Self-dual String Research Articles

Waves and strings in an interacting conformal chiral 2-form theory in six dimensions

In six dimensions there exists a unique one-parameter family of nonlinear conformal electrodynamics for a chiral 2-form gauge field that includes (in a free-field limit) the linear chiral 2-form theory and is related, by dimensional reduction, to the four-dimensional ModMax electrodynamics. In this work, we present the first exact solutions of this theory in presence of gravity. In particular, we will consider plane wave, Robinson-Trautman, and self-dual dyonic string configurations, generalizing well-known solutions of this type in six-dimensional supergravities with linear chiral 2-form multiplets. Published by the American Physical Society 2024

Constraining Spacetime Dimensions in Quantum Gravity by Scale Invariance and Electric–Magnetic Duality

Abstract We consider a low-energy effective theory of p-branequaes in a D-dimensional spacetime, and impose two conditions: (1) the theory is scale-invariant, and (2) the electric–magnetic dual $(D-p-4)$-branes exist and they obey the same type of interactions to the p-branes. (We also assume other natural conditions such as Lorentz invariance but not string theory, supersymmetry, supergravity, and so on.) We then ask what values of p and D are consistent with these conditions. Using simple dimensional analysis, we find that only two solutions are possible: $(p,D)=(2,11)$ and $(p,D)=(2n-1,4n+2)$, ($n=1,2,3,\cdots$). The first solution corresponds to M-theory, and the second solutions at $n=1$ and $n=2$ correspond to self-dual strings in little string theory and D3-branes in type IIB superstring theory, respectively, while the second solutions for $n \ge 3$ are unknown but would be higher spin theories. Thus, quantum gravity (massless spin two theory) satisfying our two conditions would only be superstring theories, and the conditions would be strong enough to characterize superstring theories in quantum gravity.

Cosmic strings arising in a self-dual Abelian Higgs model

In this note we construct self-dual cosmic strings from an Abelian Higgs model in two-dimension with a polynomial formation of the potential energy density. By integrating the Einstein equations, we obtain an equivalent form to the sources, which is a nonlinear elliptic equation with singularities and complicated exponential terms. We prove the existence of a solution governing strings in the broken symmetry category on the whole plane, and the multiple string solutions are valid under a sufficient condition imposed only on the total number of strings. The technique of upper–lower solutions and the method of regularization are employed to show the existence of a solution when there are at least two distant string centers. When all the string centers are identical, a fixed-point theorem is used to study the properties of the nonlinear elliptic equation. Finally, we establish the sharp asymptotic estimate for the solutions at infinity and derive the dependence of the total gravitational curvature on the string number.

Hyperheavenly spaces and their application in Walker and para-Kähler geometries: Part II

4-dimensional spaces equipped with congruences of null strings are considered. It is assumed that a space admits a congruence of expanding self-dual null strings and its self-dual part of the Weyl tensor is algebraically degenerate. Different Petrov-Penrose types of such spaces are analyzed. A special attention is paid to para-Kähler Einstein spaces. All para-Kähler Einstein metrics of spaces with algebraically degenerate self-dual Weyl spinor are found in all the generality.

Cardy limits of 6d superconformal theories

We explore the supersymmetric partition functions of 6d SCFTs on ℝ4 × T2 with non-vanishing charges for compatible global symmetries. We utilize the elliptic genera for self-dual strings and compute the free energy of 6d SCFTs in the Cardy limit. For a 6d (2,0) theory on N M5-brane, we obtain the free energy proportional to N3. We find that the origin of N3 comes from the condensation of the self-dual strings, whose total number is proportional to frac{N^3-N}{6} . We further extend our analysis to the general E-string theory and obtain its Cardy free energy.

6D strings and exceptional instantons

We propose new ADHM-like methods to compute the Coulomb branch instanton partition functions of 5d and 6d supersymmetric gauge theories, with certain exceptional gauge groups or exceptional matters. We study $G_2$ theories with $n_{\bf 7}\leq 3$ matters in ${\bf 7}$, and $SO(7)$ theories with $n_{\bf 8}\leq 4$ matters in the spinor representation ${\bf 8}$. We also study the elliptic genera of self-dual instanton strings of 6d SCFTs with exceptional gauge groups or matters, including all non-Higgsable atomic SCFTs with rank $2$ or $3$ tensor branches. Some of them are tested with topological vertex calculus. We also explore a D-brane-based method to study instanton particles of 5d $SO(7)$ and $SO(8)$ gauge theories with matters in spinor representations, which further tests our ADHM-like proposals.

Higgsing towards E-strings

We explore 6d (1, 0) superconformal field theories with SU(3) and SU(2) gauge symmetries which cascade after Higgsing to the E-string theory on a single M5 near an E8 wall. Specifically, we study the 2d mathcal{N} = (0, 4) gauge theories which describe self-dual strings of these 6d theories. The self-dual strings can be also viewed as instanton string solitons of 6d Yang-Mills theories. We find the 2d anomaly-free gauge theories for self-dual strings, amending the naive ADHM gauge theories which are anomalous, and calculate their elliptic genera. While these 2d theories respect the flavor symmetry of each 6d SCFT only partially, their elliptic genera manifest the symmetry fully as these functions as BPS index are invariant in strongly coupled IR limit. Our consistent 2d (0, 4) gauge theories also provide new insights on the non-linear sigma models for the instanton strings, providing novel UV completions of the small instanton singularities. Finally, we construct new 2d quiver gauge theories for the self-dual strings in 6d E-string theory for multiple M5-branes probing the E8 wall, and find their fully refined elliptic genera.

The non-abelian self-dual string

We argue that the relevant higher gauge group for the non-abelian generalization of the self-dual string equation is the string 2-group. We then derive the corresponding equations of motion and discuss their properties. The underlying geometric picture is a string structure, i.e. a categorified principal bundle with connection whose structure 2-group is the string 2-group. We readily write down the explicit elementary solution to our equations, which is the categorified analogue of the 't Hooft-Polyakov monopole. Our solution passes all the relevant consistency checks; in particular, it is globally defined on $\mathbb{R}^4$ and approaches the abelian self-dual string of charge one at infinity. We note that our equations also arise as the BPS equations in a recently proposed six-dimensional superconformal field theory and we show that with our choice of higher gauge structure, the action of this theory can be reduced to four-dimensional supersymmetric Yang-Mills theory.

Blowup equations for 6d SCFTs. Part I

We propose novel functional equations for the BPS partition functions of 6d (1, 0) SCFTs, which can be regarded as an elliptic version of Göttsche-Nakajima-Yoshioka’s K-theoretic blowup equations. From the viewpoint of geometric engineering, these are the generalized blowup equations for refined topological strings on certain local elliptic CalabiYau threefolds. We derive recursion formulas for elliptic genera of self-dual strings on the tensor branch from these functional equations and in this way obtain a universal approach for determining refined BPS invariants. As examples, we study in detail the minimal 6d SCFTs with SU(3) and SO(8) gauge symmetry. In companion papers, we will study the elliptic blowup equations for all other non-Higgsable clusters.

Wilson surfaces in M5-branes

We study Wilson surface defects in 6d mathcal{N}=left(2,0right) superconformal field theories, engineered by semi-infinite M2-branes intersecting M5-branes. Two independent approaches are used to obtain the Wilson surface observables on the T2 inside Ω-deformed R4 × T2. One approach is to compute 5d mathcal{N}={1}^{*} instanton partition functions in the presence of Wilson lines, where the instanton corrections capture the 6d Kaluza-Klein momentum modes. The other approach is to study the elliptic genera of 2d mathcal{N}=left(0,4right) gauge theories, which we propose as describing 6d self-dual strings in the presence of Wilson surface defects. We make a detailed comparison between these two independent computations, which precisely agree for Wilson surfaces in minuscule representations; for non-minuscule representations instead we only find partial agreement, due to technical problems which we comment about.

Framed M-branes, corners, and topological invariants

We uncover and highlight relations between the M-branes in M-theory and various topological invariants: the Hopf invariant over Q, Z and Z2, the Kervaire invariant, the f-invariant, and the ν-invariant. This requires either a framing or a corner structure. The canonical framing provides a minimum for the classical action and the change of framing encodes the structure of the action and possible anomalies. We characterize the flux quantization condition on the C-field and the topological action of the M5-brane via the Hopf invariant, and the dual of the C-field as (a refinement of) an element of Hopf invariant two. In the signature formulation, the contribution to the M-brane effective action is given by the Maslov index of the corner. The Kervaire invariant implies that the effective action of the M5-brane is quadratic. Our study leads to viewing the self-dual string, which is the boundary of the M2-brane on the M5-brane worldvolume, as a string theory in the sense of cobordism of manifolds with corners. We show that the dynamics of the C-field and its dual are encoded in a unified way in the 4-sphere, which suggests the corresponding spectrum as the generalized cohomology theory describing the fields. The effective action of the corner is captured by the f-invariant, which is an invariant at chromatic level two. Finally, considering M-theory on manifolds with G2 holonomy, we show that the canonical G2 structure minimizes the topological part of the M5-brane action. This is done via the ν-invariant and a variant that we introduce related to the one-loop polynomial.

Towards an M5-brane model I: A 6d superconformal field theory

We present an action for a six-dimensional superconformal field theory containing a non-abelian tensor multiplet. All of the ingredients of this action have been available in the literature. We bring these pieces together by choosing the string Lie 2-algebra as a gauge structure, which we motivated in previous work. The kinematical data contains a connection on a categorified principal bundle, which is the appropriate mathematical description of the parallel transport of self-dual strings. Our action can be written down for each of the simply laced Dynkin diagrams, and each case reduces to a four-dimensional supersymmetric Yang–Mills theory with corresponding gauge Lie algebra. Our action also reduces nicely to an M2-brane model which is a deformation of the Aharony-Bergman-Jafferis-Maldacena (ABJM) model. While this action is certainly not the desired M5-brane model, we regard it as a key stepping stone towards a potential construction of the (2, 0)-theory.

F-theory and AdS3/CFT2

We construct supersymmetric AdS3 solutions in F-theory, that is Type IIB supergravity with varying axio-dilaton, which are holographically dual to 2d mathcal{N}=left(0,4right) superconformal field theories with small superconformal algebra. In F-theory these arise from D3-branes wrapped on curves in the base of an elliptically fibered Calabi-Yau threefold Y3 and correspond to self-dual strings in the 6d mathcal{N}=left(1,0right) theory obtained from F-theory on Y3. The non-trivial fibration over the wrapped curves implies a varying coupling of the mathcal{N}=4 Super-Yang-Mills theory on the D3-branes. We compute the holographic central charges and show that these agree with the field theory and with the anomalies of self-dual strings in 6d. We complement our analysis with a discussion of the dual M-theory solutions and a comparison of the central charges.

Testing 5d-6d dualities with fractional D-branes

6d SCFTs compactified on a circle can often be studied from nonperturbative 5d super-Yang-Mills theories, using instanton solitons. However, the 5d Yang-Mills theories with 6d UV fixed points frequently have too many hypermultiplet matters, which makes it difficult to use the ADHM techniques for instantons. With the examples of 6d $\mathcal{N}=(1,0)$ SCFTs with $Sp(N)$ gauge symmetry and $2N+8$ fundamental hypermultiplets, we show that one can still make rigorous studies of these 5d-6d relations in the `fractional D-brane sectors'. We test the recently proposed 5d duals given by $Sp(N+1)$ gauge theories, and compare their instanton partition functions with the elliptic genera of 6d self-dual strings.

On some examples of para-Hermite and para-Kähler Einstein spaces with Λ≠0

Spaces equipped with two complementary (distinct) congruences of self-dual null strings and at least one congruence of anti-self-dual null strings are considered. It is shown that if such spaces are Einsteinian then the vacuum Einstein field equations can be reduced to a single nonlinear partial differential equation of the second order. Different forms of these equations are analyzed. Finally, several new explicit metrics of the para-Hermite and para-K\"{a}hler Einstein spaces with $\Lambda \ne 0$ are presented. Some relation of that metrics to a modern approach to mechanical issues is discussed.

A Lagrangian for self-dual strings

We propose a Lagrangian for the low-energy theory that resides at the (1+1)-dimensional intersection of N semi-infinite M2-branes ending orthogonally on M M5-branes in ${\mathbb R}^{1,2} \times {\mathbb C}^4/{\mathbb Z}_k$ (for arbitrary positive integers N,M,k). We formulate this theory as a 2d boundary theory with explicit ${\cal N}=(1,1)$ supersymmetry that contains two superfields in the bi-fundamental representation of $U(N)\times U(M)$ interacting with the (2+1)-dimensional $U(N)_k\times U(N)_{-k}$ ABJM Chern-Simons-matter theory in the bulk. We postulate that the boundary theory exhibits in the deep infrared supersymmetry enhancement to ${\cal N}=(4,2)$, or ${\cal N}=(4,4)$ depending on the value of k. Arguments in favor of the proposal follow from the study of the open string theory of a U-dual type IIB Hanany-Witten setup. To formulate the bulk-boundary interactions special care is taken to incorporate all the expected boundary effects on gauge symmetry, supersymmetry, and other global symmetries.

Exact M-Theory Solutions, Integrable Systems, and Superalgebras

In this paper, an overview is presented of the recent construction of fully back-reacted half-BPS solutions in 11-dimensional supergravity which correspond to near-horizon geometries of M2 branes ending on, or intersecting with, M5 and M5$'$ branes along a self-dual string. These solutions have space-time manifold ${\rm AdS}_3 \times S^3 \times S^3$ warped over a Riemann surface $\Sigma$, and are invariant under the exceptional Lie superalgebra $D(2,1;\gamma) \oplus D(2,1;\gamma)$, where $\gamma $ is a real continuous parameter and $|\gamma|$ is governed by the ratio of the number of M5 and M5$'$ branes. The construction proceeds by mapping the reduced BPS equations onto an integrable field theory on $\Sigma$ which is of the Liouville sine-Gordon type. Families of regular solutions are distinguished by the sign of $\gamma$, and include a two-parameter Janus solution for $\gamma >0$, and self-dual strings on M5 as well as asymptotically ${\rm AdS}_4/{\mathbb Z}_2$ solutions for $\gamma <0$.

Elliptic genera of ALE and ALF manifolds from gauged linear sigma models

We compute the equivariant elliptic genera of several classes of ALE and ALF manifolds using localization in gauged linear sigma models. In the sigma model computation the equivariant action corresponds to chemical potentials for U(1) currents and the elliptic genera exhibit interesting pole structure as a function of the chemical potentials. We use this to decompose the answers into polar terms that exhibit wall crossing and universal terms. We compare our results to previous results on the large radius limit of the Taub-NUT elliptic genus and also discuss applications of our results to counting of BPS world-sheet spectrum of monopole strings in the 5d N=2 super Yang-Mills theory and self-dual strings in the 6d N=(2,0) theories.

Self-dual strings and 2D SYM

We study the system of M2-branes suspended between parallel M5-branes using ABJM model with a natural half-BPS boundary condition. For small separation between M5-branes, the worldvolume theory is shown to reduce to a 2D N=(4,4) super Yang-Mills theory with some similarity to q-deformed Yang-Mills theory. The gauge coupling is related to the position of the branes in an interesting manner. The theory is considerably different from the 2D theory proposed for multiple "M-strings". We make a detailed comparison of elliptic genus of the two descriptions and find only a partial agreement.

Self-dual string and higher instanton solutions

We present and discuss explicit solutions to the non-abelian self-dual string equation as well as to the non-abelian self-duality equation in six dimensions. These solutions are generalizations of the 't Hooft-Polyakov monopole and the BPST instanton to higher gauge theory. We expect that these solutions are relevant to the effective description of M2- and M5-branes.