Flag manifold sigma-models: the 1/N-expansion and the anomaly two-form

TL;DR

This paper develops a 1/N-expansion for flag manifold sigma-models, constructs a gauged linear sigma-model representation, and analyzes the quantum anomaly affecting a classically conserved non-local charge.

Contribution

It introduces a new quantum analysis of flag manifold sigma-models, revealing the anomaly in the non-local charge conservation at the quantum level.

Findings

Explicit calculation of the anomaly in charge conservation

Development of a gauged linear sigma-model representation

Establishment of the 1/N-expansion for these models

Abstract

We construct a gauged linear sigma-model representation and develop a 1/N-expansion for flag manifold sigma-models previously proposed by the author. Classically there exists a zero-curvature representation for the equations of motion of these models, which leads in particular to the existence of a conserved non-local charge. We show that at the quantum level this charge is no longer conserved and calculate explicitly the anomaly in its conservation law.

Equations

$$\mathcal{F}(d_1,\dots ,d_m)=\{0\subset V_1\subset \dots \subset V_{m-1}\subset V_m=\text{mathds}{C}^N\},$$

$$\mathcal{F}(d_1,\dots ,d_m)=GL(N,\mathbb{C})/\mathcal{H}.$$

$${\dim }_{\text{mathds}C}\mathcal{F}(d_1,\dots ,d_m)=N^2-\sum _{i=1}^m{d}_i(d_i-d_{i-1})$$

$${\mathcal{F}}_{n_1,\dots ,n_m}=\frac{U(N)}{U(n_1)\times \dots \times U(n_m)},\sum _{i=1}^m{n}_i=N.$$

$${\dim }_{\mathbb{R}}{\mathcal{F}}_{n_1,\dots ,n_m}=N^2-\sum _{i=1}^m{n}_i^2=2\times {\dim }_{\text{mathds}C}{\mathcal{F}}_{d_1,\dots ,d_m}.$$

$$[\mathfrak{h},\mathfrak{h}]\subset \mathfrak{h},[\mathfrak{h},\mathfrak{m}]\subset \mathfrak{m}.$$

$$\mathfrak{g}:={\mathfrak{u}}_N={\oplus }_{i=1}^m{\mathfrak{u}}_{n_i}\oplus \mathfrak{m},$$

$$\mathfrak{m}\otimes \text{mathds}C={\oplus }_{i\ne j}{V}_{ij},\text{where}{V}_{ij}=\text{mathds}{C}^{n_i{n}_j}$$

$$j=[j{]}_{\mathfrak{h}}+\sum _{i\ne k}{j}_{ik},j_{ik}\in V_{ik}.$$

$$\Omega =\sum _{i<j}{a}_{ij}\mathrm{Tr}(j_{ij}\wedge j_{ji}).$$

$$\mathcal{D}{j}_{ij}=\sum _{m\ne (i,j)}{j}_{im}\wedge j_{mj}$$

$$a_{ij}+a_{jk}+a_{ki}=0\text{for all pairwise different}(i,j,k).$$

$$a_{ij}=z_i-z_j.$$

$$\Omega =\mathrm{Tr}(zj\wedge j),\text{where}z=\mathrm{Diag}(z_1\text{mathds}{1}_{n_1},\dots ,z_m\text{mathds}{1}_{n_m}).$$

$${\mathcal{F}}_{d_1,\dots ,d_m}=\{gz{g}^{-1},g\in SU(N)\}.$$

$$\mu =gz{g}^{-1}.$$

$$g_0\cdot g(x)=g(\tilde{x})\cdot h_0,g_0\in U(N),h_0\in U(n_1)\times \cdots \times U(n_m).$$

$$j(v_{a})\equiv i_{v_{a}}\,j=-g^{-1}\,\nabla_{v_{a}}g=-g^{-1}\frac{d}{dt}\left(e^{T_{a}\,t}\,g\right)\big{|}_{t=0}=-g^{-1}T_{a}g$$

$$d(\mathrm{tr}(\mu T_a))=d(\mathrm{tr}(gz{g}^{-1}{T}_a))=\mathrm{tr}(z[j,g^{-1}{T}_{a}g])=i_{v_a}\mathrm{tr}(zj\wedge j)=i_{v_a}\Omega$$

$$\mathfrak{m}\otimes \text{mathds}C={\mathfrak{m}}_{+}\oplus {\mathfrak{m}}_{-},$$

$$\text{where}[\mathfrak{h},{\mathfrak{m}}_{\pm }]\subset {\mathfrak{m}}_{\pm }.$$

$$\mathcal{J}\circ V_{pq}=\pm i{V}_{pq}\text{for}1\le p<q\le N.$$

$$[\mathcal{J}\circ X,\mathcal{J}\circ Y]-\mathcal{J}\circ ([\mathcal{J}\circ X,Y]+[X,\mathcal{J}\circ Y])-[X,Y]=0$$

$$(1-i\mathcal{J})[(1+i\mathcal{J})X,(1+i\mathcal{J})Y]=0.$$

$$[{\mathfrak{m}}_{+},{\mathfrak{m}}_{+}]\subset {\mathfrak{m}}_{+},[{\mathfrak{m}}_{-},{\mathfrak{m}}_{-}]\subset {\mathfrak{m}}_{-},$$

$$J=-g^{-1}dg=J_{\mathfrak{h}}+J_{+}+J_{-},J_{\pm }\in {\mathfrak{m}}_{\pm }.$$

$$dJ_{-}=\big{[}-J_{0}\wedge J_{0}+(\textrm{terms with}\,J_{-})-J_{+}\wedge J_{+}\big{]}_{\mathfrak{m}_{-}}\,.$$

$${\mathrm{H}}^2({\mathcal{F}}_N,\mathbf{Z})={\mathbf{Z}}^{N-1},$$

$$\stackrel{N}{\underset{i=1}{\oplus }}{L}_i={\mathcal{F}}_N\times \text{mathds}{C}^N.$$

$$\sum _{i=1}^N{\Omega }_i=0$$

$$i:{\mathcal{F}}_N\hookrightarrow \underset{N\text{factors}}{\underbracket{\text{mathds}C\text{mathds}{P}^{N-1}\times \cdots \times \text{mathds}C\text{mathds}{P}^{N-1}}}.$$

Full text

1. 1 The geometry of flag manifolds

2. 1.1 The flag manifold as a complex manifold

3. 1.2 Symplectic structures

4. 1.3 Complex structures

5. 1.4 Cohomology

6. 2 The models

7. 2.0.1 Dependence on the metric.

8. 3 Relation to models with $\mathbb{Z}_{m}$-graded target spaces

9. 4 The GLSM representation

10. 4.1 General flag manifolds

11. 5 Feynman rules for the $1\over N$-expansion

12. 6 The Wilson loop and conserved charges

13. 6.0.1 Example.

14. 6.1 The regularized charge

15. 6.2 The operator product expansion

16. 6.2.1 The commutator $[(\ast K)_{z}(z_{1}),(\ast K)_{z}(z_{2})]$

17. 6.2.2 The commutator $[(\ast K)_{z}(z_{1}),(\ast K)_{\bar{z}}(z_{2})]$

18. 6.2.3 Order $1\over N$

19. 6.2.4 The residual part

20. 6.3 The anomaly

21. 7 Conclusion and outlook

22. A Flag manifolds and elements of representation theory

23. A.1 ‘Quantization’ of the symplectic form on flag manifolds

24. B Eliminating auxiliary fields

25. C The point splitting of Lüscher

26. D The integral giving the anomaly at order $1\over\sqrt{N}$

27. E One more integral

1 The geometry of flag manifolds

1.1 The flag manifold as a complex manifold

The flag manifold in $\mathds{C}^{N}$ may be defined as the manifold of linear complex subspaces, embedded to each other:

[TABLE]

where $\mathrm{dim}\,V_{i}=d_{i}$. The group $GL(N,\mathbb{C})$ acts transitively on this manifold, and the stabilizer of any given point (i.e. of a given sequence of embedded linear spaces) is a maximal parabolic subgroup (‘a staircase’), consisting of matrices, depicted in Fig. 1.

Therefore we can view the flag manifold as a homogeneous space

[TABLE]

It follows that the flag manifold is a complex manifold of complex dimension

[TABLE]

Let us denote $\mathrm{dim}\,V_{i}/V_{i-1}=n_{i}$. Then one can also present the flag manifold as a quotient space of the unitary group:

[TABLE]

This parametrization is naturally related to a choice of Euclidian metric in $\mathds{C}^{N}$, which allows performing orthogonal decompositions of the form ${V_{i}\simeq V_{i-1}\oplus V_{i}/V_{i-1}}$. Let us compute the real dimension and make sure it is twice greater than the complex dimension computed earlier:

[TABLE]

Note that sometimes we will denote by $\mathcal{F}_{N}$ the complete flag manifold, i.e. the manifold (4), where all $n_{i}=1$.

Example. The projective space $\mathds{C}\mathds{P}^{N-1}$ may be viewed as the quotient space $(u_{1},\ldots,u_{N})\sim\lambda(u_{1},\ldots,u_{N})$ of $\mathds{C}^{N}/\{0\}$. By analogy the Grassmannian $G(k,N)$ is the quotient of the space of $k\times N$-matrices of rank $k$ w.r.t. the group $GL(k,\mathbb{C})$.

We note that the ‘holomorphic’ definition (2) already implies a choice of complex structure on the flag manifold. The definition (4), in contrast, treats the flag manifold as a real manifold (a complex structure may be defined on it separately – we will do it below).

1.2 Symplectic structures

Let us show that $\mathcal{F}_{n_{1},\ldots,n_{m}}$ is a symplectic manifold, i.e. there exists on it a closed 2-form $\omega$, $d\omega=0$, such that $\omega^{\mathrm{dim}_{\mathds{C}}\,\mathcal{F}}\neq 0$ (i.e. it is non-degenerate, $\mathrm{det}\,\omega\neq 0$). In fact, we will be only considering homogeneous (i.e. $U(N)$-invariant) symplectic forms.

More generally, let us consider the homogeneous space $G/H$ and the corresponding Lie algebra decomposition $\mathfrak{g}=\mathfrak{h}\oplus\mathfrak{m}$. A homogeneous space is called reductive if the following relations are fulfilled:

[TABLE]

The second requirement is automatically satisfied, if there is an ad-invariant metric on $\mathfrak{g}$. Indeed, let there be elements $h_{0},h_{1}\in\mathfrak{h}$ and $a,b\in\mathfrak{m}$, such that $[h_{0},a]=b+h_{1}$. Let us compute the scalar product with an arbitrary element $\gamma\in\mathfrak{h}$, then we will get $\langle\gamma,[h_{0},a]\rangle=\langle\gamma,h_{1}\rangle$. The l.h.s. is zero (which can be verified by swapping the commutator), therefore $\langle\gamma,h_{1}\rangle=0$ for all $\gamma$. It follows from the non-degeneracy of the metric that $h_{1}=0$.

There is the following general statement:

$G$-invariant tensors on a reductive homogeneous space $G/H$ are in correspondence with $H$-invariant tensors on $\mathfrak{m}$ (the latter is simply a linear space $\mathbb{R}^{\mathrm{dim}\,\mathfrak{m}}$).

In what follows we will be mainly using the unitary representation (4) and the corresponding Lie algebra decomposition

[TABLE]

where $\mathfrak{m}$ is the orthogonal complement to $\mathfrak{h}$ w.r.t. an ad-invariant metric on $\mathfrak{u}_{N}=\mathfrak{u}_{1}\oplus\mathfrak{su}_{N}$. Since $[\mathfrak{h},\mathfrak{m}]\subset\mathfrak{m}$, the subgroup $H$ is represented in the space $\mathfrak{m}$, and this representation may be decomposed into irreducibles:

[TABLE]

Moreover, $V_{ji}=\bar{V}_{ij}$. $V_{ij}$ is the vector space of the bi-fundamental representation of the group $U(n_{i})\times U(n_{j})$.

We introduce the Maurer-Cartan current $j=-g^{-1}\,dg,\,g\in U(N),$ and decompose it as follows:

[TABLE]

According to the general statement mentioned above, the most general invariant two-form is

[TABLE]

To check, in which case it is closed, we will take advantage of the flatness of the Maurer-Cartan current, $dj-j\wedge j=0\,.$ It follows that

[TABLE]

where $\mathscr{D}$ is the $H$-covariant derivative, defined as follows: $\mathscr{D}j_{ij}:=dj_{ij}-\{j_{\mathfrak{h}},j_{ij}\}\,.$ From the condition that $\Omega$ is closed it follows that

[TABLE]

The general solution to this equation is

[TABLE]

Therefore we have a family of homogeneous symplectic forms with $m-1$ real parameters. These forms may be compactly written as follows:

[TABLE]

The element $z$ may be normalized to be traceless: $\mathrm{Tr}(z)=0$. The stabilizer $H$ may now be thought of as the stabilizer of the matrix $z\in\mathfrak{u}_{N}$, and the flag manifold itself – as an adjoint orbit:

[TABLE]

The above formula gives an embedding of the flag manifold into the Lie algebra $\mathfrak{su}_{N}$. Moreover, this embedding may be identified with the image of the moment map

[TABLE]

The action of the group $U(N)$ on the flag manifold, $x\to\tilde{x}$, is as follows:

[TABLE]

The moment map (16) transforms correctly under the action of the group: $\mu(g_{0}\cdot g)=g_{0}\,\mu(g)\,g_{0}^{-1}$. To check that (16) indeed generates Hamiltonian functions for the action of the group $G$ on the flag manifold, let us assume that $v_{a}$ is a vector field on $\mathcal{F}$, corresponding to the Lie algebra element $T_{a}$. Then the following holds:

[TABLE]

Using this equality, we can check that (16) indeed gives Hamiltonian functions for the action of the group:

[TABLE]

1.3 Complex structures

A very detailed treatment of complex structures on homogeneous spaces was given as early as in the classic work [47], so here we mostly present an adaptation of some of these statements to our needs. To start with, on the manifold $U(N)\over U(1)^{N}$ of complete flags in $\mathds{C}^{N}$ there are $2^{\frac{N(N-1)}{2}}$ invariant almost complex structures, $N!\leq 2^{\frac{N(N-1)}{2}}$ of them being integrable (we note that for large $N$, according to Stirling’s formula, $e^{N\,\log(N)}<e^{\log(2)\,N^{2}\over 2}$).

Since the complex structure $\mathscr{J}$ is a certain invariant tensor on $G/H$, according to the logic mentioned in the previous section, one needs to define an $H$-invariant action of the operator $\mathscr{J}$ on the linear space $\mathfrak{m}$. Since $\mathscr{J}^{2}=-1$, the complex structure may be diagonalized over the complex numbers, in which case we need to define a decomposition

[TABLE]

Here $\mathfrak{m}_{\pm}$ play the role of holomorphic tangent spaces to $G/H$, i.e. $\mathscr{J}\circ a=\pm i\,a$ for $a\in\mathfrak{m}_{\pm}$. In (8) we decomposed $\mathfrak{m}\otimes\mathds{C}$ into irreducible components, therefore, in order to define an almost complex structure on $\mathcal{F}$, one can define the action of $\mathscr{J}$ as follows:

[TABLE]

As a result, one has exactly $2^{\frac{N(N-1)}{2}}$ possibilities. There are several equivalent definitions of integrability of a complex structure:

• •

Vanishing of the Nijenhuis tensor:

[TABLE]

for arbitrary vector fields $X,Y$.

• •

Using vector fields: the commutator of two holomorphic vector fields should be holomorphic, i.e.

[TABLE]

(The property (24) may also be stated as the condition that the distribution of holomorphic vector fields is integrable.)

• •

Using forms: the holomorphic forms should form a differential ideal, i.e. the following condition should be satisfied: $d(J_{-})_{a}\sim\sum\limits_{b}\,R_{ab}\wedge(J_{-})_{b}$ for some one-forms $R_{ab}$.

Let us demonstrate, that the last definition implies

[TABLE]

if the restriction to $\mathfrak{m}$ of the adjoint-invariant metric $\mathbb{G}$ on $\mathfrak{u}_{N}$ is Hermitian w.r.t. the chosen almost complex structure $\mathscr{J}$. In general the integrability of an almost complex structure means that $[\mathfrak{m}_{+},\mathfrak{m}_{+}]\subset\mathfrak{m}_{+}\oplus\,\mathfrak{h}$. To see this, note that an almost complex structure $\mathscr{J}$ is defined by the conditions $\mathscr{J}\circ J_{\pm}=\pm i\,J_{\pm}$, where $J_{\pm}$ are the components of a Maurer-Cartan current:

[TABLE]

Since $dJ-J\wedge J=0$, we get

[TABLE]

Therefore for the integrability of $\mathscr{J}$ one should have $[J_{+}\wedge J_{+}]_{\mathfrak{m_{-}}}=0$, i.e. $[\mathfrak{m}_{+},\mathfrak{m}_{+}]\subset\mathfrak{m}_{+}\oplus\mathfrak{h}$. We see that the conditions (25) therefore define an integrable complex structure. Conversely suppose we have an integrable complex structure on $G/H$, and $\mathfrak{m}_{\pm}$ are its respective holomorphic/anti-holomorphic subspaces. Then $[a,b]=c+\gamma$, where $a,b,c\in\mathfrak{m}_{+}$ and $\gamma\in\mathfrak{h}$. Since $\boldlangle\mathfrak{m}_{+},\mathfrak{h}\boldrangle_{\mathbb{G}}=0$, computing the scalar product with a generic element $\gamma^{\prime}\in\mathfrak{h}$, we obtain $\boldlangle\gamma^{\prime},[a,b]\boldrangle_{\mathbb{G}}=\boldlangle\gamma^{\prime},\gamma\boldrangle_{\mathbb{G}}$. Using the identity $\boldlangle[a,\gamma^{\prime}],b\boldrangle_{\mathbb{G}}+\boldlangle\gamma^{\prime},[a,b]\boldrangle_{\mathbb{G}}=0$, we get $\boldlangle\gamma^{\prime},\gamma\boldrangle_{\mathbb{G}}=-\boldlangle[a,\gamma^{\prime}],b\boldrangle_{\mathbb{G}}=\boldlangle a^{\prime},b\boldrangle_{\mathbb{G}}$, and $a^{\prime}=[\gamma^{\prime},a]\in\mathfrak{m}_{+}$. As discussed earlier, the subspace $\mathfrak{m}_{+}$ is isotropic, if the metric $\mathbb{G}$ is Hermitian, therefore $\boldlangle\gamma^{\prime},\gamma\boldrangle_{\mathbb{G}}=0$ for all $\gamma^{\prime}\in\mathfrak{h}$, which implies $\gamma=0$ due to the non-degeneracy of $\mathbb{G}$. The result $[\mathfrak{m}_{+},\mathfrak{m}_{+}]\subset\mathfrak{m}_{+}$ follows.

A typical integrable complex structure on the flag manifold defines the holomorphic/anti-holomorphic subspaces $\mathfrak{m}_{\pm}$ shown in Fig. 2.

As discussed above, one can define an almost complex structure on $\mathcal{F}_{N}$ by choosing $\frac{N(N-1)}{2}$ mutually non-conjugate forms $J_{p_{1}q_{1}},\ldots,J_{m_{\frac{N(N-1)}{2}}n_{\frac{N(N-1)}{2}}}$ and declaring them holomorphic. The remaining $\frac{N(N-1)}{2}$ forms will be therefore anti-holomorphic. To determine, which of those complex structures are integrable, it is useful to use a diagrammatic representation. We draw $N$ vertices, as well as arrows from the node $p_{1}$ to the node $q_{1}$, from $p_{2}$ to $q_{2}$ and so on (such diagrams are called ‘tournaments’, see [48]). The integrability of the almost complex structure, defined in this way, is equivalent to the acyclicity of the graph (the condition that it should not contain closed cycles). For a proof see [33], for example.

One can now establish the following fact:

Lemma. There are exactly $N!$ acyclic diagrams.

Proof. The statement of the lemma implies that there is only one combinatorial type of diagrams, and all acyclic diagrams may be obtained from any one of them by the action of the permutation group $S_{N}$. Let us describe this combinatorial type. Every acyclic diagram has a ‘source’-vertex, in which all the lines are outgoing, and a ‘sink’-vertex, in which all lines are incoming (see Fig. 3). Indeed, if that were not so, every vertex would contain at least, say, one outgoing line. Then one can start at any vertex and follow outgoing lines, until a loop is formed. Let us consider the ‘source’-vertex. The diagram formed by the remaining $N-1$ vertices together with the edges joining them can be an arbitrary acyclic diagram (as the chosen vertex is a ‘source’, there cannot be cycles containing it). Therefore we have performed the first step of the induction. The subsequent steps consist in finding the ‘source’ vertex in the reduced diagram. It is therefore clear that there always exists a vertex with $i$ outgoing lines for all $i=0,\ldots,N-1$. This statement completely describes the combinatorial structure of the diagram. Equivalently, there is a total ordering on the set of vertices. Different diagrams differ just by a relabeling of the vertices.

We have established that there are $N!$ complex structures on a complete flag manifold $U(N)\over U(1)^{N}$. Analogously there are $m!$ complex structures on a partial flag manifold $U(N)\over U(n_{1})\times\cdots\times U(n_{m})$. The number of complex structures may be easily interpreted as follows. Choosing a complex structure is equivalent to choosing a complex quotient space representation (2). In order to construct such representation one should choose a sequence of embedded linear spaces of the type (1), and the dimensions of these spaces are given by the partial sums of the integers $n_{i}$. These dimensions are therefore determined by an ordering of the set $\{n_{i}\}$, and there are $m!$ such orderings.

1.4 Cohomology

The second cohomology group of the complete flag manifold is

[TABLE]

hence there exist $N-1$ linearly independent 2-forms, which are the generators of $\mathrm{H}^{2}(\mathcal{F}_{N})$. As a model for $\mathrm{H}^{2}(\mathcal{F})$ let us consider the following construction. On $\mathcal{F}_{N}$ there are $N$ standard line bundles $L_{1},\cdots,L_{N}$, moreover their sum is a trivial bundle:

[TABLE]

The first Chern classes of these bundles give $N$ closed 2-forms: $\Omega_{i}=c_{1}(L_{i}),\;i=1\cdots N$. Due to the condition (28) and the additivity of the first Chern classes $c_{1}(E\oplus F)=c_{1}(E)+c_{1}(F)$ it is clear that the forms $\Omega_{i}$ are not independent but rather satisfy the relation

[TABLE]

The two-forms $\Omega_{i},\;i=1\,\cdots\,N$, satisfying the relation (29), generate $\mathrm{H}^{2}(\mathcal{F}_{N},\mathbf{Z})$.

One can also obtain an even more explicit description of the cohomology, which will shed some light on the relation (29). To this end we note that there exists an embedding

[TABLE]

A point $m\in(\mathds{C}\mathds{P}^{N-1})^{\times N}$ is a collection of $N$ lines in $\mathds{C}^{N}$ through the origin. Those points that correspond to $N$ pairwise orthogonal lines are the points lying on the flag manifold $\mathcal{F}_{N}$: here one should recall that $\mathcal{F}_{N}$ may be thought of as the manifold of sets of $N$ ordered mutually orthogonal lines in $\mathds{C}^{N}$. Let us now consider $(\mathds{C}\mathds{P}^{N-1})^{\times N}$ as a symplectic manifold with a product symplectic form

[TABLE]

One can check (this is proven in [41]) that (30) is a Lagrangian embedding w.r.t. this symplectic form, i.e.

[TABLE]

Identifying $\Omega_{i}=[i^{\ast}(\omega_{i})]$ and taking into account (32), we obtain the relation (29). One can analogously prove that a general flag manifold, embedded into a product of Grassmannians $\mathcal{F}_{n_{1},\ldots,n_{m}}\hookrightarrow G_{n_{1},N}\times\cdots\times G_{n_{m},N}$, is also a Lagrangian submanifold.

The above observations are tightly related to the theory of geometric quantization, i.e. to the construction of the representation theory of a Lie group $G$ using line bundles over its flag manifolds. See our paper [41] for a review.

2 The models

In this paper we will continue our study of $\sigma$-models proposed in [31]. The target spaces of these models are homogeneous complex target spaces $\mathcal{M}={G\over H}$, endowed with the ‘Killing metric’ $\mathbb{G}$ and an integrable $G$-invariant complex structure $\mathscr{J}$. The action schematically takes the form:

[TABLE]

where $X$ is the map $X:\Sigma\to\mathcal{M}$ and $\omega$ is the Kähler form corresponding to the pair $(\mathbb{G},\mathscr{J})$, defined as $\omega=\mathbb{G}\circ\mathscr{J}$. The peculiarity of such models lies in the fact that one can construct zero-curvature representations for their equations of motion. More precisely, a one-parametric family $\mathcal{A}_{u},u\in\mathds{C},$ of flat connections can be constructed from the Noether current $K$ using the formula of Pohlmeyer [2]:

[TABLE]

In the present paper we define the complex coordinates $z,\bar{z}$ as $\{z,\bar{z}\}={x^{1}\pm ix^{2}\over\sqrt{2}}$, so that the line element is $ds^{2}=2\,dz\,d\bar{z}$, and the metric is g=\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right).

The connections defined in (34) are flat, since the Noether current $K$ itself is flat for the models above. This construction is well-known for symmetric spaces [49], which e.g. for the group $G=SU(N)$ are the Grassmannians $G_{k,N}$. Our models allow for a richer class of target spaces, in particular including the so-called flag manifolds.

2.0.1 Dependence on the metric.

Let us clarify that here and in what follows when referring to the Noether current, we mean a one-form $K$. In fact, the canonical Noether procedure produces a vector field $K^{\mu}$, satisfying the conservation equation $\partial_{\mu}(\sqrt{g}K^{\mu})=0$. The one-form is obtained from this vector field by lowering the index in the standard way: $K=g_{\mu\nu}K^{\nu}\,dx^{\mu}$. The Hodge dual to this one-form is $\ast K={1\over\sqrt{g}}g_{\alpha\beta}\epsilon^{\beta\gamma}K_{\gamma}\,dx^{\alpha}$ and

[TABLE]

Let us consider a simple example of a free complex massless boson, the action being $S=\int\,dx^{1}dx^{2}\,\sqrt{g}\,g^{\alpha\beta}\,\partial_{\alpha}\bar{\psi}\,\partial_{\beta}\psi$. The corresponding one-form is then $K=i(\bar{\psi}d\psi-d{\bar{\psi}}\psi)$ and, in particular, it is independent of the worldsheet metric $g$. Suppose we now include a certain invariant $B$-field, or even a topological term, replacing the action by $\tilde{S}=S+\kappa\,\int\,i\,d\bar{\psi}\wedge d\psi$. The Noether current should now be replaced by $\tilde{K}=K+\kappa\,\ast d(\bar{\psi}\psi)$, the additional component being a topological current. Nevertheless, the full current $\tilde{K}$ now depends on the conformal class of the metric $g$, through the dependence of the Hodge star $\ast$. We come to the conclusion that, in general, both one-forms $K$ and $\ast K$ depend on the conformal class $[g]$ of the worldsheet metric. $\blacksquare$

In the present paper we will concentrate on the simplest non-symmetric (i.e. non-Grassmannian) example, when

[TABLE]

This manifold may be thought of as the space of mutually orthogonal (w.r.t. some metric on $\mathds{C}^{N}$) linear subspaces $E_{n},E_{m},E_{N-m-n}$ of $\mathds{C}^{N}$, passing through the origin, of dimensions $n,m$ and $N-m-n$. Clearly, these subspaces span all of $\mathds{C}^{N}$.

3 Relation to models with $\mathbb{Z}_{m}$-graded target spaces

Although the manifold (37) is not a symmetric space, i.e. not a space with a $\mathbb{Z}_{2}$-grading, it is a space with a $\mathbb{Z}_{3}$-grading. Before explaining the concept, we note that natural examples of $\mathbb{Z}_{3}$-graded spaces are provided by twistor spaces of symmetric spaces [51] and nearly Kähler homogeneous spaces [52]. In this context the manifold (37) may be seen as the twistor space of $G_{n,N}$, $G_{m,N}$ or $G_{N-m-n,N}$.

A homogeneous space $G\over H$ is called $\mathbb{Z}_{m}$-graded (or $m$-symmetric), if the Lie algebra $\mathfrak{g}$ of its isometry group admits the following decomposition:

[TABLE]

In this language the ordinary symmetric spaces are $2$-symmetric spaces. Then, one can show, that, similarly to what happens for symmetric spaces, the e.o.m. of certain $\sigma$-models with $\mathbb{Z}_{m}$-graded target-spaces may be rewritten as flatness conditions of a one-parametric family of connections.

A related class of models was considered in [24] and subsequently studied in [50]. The action studied in [24] has the form

[TABLE]

where $\widetilde{\omega}$ is a 2-form constructed using the $\mathbb{Z}_{m}$-decomposition of the Lie algebra (38). Note that, if  $\widetilde{\omega}$  were the Kähler form, one would obtain precisely the action (33). Now we come to the precise definition of $\widetilde{\omega}$. Decompose the current $J=-g^{-1}dg$ according to (38):

[TABLE]

The form $\widetilde{\omega}$ is defined as follows:

[TABLE]

This formula poses an important question. According to (41), the form $\widetilde{\omega}$ depends on the $\mathbb{Z}_{m}$-grading on the Lie algebra. On the other hand, generally a given Lie algebra $\mathfrak{g}$ may have many different gradings (with different, or same, values of $m$). The question is: are the models defined by (39)-(41), corresponding to different gradings of $\mathfrak{g}$, different?

Before answering this question, first we review the construction of cyclic gradings on semi-simple Lie algebras, which was completed long ago [53]. Let us consider, for simplicity, the case of $\mathfrak{g}=\mathfrak{su}(N+1)$. A cyclic grading may be constructed as follows222Here we restrict ourselves to the grading of type $A_{N}^{(1)}$.: one picks a system of $N$ simple positive roots $\alpha_{1},\ldots\alpha_{N}$, as well as the maximal negative root $\alpha_{N+1}=-\alpha_{1}-\ldots-\alpha_{N}$333In the paper of Kac [53] the roots $\alpha_{1},\ldots\alpha_{N+1}$ are seen as the positive simple roots of the corresponding affine Lie algebra $\widehat{A}_{N}$. Consider the case $N=2$. The simple positive roots of the loop algebra $\mathfrak{su}_{3}(t,t^{-1})$ may be chosen as follows:

\alpha_{1}=\left(\begin{array}[]{ccc}0&1&0\\ 0&0&0\\ 0&0&0\end{array}\right),\;\;\;\alpha_{2}=\left(\begin{array}[]{ccc}0&0&0\\ 0&0&1\\ 0&0&0\end{array}\right),\;\;\;\alpha_{0}=t\,\left(\begin{array}[]{ccc}0&0&0\\ 0&0&0\\ 1&0&0\end{array}\right)\,.

In this context the latter root $\alpha_{0}$ – the analog of $\alpha_{N+1}$ – is customarily called ‘imaginary’. In fact, the whole theory of cyclic Lie algebra gradings is formulated by Kac naturally in terms of affine Lie algebras and their Dynkin diagrams.. Then one assigns to these $N+1$ roots arbitrary (non-negative integer) gradings $m_{1},\ldots m_{N},m_{N+1}$. The gradings of all other roots are determined by the Lie algebra structure, and the value of $m$ is calculated as

[TABLE]

In usual matrix form, this grading looks as follows:

[TABLE]

The subalgebra $\mathfrak{g}_{0}$, which determines the denominator $H$ of the quotient space $G/H$, is determined by those $m_{i}$’s, which are zero. For example, if all $m_{i}>0$, the resulting space is the manifold of complete flags $SU(N+1)\over S(U(1)^{N+1})$.

In general, for a choice of grading determined by the set $m_{1},\ldots,m_{N+1}$ some of the subspaces $\mathfrak{g}_{i}$ will be identically zero. Therefore a natural restriction to adopt is to require that $\mathfrak{g}_{i}\neq 0$ for all $i\;(\mathrm{mod}\;m)$. We will call such a grading admissible. This still leaves a wide range of possibilities. For example, in the case of $SU(3)$ the following is a complete list of admissible gradings (up to the action of the Weyl group $S_{3}$):

[TABLE]

The $\mathbb{Z}_{2}$-grading and the second $\mathbb{Z}_{3}$-grading correspond to the homogeneous space $SU(3)/S(U(2)\times U(1))=\mathds{C}\mathds{P}^{2}$, and all other gradings correspond to the flag manifold $\mathcal{F}_{1,1,1}$.

We will now give an answer to the question posed above: what is the relation between the $\sigma$-models with the action (39), taken for different gradings on the corresponding Lie algebra? Our statement is [35]:

Proposition. For homogeneous spaces of the unitary group, the models defined by (39)-(41) with different $A_{N}^{(1)}$-type gradings on $\mathfrak{g}$ are classically equivalent to the model defined by the action (33) with some choice of complex structure on the target-space

In fact, one has a precise statement about the relation of the $B$-fields in the two models. To formulate it, we ‘solve’ the constraint (42) as follows444Formula (71) implies that the cyclic automorphism $\widehat{\Omega}$ of the Lie algebra, which defines the $\mathbb{Z}_{m}$ grading, can be represented as follows: $\widehat{\Omega}(a)=\Omega a\Omega^{-1}$, where $\Omega=\mathrm{diag}(e^{2\pi i\,\frac{n_{1}}{m}},\ldots,e^{2\pi i\,\frac{n_{N+1}}{m}})$.:

[TABLE]

where $n_{k}$ are integers and $n_{N+2}\equiv n_{1}-m$. We then have (see [35] for a proof):

[TABLE]

We see that, irrespective of the choice of the grading (which is now encoded in the integers $n_{i}$), the form $\tilde{\omega}$ differs from the Kähler form by a topological term. This topological term, clearly, depends on the chosen grading, but does not contribute to the equations of motion.

Comment. Although the flag manifolds (4) are $\mathbb{Z}_{m}$-graded spaces, the two classes of target spaces – $\mathbb{Z}_{m}$-graded and complex homogeneous spaces – do not coincide. For example, one has the space $\frac{G_{2}}{SU(3)}\simeq S^{6}$. The stability subgroup $SU(3)$ acts on the tangent space $\mathfrak{m}=\mathbb{R}^{6}$ via $V\oplus\bar{V}$, where $V\simeq\mathds{C}^{3}$ is the standard representation. Therefore it has a unique almost complex structure, which is not integrable (see the review [54]). On the other hand, it is a nearly Kähler manifold and is $\mathbb{Z}_{3}$-graded [52]. On the other side of the story, one has the complex manifold $S^{1}\times S^{3}\simeq U(2)$ (see [35] for a discussion), which may be viewed as a $\mathbb{T}^{2}$-bundle over $\mathds{C}\mathds{P}^{1}$ (the simplest flag manifold). This manifold is not a $\mathbb{Z}_{m}$-graded homogeneous space of the group ${G=U(2)}$.

We also note that the construction of Lax connections for models with $\mathbb{Z}_{m}$-graded spaces was explored in [50]. The fact that the corresponding integrals of motion are in involution was proven, for instance, in [30].

4 The GLSM representation

In this section we will construct a gauged linear sigma-model (GLSM) representation for the models (33) introduced earlier (for flag manifold target spaces $\mathcal{F}$). We recall that these models depend explicitly on the complex structure $\mathscr{J}$. Now, picking a complex structure $\mathscr{J}$ on $\mathcal{F}$ is equivalent to picking a total ordering of the subspaces $E_{n},E_{m},E_{N-m-n}$. For the purposes of constructing a $1\over N$-expansion any of the two orderings, where $E_{N-m-n}$ is maximal, will suffice (the reason will become clear shortly). Let us prove, first of all, that it is always possible to achieve this, without loss of generality. The point here is that the action (33), albeit depending on the complex structure, might produce the same equations of motion even for different choices of complex structure. This is due to the fact, that for certain complex structures, which we denote by $\mathscr{J}_{1}$ and $\mathscr{J}_{2}$, the difference in the two actions may just be a topological term:

[TABLE]

Let us describe precisely the situation when this happens. To this end we recall that, as was established at the end of section 1.3, the complex structures are in a one-to-one correspondence with an ordering of the mutually orthogonal spaces $\mathds{C}^{n_{1}},\ldots\mathds{C}^{n_{m}}$, that are a point in a flag manifold $U(N)\over U(n_{1})\times\cdots U(n_{m})$.

Proposition. The actions $\mathcal{S}[\mathscr{J}_{1}]$ and $\mathcal{S}[\mathscr{J}_{2}]$ differ by a topological term, as in (73), if and only if the corresponding sequences of spaces $\{\mathds{C}^{n_{1}},\ldots\mathds{C}^{n_{m}}\}$ differ by a cyclic permutation.

Proof. Let us call $\mathscr{J}$ the standard complex structure, whose holomorphic subspace $\mathfrak{m}_{+}$ is given by upper-block-triangular matrices. Then $\mathscr{J}_{1}=\sigma_{1}(\mathscr{J})$ and $\mathscr{J}_{2}=\sigma_{2}(\mathscr{J})$ for some permutations $\sigma_{1},\sigma_{2}\in S_{m}$. We recall the notation $J_{ij}$ from (9). The corresponding Kähler forms are

[TABLE]

Upon introducing the notation $\sigma(J_{ij}):=J_{\sigma(i)\sigma(j)}$, we may write the difference of the two forms as

[TABLE]

where $\tau^{-1}=\sigma_{1}^{-1}\sigma_{2}\,.$ This reduces the problem to that of $\mathscr{J}_{1}=\mathscr{J}$ and $\mathscr{J}_{2}=\tau^{-1}(\mathscr{J})$. Note that the exterior derivative commutes with the permutation $\sigma$, due to the following simple fact following from the Maurer-Cartan equation: $\sigma(dJ_{ij})=\sigma(\sum\limits_{k}\,J_{ik}\wedge J_{kj})=\sum\limits_{k}\,\sigma(J_{ik})\wedge\sigma(J_{kj})=d\sigma(J_{ij})$, where to arrive at the last equality one has to make a change of the dummy summation index $k\to\sigma(k)$.

We wish to show that $d(\omega_{1}-\omega_{2})=0$ implies that $\tau$ is a cyclic permutation. To this end we rewrite the above difference as follows:

[TABLE]

(We have made a change of dummy variables $i\to\tau(i)$ and $j\to\tau(j)$ in the second sum in (76)). Closedness of this form requires that (see (12)-(13))

[TABLE]

Let us consider the case $j>i$. Then $z_{i}=z_{j}$ if $\tau(j)>\tau(i)$ and $z_{i}=z_{j}+1$ if $\tau(j)<\tau(i)$. This means that $\{z_{i}\}_{i=1\ldots m}$ form a non-increasing sequence, and moreover the difference between any two elements is either zero or $1$. This is only possible if the set has the form $(\underbracket{z,\ldots,z}_{K},\underbracket{z-1,\ldots z-1}_{m-K})$. Accordingly the original sequence of $m$ consecutive numbers can be split into two consecutive sets:

[TABLE]

Since $\tau(j)<\tau(i)$ for ($i\leq K$, $j>K$), the permutation acts as follows:

[TABLE]

Moreover, since $\tau(i)<\tau(j)$ for $i<j\leq K$ and the image $\tau(I_{1})$ is $(m-K+1,\ldots m)$, a moment’s thought shows that $\tau(i)=m-K+i$ for $i=1\ldots K$. Analogously $\tau(j)=j-K$ for $j=K+1\ldots m$. Therefore $\tau$ is nothing but a $K$-fold cyclic permutation ‘to the left’ (or $m-K$-fold to the right).

Since for $i<j$ the non-zero $\alpha_{ij}$ are the ones, for which $\tau(j)<\tau(i)$, this implies $j=K+1\ldots m$ and $i=1\ldots K$. These $\alpha_{ij}$ are equal to $1$, therefore

[TABLE]

which is easily seen to be proportional to the (generalized) Fubini-Study form on the Grassmannian $G_{L,N}$, where $L=\sum\limits_{i=1}^{K}\,n_{i}$. Conversely, one shows that for a cyclic permutation the difference between $\omega_{1}$ and $\omega_{2}$ is the closed form written above. $\blacksquare$

Returning to the manifold (37), we may assume that we have picked an ordering of $E_{n},E_{m},E_{N-m-n}$, in which $E_{N-m-n}$ is maximal (by making a cyclic permutation if necessary). The manifold $\mathcal{F}$ with the complex structure defined by this ordering may then be thought of as the space of embedded linear complex spaces:

[TABLE]

where $V_{1}\simeq\mathds{C}^{n}$ and $V_{2}\simeq\mathds{C}^{m+n}$. As shown in our paper [34], the model (33) can be formulated in this case as a gauged linear $\sigma$-model, in terms of a ‘matter field’ $\varphi\in\mathrm{Hom}(V_{2},\mathds{C}^{N})$ and an auxiliary field $\mathcal{A}=\mathcal{A}_{z}dz+\mathcal{A}_{\bar{z}}d\bar{z}\in\mathfrak{u}_{m+n}\otimes T^{\ast}\Sigma$:

[TABLE]

The field $\varphi$ is subject to a normalization constraint $\varphi^{\dagger}\varphi={N\over g^{2}}\mathds{1}_{m+n}$, which is imposed by the Lagrange multiplier $\lambda$. Here $g$ is the coupling constant of the model and the dependence on $N$ is chosen to suit the $1\over N$-expansion, constructed in the next section. To simplify the situation even further, we will restrict to the case $m=n=1$, i.e. we will consider the target space of the form

[TABLE]

In this case we can express the matrix $\varphi$ in terms of its two column vectors $\varphi=(u,v)$, then the normalization constraint says that $\|u\|=\|v\|={\sqrt{N}\over g}$ and $\bar{u}\circ v=0$. The covariant derivates in the above Lagrangian are defined in the standard way:

[TABLE]

Note that we assume the connection $\mathcal{A}$ to be Hermitian, meaning that $\mathcal{A}^{\dagger}=\mathcal{A}$, hence $(\mathcal{A}_{z})^{\dagger}=\mathcal{A}_{\bar{z}}$. An important point is that the ‘connection’ $\mathcal{A}$ in the model above has to be taken in a non-conventional triangular form:

[TABLE]

The form of $\mathcal{A}$ is the main difference of our models from the models with symmetric target spaces. In particular, this triangular form implies that there is only the $U(1)^{2}$ gauge symmetry, for which $a$ and $b$ are the gauge fields, whereas $c$ is another auxiliary field transforming in the bi-fundamental representation of $U(1)^{2}$. Had we not imposed the constraints (87), we would have the gauge group $U(2)$, and the target space of the model would be the Grassmannian $G_{2,N}$.

Integrating out the auxiliary fields $a,b,c$ we get the following form of the Lagrangian (for details see the Appendix B):

[TABLE]

This expression may be simplified, if we introduce an additional matrix-valued field $w:\mathds{C}^{N-2}\to\mathds{C}^{N}$, defined by the completeness relation in $\mathds{C}^{N}$:

[TABLE]

This relation simply means that the matrix $w$ contains $N-2$ vectors, which are orthogonal to $u$ and $v$ and mutually orthonormal. We then obtain

[TABLE]

This is precisely the type of Lagrangian considered in [31], [32], [33]. In this way we have established the equivalence of the model (33) with its gauged linear representation (84).

Before proceeding further, let us make one more observation regarding the Lagrangian (84). A direct calculation shows that the difference

[TABLE]

The last line is clearly a total derivative, being the pull-back to the worldsheet of a Fubini-Study form on the Grassmannian $G_{2,N}$ (With the normalization $\varphi^{\dagger}\varphi={N\over g^{2}}\mathds{1}_{2}$ the Fubini-Study form is ${N\over g^{2}}\Omega_{FS}\sim i\,\mathrm{Tr}\left(d\varphi\wedge d\varphi^{\dagger}\right)$). Taking advantage of this fact, we will replace the Lagrangian (84) by a more symmetric one

[TABLE]

Above it was claimed that the Noether current $K$ derived from the action (33) is flat, i.e. $dK-K\wedge K=0$. Let us check this explicitly. To start with, the Noether current has the form

[TABLE]

Note that $K\in\mathfrak{u}_{N}$ and therefore $K^{\dagger}=-K$. Now,

[TABLE]

Let us now simplify this expression, using the normalization condition $\varphi^{\dagger}\varphi={N\over g^{2}}\mathds{1}_{2}$, as well as the equations of motion following from the Lagrangian (93):

[TABLE]

(We have used the identities $[D_{z},D_{\bar{z}}]\varphi=-i\varphi F_{z\bar{z}}$, $[D_{z},D_{\bar{z}}]\varphi^{\dagger}=iF_{z\bar{z}}\varphi^{\dagger}$).

Note an important difference with the standard case of symmetric target spaces (Grassmannians), where one has $\varphi^{\dagger}\mathscr{D}_{z}\varphi=\mathscr{D}_{\bar{z}}\varphi^{\dagger}\varphi=0$.

It follows from the two equations in (96) that $i\lambda$ is Hermitian: $(i\lambda)^{\dagger}=i\lambda$. On the other hand, multiplying the first equation by $\varphi^{\dagger}$ from the left, we get

[TABLE]

The last two terms in the r.h.s. are manifestly Hermitian, whereas the first term turns out to be upper-triangular:

[TABLE]

Hermiticity of $i\lambda$ then implies that $\mathscr{D}_{\bar{z}}(\varphi^{\dagger}\mathscr{D}_{z}\varphi)=0$ and therefore

[TABLE]

Using all this information in (95), we find that $dK-K\wedge K=0$, as claimed earlier.

4.1 General flag manifolds

Although the calculations in the following sections will concern the case of the target space (85), for completeness let us introduce the GLSM-formulation in the general case of the flag manifolds $\frac{U(N)}{U(n_{1})\times\cdots\times U(n_{m})}$, $\sum\limits_{i=1}^{m}\,n_{i}=N$. We assume that an arbitrary complex structure $\mathscr{J}$ has been chosen, which is tantamount to choosing an ordering of the factors

[TABLE]

as was explained in Section 1.3. We can always reshuffle $n_{1},\ldots,n_{m}$ in such a way that the ordering is the literal ordering of factors in (105). In this case we are dealing with the manifold of complex flags

[TABLE]

We define $M:=\mathrm{dim}\,V_{m-1}$ and introduce the matter field $\varphi\in\mathrm{Hom}(\mathds{C}^{M},\mathds{C}^{N})$. Its columns parametrize $M$ vectors that define the flag. They are orthonormal:

[TABLE]

This is an analog of the moment map constraint from the theory of Kähler quotients. Now we introduce an analogue of the gauge field

[TABLE]

and the covariant derivative

[TABLE]

In (113) the diagonal blocks $(A_{jj})_{z}$ are of size $n_{j}\times n_{j}$ and represent gauge fields for the respective gauge groups $U(n_{j})$. The off-diagonal blocks $(A_{ij})_{z},i\neq j$ are additional ‘matter fields’ transforming in bi-fundamental representations of the corresponding gauge groups $U(n_{i})\times U(n_{j})$. Note that the gauge group $U(n_{m})$ does not appear in the above formulas – this is analogous to the situation familiar from the $\mathds{C}\mathds{P}^{N-1}$-model, where the standard GLSM-representation involves only the $U(1)$ gauge field, despite the fact that the quotient space form $\mathds{C}\mathds{P}^{N-1}\simeq\frac{U(N)}{U(1)\times U(N-1)}$ involves a product of gauge groups $U(1)\times U(N-1)$.

The Lagrangian of the model is a direct generalization of (93):

[TABLE]

As a result, we have obtained a theory that is very similar to the Grassmannian $G_{M,N}$ sigma-model, but with a ‘reduced’ gauge field.

5 Feynman rules for the $1\over N$-expansion

We now return to the case of the target space (85). In the previous section we introduced the Lagrangian (93) of our model. The Lagrangian in the form (84) is not the most convenient one for carrying out the $1\over N$-expansion, as it would lead to mixed $\langle\mathcal{A}\,\lambda\rangle$ propagators. For this reason we will be using the Lagrangian in the form (93):

[TABLE]

One should keep in mind the normalization of the metric that leads to $D_{\mu}\otimes D_{\mu}=D_{\bar{z}}\otimes D_{z}+D_{z}\otimes D_{\bar{z}}$. Note that we have also normalized the fields differently now:

[TABLE]

This is the proper normalization for carrying out the $1\over N$-expansion. We see that in the large-$N$ limit the coupling constant $\hat{g}\sim{1\over\sqrt{N}}$ vanishes, just like in the large-$N$ limit of gauge theories. The renormalized coupling constant $g$ stays fixed in the large-$N$ limit.

Quantization of the theory (116) as it stands will lead to one-loop divergences. To get rid of them, we will introduce additional fermionic ghost fields $\chi$ – the Pauli-Villars regulators:

[TABLE]

Integrating over the ‘matter fields’ $\varphi,\chi$ in the path integral, we obtain the effective action:

[TABLE]

We will now find the saddle point w.r.t. $\Lambda$, assuming that at the saddle point $\Lambda_{0}=-i\,m^{2}\,\mathds{1}_{2}$ and $\langle\mathcal{A}_{z}\rangle=\langle\mathcal{A}_{\bar{z}}\rangle=0$:

[TABLE]

This has the solution

[TABLE]

At the end of the day we wish to get rid of the ghost fields $\chi$, so we will take the limit $M\to\infty$, in such a way that the mass $m$ of the field $\varphi$ stays finite, therefore in this limit we would need to take $g\to 0_{+}$ accordingly.

The Lagrangian (5) may be rewritten in the form of two coupled $\mathds{C}\mathds{P}^{N-1}$-model Lagrangians:

[TABLE]

The two $\mathds{C}\mathds{P}^{N-1}$ models, parametrized by vectors $u$ and $v$, interact via the matter field $c$, as well as through the orthogonality constraint $\bar{u}\circ v=0$, implemented with the help of the Lagrange multiplier $\tau$. The first line in (122) encodes the propagators and interaction vertices of two $\mathds{C}\mathds{P}^{N-1}$ models – these are shown in Fig. 4 and Fig. 5 respectively. Apart from these, our model has the additional complex fields $\tau$ and $c_{z}$ transforming in the bi-fundamental representation of $U(1)\times U(1)$. Their respective propagators and interactions are shown in Fig. 6.

In implementing the steepest descent method for the large-$N$ limit, we will parametrize $\Lambda=\Lambda_{0}+\frac{1}{\sqrt{N}}\widetilde{\Lambda}$, $\mathcal{A}=\frac{1}{\sqrt{N}}\widetilde{\mathcal{A}}$. The expansion of the effective action $\mathcal{S}^{\mathrm{eff}}$ starts at the quadratic order in $\widetilde{\lambda},\widetilde{\mathcal{A}}$, and all subsequent orders are suppressed by powers of $\sqrt{N}$. The propagators shown in the figures are obtained by expanding the effective action in (123) to quadratic order in the fields, which yields in momentum space:

[TABLE]

The functions $\Gamma_{\mu\nu}(\vec{p})$ and $\Gamma_{\lambda}(\vec{p})$ were computed already in [10]. The function $\widetilde{\Gamma}(p)$ is found in a similar way, and altogether we have:

[TABLE]

In the effective action there are no mixed terms of the form $\int\,d^{2}p\,\Gamma_{\mu}(p)\,a_{\mu}(p)\,\lambda(p)$ due to the fact that the corresponding one-loop diagram vanishes:

[TABLE]

In the Lorenz gauge the propagators featuring in the figures have the following expressions:

[TABLE]

6 The Wilson loop and conserved charges

The main object of interest that we wish to be able to calculate is the following correlation function, with a Wilson loop of the flat connection inserted:

[TABLE]

where $\mathscr{A}_{u}$ is given by formula (34). The notation $\mathrm{P}_{0}$ is meant to emphasize that the Wilson loop depends on the choice of a base point on the contour, which we call $p_{0}\in\Gamma$. On the worldsheet $\Sigma\simeq\mathbb{R}^{2}$, when the contour $\Gamma$ is infinite, it is often convenient to choose infinity as the base point. The Noether current $K$ entering this formula is the one corresponding to the $U(N)$-symmetry acting on the $\varphi$ fields: $\varphi\to g\circ\varphi,\,g\in U(N)$. Therefore the Noether current one-form is (see (94))

[TABLE]

The notation $Q(\Gamma)$ is meant to emphasize that we will be interested in the dependence on the contour $\Gamma$. Note that classically the connection $\mathscr{A}_{u}$ is flat, hence $\mathrm{P}_{0}\,e^{-\int_{\Gamma}\,\mathscr{A}_{u}}$ is independent of $\Gamma$, provided one fixes the homology class of $\Gamma$ (and assuming the base point $p_{0}$ is left intact). Nevertheless, the quantity $Q(\Gamma)$ will in general depend on $\Gamma$ due to the UV-divergences that will need to be regularized, and this gives rise to the quantum anomalies in the higher conservation laws of the model, as first calculated in [17].

6.0.1 Example.

It is useful to consider a correlation function with an insertion of a conserved charge $\mathcal{Q}$ in a simpler model. In fact, the model will be as simple as possible, namely a two-dimensional theory with the Lagrangian

[TABLE]

This Lagrangian exhibits a shift symmetry of the form $\phi\to\phi+a$, where $a$ is a constant. The corresponding Noether current is $J=\partial_{z}\phi\,dz+\partial_{\bar{z}}\phi\,d\bar{z}\,$ and the conserved charge is

[TABLE]

The only connected correlation function to consider is $\langle\mathcal{Q}\,\phi(0)\rangle$, where we have chosen to place the second operator at the origin. The contour $\Gamma$ will run along the $y$-axis at a fixed value of $x$, though one could as well consider a curved contour, as shown in Fig. 7. Using the fact that the two-dimensional propagator is $\langle\phi(z,\bar{z})\,\phi(0)\rangle=-{1\over 4\pi}\,\log{|z|^{2}}$, we get for the correlation function

[TABLE]

The value is independent of the position of the contour $\Gamma$, as long as it does not cross the point of insertion of the local operator (in this case the origin). The discontinuity of the correlation function that arises when the contour is moved past a local operator is compatible with the Ward identity for the symmetry in question. $\blacksquare$

The first non-local charge (as well as the usual local charge) may be calculated by expanding the path-ordered exponent $\mathscr{W}_{0}(u):=\mathrm{P}_{0}\,e^{-\int_{\Gamma}\,\mathscr{A}_{u}}$ around the point $u=1$ to second order. In order to obtain charges that are manifestly (anti)-Hermitian, it will be useful to parametrize $u=e^{-i\theta}$ and expand to second order in $\theta$. First we note that, to this order,

[TABLE]

Here $\ast$ is the Hodge star, defined by $\ast dz=i\,dz$. We are now in a position to expand the path-ordered exponent:

[TABLE]

The two first conserved charges are therefore

[TABLE]

$\mathcal{Q}_{1}$ is the usual Noether charge corresponding to the $SU(N)$-symmetry. The ‘conservation’ of the charges means in this context that they are independent of smooth deformations of the contour $\Gamma$. For the charge $\mathcal{Q}_{1}$ this follows from the Stokes theorem and the conservation of the current, i.e. $d\ast K=0$. We will follow the conventions of [17] and further simplify the charge $\widehat{\mathcal{Q}}_{2}$, using the fact that we are free to add to it an arbitrary function of the charge $\mathcal{Q}_{1}$. The square of this charge is

[TABLE]

Therefore we introduce the charge

[TABLE]

The charge so defined also has the nice property that it lies in the Lie algebra of $G$, $\mathcal{Q}_{2}\in\mathfrak{g}$. In order to automatically obtain charges with values in the Lie algebra, one should consider the ‘Berry connection’ $\mathscr{B}=(\mathscr{W}_{0}(u))^{-1}\frac{d\mathscr{W}_{0}(u)}{d\theta}$ and expand it around $\theta=0$.

Let us check that, classically, the charge $\mathcal{Q}_{2}$ is independent of $\Gamma$. To this end we introduce the one-form (the lower limit of integration – point $p_{0}$ – is again the base point, which enters the definition of the Wilson loop)

[TABLE]

Here $\int\limits_{p_{0}}^{p}\,\ast K$ is the primitive of the one-form $\ast K$, which is well-defined due to the fact that $\ast K$ is closed (conservation of the current). In fact, if the point $p$ was always restricted to lie on $\Gamma$, the integral $\int\limits_{p_{0}}^{p}\,\ast K$ could be understood as an integral along $\Gamma$, even without the closedness of $\ast K$. However we will now wish to appeal again to Stokes’ theorem in the interior of $D_{\delta\Gamma}$ (this is the space between the two contours, see Fig. 11), and therefore we will need a one-form well-defined inside $D_{\delta\Gamma}$.

With this said, we may now write the charge $\mathcal{Q}_{2}$ as

[TABLE]

The variation again may be computed using Stokes’ theorem. Taking into account that $dS=2\,\ast K\wedge\ast K=2\,K\wedge K$ and that the current $K$ is flat, $dK-K\wedge K=0$,we find that the variation is zero:

[TABLE]

6.1 The regularized charge

In the quantum theory, due to the short-distance singularities in the product of two currents, one needs to define a regularized version of the non-local charge $Q_{2}$. This will involve the splitting of coincident points in the definition of the one-form $S$ in (142). One possibility is to define the following $\epsilon$-regularized form:

[TABLE]

Here it is implied that $\epsilon$ is a fixed two-vector, and we are considering the case when $\Sigma_{2}\simeq\mathds{R}^{2}$, equipped with a flat metric, when the notion of addition ‘$p+\epsilon$’ makes sense. The one-form $S_{\epsilon}$ depends on the contour of integration in (145) in the topological sense, i.e. it depends on whether the contour winds around the point $p$. Therefore we will fix a certain value of $\epsilon$ (such as the one shown in Fig. 8) and define the auxiliary contours in (145) for this value of $\epsilon$. For instance, for contours $\Gamma$ of the type shown in Fig. 8 one can define the auxiliary contour by requiring that it lay on one side of $\Gamma$. Then we can gradually change the value of $\epsilon$ so that it circles around the origin, $\epsilon\to e^{2\pi i}\epsilon$. As a result, $S_{\epsilon}$ will gain a monodromy of the form

[TABLE]

where the integral is around a loop centered at $p$. Since the current $K$ is conserved, $d\ast K=0$, the value of the integral does not depend on the size of the loop. Yet it is not zero, due to the presence of the operator insertion of $\ast K(p)$ at the center of the loop. However, the loop may be shrunk to be arbitrarily small, and the value of the commutator may be calculated from the most singular term in the OPE of the two currents. Such analysis below will bring us to the following answer:

[TABLE]

In order to cancel this monodromy, we will consider the operator

[TABLE]

where $\mathbf{a}$ is a constant, independent of $\epsilon$, to be chosen later. It is clear that the operator defined above is invariant as the regularization parameter $\epsilon$ rotates around the origin: $\epsilon\to e^{2\pi i}\,\epsilon$. This operator $\mathcal{Q}_{\epsilon}$ is our candidate for the regularized version of the charge $\mathcal{Q}_{2}$.

We wish to prove the following:

• •

There exists a limit $\underset{\epsilon\to 0}{\mathrm{lim}}\,\mathcal{Q}_{\epsilon}$

• •

The limit depends on the curve $\Gamma$ through an anomaly 2-form $\Omega_{A}$, namely

[TABLE]

where $D_{\delta\Gamma}$ is a two-dimensional domain bounded by the original and final curves $\Gamma_{1},\Gamma_{2}$ (see Fig. 11).

Let us start from the second point by calculating the difference $\mathcal{Q}_{\epsilon}(\Gamma_{1})-\mathcal{Q}_{\epsilon}(\Gamma_{2})$ of the values of the charge on two different contours. We use Stokes’ theorem and the conservation of the current to obtain:

[TABLE]

Therefore one of the goals will be the calculation of the OPE of the two currents $[K_{z}(p+\epsilon),K_{\bar{z}}(p)]$ as $\epsilon\to 0$.

6.2 The operator product expansion

We start by recalling the definition of the current (94):

[TABLE]

First we will calculate the OPE between two holomorphic components of the current, and as a result we will be able to prove the first point above – that there exists a limit $\underset{\epsilon\to 0}{\mathrm{lim}}\,\mathcal{Q}_{\epsilon}$.

In what follows we will encounter the function $D(z)$ – the massive scalar propagator in two dimensions (recalling the definition $\{z,\bar{z}\}={x^{1}\pm ix^{2}\over\sqrt{2}}$ of the complex coordinates):

[TABLE]

The final line captures the first few terms in the expansion of $D(z)$ for $z\to 0$.

Let us also make a reservation regarding the technique that we use in the present section. Most of the diagrams that we will be calculating (see Figs. 12 and 13) really correspond to the theory of a scalar field $\varphi$ in a background gauge field $\mathcal{A}$. Therefore these diagrams are the same as the ones that would arise in the corresponding Grassmannian sigma-model with target space $G_{M,N}$. For this reason in these calculations we will simply treat $\mathcal{A}$ as a non-abelian external gauge field (which can later be restricted to the form (113)). To simplify the figures, we will still be drawing the same diagrams as for a single copy of the $\mathds{C}\mathds{P}^{N-1}$-model, but with the understanding that the fields carry an additional index. The sole role of this index is to make sure the ordering of the fields is taken into account. The only diagrams where the gauge field enters in the internal lines are the ones of Fig. 14, and for these diagrams we carry out a more thorough analysis in Section 6.2.4.

6.2.1 The commutator $[(\ast K)_{z}(z_{1}),(\ast K)_{z}(z_{2})]$

We now pass to the calculation of the OPE of the commutator of two $(\ast K)_{z}$-components. To this end we consider the product

[TABLE]

We write

[TABLE]

Note that in the commutator $[(\ast K)_{z}(z_{1}),(\ast K)_{z}(z_{2})]$ the logarithmic (as well as finite) terms will disappear as they are symmetric under $z_{1}\leftrightarrow z_{2}$, therefore we have:

[TABLE]

We have inserted a normal ordered product in the r.h.s., since the self-contractions of the field $\varphi$ would give a contribution proportional to the unit operator $\mathds{1}_{N}$, which clearly cannot enter in the expansion of a commutator.

From (6.2) it follows that $\partial_{z_{1}}D(z_{1}-z_{2})=-{1\over 4\pi}\frac{1}{z_{1}-z_{2}}+O(z_{1}-z_{2})$, where $O(z_{1}-z_{2})$ denotes terms that vanish in the limit $z_{1}\to z_{2}$. Therefore

[TABLE]

Analogously for the $\bar{z}$-components we have:

[TABLE]

The above two equalities are already sufficient to prove (147). Indeed, in the limit of an infinitesimally small contour around $w$ we have

[TABLE]

A similar derivation is used to find the behavior of $S_{\epsilon}$ for $\epsilon\to 0$:

[TABLE]

where $\ldots$ denotes finite terms. It is now obvious that the integrand in (148) has a finite limit for $\epsilon\to 0$. This limit is, in particular, independent of the angle at which $\epsilon$ approaches zero.

6.2.2 The commutator $[(\ast K)_{z}(z_{1}),(\ast K)_{\bar{z}}(z_{2})]$

First of all, using the above definition, we calculate

[TABLE]

therefore

[TABLE]

Classically $\varphi^{\dagger}(z_{1})\varphi(z_{1})={N\over g^{2}}\,\mathds{1}_{2}$. Here however we need to write down the OPE of the following form

[TABLE]

where $\ldots$ denotes terms that vanish as $z_{2}\to z_{1}$. We will now show that $:\varphi^{\dagger}(z_{1})\varphi(z_{1}):=0$. Indeed, this follows from the cancellation of two diagrams shown in Fig. 9.

Note that, in principle, one should also consider the correlation functions $\langle\varphi^{\dagger}(z_{1})\varphi(z_{2})\mathcal{A}(z_{3})\rangle$ and $\langle\varphi^{\dagger}(z_{1})\varphi(z_{2})\lambda(z_{3})\rangle$ (see Fig. 10). One can show, however, that

[TABLE]

The diagram $(A)$ vanishes for symmetry reasons, as its value is equal to $\Gamma_{\mu}$ from (127). As for the second diagram, the loop integral is precisely equal to the inverse of the $\lambda$-field propagator $Q(p)$, and therefore in coordinate space the diagram $(B)$ is proportional to $\delta(z_{1}-z_{3})$. Such a term does not contribute to the anomaly, as at the level of correlation functions we have the following result (see Fig. 11):

[TABLE]

Let us also recall that $\varphi:\mathds{C}^{2}\to\mathds{C}^{N}$ is a $(N\times 2)$-matrix. Let us denote the $i$-th row by $\varphi^{i}$ and, following [17], consider the off-diagonal entries of $\left((\ast K)_{z}(z_{1})\cdot(\ast K)_{\bar{z}}(z_{2})\right)^{ij}$ ($i\neq j$) in the OPE. This will slightly simplify things, as in this case the contractions between the outer fields $\varphi^{i}$ and $(\varphi^{\dagger})^{j}$ are prohibited. Then we get

[TABLE]

where it is understood that one should only keep the non-vanishing terms in the expansion of $D(z_{1}-z_{2})$ as $z_{2}\to z_{1}$. Using the definition of the current $K$, we find that

[TABLE]

where in the last line we have used the e.o.m. (96). Substituting this in the above OPE, we get

[TABLE]

Multiplying the currents in the opposite order, we obtain

[TABLE]

Classically we see from (104) that we could replace $\underset{z_{2}\to z_{1}}{\mathrm{lim}}{g^{2}\over N}\mathscr{D}_{\bar{z}}\varphi^{\dagger}(z_{1})\cdot\mathscr{D}_{z}\varphi(z_{2})=-{1\over 2}i\lambda(z_{1})-{1\over 2}i\,F_{z\bar{z}}(z_{1})$. At the quantum level the corresponding statement is that the following OPE holds:

[TABLE]

where $\kappa_{\lambda}(z)$ and $\kappa_{F}(z)$ are two coefficient functions to be found, at leading order $1\over\sqrt{N}$, from the diagrams shown in Fig. 12.

The sum of diagrams $(a)$, $(b)$ and $(c)$ leads to the following integral 555Note that here, and in what follows, we use the non-rescaled fields $\mathcal{A},\lambda$ in the external lines of various diagrams. As explained in Section 5, to justify the result within the $1\over N$-expansion, one should keep in mind that $\lambda=\frac{1}{\sqrt{N}}\widetilde{\lambda}$, $\mathcal{A}=\frac{1}{\sqrt{N}}\widetilde{\mathcal{A}}$. It is in this sense that we count the orders of the $1\over N$-expansion in Figs. 12, 13. 666In writing out explicit expressions for the Feynman integrals we will suppress factors of $N$, coming from loops of matter fields, and reinstate them only in the final expressions for the OPE’s.:

[TABLE]

By a direct calculation one can show (see Appendix D) that, up to terms which vanish in the limit $\epsilon\to 0$, it is equal to

[TABLE]

where $\Gamma_{z\bar{z}}$ is a component of the $\Gamma_{\mu\nu}$-tensor defined in the effective action (123). The contribution of $\Gamma_{z\bar{z}}(p)$ may be discarded in the OPE, as again it leads to contact terms in the correlation functions (analogously to the situation described in Fig. 11).

Let us now evaluate the diagram $(d)$. It corresponds to the integral

[TABLE]

Using manipulations similar to the ones described in Appendix D, one can show that up to terms vanishing in the limit $\epsilon\to 0$ this integral is equal to

[TABLE]

Once again, the terms proportional to $\Gamma_{\lambda}(p)\,i\lambda(p)$ are contact terms and may be dropped in the OPE.

As a result we obtain the following OPE:

[TABLE]

Using the expansions (6.2.2) and (175), we find the following OPE for the commutator of two currents (for $i\neq j$):

[TABLE]

In the last two formulas by $F_{z\bar{z}}$ we mean the linear part of the full non-abelian field strength. In the next subsection we will see that the calculation of the OPE at the next order in the $1\over N$-expansion amounts to completing the field strength to the non-abelian form.

6.2.3 Order $1\over N$

So far we have calculated the OPE to leading order, $1\over\sqrt{N}$. At the next order, $1\over N$, we have the diagrams depicted in Figs. 13 and 14.

First of all, in the integrand corresponding to Fig. $(a^{\prime})$ we will do the following rewriting:

[TABLE]

Let us start by considering the two terms in square brackets (the contribution of the residual part will be analyzed in the next subsection). They may be used to cancel the leftmost or the rightmost propagator in Fig. $(a^{\prime})$. Cancelling the propagator $1\over k^{2}+m^{2}$, for example, one is left with the following integral (after the change of variable $k\to k-p_{1}$):

[TABLE]

Here $p_{1}$ and $p_{2}$ are the momenta carried by the external gauge fields, and $\simeq$ means ‘up to terms that vanish in the limit $\epsilon\to 0$’. It is easily seen that in the bracket $(2k-p_{1})_{\mu}$ one can drop the term, proportional to $p_{1}$, as the integral multiplying this term vanishes by symmetry in the limit $\epsilon\to 0$. In this limit the integral is indistinguishable from

[TABLE]

where we have introduced the notation

[TABLE]

Apart from the contribution $I_{a^{\prime}}^{(1)}$, there will be a contribution $I_{a^{\prime}}^{(2)}$, where instead of cancelling the propagator $1\over k^{2}+m^{2}$ one cancels the propagator $1\over(k+p)^{2}+m^{2}$. The only difference in the resulting expression will be that in the argument of the function $\tau$ one will have to replace $p_{1}\to p_{2}$.

We now pass to the calculation of diagrams $(b^{\prime})$ and $(c^{\prime})$. Their sum is

[TABLE]

Altogether (6.2.3), a similar expression with the replacement $\tau(p_{2})\to\tau(p_{1})$ (which we call $I_{a^{\prime}}^{(2)}$) and (181) give the following result:

[TABLE]

We will now relate the functions $\tau_{\mu\nu}$ to the effective action $\Gamma_{\mu\nu}$, which is defined as the limit $M\to\infty$ of the following integral:

[TABLE]

Its $(z\bar{z})$-component may be equivalently rewritten as

[TABLE]

Note that the last term, in the limit $M\to\infty$, is equal to ${\int\frac{d^{2}k}{(2\pi)^{2}}\,\frac{1}{(k^{2}+1)^{2}}={1\over 4\pi}}$. Introducing momentarily a factor $e^{ik\epsilon}$ in the (convergent) integral in (184), recalling the definition of $\tau_{z\bar{z}}$ and taking the limit $M\to\infty$, we may rewrite (184) in the following way:

[TABLE]

In other words, as $\epsilon$ goes to 0, we have the asymptotic behavior

[TABLE]

Analogously one can show that

[TABLE]

(For completeness we prove this relation in the Appendix E.)

Substituting the above relations in (188), we arrive at the expression

[TABLE]

We will now argue that, just as we did earlier, the parts proportional to the effective action (i.e. the last two lines) may in fact be dropped. First we rewrite them as

[TABLE]

If one takes for $\mathcal{A}_{\mu}(p_{1})$ the diagonal (gauge) part, say $a_{\mu}(p_{1})$, and computes a correlation function with the field strength operator $(p_{1})_{\mu}a_{\nu}(p_{1})-(p_{1})_{\nu}a_{\mu}(p_{1})$, the effective action part will fully cancel, leaving contributions which in coordinate space are again local (w.r.t. the point where the field strength operator is inserted). Such contributions can be dropped according to the logic explained earlier. On the other hand, if $\mathcal{A}_{\mu}(p_{1})$ is one of the off-diagonal ‘matter’ fields, say $c_{\bar{z}}(p_{1})$, and one computes a correlation function with the insertion of an operator $c_{z}$, there will be an additional non-local piece proportional (in momentum space) to ${\Gamma_{zz}(p)\over\Gamma_{z\bar{z}}(p)}=-{p_{z}\over p_{\bar{z}}}$. In the next subsection we will see, however, that in this case there is an additional contribution from the ‘residual part’ in (177) which completely cancels out the $\Gamma_{zz}$-piece in the formula (189).

Anyway (188) is not yet the final answer, as we still have to analyze the diagrams $(d^{\prime})$ and $(e^{\prime})$. The integrand of the diagram $(d^{\prime})$ has in the numerator the product $k_{\bar{z}}(k_{z}+p_{z})$, for which we can use the decomposition (177). The first two terms can again be used to cancel the propagators, which results in the following contribution (the residual part is again left for a later analysis):

[TABLE]

The diagram $(e^{\prime})$ gives $I_{e^{\prime}}=D(\epsilon)A_{\bar{z}}(p_{1})A_{z}(p_{2})+(p_{1}\leftrightarrow p_{2})$, so that the sum of two diagrams is

[TABLE]

Together, (188) and (191) amount to (after dropping the $\Gamma$-terms)

[TABLE]

Comparing with the leading order OPE (175), we see that the quadratic terms in $\mathcal{A}$ simply amount to completing the linearized field strength to the full non-abelian field strength $F_{z\bar{z}}=\partial_{z}\mathcal{A}_{\bar{z}}-\partial_{\bar{z}}\mathcal{A}_{z}+i\,[\mathcal{A}_{z},\mathcal{A}_{\bar{z}}]$.

6.2.4 The residual part

Finally we need to analyze the contributions that the residual part of the decomposition (177) makes to the values of the diagrams ($a^{\prime}$) and ($d^{\prime}$). Our claim is that these contributions cancel partially against the diagrams obtained by the procedure described in Fig. 14, and partially against the $\Gamma$-terms in (188), producing contributions that are already included in the linear terms (175) of the OPE. To start with, it is clear that the Feynman integrals corresponding to the diagrams in Fig. 14 are essentially products

[TABLE]

Since the lower diagrams are the diagrams of Fig. 12, we already know the corresponding expressions: these are given by (172) and (174). If we dropped in the latter expressions the parts with the effective action $\Gamma$, we would obtain exactly the correlation functions of the r.h.s. of (175) with two additional $\mathcal{A}$-operators. Therefore, up to the $\Gamma$-terms, the values of the diagrams in Fig. 14 are already included in the linear terms of the OPE (175). As for the $\Gamma$-parts of the diagrams on Fig. 14, they cancel with the ‘residual parts’ of the diagrams $(a^{\prime})$ and $(d^{\prime})$, as well as with certain $\Gamma$-terms in (188). The mechanism is as follows: the functions $\Gamma_{\lambda}(p)$ and $\Gamma_{z\bar{z}}(p)$ cancel against the corresponding intermediate propagators of Fig. 14, leaving truncated diagrams whose associated Feynman integrals cancel against the ‘residual parts’ of the diagrams $(a^{\prime})$ and $(d^{\prime})$, possibly up to contact terms.

Let us elaborate this phenomenon at the example of the rightmost two diagrams in Fig. 14. We will be using the Feynman rules derived in Section 5.

Diagram $(c^{\prime\prime})$. The $\Gamma$-dependent part of the short-distance expansion of the lower part of the diagram is given by (172) – it is equal to $(p_{\bar{z}}a_{z}-p_{z}a_{\bar{z}})\,\Gamma_{z\bar{z}}$, where $a$ is the $U(1)$ gauge field corresponding to the blue line. Multiplying by the propagators of the gauge fields, we see that the coupling to the upper part of the diagram is given by $(p_{\bar{z}}D_{z\bar{z}}(2k-p)_{z}+p_{\bar{z}}D_{zz}(2k-p)_{\bar{z}}-p_{z}D_{z\bar{z}}(2k-p)_{\bar{z}}-p_{z}D_{\bar{z}\bar{z}}(2k-p)_{z})\,\Gamma_{z\bar{z}}$. Recalling that $D_{z\bar{z}}={1\over 8}\Gamma_{z\bar{z}}^{-1}$ and $D_{zz}=-{1\over 8}\frac{p_{z}}{p_{\bar{z}}}\Gamma_{z\bar{z}}^{-1}$, we get ${1\over 2}(p_{\bar{z}}k_{z}-p_{z}k_{\bar{z}})$. This cancels exactly against the analogous contribution in (177).

Diagram $(d^{\prime\prime})$. Just like above, we read off the $\Gamma$-dependent part of the short-distance expansion of the lower part of the diagram from (172) – in this case it is equal to $-p_{z}c_{\bar{z}}\,\Gamma_{z\bar{z}}$, where $c_{\bar{z}}$ is the off-diagonal auxiliary field in (87). We now supply the propagator of the $c$-field, which is $D_{c}={1\over 2}\Gamma_{z\bar{z}}^{-1}$ and the coupling to the upper vertex, which is $(2k-p)_{\bar{z}}$. Altogether this gives $-{1\over 2}p_{z}(2k-p)_{\bar{z}}$. This does not completely cancel the contribution of (177), the sum of the two terms being instead ${1\over 4}(p^{2}-2kp)={1\over 4}((k-p)^{2}+m^{2})-{1\over 4}(k^{2}+m^{2})$. This allows canceling the propagators in the upper diagram, leading to the following expression:

[TABLE]

To obtain the latter expression, one needs to subtract the analogous integral with the replacement $m^{2}\to m^{2}+M^{2}$ (which in itself vanishes in the limit $M\to\infty$), split the resulting integral into two parts, use the definition (183) of $\Gamma_{\mu\nu}^{(M^{2})}$ and then take the limit $M\to\infty$. Now, the above expression is to be multiplied by $a_{\mu}(p+p^{\prime})c_{\bar{z}}(p^{\prime})$ and it has to be added to the terms (189) obtained from the diagram $(a^{\prime})$ of Fig. 13. In (189) we will assume that $\mathcal{A}_{\mu}(p_{1})$ corresponds to the field $c_{\bar{z}}(p_{1})$ (hence $p_{1}=p^{\prime}$) and $\mathcal{A}_{\mu}(p_{2})$ corresponds to $a_{\mu}(p+p^{\prime})$ (hence $p_{2}=p+p^{\prime}$). Adding the above expression to (189) and restricting the fields as described, we get

[TABLE]

This is again a contact term: computing a correlation function with an external $c_{z}$-operator produces a local contribution, which may be dropped.

6.3 The anomaly

We are now ready to collect the results that concern the OPE’s of the Noether currents taken at two nearby points obtained in the previous section and calculate the anomaly two-form. To this end, we recall the formulas (150) (the definition of the anomaly) and (176) (the OPE):

[TABLE]

[TABLE]

Since, according to (6.2), $D(\epsilon)=-{1\over 4\pi}\log{\left(m^{2}|\epsilon|^{2}\over 2\right)}-{\gamma\over 2\pi}+\ldots$, we will set $\mathrm{Re}(\mathbf{a})={1\over 4\pi}\log{\left(m^{2}\over 2\right)}+{\gamma\over 2\pi}$. Then in the limit $\epsilon\to 0$ we obtain the following result:

[TABLE]

and the ‘gauge field’ $\mathcal{A}$ is in general of the restricted form (113) (for the particular case of the flag manifold (85) it is of the form (87)). Note that formally all our calculations were performed for the off-diagonal matrix elements ($i\neq j$), therefore since $\mathcal{Q}_{\epsilon}\in\mathfrak{su}_{N}$, one would need to subtract the trace-part of the respective expressions (which is proportional to the identity matrix $\mathds{1}_{N}$). Instead, we have inserted the normal ordering in the r.h.s. of (192), as in this case the trace-part vanishes due to the property $:\varphi^{\dagger}(z)\varphi(z):=0$ discussed earlier in Section 6.2.2 (see Fig. 9).

7 Conclusion and outlook

In the present paper we investigated the classical and quantum properties of the flag manifold sigma-models, introduced in our work [31, 32]. In fact, the models of [31] constitute a slightly broader class, allowing for complex homogeneous target spaces, however the subclass of flag manifold models is representative and interesting enough for an elaborate study. As we showed in Section 3, models of this class may also be understood in terms of the theory of sigma-models with $\mathbb{Z}_{m}$-symmetric target spaces considered in [24]. Throughout the paper for simplicity we restricted to the case when the isometry group is $SU(N)$, although we expect that similar analysis might be carried out for other compact semi-simple groups. With the aim of constructing the $1\over N$-expansion for the flag manifold sigma-models, we first reformulated these models as gauged linear sigma-models in Section 4. The construction is non-trivial, as the target spaces in question are not Kähler. Nevertheless, formally the result is akin to the GLSM-formulation for Grassmannian (Kähler) target spaces, albeit with a ‘gauge field’ of a special (restricted) form. In the next section, using this formulation, we derived the Feynman rules for the simplest non-symmetric-space model. Subsequently we introduced a regularized version of the non-local charge, which is different from the original definition of Lüscher [15]. As opposed to the definition in [15], our version of the regularized charge depends on the integration contour $\Gamma$ through an anomaly two-form and does not depend on the parametrization of the contour. We compute this anomaly two-form explicitly in Sections 6.2, 6.3. It formally coincides with the anomaly for the Grassmannian models [18], but the gauge fields entering the anomaly should be taken in restricted form, of the type mentioned earlier. It is an important and interesting question, how the anomaly may be cancelled. Most likely this can be done by introducing fermions, i.e. by a mechanism similar to the one of [19, 22]. The investigation of the peculiarities of this mechanism in application to the present models will be a subject of future work.

Acknowledgements. I would like to thank I. Aref’eva, S. Frolov, E. Ivanov, S. Ketov, O. Lechtenfeld, A. Maltsev, M. Semenov-Tian-Shansky, A. Tseytlin, K. Zarembo, P. Zinn-Justin for valuable comments. I am indebted to Prof. A.A.Slavnov and to my parents for support and encouragement. I would like to thank the Institut des Hautes Études Scientifiques, where part of this work was done, and in particular V. Pestun for hospitality.

Appendix

Appendix A Flag manifolds and elements of representation theory

This Appendix lies somewhat outside the main line of exposition in the present paper. Yet we have included it to demonstrate, how flag manifolds arise in a well-known physical situation. Incidentally this makes a neat connection to the applications of flag manifolds in representation theory, discussed below in Section A.1.

It is well-known, how one can describe a classical particle, interacting with an external electromagnetic field $A_{\mu}$. The action has the form

[TABLE]

The question is, how to write an analogous action for the case when the gauge field is non-abelian, or, simply speaking, when it has additional gauge indices $A_{\mu}^{ab}$. The answer is that the particle should possess additional degrees of freedom, taking values in a certain flag manifold, corresponding to the representation, in which the particle transforms. In other words, one should enlarge the phase space as follows [55] (here $\mathcal{M}$ is the configuration space):

[TABLE]

We start by rewriting the standard action of a particle in first-order form:

[TABLE]

Upon enlarging the phase space we can analogously write down the non-abelian action as follows ($A$ is assumed Hermitian):

[TABLE]

where $\theta$ is the canonical one-form, defined by the condition

[TABLE]

and $\mu$ is the moment map for the action of the group $G$ on $\mathcal{F}$. We note that the form $\theta$ is only defined up to the addition of a total derivative, $\theta\to\theta+dh$, but the difference only affects the boundary terms in the action. In the case of periodic boundary conditions one may even write

[TABLE]

where $D$ is a disc, whose boundary is the curve $\Gamma$: $\partial D=\Gamma$. In fact this term is nothing but the one-dimensional version of the Wess-Zumino-Novikov-Witten term [56, 57, 58].

One needs to show that the expression so obtained is gauge-invariant. For simplicity let us consider the standard representation of $SU(N)$ (when the representation space is $\mathds{C}^{N}$): in this case the relevant flag manifold is the projective space $\mathds{C}\mathds{P}^{N-1}$. Let us normalize the homogeneous coordinates $(z_{1},\ldots,z_{N})$ on $\mathds{C}\mathds{P}^{N-1}$:

[TABLE]

One still has the remaining gauge group $U(1)$, which acts by multiplication of all coordinates $z_{k}$ by a common phase. The Fubini-Study form on $\mathds{C}\mathds{P}^{N-1}$, when written in homogeneous coordinates, looks as follows:

[TABLE]

but it may be simplified, if one uses the above normalization:

[TABLE]

Then we have the following expressions for $\theta$ and $\mu$:

[TABLE]

The part of the action, corresponding to the motion in the ‘internal’ space (in this case the projective space), has the form

[TABLE]

and one should take into account that the normalization condition (199) is also implied. From the second form of the action it is evident that it is gauge-invariant w.r.t. the transformations

[TABLE]

To make it even more obvious, we note that the exterior derivative of the one-form $\theta-\mathrm{tr}\,(A\,\mu)$ (viewed as a form on the enlarged phase space (194)) produces a two-form, which is explicitly gauge-invariant:

[TABLE]

Each of the two terms in (206) is separately gauge-invariant, however (206) is the only linear combination of them, which is closed (and therefore locally is an exterior derivative of a one-form).

Next we wish to write out the equations of motion on the flag manifold, which follow from the action above. Let us concentrate for simplicity on the $SU(2)$-case, where the sphere $S^{2}$ plays the role of a flag manifold. Instead of using the spinor $(z_{1},z_{2})$, we can parametrize it in a more standard way, with the help of a unit vector $\vec{n}\in\mathbb{R}^{3}$. The equations take the form

[TABLE]

is a vector of components of the gauge field in the basis of Pauli matrices. We see that the equations are linear in $\vec{n}$, and the condition

[TABLE]

itself is a consequence of the equations, i.e. the motion takes place on a sphere in $\mathbb{R}^{3}$. This is a general fact. Indeed, in the case of a general compact simple Lie algebra $\mathfrak{g}$ with basis $\{\tau_{i}\}$ we can introduce a variable $n=\sum\,x^{i}\,\tau_{i}\in\mathfrak{g}$, and the equations will then take the form

[TABLE]

or, in terms of the variables $x^{i}$,

[TABLE]

It is in this form that this system of equations was discovered in [59]. The motion defined by these equations in reality takes place on flag manifolds embedded in $\mathfrak{g}$, since the ‘Casimirs’

[TABLE]

are integrals of motion of the system (209). We have thus established a connection with the formulation through flag manifolds used earlier.

A.1 ‘Quantization’ of the symplectic form on flag manifolds

One of the approaches to quantization is related to considering path integrals of the following form777Another approach to the quantization of coadjoint orbits, which is also based on the path integral, was developed in [60].:

[TABLE]

where the exponent contains the action (196). The connection $\theta$ is not a globally-defined one-form on the flag manifold. Indeed, let us consider the simplest case of $\mathcal{F}=\mathds{C}\mathds{P}^{1}=S^{2}$. The most general invariant symplectic form is as follows:

[TABLE]

with an arbitrary constant $a$. It can be also written in the form $\omega=-\frac{a}{2}\,dz\wedge d\phi$, where $z=\cos{\vartheta}$ is the $z$-coordinate of a given point on the sphere. Since the latter form is nothing but the area element of a cylinder, it implies that the projection of a sphere to the cylinder preserves the area. Since the action $\mathcal{S}$ entering the exponent in (212) involves a term $\int\,\theta$, where $\theta$ is a connection satisfying $d\theta=\omega$, standard arguments familiar from Wess-Zumino-Novikov-Witten theory [56, 57, 58] lead to the requirement that the coefficient $a$ is quantized according to $\int\omega\in 2\pi n,n\in\mathbb{Z}$. By analogy, in the case when the manifold $\mathcal{F}$ has several non-trivial basic cycles $C_{1},\ldots,C_{N-1}$, in order for the exponent $e^{i\,\int\limits_{\Gamma}\,\theta}$ to be well-defined for any contour $\Gamma$, the integral of the symplectic form $\Omega$ over any 2-cycle $C_{i}$ (which in the case of a flag manifold is always a sphere $S^{2}$) should be quantized:

[TABLE]

Let us construct these 2-cycles explicitly for the case when $\mathcal{F}$ is a complete flag manifold

[TABLE]

It can be parametrized using $N$ orthonormal vectors $u_{i},i=1\ldots N$, $u_{i}\circ\bar{u}_{j}=\delta_{ij}$, defined modulo phase transformations: $u_{k}\sim e^{i\alpha_{k}}\,u_{k}$. As we showed in Section 1.2, the most general symplectic form on $\mathcal{F}({1,\ldots,N-1,N})$ may be written as follows:

[TABLE]

If one fixes $N-2$ out of $N$ lines defined by the vectors $u_{1},\ldots,u_{N}$, the remaining free parameters define the configuration space of ordered pairs of mutually orthogonal lines, passing through the origin and laying in a plane, orthogonal to the $N-2$ fixed lines. This configuration space is nothing but the sphere $\mathds{C}\mathds{P}^{1}$:

[TABLE]

Let us now fix the permutation $(i_{1},\ldots,i_{N})$ in such a way that $z_{i_{m}}$ would form a non-increasing sequence, i.e. $z_{i_{m}}\geq z_{i_{n}}$ for $m<n$ (in the case of the complete flag manifold (215) the sequence should be strictly decreasing, whereas the ‘non-increasing’ case in general corresponds to partial flag manifolds in a natural way). We can rewrite the symplectic form as follows: $\Omega=i\,\sum\limits_{m<n}\,a_{i_{m}i_{n}}\,J_{i_{m}i_{n}}\wedge J_{i_{n}i_{m}}$, and uniquely fix a complex structure, in which the one-forms $J_{i_{m}i_{n}},m<n,$ are holomorphic. After such a permutation we may choose $(\mathds{C}\mathds{P}^{1})_{{i_{m}},i_{m+1}}$ as a basis in the homology group $H_{2}(\mathcal{F}_{1,\ldots,N-1,N},\mathbb{Z})$, with the orientation of the spheres induced by the complex structure. Then the integrals of the symplectic form over these cycles will be positive:

[TABLE]

In order for the value of the integral to be an integer, one should choose $z_{i}$ in the form

[TABLE]

We may view $(\mathds{C}\mathds{P}^{1})_{{i_{m}},i_{m+1}}$ as the positive simple roots, and $\Omega$ is, in this case, a dominant weight. Adding to $\{\tilde{z}_{i}\}$ a vector, proportional to $\lambda(1,\ldots,1)$, does not change the values of the integrals. Using this property, we may normalize the values $\{\tilde{z}_{i}\}$ in such a way that their sum would be zero. According to the general theory of adjoint orbits, the flag manifold under consideration is the orbit of the element

[TABLE]

The fundamental weights are, by definition, the ones, whose value on simple positive roots is equal to zero, apart from a single root, on which the fundamental weight has value one: $\langle\Omega_{F_{[j]}},\lambda_{[i]}\rangle=\delta_{ij}$. These weights correspond to the highest weights of fundamental representations. According to the theory described above, they correspond to orbits of the elements

[TABLE]

According to (218), in this case we have

[TABLE]

The adjoint orbit in question is the Grassmannian $G_{i,N}$. The general theory that we have described is nothing but ‘geometric quantization’ for the case of flag manifolds.

Appendix B Eliminating auxiliary fields

In this Appendix we demonstrate explicitly how the auxiliary fields $a,b,c$ may be eliminated from the Lagrangian (84) with $m+n=2$ and the gauge field given by (87). To this end, we write the Lagrangian as follows:

[TABLE]

Here we have taken advantage of the orthonormality relations $\|u\|=\|v\|={\sqrt{N}\over g}$ and $\bar{u}\circ v=0$. Completing the squares, we can obtain relations of the form

[TABLE]

Therefore upon setting the fields $a,b,c$ equal to their stationary values, we get the Lagrangian in the form (88).

Appendix C The point splitting of Lüscher

In this section we recall the point splitting method for the non-local charge, introduced in the original paper [15]. In order to measure this splitting one needs to introduce a metric $g$ on the worldsheet $\Sigma_{2}$ (note that classically everything depended only on the conformal class $[g]$ of the metric). For simplicity we will now consider the situation when $\Sigma_{2}\simeq\mathds{R}^{2}$, equipped with a flat metric. As in the body of the paper, we will assume that the contour $\Gamma$ is infinite (a straight line, for example) and we will fix the reparametrization invariance on the worldline by choosing a gauge

[TABLE]

Here $x^{\mu}:\Gamma\simeq\mathds{R}_{t}\to\Sigma_{2}$ is the parametrization of $\Gamma$ by a variable $t$ taking values from minus infinity to plus infinity. If $\Gamma$ was a contour of finite length, one could still parametrize it by a variable $t$ with infinite range, however one would need to choose a gauge $(\dot{x}^{\mu})^{2}=f^{2}(t)$, so that the length $\int\limits_{-\infty}^{\infty}\,|f(t)|\,dt<\infty$ was finite.

The regularized non-local charge of [15] has the form:

[TABLE]

where $\epsilon$ is now a positive real number. The factor $Z(\epsilon)$ has been inserted to cancel the divergence in the short-distance expansion of the two currents in the second term. Next we compute the variation of this charge under deformations of the contour $\delta x^{\mu}$ (sometimes we will denote this variation by $\delta_{\Gamma}$). Upon integrating by parts in the terms involving $\delta\dot{x}^{\mu}$, we obtain:

[TABLE]

The ‘bulk term’ (the one in the last line) vanishes, since due to the antisymmetry of $\delta x^{\mu}(t)\dot{x}^{\lambda}(t)-\delta x^{\lambda}(t)\dot{x}^{\mu}(t)$ it turns out to be proportional to $\partial_{\mu}K_{\mu}=0$ (conservation of the current). The final result is therefore

[TABLE]

Now in the last line we will use the OPE’s of two currents obtained in Section 6.2. If, say, $\mu=z$ and $\alpha=\bar{z}$, one needs the OPE $[K_{z}(p+\epsilon),K_{\bar{z}}(p)]$, which, according to (176), is logarithmic in $\epsilon$. Therefore in this case in the bracket $\left(\dot{z}(t-\epsilon)\delta\bar{z}(t)-\delta z(t-\epsilon)\dot{\bar{z}}(t)\right)$ we can pass to the limit $\epsilon\to 0$. The logarithmic divergence itself is canceled by tuning the dependence on $\epsilon$ of the factor $Z(\epsilon)$.

If however $\mu=\alpha=z$, we have to consider the OPE $[K_{\bar{z}}(p+\epsilon),K_{\bar{z}}(p)]$, which according to (157) has a pole term $\sim{1\over\epsilon}$ at the leading order. This term is multiplied by $\left(\dot{z}(t-\epsilon)\delta z(t)-\delta z(t-\epsilon)\dot{z}(t)\right)\sim\epsilon\left(\dot{\delta z}\,\dot{z}-\ddot{z}\,\delta z\right)$, producing an undesirable extra contribution in addition to (149). In other words, if one assumes the definition (226), the anomaly is not given by a 2-form.

Appendix D The integral giving the anomaly at order $1\over\sqrt{N}$

In this Appendix we wish to consider the integral (171):

[TABLE]

and to bring it to the form (172). We start by rewriting the factors in the numerators of the first two summands as follows:

[TABLE]

The first terms here allow cancelling one of the propagators each. The remaining terms are multiplied by a factor $e^{ik\epsilon}\left(2k_{\mu}+p_{\mu}\over(k^{2}+m^{2})((k+p)^{2}+m^{2})\right)$, which in the limit $\epsilon\to 0$ is odd under the interchange $k\to-k-p$. As a result, we are free to add to (228)-(229) arbitrary terms that are independent of the momentum $k$, since such terms will vanish in the integral in the limit $\epsilon\to 0$ due to anti-symmetry. We use this arbitrariness to bring the expressions (228)-(229) to the form

[TABLE]

When substituted into the integral, this gives

[TABLE]

Again, up to terms that vanish in the limit $\epsilon\to 0$ (which arise due to the change of variables $k\to-k-p$ in the first bracket), we obtain:

[TABLE]

We now wish to relate this integral to the $\Gamma_{z\bar{z}}$ function entering the effective action (123). The analogous function $\Gamma_{z\bar{z}}^{(M^{2})}$ in the presence of a Pauli-Villars regulator of mass $M$ can be written as a difference of two convergent terms:

[TABLE]

The limit of the last term $\left(m^{2}\to m^{2}+M^{2}\right)$ as $M\to\infty$ is $1\over 4\pi$. Therefore we can rewrite $I$ as follows:

[TABLE]

which is the expression (172).

Appendix E One more integral

Here we will prove the relation

[TABLE]

between the $\Gamma_{zz}$-component of the effective action and the integral

[TABLE]

which we encountered in Section 6.2.3. First, we formally introduce a factor $e^{ik\epsilon}$ in the definition of $\Gamma_{zz}$, which allows us to write:

[TABLE]

Therefore we need to construct the asymptotics of the second (simpler) integral $\mathcal{I}$ as $\epsilon\to 0$. We recall that, by definition, $(k\cdot\epsilon)=k_{z}\epsilon+k_{\bar{z}}\bar{\epsilon}$. As a consequence of rotational symmetry $\epsilon\to e^{i\gamma}\epsilon$, the integral is equal to

[TABLE]

To calculate $f$, we may set $\epsilon$ to be real and positive. We pass to polar coordinates setting $k_{z}={\rho\over\sqrt{2}}e^{i\psi}$, then $f=\int\limits_{0}^{\infty}\,\frac{\rho d\rho}{2\pi}\,\frac{\rho^{2}}{(\rho^{2}+m^{2})^{2}}\,\int\limits_{0}^{2\pi}\,\frac{d\psi}{2\pi}\,e^{i(2\psi+\rho\epsilon\cos(\psi))}=-\int\limits_{0}^{\infty}\,\frac{\rho d\rho}{2\pi}\,\frac{\rho^{2}\,J_{2}(\rho\epsilon)}{(\rho^{2}+m^{2})^{2}}$. Rescaling $\rho\to{\rho\over\epsilon}$ and passing to the limit $\epsilon\to 0$ (which does not introduce divergences in the integral, thanks to the properties of the Bessel function at [math] and $\infty$), we get $\underset{\epsilon\to 0}{\textrm{lim}}\,f(\epsilon)=-\int\limits_{0}^{\infty}\,\frac{d\rho}{2\pi\rho}\,J_{2}(\rho)=-{1\over 4\pi}$. This leads to the relation (234).