F-theory and heterotic duality, Weierstrass models from Wilson lines

Wilson loops in gauge theories pose a fundamental challenge for dualities. Wilson loops are labeled by a representation of the gauge group and should map under duality to loop operators labeled by the same data, yet generically, dual theories have completely different gauge groups. In this paper we resolve this conundrum for three dimensional mirror symmetry. We show that Wilson loops are exchanged under mirror symmetry with Vortex loop operators, whose microscopic definition in terms of a supersymmetric quantum mechanics coupled to the theory encode in a non-trivial way a representation of the original gauge group, despite that the gauge groups of mirror theories can be radically different. Our predictions for the mirror map, which we derive guided by branes in string theory, are confirmed by the computation of the exact expectation value of Wilson and Vortex loop operators on the three-sphere.

We construct non-geometric string compactifications by using the F-theory dual of the heterotic string compactified on a two-torus with two Wilson line parameters, together with a close connection between modular forms and the equations for certain K3 surfaces of Picard rank $16$. We construct explicit Weierstrass models for all inequivalent Jacobian elliptic fibrations supported on this family of K3 surfaces and express their parameters in terms of modular forms generalizing Siegel modular forms. In this way, we find a complete list of all dual non-geometric compactifications obtained by the partial higgsing of the heterotic string gauge algebra using two Wilson line parameters.

Eight-dimensional nongeometric heterotic strings were constructed as duals of F-theory on Λ1,1 ⊕ E8 ⊕ E7 lattice polarized K3 surfaces by Malmendier and Morrison. We study the structure of the moduli space of this construction. There are special points in this space at which the ranks of the non-Abelian gauge groups on the 7-branes in F-theory are enhanced to 18. We demonstrate that the enhanced rank-18 non-Abelian gauge groups arise as a consequence of the coincident 7-branes, which deform stable degenerations on the F-theory side. This observation suggests that the non-geometric heterotic strings include nonperturbative effects of the coincident 7-branes on the F-theory side. The gauge groups that arise at these special points in the moduli space do not allow for perturbative descriptions on the heterotic side.We also construct a family of elliptically fibered Calabi-Yau 3-folds by fibering K3 surfaces with enhanced singularities over ℙ1. Highly enhanced gauge groups arise in F-theory compactifications on the resulting Calabi-Yau 3-folds.

Starting with some known localization (matrix model) representations for correlators involving 1/2 BPS circular Wilson loop mathcal{W} in mathcal{N} = 4 SYM theory we work out their 1/N expansions in the limit of large ’t Hooft coupling λ. Motivated by a possibility of eventual matching to higher genus corrections in dual string theory we follow arXiv:2007.08512 and express the result in terms of the string coupling {g}_{mathrm{s}}sim {g}_{mathrm{YM}}^2sim lambda /N and string tension Tsim sqrt{lambda } . Keeping only the leading in 1/T term at each order in gs we observe that while the expansion of leftlangle mathcal{W}rightrangle is a series in {g}_{mathrm{s}}^2/T , the correlator of the Wilson loop with chiral primary operators {mathcal{O}}_J has expansion in powers of {g}_{mathrm{s}}^2/{T}^2 . Like in the case of leftlangle mathcal{W}rightrangle where these leading terms are known to resum into an exponential of a “one-handle” contribution sim {g}_{mathrm{s}}^2/T , the leading strong coupling terms in leftlangle {mathcal{WO}}_Jrightrangle sum up to a simple square root function of {g}_{mathrm{s}}^2/{T}^2 . Analogous expansions in powers of {g}_{mathrm{s}}^2/T are found for correlators of several coincident Wilson loops and they again have a simple resummed form. We also find similar expansions for correlators of coincident 1/2 BPS Wilson loops in the ABJM theory.

We study the half-BPS circular Wilson loop in mathcal{N} = 4 super Yang-Mills with orthogonal gauge group. By supersymmetric localization, its expectation value can be computed exactly from a matrix integral over the Lie algebra of SO(N). We focus on the large N limit and present some simple quantitative tests of the duality with type IIB string theory in AdS5× ℝℙ5. In particular, we show that the strong coupling limit of the expectation value of the Wilson loop in the spinor representation of the gauge group precisely matches the classical action of the dual string theory object, which is expected to be a D5-brane wrapping a ℝℙ4 subspace of ℝℙ5. We also briefly discuss the large N, large λ limits of the SO(N) Wilson loop in the symmetric/antisymmetric representations and their D3/D5-brane duals. Finally, we use the D5-brane description to extract the leading strong coupling behavior of the “bremsstrahlung function” associated to a spinor probe charge, or equivalently the normalization of the two-point function of the displacement operator on the spinor Wilson loop, and obtain agreement with the localization prediction.

We consider the small deformation of the point-like Wilson loop in the 3-dimensional $\mathcal{N}=6$ superconformal Chern-Simons theory. By Taylor expansion of the point-like Wilson loop in powers of the loop variables, we obtain the BPS operators that correspond to the excited string states of the dual IIA string theory on the pp wave background. The BPS conditions of the Wilson loop constrain both the loop variables and the forms of the operators obtained in the Taylor expansion.

The primary purpose of these lecture notes is to explore the moduli space of type IIA, type IIB, and heterotic string compactified on a K3 surface. The main tool which is invoked is that of string duality. K3 surfaces provide a fascinating arena for string compactification as they are not trivial spaces but are sufficiently simple for one to be able to analyze most of their properties in detail. They also make an almost ubiquitous appearance in the common statements concerning string duality. We review the necessary facts concerning the classical geometry of K3 surfaces that will be needed and then we review old string theory on K3 surfaces in terms of conformal field theory. The type IIA string, the type IIB string, the E8 x E8 heterotic string, and Spin(32)/Z2 heterotic string on a K3 surface are then each analyzed in turn. The discussion is biased in favour of purely geometric notions concerning the K3 surface itself. These are an extended form of the notes from lectures given at TASI 96.

We compute the supersymmetric Rényi entropies across a spherical entanglement surface in $$ \mathcal{N}=4 $$ N = 4 SU(N) SYM theory using localization on the four-dimensional ellipsoid. We extract the leading result at large N and λ and match its universal part to a gravity calculation involving a hyperbolically sliced supersymmetric black hole solution of $$ \mathcal{N}={4}^{+} $$ N = 4 + SU(2) × U(1) gauged supergravity in five dimensions. We repeat the analysis in the presence of a Wilson loop insertion and find again a perfect match with the dual string theory. Understanding the Wilson loop operator requires knowledge of the full ten-dimensional IIB supergravity solution which we elaborate upon.

F-theoretic constructions can alternatively be understood as consequences of certain N = 2 Seiberg-Witten theories via type IIB r D3s probing the quantum corrected orientifold backgrounds. We present four models that come out from such consideration. In Model 1, the 7-branes wrap the flat R^4 directions, leading to the well known Sp(2r) theories. We study singularity structure of moduli space of Seiberg-Witten curve, such as maximal Argyres-Douglas loci, in order to construct 1-1 map between moduli spaces. In Model 2, the 7-branes are wrapped on Taub-NUT and multi Taub-NUT spaces instead of R^4. These configurations may explain many of the Gaiotto-type constructions including possible extensions to non-conformal models with cascading behaviors. In this model the UV is described by the probe D3s decomposed into D5-anti D5 pairs wrapped on multi Taub-NUT space, while the IR remains a 4d theory. For certain arrangements of the 7-branes, this model may be dualized to the brane networks of Benini, Benvenuti and Tachikawa. The Gaiotto dualities in Model 2 are explained by chiral anomaly cancellations, anti-GSO projections and brane transmutations. Model 3 is described by seven-branes wrapped on a K3 manifold and D3 and anti-D3 probes whose number may differ at most by 24. These constructions could lead to new N = 2 models with possible dualities to both type IIB and heterotic theories on non-Kahler manifolds. In the limit where the number of probes becomes very large, the physics is captured by M(atrix) theory on K3 x K3 manifold. Finally, Model 4 is described by k D3s probing intersecting seven-brane backgrounds with Sp(2k) x Sp(2k) gauge group. These constructions could produce new N =1 heterotic dual on a non-Kahler K3 manifold that is no longer conformally Calabi-Yau. We discuss possible constraints on these models coming from global charge and anomaly cancellations in F-theory.

We explore the moduli space of heterotic strings in two dimensions. In doing so, we introduce new lines of compactified theories with Spin(24) gauge symmetry and discuss compactifications with Wilson lines. The phase structure of $d=2$ heterotic string theory is examined by classifying the hypersurfaces in moduli space which support massless quanta or discrete states. Finally, we compute the torus amplitude over much of the moduli space.

In this thesis we apply the technique of localization to three-dimensional N=2 superconformal field theories. We consider both theories which are exactly superconformal, and those which are believed to flow to nontrivial superconformal fixed points, for which we consider implicitly these fixed points. We find that in such theories, the partition function and certain supersymmetric observables, such as Wilson loops, can be computed exactly by a matrix model. This matrix model consists of an integral over g, the Lie algebra of the gauge group of the theory, of a certain product of 1-loop factors and classical contributions. One can also consider a space of supersymmetric deformations of the partition function corresponding to the set of abelian global symmetries. In the second part of the thesis we apply these results to test dualities. We start with the case of ABJM theory, which is dual to M-theory on an asymptotically AdS_4xS^7 background. We extract strong coupling results in the field theory, which can be compared to semiclassical, weak coupling results in the gravity theory, and a nontrivial agreement is found. We also consider several classes of dualities between two three-dimensional field theories, namely, 3D mirror symmetry, Aharony duality, and Giveon-Kutasov duality. Here the dualities are typically between the IR limits of two Yang-Mills theories, which are strongly coupled in three dimensions since Yang-Mills theory is asymptotically free here. Thus the comparison is again very nontrivial, and relies on the exactness of the localization computation. We also compare the deformed partition functions, which tests the mapping of global symmetries of the dual theories. Finally, we discuss some recent progress in the understanding of general three-dimensional theories in the form of the F-theorem, a conjectured analogy to the a-theorem in four dimensions and c-theorem in two dimensions, which is closely related to the localization computation.

We extend our study of the large-$N$ expansion of general nonequilibrium many-body systems with matrix degrees of freedom $M$, and its dual description as a sum over surface topologies in a dual string theory, to the Keldysh-rotated version of the Schwinger-Keldysh formalism. The Keldysh rotation trades the original fields ${M}_{\ifmmode\pm\else\textpm\fi{}}$---defined as the values of $M$ on the forward and backward segments of the closed time contour---for their linear combinations ${M}_{\mathrm{cl}}$ and ${M}_{\mathrm{qu}}$, known as the ``classical'' and ``quantum'' fields. First we develop a novel ``signpost'' notation for nonequilibrium Feynman diagrams in the Keldysh-rotated form, which simplifies the analysis considerably. Before the Keldysh rotation, each world-sheet surface $\mathrm{\ensuremath{\Sigma}}$ in the dual string theory expansion was found to exhibit a triple decomposition into the parts ${\mathrm{\ensuremath{\Sigma}}}^{\ifmmode\pm\else\textpm\fi{}}$ corresponding to the forward and backward segments of the closed time contour, and ${\mathrm{\ensuremath{\Sigma}}}^{\ensuremath{\wedge}}$ which corresponds to the instant in time where the two segments meet. After the Keldysh rotation, we find that the world-sheet surface $\mathrm{\ensuremath{\Sigma}}$ of the dual string theory undergoes a very different natural decomposition: $\mathrm{\ensuremath{\Sigma}}$ consists of a ``classical'' part ${\mathrm{\ensuremath{\Sigma}}}^{\mathrm{cl}}$ and a ``quantum embellishment'' part ${\mathrm{\ensuremath{\Sigma}}}^{\mathrm{qu}}$. We show that both parts of $\mathrm{\ensuremath{\Sigma}}$ carry their own independent genus expansion. The nonequilibrium sum over world-sheet topologies is naturally refined into a sum over the double decomposition of each $\mathrm{\ensuremath{\Sigma}}$ into its classical and quantum part. We apply this picture to the classical limits of the quantum nonequilibrium system (with or without interactions with a thermal bath), and find that in these limits, the dual string perturbation theory expansion reduces to its appropriately defined classical limit.

We explore a novel type of transition in certain 6D and 4D quantum field theories, in which the matter content of the theory changes while the gauge group and other parts of the spectrum remain invariant. Such transitions can occur, for example, for SU(6) and SU(7) gauge groups, where matter fields in a three-index antisymmetric representation and the fundamental representation are exchanged in the transition for matter in the two-index antisymmetric representation. These matter transitions are realized by passing through superconformal theories at the transition point. We explore these transitions in dual F-theory and heterotic descriptions, where a number of novel features arise. For example, in the heterotic description the relevant 6D SU(7) theories are described by bundles on K3 surfaces where the geometry of the K3 is constrained in addition to the bundle structure. On the F-theory side, non-standard representations such as the three-index antisymmetric representation of SU(N) require Weierstrass models that cannot be realized from the standard SU(N) Tate form. We also briefly describe some other situations, with groups such as Sp(3), SO(12), and SU(3), where analogous matter transitions can occur between different representations. For SU(3), in particular, we find a matter transition between adjoint matter and matter in the symmetric representation, giving an explicit Weierstrass model for the F-theory description of the symmetric representation that complements another recent analogous construction.

In recent work Kachru, Kumar, and Silverstein introduced a special class of nonsupersymmetric type II string theories in which the cosmological constant vanishes at the first two orders of perturbation theory. Heuristic arguments suggest the cosmological constant may vanish in these theories to all orders in perturbation theory leading to a flat potential for the dilaton. A slight variant of their model can be described in terms of a dual heterotic theory. The dual theory has a nonzero cosmological constant which is nonperturbative in the coupling of the original type II theory. The dual theory also predicts a mismatch between Bose and Fermi degrees of freedom in the nonperturbative D-brane spectrum of the type II theory.

Motivated by the Swampland Distance and the Emergent String Conjecture of Quantum Gravity, we analyse the infinite distance degenerations in the complex structure moduli space of elliptic K3 surfaces. All complex degenerations of K3 surfaces are known to be classified according to their associated Kulikov models of Type I (finite distance), Type II or Type III (infinite distance). For elliptic K3 surfaces, we characterise the underlying Weierstrass models in detail. Similarly to the known two classes of Type II Kulikov models for elliptic K3 surfaces we find that the Weierstrass models of the more elusive Type III Kulikov models can be brought into two canonical forms. We furthermore show that all infinite distance limits are related to degenerations of Weierstrass models with non-minimal singularities in codimension one or to models with degenerating generic fibers as in the Sen limit. We explicitly work out the general structure of blowups and base changes required to remove the non-minimal singularities. These results form the basis for a classification of the infinite distance limits of elliptic K3 surfaces as probed by F-theory in the companion paper [1]. The Type III limits, in particular, are (partial) decompactification limits as signalled by an emergent affine enhancement of the symmetry algebra.