fixed plane breaks supersymmetry. In the IYIT model, we show that the brane exhibits runaway behavior when the flavor group is gauged. On the other hand, if the flavor group is not gauged, we find that the brane does not run away. We suggest that a stable supersymmetry-breaking vacuum is realized in the region beyond the reach of the supergravity approximation.

Confining phase of N=1 Sp(Nc) gauge theories via M theory fivebrane

The moduli space of vacua for the confining phase of N=1 $Sp(N_c)$ supersymmetric gauge theories in four dimensions is studied by M theory fivebrane. We construct M theory fivebrane configuration corresponding to the perturbation of superpotential in which the power of adjoint field is related to the number of NS'5 branes in type IIA brane configuration. We interpret the dyon vacuum expectation values in field theory results as the brane geometry and the comparison with meson vevs will turn out that the low energy effective superpotential with enhanced gauge group SU(2) is exact.

The threebrane soliton of the M-fivebrane

We discuss the supersymmetry algebra of the M theory fivebrane and obtain a new threebrane soliton preserving half of the six-dimensional supersymmetry. This solution is dimensionally reduced to various D- p -branes.

M theory fivebrane interpretation for strong coupling dynamics of SO(N) gauge theories

We study M theory fivebranes to understand the moduli space of vacua of N=1 supersymmetric SO(N_c) gauge theory with N_f flavors in four dimensions. We discuss how the various branches of this theory arise in the string/M theory brane configurations and compare our results with the ones obtained earlier by Intriligator and Seiberg in the context of field theory. In the M theory approach, we explain the various branches from the asymptotic position of semi-infinite D4 branes in the $w=x^8+i x^9$ direction, which is closely related to the eigenvalues of the meson matrix $M^{ij} = Q^{i}_a Q^{j}_a$ where $Q^{i}_a$ is a squark multiplet ( $i=1, \cdots, 2N_f$ and $ a=1, \cdots, N_c$).In M theory, these branches are explained by observing a new phenomena which did not occur for the gauge groups SU(N_c) or Sp(N_c).

Gravitational anomaly cancellation for $M$-theory fivebranes

We study gravitational anomalies for fivebranes in M theory. We show that an apparent anomaly in diffeomorphisms acting on the normal bundle is cancelled by a careful treatment of the M theory Chern-Simons coupling in the presence of fivebranes. One interesting aspect of our treatment is the way in which a magnetic object (the fivebrane) is smoothed out through coupling to gravity and the resulting relation between antisymmetric tensor gauge transformations and diffeomorphisms in the presence of a fivebrane.