Moduli Space Of Flat Connections Research Articles

Shifted geometric quantization

We introduce geometric quantization in the setting of shifted symplectic structures. We define Lagrangian fibrations and prequantizations of shifted symplectic stacks and their geometric quantization. In addition, we study many examples including symplectic groupoids, Hamiltonian spaces and moduli spaces of flat connections. In the case of Hamiltonian spaces we prove a derived analog of the “quantization commutes with reduction” principle.

Resurgence, conformal blocks, and the sum over geometries in quantum gravity

In two dimensional conformal field theories the limit of large central charge plays the role of a semi-classical limit. Certain universal observables, such as conformal blocks involving the exchange of the identity operator, can be expanded around this classical limit in powers of the central charge c. This expansion is an asymptotic series, so — via the same resurgence analysis familiar from quantum mechanics — necessitates the existence of non-perturbative effects. In the case of identity conformal blocks, these new effects have a simple interpretation: the CFT must possess new primary operators with dimension of order the central charge. This constrains the data of CFTs with large central charge in a way that is similar to (but distinct from) the conformal bootstrap. We study this phenomenon in three ways: numerically, analytically using Zamolodchikov’s recursion relations, and by considering non-unitary minimal models with large (negative) central charge. In the holographic dual to a CFT2, the expansion in powers of c is the perturbative loop expansion in powers of ћ. So our results imply that the graviton loop expansion is an asymptotic series, whose cure requires the inclusion of new saddle points in the gravitational path integral. In certain cases these saddle points have a simple interpretation: they are conical excesses, particle-like states with negative mass which are not in the physical spectrum but nevertheless appear as non-manifold saddle points that control the asymptotic behaviour of the loop expansion. This phenomenon also has an interpretation in SL(2, ℝ) Chern-Simons theory, where the non-perturbative effects are associated with the non-Teichmüller component of the moduli space of flat connections.

Genus-one complex quantum Chern–Simons theory

We consider the geometric quantisation of Chern--Simons theory for closed genus-one surfaces and semisimple complex groups. First we introduce the natural complexified analogue of the Hitchin connection in K\{a}hler quantisation, with polarisations coming from the nonabelian Hodge hyper-K\{a}hler geometry of the moduli spaces of flat connections, thereby complementing the real-polarised approach of Witten. Then we consider the connection of Witten, and we identify it with the complexified Hitchin connection using a version of the Bargmann transform on polarised sections over the moduli spaces.

Superintegrable Systems on Moduli Spaces of Flat Connections

The main result of this paper is the construction of a family of superintegrable Hamiltonian systems on moduli spaces of flat connections on a principal G-bundle on a surface. The moduli space is a Poisson variety with Atiyah–Bott Poisson structure. Among particular cases of such systems are spin generalizations of Ruijsenaars–Schneider models.

Poisson–Lie Groups and Gauge Theory

We review Poisson–Lie groups and their applications in gauge theory and integrable systems from a mathematical physics perspective. We also comment on recent results and developments and their applications. In particular, we discuss the role of quasitriangular Poisson–Lie groups and dynamical r-matrices in the description of moduli spaces of flat connections and the Chern–Simons gauge theory.

Boundary $\mathcal{N} = 2$ theory, Floer homologies, affine algebras, and the Verlinde formula

Generalizing our ideas in [arXiv:1006.3313], we explain how topologically-twisted N=2 gauge theory on a four-manifold with boundary, will allow us to furnish purely physical proofs of (i) the Atiyah-Floer conjecture, (ii) Munoz's theorem relating quantum and instanton Floer cohomology, (iii) their monopole counterparts, and (iv) their higher rank generalizations. In the case where the boundary is a Seifert manifold, one can also relate its instanton Floer homology to modules of an affine algebra via a 2d A-model with target the based loop group. As an offshoot, we will be able to demonstrate an action of the affine algebra on the quantum cohomology of the moduli space of flat connections on a Riemann surface, as well as derive the Verlinde formula.

Semiclassical limit of topological R\xe9nyi entropy in 3d Chern-Simons theory

We study the multi-boundary entanglement structure of the state associated with the torus link complement S3\\Tp,q in the set-up of three-dimensional SU(2)k Chern-Simons theory. The focal point of this work is the asymptotic behavior of the Rényi entropies, including the entanglement entropy, in the semiclassical limit of k → ∞. We present a detailed analysis of several torus links and observe that the entropies converge to a finite value in the semiclassical limit. We further propose that the large k limiting value of the Rényi entropy of torus links of type Tp,pn is the sum of two parts: (i) the universal part which is independent of n, and (ii) the non-universal or the linking part which explicitly depends on the linking number n. Using the analytic techniques, we show that the universal part comprises of Riemann zeta functions and can be written in terms of the partition functions of two-dimensional topological Yang-Mills theory. More precisely, it is equal to the Rényi entropy of certain states prepared in topological 2d Yang-Mills theory with SU(2) gauge group. Further, the universal parts appearing in the large k limits of the entanglement entropy and the minimum Rényi entropy for torus links Tp,pn can be interpreted in terms of the volume of the moduli space of flat connections on certain Riemann surfaces. We also analyze the Rényi entropies of Tp,pn link in the double scaling limit of k → ∞ and n → ∞ and propose that the entropies converge in the double limit as well.

Kauffman skein algebras and quantum Teichmüller spaces via factorization homology

We compute the factorization homology of the four-punctured sphere and punctured torus over the quantum group [Formula: see text] explicitly as categories of equivariant modules using the framework developed by Ben-Zvi et al. We identify the algebra of [Formula: see text]-invariants (quantum global sections) with the spherical double affine Hecke algebra of type [Formula: see text], in the four-punctured sphere case, and with the “cyclic deformation” of [Formula: see text] in the punctured torus case. In both cases, we give an identification with the corresponding quantum Teichmüller space as proposed by Teschner and Vartanov as a quantization of the moduli space of flat connections.

On orientations for gauge-theoretic moduli spaces

Let X be a compact manifold, D:Γ∞(E0)→Γ∞(E1) a real elliptic operator on X, G a Lie group, P→X a principal G-bundle, and BP the infinite-dimensional moduli space of all connections ∇P on P modulo gauge, as a topological stack. For each [∇P]∈BP, we can consider the twisted elliptic operator D∇Ad(P):Γ∞(Ad(P)⊗E0)→Γ∞(Ad(P)⊗E1) on X. This is a continuous family of elliptic operators over the base BP, and so has an orientation bundle OPE•→BP, a principal Z2-bundle parametrizing orientations of KerD∇Ad(P)⊕CokerD∇Ad(P) at each [∇P]. An orientation on (BP,E•) is a trivialization OPE•≅BP×Z2.In gauge theory one studies moduli spaces MPga of connections ∇P on P satisfying some curvature condition, such as anti-self-dual instantons on Riemannian 4-manifolds (X,g). Under good conditions MPga is a smooth manifold, and orientations on (BP,E•) pull back to orientations on MPga in the usual sense of differential geometry under the inclusion MPga↪BP. This is important in areas such as Donaldson theory, where one needs an orientation on MPga to define enumerative invariants.We explain a package of techniques, some known and some new, for proving orientability and constructing canonical orientations on (BP,E•), after fixing some algebro-topological information on X. We use these to construct canonical orientations on gauge theory moduli spaces, including new results for moduli spaces of flat connections on 2- and 3-manifolds, instantons, the Kapustin–Witten equations, and the Vafa–Witten equations on 4-manifolds, and the Haydys–Witten equations on 5-manifolds.

Q-deformed 3D loop gravity on the torus

The q-deformed loop gravity framework was introduced as a canonical formalism for the Turaev–Viro model (with ), allowing to quantize 3D Euclidean gravity with a (negative) cosmological constant using a quantum deformation of the gauge group. We describe its application to the 2-torus, explicitly writing the q-deformed gauge symmetries and deriving the reduced physical phase space of Dirac observables, which leads back to the Goldman brackets for the moduli space of flat connections. Furthermore it turns out that the q-deformed loop gravity can be derived through a gauge fixing from the Fock–Rosly bracket, which provides an explicit link between loop quantum gravity (for q real) and the combinatorial quantization of 3D gravity as a Chern–Simons theory with non-vanishing cosmological constant . A side-product is the reformulation of the loop quantum gravity phase space for vanishing cosmological constant , based on holonomies and fluxes, in terms of Poincaré holonomies. Although we focus on the case of the torus as an example, our results outline the general equivalence between 3D q-deformed loop quantum gravity and the combinatorial quantization of Chern–Simons theory for arbitrary graph and topology.

Toda Conformal Blocks, Quantum Groups, and Flat Connections

This paper investigates the relations between the Toda conformal field theories, quantum group theory and the quantisation of moduli spaces of flat connections. We use the free field representation of the {{mathcal {W}}}-algebras to define natural bases for spaces of conformal blocks of the Toda conformal field theory associated to the Lie algebra {mathfrak {s}}{mathfrak {l}}_3 on the three-punctured sphere with representations of generic type associated to the three punctures. The operator-valued monodromies of degenerate fields can be used to describe the quantisation of the moduli spaces of flat mathrm {SL}(3)-connections. It is shown that the matrix elements of the monodromies can be expressed as Laurent polynomials of more elementary operators which have a simple definition in the free field representation. These operators are identified as quantised counterparts of natural higher rank analogs of the Fenchel–Nielsen coordinates from Teichmüller theory. Possible applications to the study of the non-Lagrangian SUSY field theories are briefly outlined.

Modular Group Representations in Combinatorial Quantization with Non-Semisimple Hopf Algebras

Let $\Sigma_{g,n}$ be a compact oriented surface of genus $g$ with $n$ open disks removed. The algebra $\mathcal{L}_{g,n}(H)$ was introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche and is a combinatorial quantization of the moduli space of flat connections on $\Sigma_{g,n}$. Here we focus on the two building blocks $\mathcal{L}_{0,1}(H)$ and $\mathcal{L}_{1,0}(H)$ under the assumption that the gauge Hopf algebra $H$ is finite-dimensional, factorizable and ribbon, but not necessarily semisimple. We construct a projective representation of $\mathrm{SL}_2(\mathbb{Z})$, the mapping class group of the torus, based on $\mathcal{L}_{1,0}(H)$ and we study it explicitly for $H = \overline{U}_q(\mathfrak{sl}(2))$. We also show that it is equivalent to the representation constructed by Lyubashenko and Majid.

Polysymplectic reduction and the moduli space of flat connections

A polysymplectic structure is a vector-valued symplectic form, that is, a closed nondegenerate 2-form with values in a vector space. We first outline the polysymplectic Hamiltonian formalism with coefficients in a vector space , we then apply this framework to show that the moduli space of flat connections on a principal bundle over a compact manifold M is a polysymplectic reduction of the space of all connections by the action of the gauge group with respect to a natural polysymplectic structure with values in an infinite dimensional Banach space. As a consequence, the moduli space inherits a canonical H2(M)-valued presymplectic structure.Along the way, we establish various properties of polysymplectic manifolds. For example, a Darboux-type theorem asserts that every -symplectic manifold locally symplectically embeds in a standard polysymplectic manifold . We also show that both the Arnold conjecture and the well-known convexity properties of the classical moment map fail to hold in the polysymplectic setting.

Projective Representations of Mapping Class Groups in Combinatorial Quantization

Let $\Sigma_{g,n}$ be a compact oriented surface of genus $g$ with $n$ open disks removed. The graph algebra $\mathcal{L}_{g,n}(H)$ was introduced by Alekseev--Grosse--Schomerus and Buffenoir--Roche and is a combinatorial quantization of the moduli space of flat connections on $\Sigma_{g,n}$. We construct a projective representation of the mapping class group of $\Sigma_{g,n}$ using $\mathcal{L}_{g,n}(H)$ and its subalgebra of invariant elements. Here we assume that the gauge Hopf algebra $H$ is finite-dimensional, factorizable and ribbon, but not necessarily semi-simple. We also give explicit formulas for the representation of the Dehn twists generating the mapping class group; in particular, we show that it is equivalent to a representation constructed by V. Lyubashenko using categorical methods.

Moduli space of noncommutative flat connections and finite-fold nomcommutative coverings

If we regard classical differential geometry then the moduli space of flat connections depends on the fundamental group. Here we consider the noncommutative generalization of this result. This work states the correspondence between noncommutative coverings and flat connections.

Asymptotic expansions of the Witten–Reshetikhin–Turaev invariants of mapping tori I

In this paper we engage in a general study of the asymptotic expansion of the Witten–Reshetikhin–Turaev invariants of mapping tori of surface mapping class group elements. We use the geometric construction of the Witten–Reshetikhin–Turaev topological quantum field theory via the geometric quantization of moduli spaces of flat connections on surfaces. We identify assumptions on the mapping class group elements that allow us to provide a full asymptotic expansion. In particular, we show that our results apply to all pseudo-Anosov mapping classes on a punctured torus and show by example that our assumptions on the mapping class group elements are strictly weaker than hitherto successfully considered assumptions in this context. The proof of our main theorem relies on our new results regarding asymptotic expansions of oscillatory integrals, which allows us to go significantly beyond the standard transversely cut out assumption on the fixed point set. This makes use of the Picard–Lefschetz theory for Laplace integrals.

Conditions of smoothness of moduli spaces of flat connections and of character varieties

We use gauge theoretic and algebraic methods to examine sufficient conditions for smooth points on the moduli space of flat connections on a compact manifold and on the character variety of a finitely generated and presented group. We give a complete proof of the slice theorem for the action of the group of gauge transformations on the space of flat connections. Consequently, the slice is smooth if the second cohomology of the manifold with coefficients in the semisimple part of the adjoint bundle vanishes. On the other hand, we find that the smoothness of the slice for the character variety of a finitely generated and presented group depends not only on the second group cohomology but also on the relation module of the presentation. However, when there is a single relator or if there is no relation among the relators in the presentation, our condition reduces to the minimality of the second group cohomology. This is also verified using Fox calculus. Finally, we compare the conditions of smoothness in the two approaches.

Differential characters and cohomology of the moduli of flat connections

Let $\pi\colon P\to M$ be a principal bundle and $p$ an invariant polynomial of degree r on the Lie algebra of the structure group. The theory of Chern-Simons differential characters is exploited to define an homology map $\chi^{k} : H_{2r-k-1}(M)\times H_{k}(\mathcal{F}/\mathcal{G})\to \mathbb{R}/\mathbb{Z}$, for $k<r-1$, where $\mathcal{F} /\mathcal{G}$ is the moduli space of flat connections of $\pi$ under the action of a subgroup $\mathcal{G}$ of the gauge group. The differential characters of first order are related to the Dijkgraaf-Witten action for Chern-Simons Theory. The second order characters are interpreted geometrically as the holonomy of a connection in a line bundle over $\mathcal{F}/\mathcal{G})$. The relationship with other constructions in the literature is also analyzed.

Geometric U-folds in four dimensions

We describe a general construction of geometric U-folds compatible with a non-trivial extension of the global formulation of four-dimensional extended supergravity on a differentiable spin manifold. The topology of geometric U-folds depends on certain flat fiber bundles which encode how supergravity fields are globally glued together. We show that smooth non-trivial U-folds of this type can exist only in theories where both the scalar and space-time manifolds have non-trivial fundamental group and in addition the scalar map of the solution is homotopically non-trivial. Consistency with string theory requires smooth geometric U-folds to be glued using subgroups of the effective discrete U-duality group, implying that the fundamental group of the scalar manifold of such solutions must be a subgroup of the latter. We construct simple examples of geometric U-folds in a generalization of the axion–dilaton model of supergravity coupled to a single vector multiplet, whose scalar manifold is a generally non-compact Riemann surface of genus at least two endowed with its uniformizing metric. We also discuss the relation between geometric U-folds and a moduli space of flat connections defined on the scalar manifold, which involves certain character varieties not studied in the literature.

Moduli spaces of Λ-modules on abelian varieties

We study the moduli space MX(Λ,n) of semistable Λ-modules of vanishing Chern classes over an abelian variety X, where Λ belongs to a certain subclass of D-algebras. In particular, for Λ=DX (resp. Λ=Sym•TX) we obtain a description of the moduli spaces of flat connections (resp. Higgs bundles).We give a description of MX(Λ,n) in terms of a symmetric product of a certain fibre bundle over the dual abelian variety Xˆ. We also give a moduli interpretation to the associated Hilbert scheme as the classifying space of Λ-modules with extra structure. Finally, we study the non-abelian Hodge theory associated to these new moduli spaces.