Holomorphic Chern-Simons Theory Research Articles

Quantum chaos in 2D gravity

We present a quantitative and fully non-perturbative description of the ergodic phase of quantum chaos in the setting of two-dimensional gravity. To this end we describe the doubly non-perturbative completion of semiclassical 2D gravity in terms of its associated universe field theory. The guiding principle of our analysis is a flavor-matrix theory (fMT) description of the ergodic phase of holographic gravity, which exhibits \mathrm{U}(n|n)U(n|n) causal symmetry breaking and restoration. JT gravity and its 2D-gravity cousins alone do not realize an action principle with causal symmetry, however we demonstrate that their universe field theory, the Kodaira-Spencer (KS) theory of gravity, does. After directly deriving the fMT from brane-antibrane correlators in KS theory, we show that causal symmetry breaking and restoration can be understood geometrically in terms of different (topological) D-brane vacua. We interpret our results in terms of an open-closed string duality between holomorphic Chern-Simons theory and its closed-string equivalent, the KS theory of gravity. Emphasis will be put on relating these geometric principles to the characteristic spectral correlations of the quantum ergodic phase.

Twistors, the ASD Yang-Mills equations and 4d Chern-Simons theory

We show that the approaches to integrable systems via 4d Chern-Simons theory and via symmetry reductions of the anti-self-dual Yang-Mills equations are closely related, at least classically. Following a suggestion of Kevin Costello, we start from holomorphic Chern-Simons theory on twistor space, defined with the help of a meromorphic (3,0)-form Ω. If Ω is nowhere vanishing, it descends to a theory on 4d space-time with classical equations of motion equivalent to the anti-self-dual Yang-Mills equations. Examples include a 4d analogue of the Wess-Zumino-Witten model and a theory of a Lie algebra valued scalar with a cubic two derivative interaction. Under symmetry reduction, these yield actions for 2d integrable systems. On the other hand, performing the symmetry reduction directly on twistor space reduces holomorphic Chern-Simons theory to the 4d Chern-Simons theory with disorder defects studied by Costello & Yamazaki. Finally we show that a similar reduction by a single translation leads to a 5d partially holomorphic Chern-Simons theory describing the Bogomolny equations.

Toward a twistor action for chiral higher-spin gravity

A covariant twistor action for chiral higher-spin theory in (A)dS and flat space is constructed in terms of a holomorphic Chern-Simons theory on twistor space. The action reproduces all known cubic vertices of chiral higher-spin theory in flat space. The spacetime action of the holomorphic Chern-Simons theory in flat space is also obtained.

D=5 holomorphic Chern-Simons and the pure spinor superstring

The physical states of D=5 holomorphic Chern-Simons theory correspond to on-shell D=10 open superstring states in the cohomology of q+, where q+ is one of the 16 spacetime supersymmetry generators. Scattering amplitudes of these states can be computed either using the usual Ramond-Neveu-Schwarz (RNS) superstring prescription with N=1 worldsheet supersymmetry, or using a topological ĉ=5 string theory with twisted N=2 worldsheet supersymmetry.It will be argued that the relation between D=5 holomophic Chern-Simons and the RNS superstring is identical to the relation between the pure spinor superstring and the recently constructed B-RNS-GSS superstring which has both N=1 worldsheet supersymmetry and D=10 spacetime supersymmetry. Physical states of the pure spinor superstring correspond to on-shell B-RNS-GSS states which are in the cohomology of λαqα, where λα is a D=10 pure spinor. And scattering amplitudes of these states can be computed either using the full B-RNS-GSS superstring prescription with N=1 worldsheet supersymmetry, or using the pure spinor superstring amplitude prescription with twisted N=2 worldsheet supersymmetry. This should be useful for proving equivalence of the RNS and pure spinor amplitude prescriptions and for clarifying the relation of their multiloop subtleties.

Deformed integrable models from holomorphic Chern-Simons theory

We study the approaches to two-dimensional integrable field theories via a six-dimensional(6D) holomorphic Chern-Simons theory defined on twistor space. Under symmetry reduction, it reduces to a four-dimensional Chern-Simons theory, while under solving along fibres it leads to four-dimensional(4D) integrable theory, the anti-self-dual Yang-Mills or its generalizations. From both four-dimensional theories, various two-dimensional integrable field theories can be obtained. In this work, we try to investigate several two-dimensional integrable deformations in this framework. We find that the $\lambda$-deformation, the rational $\eta$-deformation and the generalized $\lambda$-deformation can not be realized from 4D integrable model approach, even though they could be obtained from 4D Chern-Simons theory. The obstacle stems from the incompatibility between the symmetry reduction and the boundary conditions. Nevertheless, we show that a coupled theory of $\lambda$-deformation and the $\eta$-deformation in the trigonometric description could be obtained from the 6D theory in both ways, by considering the case that $(3,0)$-form in the 6D theory is allowed to have zeros.

3-Dimensional mixed BF theory and Hitchin’s integrable system

The affine Gaudin model, associated with an untwisted affine Kac–Moody algebra, is known to arise from a certain gauge fixing of 4-dimensional mixed topological–holomorphic Chern–Simons theory in the Hamiltonian framework. We show that the finite Gaudin model, associated with a finite-dimensional semisimple Lie algebra, or more generally the tamely ramified Hitchin system on an arbitrary Riemann surface, can likewise be obtained from a similar gauge fixing of 3-dimensional mixed BF theory in the Hamiltonian framework.

Symmetric space \u03bb-model exchange algebra from 4d holomorphic Chern-Simons theory

We derive, within the Hamiltonian formalism, the classical exchange algebra of a lambda deformed string sigma model in a symmetric space directly from a 4d holomorphic Chern-Simons theory. The explicit forms of the extended Lax connection and R-matrix entering the Maillet bracket of the lambda model are explained from a symmetry principle. This approach, based on a gauge theory, may provide a mechanism for taming the non-ultralocality that afflicts most of the integrable string theories propagating in coset spaces.

A twistor space action for Yang-Mills theory

We consider the twistor space ${\cal P}^6\cong{\mathbb R}^4{\times}{\mathbb C}P^1$ of ${\mathbb R}^4$ with a non-integrable almost complex structure ${\cal J}$ such that the canonical bundle of the almost complex manifold $({\cal P}^6, {\cal J})$ is trivial. It is shown that ${\cal J}$-holomorphic Chern-Simons theory on a real $(6|2)$-dimensional graded extension ${\cal P}^{6|2}$ of the twistor space ${\cal P}^6$ is equivalent to self-dual Yang-Mills theory on Euclidean space ${\mathbb R}^4$ with Lorentz invariant action. It is also shown that adding a local term to a Chern-Simons-type action on ${\cal P}^{6|2}$, one can extend it to a twistor action describing full Yang-Mills theory.

Twisted Supergravity and Koszul Duality: A Case Study in AdS$$_3$$

In this note, we study a simplified variant of the familiar holographic duality between supergravity on AdS$_3\times S^3\times T^4$ and the SCFT (on the moduli space of) the symmetric orbifold theory $Sym^N(T^4)$ as $N \rightarrow \infty$. This variant arises conjecturally from a twist proposed by the first author and Si Li. We recover a number of results concerning protected subsectors of the original duality working directly in the twisted bulk theory. Moreover, we identify the symmetry algebra arising in the $N\rightarrow \infty$ limit of the twisted gravitational theory. We emphasize the role of $\textit{Koszul duality}$---a ubiquitous mathematical notion to which we provide a friendly introduction---in field theory and string theory. After illustrating the appearance of Koszul duality in the "toy" example of holomorphic Chern-Simons theory, we describe how (a deformation of) Koszul duality relates bulk and boundary operators in our twisted setup, and explain how one can compute algebra OPEs diagrammatically using this notion. Further details, results, and computations will appear in a companion paper.

Holomorphic Chern-Simons theory and lambda models: PCM case

In this note we consider the symplectic reduction of a four-dimensional holomorphic Chern-Simons theory recently introduced in [1] for describing integrable field theories. We work out explicitly the case of the lambda deformed Principal Chiral Model (PCM) and show that the symplectic reduction works as a localization mechanism. The reduced Chern-Simons theory restricts to the set of poles of the twist function underlying the theory, where the known classical integrability of the lambda deformed PCM can be reconstructed from the phase space data associated to this set of points in the spectral space.

Anomaly cancellation in the topological string

We describe the coupling of holomorphic Chern-Simons theory at large N with Kodaira-Spencer gravity. We explain a new anomaly cancellation mechanism at all loops in perturbation theory for open-closed topological B-model. At one loop this anomaly cancellation is analogous to the Green-Schwarz mechanism. As an application, we introduce a type I version of Kodaira-Spencer theory in complex dimensions 3 and 5. In complex dimension 5, we show that it can only be coupled consistently at the quantum level to holomorphic Chern-Simons theory with gauge group SO(32). This is analogous to the Green-Schwarz mechanism for the physical type I string. This coupled system is conjectured to be a supersymmetric localization of type I string theory. In complex dimension 3, the required gauge group is SO(8).

Finite deformations from a heterotic superpotential: holomorphic Chern-Simons and an L\u221e algebra

We consider finite deformations of the Hull-Strominger system. Starting from the heterotic superpotential, we identify complex coordinates on the off-shell parameter space. Expanding the superpotential around a supersymmetric vacuum leads to a thirdorder Maurer-Cartan equation that controls the moduli. The resulting complex effective action generalises that of both Kodaira-Spencer and holomorphic Chern-Simons theory. The supersymmetric locus of this action is described by an L3 algebra.

Supersymmetric Yang-Mills theory as higher Chern-Simons theory

We observe that the string field theory actions for the topological sigma models describe higher or categorified Chern-Simons theories. These theories yield dynamical equations for connective structures on higher principal bundles. As a special case, we consider holomorphic higher Chern-Simons theory on the ambitwistor space of four-dimensional space-time. In particular, we propose a higher ambitwistor space action functional for maximally supersymmetric Yang-Mills theory.

Holomorphic Chern-Simons theory coupled to off-shell Kodaira-Spencer gravity

We construct an action for holomorphic Chern-Simons theory that couples the gauge field to off-shell gravitational backgrounds, comprising the complex structure and the (3,0)-form of the target space. Gauge invariance of the off-shell action is achieved by enlarging the field space to include an appropriate system of Lagrange multipliers, ghost and ghost-for-ghost fields. Both the BRST transformations and the BV action are compactly and neatly written in terms of superfields which include fields, backgrounds and their antifields. We show that the anti-holomorphic target space derivative can be written as a BRST-commutator on a functional space containing the anti-fields of both the dynamical fields and the gravitational backgrounds. We derive from this result a Ward identity that determines the anti-holomorphic dependence of physical correlators.

Taming open/closed string duality with a Losev trick

A target space string field theory formulation for open and closed B-model is provided by giving a Batalin-Vilkovisky quantization of the holomorphic Chern-Simons theory with off-shell gravity background. The target space expression for the coefficients of the holomorphic anomaly equation for open strings are obtained. Furthermore, open/closed string duality is proved from a judicious integration over the open string fields. In particular, by restriction to the case of independence on continuous open moduli, the shift formulas of [8] are reproduced and shown therefore to encode the data of a closed string dual.

Yukawa couplings in F-theory and non-commutative geometry

We consider Yukawa couplings generated by a configuration of intersecting sevenbranes in F-theory. In configurations with a single interaction point and no fluxes turned on, the Yukawa matrices have rank one. This is no longer true when the three-form H-flux is turned on, which is generically the case for F-theory compactifications on Calabi-Yau fourfolds. In the presence of H-fluxes, the Yukawa coupling is computed using a non-commutative deformation of holomorphic Chern-Simons theory (and its reduction to seven-branes) and subsequently the rank of the Yukawa matrix changes. Such fluxes give rise to a hierarchical structure in the Yukawa matrix in F-theory GUTs of the type which has recently been proposed as a resolution of the flavor hierarchy problem.

Gromov–Witten theory and Donaldson–Thomas theory, I

We conjecture an equivalence between the Gromov–Witten theory of 3-folds and the holomorphic Chern–Simons theory of Donaldson and Thomas. For Calabi–Yau 3-folds, the equivalence is defined by the change of variables is the Euler characteristic parameter of Donaldson–Thomas theory. The conjecture is proven for local Calabi–Yau toric surfaces.

Geometric transitions and integrable systems

We consider B-model large N duality for a new class of noncompact Calabi–Yau spaces modeled on the neighborhood of a ruled surface in a Calabi–Yau threefold. The closed string side of the transition is governed at genus zero by an A 1 Hitchin integrable system on a genus g Riemann surface Σ. The open string side is described by a holomorphic Chern–Simons theory which reduces to a generalized matrix model in which the eigenvalues lie on the compact Riemann surface Σ. We show that the large N planar limit of the generalized matrix model is governed by the same A 1 Hitchin system therefore proving genus zero large N duality for this class of transitions.

Computation of Superpotentials for D-Branes

We present a general method for the computation of tree-level superpotentials for the world-volume theory of B-type D-branes. This includes quiver gauge theories in the case that the D-brane is marginally stable. The technique involves analyzing the A-infinity structure inherent in the derived category of coherent sheaves. This effectively gives a practical method of computing correlation functions in holomorphic Chern-Simons theory. As an example, we give a more rigorous proof of previous results concerning 3-branes on certain singularities including conifolds. We also provide a new example.

Marginal deformations of [formula omitted] SYM from open/closed twistor strings

We investigate how the marginal deformations of N = 4 supersymmetric Yang–Mills theory (analysed in particular by Leigh and Strassler) arise within B-model topological string theory on supertwistor space C P 3 | 4 . This is achieved by turning on a certain closed string background in the fermionic directions. Through a specific open/closed correlation function, this mode induces a correction to holomorphic Chern–Simons theory, corresponding to the self-dual part of the chiral operator added on the gauge theory side. The effect of the deformation is interpreted as non-anticommutativity between some of the odd coordinates of C P 3 | 4 . Motivated by this, we extend the twistor formalism for calculating MHV amplitudes in N = 4 SYM to these N = 1 theories by introducing a suitable star product between the wavefunctions. We check that our prescription yields the expected results to linear order in the deformation parameter.