We study a special sort of 2-dimensional extended Topological Quantum Field Theories (TQFTs) which we call open-closed TQFTs. These are defined on open-closed cobordisms by which we mean smooth compact oriented 2-manifolds with corners that have a particular global structure in order to model the smooth topology of open and closed string worldsheets. We show that the category of open-closed TQFTs is equivalent to the category of knowledgeable Frobenius algebras. A knowledgeable Frobenius algebra (A,C,i,i^*) consists of a symmetric Frobenius algebra A, a commutative Frobenius algebra C, and an algebra homomorphism i:C->A with dual i^*:A->C, subject to some conditions. This result is achieved by providing a generators and relations description of the category of open-closed cobordisms. In order to prove the sufficiency of our relations, we provide a normal form for such cobordisms which is characterized by topological invariants. Starting from an arbitrary such cobordism, we construct a sequence of moves (generalized handle slides and handle cancellations) which transforms the given cobordism into the normal form. Using the generators and relations description of the category of open-closed cobordisms, we show that it is equivalent to the symmetric monoidal category freely generated by a knowledgeable Frobenius algebra. Our formalism is then generalized to the context of open-closed cobordisms with labeled free boundary components, i.e. to open-closed string worldsheets with D-brane labels at their free boundaries.
To open-closed cobordism surfaces, open-closed string topology associates topological quantum field theory (TQFT) operations, namely string operations, which depend only on homeomorphism types of surfaces and which satisfy the sewing property. We show that most TQFT string operations vanish in open-closed string topology. We describe those open-closed cobordisms with vanishing string operations, and give a short list of open-closed cobordisms with possibly nontrivial string operations.
Previously, we showed that most of the open-closed topological quantum field theory (TQFT) string operations vanish including all the higher genus TQFT operations, and we described a small list of genus zero open-closed TQFT string operations which can be nontrivial. In this paper, we consider open-closed string operations associated to open-closed cobordisms homeomorphic to discs. These operations constitute the main part of genus zero string operations, and they include the saddle string operation of two open strings interacting at their internal points. We show not only that disc string ope…
Cohen and Godin constructed positive boundary topological quantum field theory (TQFT) structure on the homology of free loop spaces of oriented closed smooth manifolds by associating a certain operations called string operations to orientable surfaces with parametrized boundaries. We show that all TQFT string operations associated to surfaces of genus at least one vanish identically. This is a simple consequence of properties of the loop coproduct which will be discussed in detail. One interesting property is that the loop coproduct is nontrivial only on the degree $d$ homology group of the co…
In this expository paper written for physicists and geometers we introduce the notions of TQFT and of orbifold. Then we survey the construction of TQFT's originating from orbifolds such as Chen-Ruan theory and Orbifold String Topology.
We give some new examples of Frobenius objects in the category of sets and relations $\textbf{Rel}$. One example is a groupoid with a twisted counit. Another example is the set of conjugacy classes of a group. We also classify Frobenius objects in $\textbf{Rel}$ with two or three elements, and we compute the associated surface invariants using the partition functions of the corresponding TQFTs.
In this paper, we describe the relation between the Turaev--Viro TQFT and the string-net space introduced in the papers of Levin and Wen. In particular, the case of surfaces with boundary is considered in detail.
Given a smooth closed manifold M with a family {L_i} of closed submanifolds, we consider the free loop space LM and the spaces PM(L_i,L_j) of open strings (paths g:[0,1]->M with g(0) in L_i, and g(1) in L_j). We construct string topology operations resulting in an open-closed TQFT on the family (h_*(LM),h_*(PM(L_i,L_j)) which extends the known string topology TQFT on h_*(LM). Here, h_* is a multiplicative generalized homology theory supporting orientations for M and the L_i. To construct the operations, we introduce the notion of open-closed fat graph, generalizing fat graphs to the open-close…
We construct a family of 3d quantum field theories $\mathcal T_{n,k}^A$ that conjecturally provide a physical realization -- and derived generalization -- of non-semisimple mathematical TQFT's based on the modules for the quantum group $U_q(\mathfrak{sl}_n)$ at an even root of unity $q=\text{exp}(iπ/k)$. The theories $\mathcal T_{n,k}^A$ are defined as topological twists of certain 3d $\mathcal N=4$ Chern-Simons-matter theories, which also admit string/M-theory realizations. They may be thought of as $SU(n)_{k-n}$ Chern-Simons theories, coupled to a twisted $\mathcal N=4$ matter sector (the so…
Let G be a compact Lie group. Let E be a principal G-bundle over a closed manifold M, and Ad(E) its adjoint bundle. In this paper we describe a new Frobenius algebra structure on h_*(Ad(E)), where h_* is an appropriate generalized homology theory. Recall that a Frobenius algebra has both a product and a coproduct. The product in this new Frobenius algebra is induced by the string topology product. In particular, the product can be defined when G is any topological group and in the case that E is contractible it is precisely the Chas-Sullivan string product on H_*(LM). We will show that the cop…
We establish the general machinery of string topology for differentiable stacks. This machinery allows us to treat on an equal footing free loops in stacks and hidden loops. In particular, we give a good notion of a free loop stack, and of a mapping stack $\map(Y,\XX)$, where $Y$ is a compact space and $\XX$ a topological stack, which is functorial both in $\XX$ and $Y$ and behaves well enough with respect to pushouts. We also construct a bivariant (in the sense of Fulton and MacPherson) theory for topological stacks: it gives us a flexible theory of Gysin maps which are automatically compatib…
Given a differential graded (dg) symmetric Frobenius algebra $A$ we construct an unbounded complex $\mathcal{D}^{*}(A,A)$, called the Tate-Hochschild complex, which arises as a totalization of a double complex having Hochschild chains as negative columns and Hochschild cochains as non-negative columns. We prove that the complex $\mathcal{D}^*(A,A)$ computes the singular Hochschild cohomology of $A$. We construct a cyclic (or Calabi-Yau) $A$-infinity algebra structure, which extends the classical Hochschild cup and cap products, and an $L$-infinity algebra structure extending the classical Gers…
We prove that the modular operad of diffeomorphism classes of Riemann surfaces with both `open' and `closed' boundary components, in the sense of string field theory, is the modular completion of its genus 0 part quotiented by the Cardy condition. We also provide a finitary presentation of a version of this modular two-colored operad and characterize its algebras via morphisms of Frobenius algebras, recovering some previously known results of Kaufmann, Penner and others. As an important auxiliary tool we characterize inclusions of cyclic operads that induce inclusions of their modular completi…
In this paper we extend our correlation functions to the open/closed case. This gives rise to actions of an open/closed version of the Sullivan PROP as well as an action of the relevant moduli space. There are several unexpected structures and conditions that arise in this extension which are forced upon us by considering the open sector. For string topology type operations, one cannot just consider graphs, but has to take punctures into account and one has to restrict the underlying Frobenius algebras. In the moduli space, one first has to pass to a smaller moduli space which is closed under …
We study cohomology theories of strongly homotopy algebras, namely $A_\infty, C_\infty$ and $L_\infty$-algebras and establish the Hodge decomposition of Hochschild and cyclic cohomology of $C_\infty$-algebras thus generalising previous work by Loday and Gerstenhaber-Schack. These results are then used to show that a $C_\infty$-algebra with an invariant inner product on its cohomology can be uniquely extended to a symplectic $C_\infty$-algebra (an $\infty$-generalisation of a commutative Frobenius algebra introduced by Kontsevich). As another application, we show that the `string topology' oper…
Let M be a closed, connected manifold, and LM its loop space. In this paper we describe closed string topology operations in h_*(LM), where h_* is a generalized homology theory that supports an orientation of M. We will show that these operations give h_*(LM) the structure of a unital, commutative Frobenius algebra without a counit. Equivalently they describe a positive boundary, two dimensional topological quantum field theory associated to h_*(LM). This implies that there are operations corresponding to any surface with p incoming and q outgoing boundary components, so long as q >0. The abse…
We provide a complete generators and relations presentation of the 2-dimensional extended unoriented and oriented bordism bicategories as symmetric monoidal bicategories. Thereby we classify these types of 2-dimensional extended topological field theories with arbitrary target bicategory. As an immediate corollary we obtain a concrete classification when the target is the symmetric monoidal bicategory of algebras, bimodules, and intertwiners over a fixed commutative ground ring. In the oriented case, such an extended topological field theory is equivalent to specifying a (non-commutative) sepa…
