Space Of Conformal Blocks Research Articles

The Hilbert space of large N Chern-Simons matter theories

We demonstrate that the known expressions for the thermal partition function of large N Chern-Simons matter theories admit a simple Hilbert space interpretation as the partition function of an associated ungauged large N matter theory with one additional condition: the Fock space of this associated theory is projected down to the subspace of its quantum singlets i.e. singlets under the Gauss law for Chern-Simons gauge theory. Via the Chern-Simons / WZW correspondence, the space of quantum singlets are equivalent to the space of WZW conformal blocks. One step in our demonstration involves recasting the Verlinde formula for the dimension of the space of conformal blocks in SU(N)k and U(N)k,k′ WZW theories into a simple and physically transparent form, which we also rederive by evaluating the partition function and superconformal index of pure Chern-Simons theory in the presence of Wilson lines. A particular consequence of the projection of the Fock space of Chern-Simons matter theories to quantum (or WZW) singlets is the ‘Bosonic Exclusion Principle’: the number of bosons occupying any single particle state is bounded above by the Chern-Simons level. The quantum singlet condition (unlike its Yang-Mills Gauss Law counterpart) has a nontrivial impact on thermodynamics even in the infinite volume limit. In this limit the projected Fock space partition function reduces to a product of partition functions, one for each single particle state. These single particle state partition functions are q-deformations of their free boson and free fermion counterparts and interpolate between these two special cases. We also propose a formula for the large N partition function that is valid for arbitrary finite volume of the spatial S2 and not only at large volume.

Vertex Algebras for S-duality

We define new deformable families of vertex operator algebras $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ associated to a large set of S-duality operations in four-dimensional supersymmetric gauge theory. They are defined as algebras of protected operators for two-dimensional supersymmetric junctions which interpolate between a Dirichlet boundary condition and its S-duality image. The $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ vertex operator algebras are equipped with two $$\mathfrak {g}$$ affine vertex subalgebras whose levels are related by the S-duality operation. They compose accordingly under a natural convolution operation and can be used to define an action of the S-duality operations on a certain space of vertex operator algebras equipped with a $$\mathfrak {g}$$ affine vertex subalgebra. We give a self-contained definition of the S-duality action on that space of vertex operator algebras. The space of conformal blocks (in the derived sense, i.e. chiral homology) for $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ is expected to play an important role in a broad generalization of the quantum Geometric Langlands program. Namely, we expect the S-duality action on vertex operator algebras to extend to an action on the corresponding spaces of conformal blocks. This action should coincide with and generalize the usual quantum Geometric Langlands correspondence. The strategy we use to define the $$\mathfrak {A}[\mathfrak {g}, \Psi , \sigma ]$$ vertex operator algebras is of broader applicability and leads to many new results and conjectures about deformable families of vertex operator algebras.

Conformal Nets II: Conformal Blocks

Conformal nets provide a mathematical formalism for conformal field theory. Associated to a conformal net with finite index, we give a construction of the ‘bundle of conformal blocks’, a representation of the mapping class groupoid of closed topological surfaces into the category of finite-dimensional projective Hilbert spaces. We also construct infinite-dimensional spaces of conformal blocks for topological surfaces with smooth boundary. We prove that the conformal blocks satisfy a factorization formula for gluing surfaces along circles, and an analogous formula for gluing surfaces along intervals. We use this interval factorization property to give a new proof of the modularity of the category of representations of a conformal net.

Hecke Transformations of Conformal Blocks in WZW Theory. I. KZB Equations for Non-Trivial Bundles

We describe new families of the Knizhnik-Zamolodchikov-Bernard (KZB) equations related to the WZW-theory corresponding to the adjoint $G$-bundles of different topological types over complex curves $\Sigma_{g,n}$ of genus $g$ with $n$ marked points. The bundles are defined by their characteristic classes - elements of $H^2(\Sigma_{g,n},\mathcal{Z}(G))$, where $\mathcal{Z}(G)$ is a center of the simple complex Lie group $G$. The KZB equations are the horizontality condition for the projectively flat connection (the KZB connection) defined on the bundle of conformal blocks over the moduli space of curves. The space of conformal blocks has been known to be decomposed into a few sectors corresponding to the characteristic classes of the underlying bundles. The KZB connection preserves these sectors. In this paper we construct the connection explicitly for elliptic curves with marked points and prove its flatness.

Logarithmic Intertwining Operators and the Space of Conformal Blocks Over the Projective Line

We show that the space of logarithmic intertwining operators among logarithmic modules for a vertex operator algebra is isomorphic to the space of 3-point conformal blocks over the projective line. This is considered as a generalization of Zhu's result for ordinary intertwining operators among ordinary modules.

Conformal blocks in the tensor product of vector representations and localization formulas

Using equivariant localization formulas we give a formula for conformal blocks at level one on the sphere as suitable polynomials. Using this presentation we give a generating set in the space of conformal blocks at any level if the marked points on the sphere are generic.

Rank-level duality for conformal blocks of the linear group

We generalise the rank-level duality (also known as strange duality) proved by P. Belkale [J. Amer. Math. Soc. 21 (2008), no. 1, 235–258] for generic curves and A. Marian and D. Oprea [Invent. Math. 168 (2007), no. 2, 225–247] for every smooth curve to the case of spaces of conformal blocks related to moduli spaces of parabolic bundles on a smooth pointed projective curve.

The classifying algebra for defects

We demonstrate that topological defects in a rational conformal field theory can be described by a classifying algebra for defects – a finite-dimensional semisimple unital commutative associative algebra whose irreducible representations give the defect transmission coefficients. We show in particular that the structure constants of the classifying algebra are traces of operators on spaces of conformal blocks and that the defect transmission coefficients determine the defect partition functions.

The three-dimensional origin of the classifying algebra

It is known that reflection coefficients for bulk fields of a rational conformal field theory in the presence of an elementary boundary condition can be obtained as representation matrices of irreducible representations of the classifying algebra, a semisimple commutative associative complex algebra. We show how this algebra arises naturally from the three-dimensional geometry of factorization of correlators of bulk fields on the disk. This allows us to derive explicit expressions for the structure constants of the classifying algebra as invariants of ribbon graphs in the three-manifold S 2 × S 1 . Our result unravels a precise relation between intertwiners of the action of the mapping class group on spaces of conformal blocks and boundary conditions in rational conformal field theories.

TRIGONOMETRIC DEGENERATION AND ORBIFOLD WESS-ZUMINO-WITTEN MODEL I

Trigonometric degeneration of the Baxter-Belavin elliptic r matrix is described by the degeneration of the twisted WZW model on elliptic curves. The spaces of conformal blocks and conformal coinvariants of the degenerate model are factorised into those of the orbifold WZW model.

On Bohr-Sommerfeld bases

This paper combines algebraic and Lagrangian geometry to construct a special basis in every space of conformal blocks, the Bohr-Sommerfeld (BS) basis. We use the method of Borthwick-Paul-Uribe [3], in which every vector of a BS basis is determined by some half-weight Legendrian distribution coming from a Bohr-Sommerfeld fibre of a real polarization of the underlying symplectic manifold. The advantage of BS bases (compared to the bases of theta functions in [23]) is that we can use the powerful methods of asymptotic analysis of quantum states. This shows that Bohr-Sommerfeld bases are quasiclassically unitary. Thus we can apply these bases to compare the Hitchin connection [11] and the KZ connection defined by the monodromy of the Knizhnik-Zamolodchikov equation in the combinatorial theory (see, for example, [14] and [15]).

On inner product in modular tensor categories II: Inner product on conformal blocks and affine inner product identities

This is the second part of the paper (the first part is published in Jour. of AMS, vol.9, 1135--1170, q-alg/9508017). In the first part, we defined for every modular tensor category (MTC) inner products on the spaces of morphisms and proved that the inner product on the space $\Hom (\bigoplus X_i\otimes X^*_i, U)$ is modular invariant. Also, we have shown that in the case of the MTC arising from the representations of the quantum group $U_q \sln$ at roots of unity and $U$ being a symmetric power of the fundamental representation, this inner product coincides with so-called Macdonald's inner product on symmetric polynomials. In this paper, we apply the same construction to the MTC coming from the integrable representations of affine Lie algebras. In this case our construction immediately gives a hermitian form on the spaces of conformal blocks, and this form is modular invariant (Warning: we cannot prove that it is positive definite). We show that this form can be rewritten in terms of asymptotics of KZ equations, and calculate it for $sl_2$, in which case the formula is a natural affine analogue of Macdonald's inner product identities. We also formulate as a conjecture similar formula for $sl_n$.

Quantization of the Space of Conformal Blocks

We consider the discrete Knizhnik–Zamolodchikov connection (qKZ) associated to gl(N), defined in terms of rational R-matrices. We prove that under certain resonance conditions, the qKZ connection has a non-trivial invariant subbundle which we call the subbundle of quantized conformal blocks. The subbundle is given explicitly by algebraic equations in terms of the Yangian Y(gl(N)) action. The subbundle is a deformation of the subbundle of conformal blocks in CFT. The proof is based on an identity in the algebra with two generators x,y and defining relation xy=yx+yy.

Analytic expression for the dimension of the space of conformal blocks in the Wess?Zumino?Novikov?Witten model with gauge groupSU(2)

Analytic expression for the dimension of the space of conformal blocks in the Wess?Zumino?Novikov?Witten model with gauge groupSU(2)

Conformal blocks and generalized theta functions

Let M(r) be the moduli space of rank r vector bundles with trivial determinant on a Riemann surface X . This space carries a natural line bundle, the determinant line bundle L . We describe a canonical isomorphism of the space of global sections of L^k with a space known in conformal field theory as the ``space of conformal blocks, which is defined in terms of representations of the Lie algebra sl(r, C((z))).

RESHETIKHIN-TURAEV AND CRANE-KOHNO-KONTSEVICH 3-MANIFOLD INVARIANTS COINCIDE

The coincidence of two different presentations of Witten 3-manifold invariants is proved. One of them, invented by Reshetikhin and Turaev, is based on the surgery presentation a of 3-manifold and the representation theory of quantum groups; another one, invented by Kohno and Crane and, in slightly different language by Kontsevich, is based on a Heegaard decomposition of a 3-manifold and representations of the Teichmuller group, arising in conformal field theory. The explicit formula for the matrix elements of generators of the Teichmuller group in the space of conformal blocks in the SU(2) k, WZNW-model is given,using the Jones polynomial of certain links.

Duality and quantum groups

We show that the duality properties of Rational Conformal Field Theories follow from the defining relations and the representation theory of quantum groups. The fusion and braiding matrices are q-analogues of the 6 j-symbols and the modular transformation matrices are obtained from the properties of the co-multiplication. We study in detail the Wess-Zumino-Witten models and the rational gaussian models as examples, but carry out the arguments in general. We point out the connections with the Chern-Simons approach. We give general arguments of why the general solution to the polynomial equations of Moore and Seiberg describing the duality properties of Rational Conformal Field Theories defines a Quantum Group acting on the space of conformal blocks. A direct connection between Rational Theories and knot invariants is also presented along the lines of Jones' original work.

Instantons and affine algebras I: The Hilbert scheme and vertex operators

This is the first in a series of papers which describe the action of an affine Lie algebra with central charge $n$ on the moduli space of $U(n)$-instantons on a four manifold $X$. This generalises work of Nakajima, who considered the case when $X$ is an ALE space. In particular, this describes the combinatorial complexity of the moduli space as being precisely that of representation theory, and thus will lead to a description of the Betti numbers of moduli space as dimensions of weight spaces. This Lie algebra acts on the space of conformal blocks (i\.e\., the cohomology of a determinant line bundle on the moduli space) generalising the ``insertion'' and ``deletion'' operations of conformal field theory, and indeed on any cohomology theory. In the particular case of $U(1)$-instantons, which is essentially the subject of this present paper, the construction produces the basic representation after Frenkel-Kac. Then the well known quadratic nature of $ch_2$, $$ch_2 = \frac{1}{2} c_1\cdot c_1 - c_2 $$ becomes precisely the formula for the eigenvalue of the degree operator, i\.e\. the well known quadratic behaviour of affine Lie algebras.