Spin from defects in two-dimensional quantum field theory
We build two-dimensional quantum field theories on spin surfaces starting from theories on oriented surfaces with networks of topological defect lines and junctions. The construction uses a combinatorial description of the spin structure in terms of a triangulation equipped with extra data. The amplitude for the spin surfaces is defined to be the amplitude for the underlying oriented surface together with a defect network dual to the triangulation. Independence of the triangulation and of the other choices follows if the line defect and junctions are obtained from a -separable Frobenius algebra with involutive Nakayama automorphism in the monoidal category of topological defects. For rational conformal field theory we can give a more explicit description of the defect category, and we work out two examples related to free fermions in detail: the Ising model and the WZW model at level .
- 1 Introduction
- 2 A combinatorial model for spin surfaces
- 3 Two-dimensional QFT with defects
- 4 Amplitudes on spin surfaces from amplitudes on defect surfaces
- 5 Application to two-dimensional rational conformal field theory
Consider a two-dimensional quantum field theory which is defined on oriented surfaces with metric, and which allows for the presence of line defects. We will be interested in topological line defects, that is, line defects that can be deformed on the surface without changing the amplitude which assigns to the surface. Line defects can meet in junction points, and also here we will be interested only in topological junctions. Altogether, the QFT assigns amplitudes to surfaces with defect networks, and these amplitudes are invariant under deformations which move the defect lines and junctions without generating new intersections. We will briefly review this framework in Section 3.
In this paper we show how given a topological line defect of the QFT , together with 1- and 3-valent topological junctions for this defect – all subject to conditions we list below – one can construct a new QFT which is defined on surfaces with metric and with spin structure.
Quite generally, to construct new -dimensional QFTs from a given -dimensional QFT, one can try to implement the following idea:
Carry out a state sum construction of an -dimensional topological field theory inside an -dimensional QFT with defects.
That is, try to model the cell decomposition of the -dimensional manifolds used in the state sum construction with defects of dimensions in the QFT with defects, and translate the invariance conditions of the state sum construction into requirements on these defects. This idea generalises the state sum construction of topological field theories in the sense that the latter can be thought of as being internal to the trivial QFT.
So far the above idea has been applied only in two dimensions, and we will describe this case in more detail now.
The topological line defects and topological junctions of an oriented two-dimensional QFT form a -linear monoidal category [DKR]. The objects in are the possible line defects, the tensor product amounts to fusion of line defects, and the morphisms are given by junctions. Since junctions can be thought of as fields inserted at the point were the defect lines meet, they form a -vector space. In addition, has two-sided duals (given by orientation reversal of the defect line) and is pivotal.
The category contains much interesting information about the QFT . For example, given a -separable symmetric Frobenius algebra in (we will explain these notions in the table below), one can construct a new QFT , the orbifold of by , which is again defined on oriented surfaces with metric [FrFRS, CRu]. This amounts to applying the above idea to the state sum construction of oriented two-dimensional topological field theories [BP, FHK]. The name “orbifold” is justified since the construction of reduces to the standard orbifold construction if comes from a group symmetry of . But there are ’s which do not come from group symmetries (e.g. the exceptional cases in [FRS1, CRCR]), so in this sense could be called a generalised orbifold.
Another example is given by the main result of this paper: a -separable Frobenius algebra whose Nakayama automorphism (see table below) squares to the identity allows one to define a QFT on spin surfaces starting from the QFT , which was defined on oriented surfaces with defects. This is again an instance of the above idea, applied now to the state sum construction of 2d spin TFTs given in [NR]. Let us describe the resulting construction in more detail.
To evaluate on a spin surface , one first encodes the spin structure in terms of
a triangulation of the underlying oriented surface ,
a choice of an orientation for each edge of the triangulation and of a preferred edge for each triangle (a marking),
a sign for each edge (the edge signs).
This combinatorial model for spin structures was introduced in [NR] and will be reviewed in Section 2. Other closely related combinatorial models can be found in [Ku, CRe, Bu]. An extension of the work in [NR] to -spin surfaces can be found in [No].
Next, one places a network of line defects and junctions labelled by the structure maps of the Frobenius algebra on the graph dual to the triangulation. The precise type of the junctions depends on the marking and the edge signs. The amplitude is defined as the amplitude that assigns to with the above defect network. This is described in detail in Section 4.
Frobenius algebras as above also appear in another state sum construction of two-dimensional spin topological field theory in [BT] and in the description of “generalised twisted sectors” in orbifolds in [BCP] (there without restriction on the order of the Nakayama automorphism).
The following table gives the definitions of the various algebraic notions above, as well as their meaning in the context of topological defects. More details can be found in Section 4. The identities of topological defect networks in the table are to be understood as follows. Given two surfaces with metric, and , equipped with defect networks such that the defect networks only differ in a small patch as shown in the table, the amplitudes of and agree, i.e. .
|concept||algebraic description||description in QFT with defects|
|object in the pivotal monoidal category||topological defect of|
|algebra||morphisms and such that is a unit for and is associative.||
a 1-valent and a 3-valent topological defect junction,
, ,such that we have the identities
|algebra||An algebra and a coalgebra (with counit and coproduct ) such that the Frobenius condition holds: .||
Additional 1- and 3-valent defect junctions,
, ,such that and satisfy the same identities as and but with reversed orientations on the defect lines, as well as
|-separable||The product and coproduct of the Frobenius algebra satisfy .||
The defect junctions and satisfy
|symmetric||The pairing satisfies .||
The defect junctions and satisfy
|autom.||Let be as above and set . Then is given by . is an automorphism of the Frobenius algebra .||
abbreviates the following combination of defect lines and junctions
We remark that a Frobenius algebra is symmetric if and only if its Nakayama automorphism is the identity.
The description of QFTs on spin surfaces in terms of QFTs on oriented surfaces with defect networks is useful if one has good control over the defect category . One class of theories where this is the case are rational two-dimensional conformal field theories. In fact, the study of rational CFTs on spin surfaces is one of the main motivations for us to develop the present formalism. Let us have a closer look at this case, which we also discuss in detail in Section 5.
Fix a vertex operator algebra with the property that its category of representations, , is a modular tensor category. We will call such vertex operator algebras rational. Rational CFTs are then those with left/right symmetry given by a rational . The simplest rational CFT with a given symmetry is called the charge-conjugate theory, or the “Cardy case”. In the Cardy case CFT for , the category of topological defects, which are in addition transparent to the left/right symmetry , is given by itself.
In the free fermion examples which we investigate in Sections 5.3 and 5.4, it turns out that instead of the CFT on its own, one should consider the product of the CFT with the trivial 2d TFT that takes values in super vector spaces (see Section 3.2). The relevant category of topological defects is then , and it is this category in which we need to find -separable Frobenius algebras whose Nakayama automorphism squares to the identity.
There are many questions still unanswered in the present paper. For example, in the context of rational CFTs: Can every 2d spin CFT be obtained from a suitable oriented CFT in this way? Can one explicitly classify all -separable Frobenius algebras with (or suitable Morita classes thereof) in in interesting examples? For supersymmetric CFTs, what characterises those which give world sheet and/or spacetime supersymmetric theories?
Finally, it should be straightforward to extend the constructions in this paper to spin surfaces with boundaries and defects, as well as to -spin surfaces (following [No]). We hope to return to these points in the future.
Acknowledgments: The authors would like to thank Nils Carqueville for helpful discussions and comments on a draft version of this paper. IR thanks the Erwin-Schrödinger Institute in Vienna for hospitality during the programme “Modern trends in topological quantum field theory” (February and March 2014) where part of this work was completed. SN was supported by the DFG funded Research Training Group 1670 “Mathematics inspired by string theory and quantum field theory”. IR is supported in part by the DFG funded Collaborative Research Center 676 “Particles, Strings, and the Early Universe”.
2. A combinatorial model for spin surfaces
In this section we briefly review the combinatorial model for spin surfaces introduced in [NR].
By a surface we mean an oriented two-dimensional smooth compact real manifold with parametrised boundary. The boundary parametrisation consists of, firstly, an ordering of the connected components of , and, secondly, for each an orientation preserving smooth embedding . Here, is a half-open annulus for some and maps the unit circle to the ’th boundary component.
Denote by the -matrices with positive determinant. Let be the connected double cover of . A spin surface is a surface together with a -principal bundle, which is a double cover of the oriented frame bundle such that the action is compatible with the action on the frame bundle. Furthermore, is equipped with a lift of the boundary parametrisation maps to the spin bundle. In more detail, allows for two non-isomorphic spin structures: the Neveu-Schwarz-type (-type) spin structure extends to the disc, the Ramond-type (-type) spin structure does not. Write for , equipped with one of these two spin structures. Then is a map of spin surfaces. We say the ’th boundary component of is of -type (-type) if the spin structure of is of -type (-type). More details can be found in [NR, Sect. 2], see in particular [NR, Def. 2.14].
Let be a surface. Fix the standard triangulation of the unit circle to be given by the three arcs between the points , and . Via the boundary parametrisations , this gives a triangulation of the boundary of . Choose an extension of this triangulation from the boundary to the interior. A marking on this triangulation is an assignment of a preferred edge to each triangle and of an orientation to each edge. The orientation of the boundary edges is defined to be that induced by the counter-clockwise orientation of the unit circle.
By a choice of edge signs we mean a map from the set of edges of the triangulation to . For a given choice of edge signs on a marked triangulation of one obtains a spin structure on minus the vertices of the triangulation [NR, Sect. 3.7]. The spin structure extends to the vertices if and only if the following condition holds at each vertex [NR, Cor. 3.14 & Lem. 3.15]:
is an inner vertex: Let be the number of triangles containing such that the preferred edge of is the first one, counting the edges of counter-clockwise starting from the vertex . Let be the number of edges containing and pointing away from . The edge signs must satisfy
where the product is over all edges containing the vertex .
is a boundary vertex: Let and be as above. For include the boundary edge pointing away from . If the boundary component containing is of -type, and if the vertex is the image of the point 1 under the corresponding parametrisation map , set . Otherwise set . The edge signs must satisfy
where again the product is over all edges containing the vertex , including the two boundary edges adjacent to .
If this condition holds at each vertex, we call the edge signs admissible. By definition, admissible edge signs turn the surface into a spin surface . This combinatorial model for spin structures is the first main ingredient in this paper.
Given a spin surface, one can ask whether or not a given closed path in the frame bundle lifts to the spin bundle. Since we are working in two dimensions, we can turn a smooth closed path with nowhere vanishing derivative into an – up to homotopy unique – path in the frame bundle. Namely, at each point of the path pick a second vector, which together with the velocity vector of the path is an oriented frame, and which depends continuously on the parametrisation of the path. Thus, given we can ask whether or not lifts to a closed path in . We now explain how the lifting property can be read off from the edge signs.
Assume that in each triangle of the triangulation, the path looks as in one of the six configurations shown in Figure 2.1 a. Now multiply together the signs given in Figure 2.1 a for all triangles transversed by , and the signs given in Figure 2.1 b for all edges crossed by . Let be the result. Then the corresponding path in the frame bundle has a closed lift if and only if
where the product is over all edges crossed by [NR, Lem. 3.13].
Suppose the path is the boundary of a small disc containing a vertex of the triangulation. Then the spin structure extends to the vertex if and only if does not have a closed lift. One quickly checks that in this way (2.3) reproduces the admissibility condition for edge signs at an inner vertex as given in (2.1).
3. Two-dimensional QFT with defects
The second ingredient in our construction are surfaces with defects and field theories on such surfaces. We collect some material from [DKR] to fix our conventions.
3.1. Surfaces with defects, state spaces, amplitudes
A surface with defects is a surface together with a compact oriented one-dimensional submanifold , such that and meets transversally. We also fix a set of defect conditions. Each connected component of is labelled by an element of .
So far the underlying surface was an oriented smooth manifold. To discuss non-topological quantum field theories, we equip with a metric or a conformal structure. For the purpose of this paper, by a quantum field theory on surfaces with defects we shall mean the following:
For each surface with defects and for all boundary components of , an assignment of a -super vector space (the state space).
For each surface with defects an even linear map (the amplitude).
For a permutation of , let be identical to except that the ordering of the boundary components of is changed: The ’th boundary of is the ’th boundary component of . Then
where is the parity sign arising from applying the permutation to the vectors . (For example, a transposition of two adjacent arguments gives a sign if and only if both are odd.)
Compatibility with glueing. Since we will not use the glueing property explicitly in this paper, we will omit its description and refer to [DKR].
Consider a surface with defects which is diffeomorphic to such that each connected component of the defect submanifold in gets mapped to an interval . That is, up to diffeomorphism is a cylinder with parallel defect lines connecting the two boundary circles.
We will say is non-degenerate if for all such cylinders the pairing is non-degenerate in the sense that for all there exists a such that , and vice versa.
We assume that all defect conditions in describe topological defects. This is a requirement on the state spaces and amplitudes :
: Consider the ’th boundary component of a surface with defects with parametrisation map . For each defect ending on that boundary component write , where is the defect condition and if the defect is oriented away from the boundary, and otherwise. Starting from let be the sequence of defects encountered when following clockwise. Then depends on the defects only through the ordered list , but not on where and how precisely the defect lines end on the boundary component.
: Let and be two surfaces with defects with the same underlying surface , such that the defect submanifold of (with its orientation and defect condition for each component) can be isotoped to that of . The isotopy is allowed to move the endpoints of defect lines as long as no such endpoint crosses the points , . Then .
Each state space contains a subspace of topological junction fields (which may be 0). When inserting a topological junction field in the ’th argument (corresponding to the ’th boundary component) the map does not change when changing the position or size of the ’th boundary component (by glueing on appropriate cylinders, see [DKR] for more details).
We also demand that topological junction fields are parity-even.
Below we will consider surfaces with defect networks. In such a network, the junction points are labelled by a topological junction field and it is understood that such a junction point represents a small circular hole, parametrised in a way compatible with the linear order of the defect lines attached to the junction point. Let be such a surface, and let be a junction point labelled by a topological junction field . When evaluating on it is understood that the argument corresponding to the boundary component replacing the point is taken to be . Since is a topological junction field, the precise shape of the small circular hole and the choice of parametrisation map are irrelevant. Since a topological junction field is parity-even, we do not need to remember the ordering of topological junction fields on surfaces with defect networks.
The topological junction fields can be used to turn the set of defect fields into a -linear pivotal monoidal category . The objects are sequences of defect conditions with orientations, and morphisms are topological junction fields on a boundary circle to which the defects from and are attached. The tensor product is concatenation of sequences; duality is orientation reversal and reversion of the order of the sequence. Pivotality follows from rotation invariance of topological junction fields. For details we refer to [DKR, Sect. 2.4] and to [CRu, Thm. 3.3].
We will assume the category to be additive. This amounts to completing the set of (sequences of) defect conditions with respect to taking finite sums. The -linear additive pivotal monoidal category of defect conditions and topological junction fields is the second central ingredient in this paper.
Below, we will often draw parts of defect surfaces to illustrate a configuration of defect lines and junction labels. We will use a notation for junction fields that is close to string diagram notation for morphisms in monoidal categories. Namely, we draw a topological junction field inserted on the surface as
here in the example , etc. To turn this into a surface with defects, the box has to be replaced by a circular boundary, parametrised such that the image of lies on the left vertical side of the box. Again, the precise way in which this replacement is done does not matter as is topological.
3.2. The trivial defect TFT and products of defect theories
Let be the category of -super vector spaces. This is a symmetric monoidal category with symmetric braiding given by exchange of factors with a parity sign: For and , homogeneous elements of degrees :
We will denote by the trivial two-dimensional topological field theory with defects which takes values in . By this we mean the lattice TFT with defects constructed from the trivial algebra via the procedure in [DKR, Sect. 3].
Let us stress this point again: since the morphisms in are topological junctions, they are given by even linear maps (i.e. by morphisms in ). The state space assigned to a boundary circle, on the other hand, may contain even and odd elements.
We will make use of to introduce a -grading on the state spaces of ungraded defect theories in Sections 5.2–5.4. This will be done by defining a product theory . The necessity of such products will be illustrated in the free fermion example in Section 5.3.
Let , be two QFTs on surfaces with defects. The product theory is defined as follows:
The defect category is . Here “” is the product of -linear additive categories: the objects are direct sums of pairs of objects and . The morphism space between two such pairs is . For direct sums of such pairs, the morphism spaces are the corresponding direct sums.
The state space of the product theory for the ’th boundary component of a defect surface is given by (tensor product of -super vector spaces).
The amplitudes of the product theory are , where is the permutation in (i.e. the permutation with parity signs) which reorders the argument from to .
We will only apply the product construction in the case that all state spaces of are purely even and is the trivial theory with values in defined above. In this situation there are no parity signs in the permutation map above. The topological theory has two fundamental defects, labelled by the one-dimensional even and odd super vector spaces and . The state spaces of for a boundary circle with a defect line or starting at it are
where for a super vector space , denotes the parity shifted super vector space, i.e. and . The last isomorphism in the two sets of identities above holds because by definition and .
4. Amplitudes on spin surfaces from amplitudes on defect surfaces
Fix a QFT on surfaces with defects, possibly equipped with a metric or a conformal structure. Let be the corresponding monoidal category of topological defects and topological junction fields. Fix furthermore a defect condition and topological junction fields which are morphisms
in . The aim of this section is to explain how to obtain from the data , subject to certain conditions, a QFT which assigns amplitudes to spin surfaces.
4.1. From spin structures to defect networks
Let be a spin surface. Our first task is to produce from a surface with defects . The construction is as follows:
Pick a triangulation of . Equip the triangulation with a marking and with edge signs encoding the spin structure of .
At each boundary component of , the slightly more complicated looking defect network shown in Figure 4.1 c is inserted.
We now need to make sure that the amplitude does not depend on the choices made in (a). This is guaranteed by a number of identities on . We will write these as identities of morphisms in the defect category , using string diagram notation. By definition of , we may then as well think of these string diagrams as identities satisfied by amplitudes of surfaces with defects which only differ locally as indicated in the string diagram.
The derivation of the identities below is essentially the same as in [NR] with one important exception: In [NR] we worked with a symmetric monoidal category, and so one is allowed to use the symmetric braiding but no duals; here we work in a pivotal non-braided category, so one is allowed to use duals but no braiding. We omit the calculations and just state the results, using the same ordering of the conditions as in [NR, Sect. 4.2].
Edge orientation change. For ,
(The first and second equality are equivalent by pivotality.)
Leaf exchange automorphism on a single triangle. For ,
Cyclic permutation of boundary edges for a single triangle. For ,
Pachner 2-2 move. For ,
Pachner 3-1 move and its inverse. For subject to ,
Using these conditions, we can now state:
Suppose and satisfy conditions (1)–(5) above. Let be a spin surface and , surfaces with defects obtained by steps (a)–(c) above. Then .
The proof is analogous to that of [NR, Prop. 4.1] and we omit it here.
4.2. Algebraic description
A Frobenius algebra in a monoidal category is an object together with a unital algebra and counital coalgebra structure, such that the coproduct is a bimodule map. We denote the structure morphisms as
A Frobenius algebra is called -separable if . In a pivotal monoidal category, the Nakayama automorphism of a Frobenius algebra takes the form
where in the above string diagram we use the graphical shorthands for morphisms as collected in Figure 4.2. One checks that is an algebra and coalgebra automorphism.
After these preliminaries, we formulate the data subject to conditions (1)–(5) above in terms of a Frobenius algebra with extra properties. This is done by the same procedure as [NR, Sect. 4.3] and we state the following results without proof.
Define the product as
As in [NR, Sect. 4.3] we make the
Assumptions: 1) The coparing is non-degenerate. 2) The product has a unit .
We denote the pairing dual to the copairing by . Using the graphical shorthands in Figure 4.2, we set
That the two distinct expressions for and are indeed equal is straightforward to check, see e.g. [NR, Sect. 4.3]. We have the following theorem, the proof of which is easily adapted from [NR, Sect. 4.3].
4.3. State spaces
Consider a surface with defects which is diffeomorphic to a cylinder with one -defect connecting the two parametrised boundaries. Let be the state space assigned to the boundary component from which the -defect points away. In this section we will define two subspaces and and investigate their properties.
To start with, we will need two more assumption on the behaviour of . The first is easy:
Assumption: 3) The QFT is non-degenerate.
The second additional assumption is slightly more elaborate. Let and a morphism in , i.e. a topological junction field. Consider the two cylinders
Here we suppose that the underlying surface of both cylinders is the same, and that the defect graph (but not the labelling) agrees in a neighbourhood of both boundary components. Note that we can then write .
Let be the state space assigned to boundary component 2 of and , and let (resp. ) be that assigned to boundary component 1 of (resp. ). We would like to encode the difference between the value of on the two cylinders in a linear map . That is, we assume:
Assumption: 4) In the situation (4.11) there exists a linear map such that for all and we have
The linear map is unique by non-degeneracy in Assumption 3. The symbol “” is chosen as behaves like a partial trace:
Let . Then
for all and we have
for all and we have
For part 1 consider the equalities
Step 1 is the definition of , step 2 uses invariance of topological defects and junctions under isotopies, and step 3 is again the definition of . Non-degeneracy of (Assumption 3) implies the identity in part 1. Part 2 works along the same lines:
In the following computation we fix an underlying cylinder as in (4.11) and just vary the defect network and junction fields. In all cases a defect of type will emanate from boundary 1 (the “inner” boundary in the picture), and we denote by the corresponding state space. Using Assumption 4, we define the following endomorphisms of :
Here, in the first definition is understood as a morphism . In terms of cylinders with defect networks, in the case of for example, we have
is an involution.
and are projectors.
For , .
Part 1 is immediate from Lemma 4.3 (2). For part 2 compute
Step 1 is Lemma 4.3 (2), step 2 is associativity, coassociativity and the Frobenius property of , step 3 is Lemma 4.3 (1) applied to the coproduct, and in step 4 -separability of is used and the definition of is substituted. The calculation for is similar. Part 3 for follows from
Steps 1 and 2 are Lemma 4.3, parts (2) and (1), respectively. Step 3 follows as is an automorphism of the Frobenius algebra , and step 4 is as step 1. The argument for is the same. ∎
We can now define the quantum field theory on spin surfaces in terms of . Let be a spin surface and a corresponding defect surface obtained by steps (a)–(c) in Section 4.1. We set
Let be a spin surface and let be the type of the ’th boundary component of . Then, for , , we have
The proof uses invariance of under the choice of triangulation and is analogous to that of [NR, Prop. 4.14]; we omit the details.
The above lemma shows that will in general be degenerate, but that it can easily be made non-degenerate by restricting to the image of . In more detail, one proceeds as follows. Define
Let and let be a spin cylinder with boundary components of type . Then is non-degenerate.
Let be a surface with defects such that . On place the following defect network:
By Assumption 4 we can write