Self-dual 6d 2-form fields coupled to non-abelian gauge field: quantum corrections

We study a 6d model of a set of self-dual 2-form B-fields interacting with a non-abelian vector A-field which is restricted to a 5d subspace. One motivation is that if the gauge vector could be expressed in terms of the B-field or integrated out, this model could lead to an interacting theory of B-fields only. Treating the 5d gauge vector as a background field, we compute the divergent part of the corresponding one-loop effective action which has the (DF)2 + F3 structure and compare it with similar contributions from other 6d fields. We also discuss a 4d analog of the non-abelian self-dual model, which turns out to be UV finite.


Introduction
The possible existence of interacting theories of non-abelian 2-form fields in 6 dimensions possessing some unusual properties such as lack of manifest Lorentz symmetry and/or locality is an important open problem (for a recent review and references see, e.g., [1]). As a first step, one may study a system of 2-form potentials B in some representation of gauge group G coupled "minimally" to a non-abelian gauge vector A. Similar couplings appeared, e.g., in the context of attempts to construct an interacting theory of 6d (2,0) tensor multiplets in [2][3][4] (see also [5][6][7]).
Here we shall consider a simple bosonic model of interacting (B, A) fields following [8][9][10][11]. We shall study both the model with self-dual B-field strength and the non-chiral B-field model. It turns out that a consistent gauge-invariant coupling is possible provided JHEP06(2018)134 one keeps only the 5d part of the 6d Lorentz symmetry. 1 The action is quadratic in B and takes a local form in a particular gauge, with the A-field restricted to "live" only in 5d subspace of the 6d space. More generally, one may attempt to consider an extension where A is expressed in terms of B leading to a non-local interacting theory of B-fields only.
Our aim will be to study this (B, A) model at the quantum level. 2 We shall concentrate on the one-loop approximation where B is integrated out and A is treated as a background. As is well known, quantizing free scalar, spinor or Yang-Mills (YM) fields coupled to an external vector in 6 dimensions produces (DF ) 2 + F 3 logarithmic UV divergences in the effective action (see, e.g., [12]). We shall find that similar divergences appear also from the B-field loop, implying, in particular, the breaking of the classical scale invariance. One may hope to cancel these divergences by adding other fields (e.g., imposing supersymmetry) but so far we did not find such a finite model.
As in the case of the 6d Weyl fermions [13,14], one could expect that the chiral nature of the self-dual B-field model implies the presence of anomalous (gauge-symmetry breaking) terms in the parity-odd part of the effective action (which would be a gaugefield counterpart of the familiar gravitational anomaly in the case of a single self-dual tensor [14,15]). However, this does not happen in the present case: as the A-field is restricted to 5 dimensions, the effective action has no parity-odd part, i.e. there is no gauge anomaly as in any 5d theory.
We start in section 2 with a description of the gauge symmetries and the classical action of the (B, A) model -both its non-chiral version and the chiral version with the self-dual B-field strength. In the B i6 = 0 gauge the corresponding actions take simple form (2.14) and (2.27). We shall argue that the one-loop effective action of non-chiral model (2.31) should be twice the effective action of the self-dual model. 3 The general (DF ) 2 + F 3 structure (3.2) of the UV divergent part of the 6d effective action in a gauge field background will be discussed in section 3. We shall summarize the results for the corresponding two coefficients β 2 and β 3 for a collection of 6d fields (see (3.6), (3.7)).
The values of β 2 and β 3 for the self-dual and the non-chiral models will be derived in detail in section 4 by computing the divergent parts of the A 2 and A 3 terms in the effective action. We shall use dimensional regularization procedure (applied only with respect to 5-momenta) that preserves background gauge invariance.
Some concluding remarks will be made in section 5. In appendix A we review the structure of the free B-field partition function. Some standard integrals are summarized in appendix B. The same values of β 2 (4.12) and β 3 (4.19) in the self-dual model are independently obtained in appendix C from the A 6 term in the effective action. In appendix D we discuss a non-local effective action of a 4d analog of the 6d self-dual (B, A) model.

Gauge symmetry and field strength
The abelian antisymmetric rank 2 tensor field has a familiar gauge symmetry There is the residual gauge symmetry, δ µ = ∂ µ η, which allows one to remove one component from µ and is thus important for the correct degrees of freedom count. A non-abelian generalization of (2.1) should also admit some non-abelian analog of this residual gauge symmetry. The abelian gauge-invariant 3-form field strength is To write down a gauge-invariant action in a non-abelian case, there should exist a generalized field strength that transforms covariantly. It turns out that it is possible to construct such a model if one relaxes the condition of 6d Lorentz covariance (and locality). Our starting point will be a model involving a 6d 2-form field B µν in some representation of gauge group G and a gauge vector field A µ . For the simplicity, we assume that both B µν and A µ are taken in the adjoint representation of G and use the following notation: One can define the non-abelian gauge transformations as [8] Here λ is the parameter of A µ gauge transformations under which B µν transforms covariantly; µ is the parameter of the gauge transformations of B µν , which, like λ, is now taking values in the algebra of G. The vector n µ is a fixed constant unit vector which selects a particular direction in 6d space breaking O(6) symmetry to O(5). In the abelian limit, the gauge transformation (2.3) reduces to (2.1). The structure of the non-local term in (2.3) is chosen to be such that if we further assume that n µ A µ = 0 then there is a non-abelian generalization of the residual gauge symmetry of the parameter µ in (2.1) under which δB µν is invariant: If we impose the additional condition that n µ ∂ µ A ν = 0, i.e. that A µ depends only on 5 of the 6 coordinates (so that, in particular, [(n µ ∂ µ ) −1 , D ν ]f = 0 for a 6d function f (x λ )) then one can check that the gauge algebra closes: The corresponding field strength of B µν is defined as where the non-local terms ensure that H transforms covariantly:

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Thus one can consistently couple the non-abelian antisymmetric tensor to a non-abelian gauge field restricted to a codimension-1 ("boundary") subspace, i.e. with an effective nonlocality along the "bulk" direction (see figure 1). This non-locality may be viewed as a gauge artifact as there is a gauge in which the corresponding action is local (see below). Note also that we do not impose any boundary condition at x 6 = 0. Without loss of generality, one can always choose n µ to point in the 6th direction, i.e. n µ = (0, 0, 0, 0, 0, 1), so that the vector field restricted as above "lives" in 5d subspace 4 This (B, A) model may be viewed as an intermediate step towards constructing an interacting model of B-fields only. For example, one may interpret A not as an independent field but as related to B by some non-local condition. In [8] the x 6 direction was assumed to be compactified to a circle of radius R and A i was related to the zero mode of . Moreover, the zero mode of the 3-form field strength was defined directly via the Hodge duality: H ijk (x k , 0) ≡ i 4πR ijkmn F mn . 5 One may also treat A i first as an independent quantum field and then integrate it out in the path integral obtaining an effective non-local model of self-coupled non-abelian B-fields.

Classical action and gauge fixing
Our starting point will be the following gauge invariant action describing the non-abelian 6d field B µν (x µ ) coupled to the 5d gauge field A i (x j ): Here Tr is in some representation R of gauge group, with Tr (H a t a ) 2 = T R H a H a (a = 1, . . . , dim G), with t a being hermitian generators and T R = 1 2 or T R = C 2 if H is a matrix with indices in fundamental or adjoint representation.
The overall (dimensionless) normalization constant in the action (2.9) will not be important as in this paper we will only consider the 1-loop approximation treating A i as a background field. In general, to make the model renormalizable one would need to introduce also A-dependent counterterms d 6 x [c 1 (DF ) 2 +c 2 F 3 ] (see below), i.e. two extra dimensionless coupling constants. 6 From (2.6) and (2.8) we have The action (2.9) is invariant under gauge transformations (2.2) and (2.3), i.e.
where λ = λ(x i ) and we redefined the gauge parameter i →¯ i as We can fix the¯ i gauge freedom by the natural gauge B i6 = 0 in which the field strength becomes manifestly local The gauge-fixed action (2.9) is then given by where [ij] stands for antisymmetrisation with weight 1 2 and In general, one could also consider adding the 5d Chern-Simons action for A: , but it will not be naturally induced in the model based on (2.9) and also in its self-dual version discussed below. 7 Note that the 6d action (2.9) or (2.14) is manifestly scale invariant. Starting with such an (dB + AB) 2 action and integrating out Ai should give a local, scale-invariant but non-polynomial and non-Lorentzinvariant action ∼ (BB) −1 dBdB for the non-abelian B-fields. It will not have free quadratic part and will thus require some non-trivial (scale-invariance breaking) B-field background to define a perturbation theory (cf. [16]).

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Our aim will be to compute the logarithmic divergences in the A i -dependent 1-loop effective action found by integrating out the B-field in (2.14) 8 The operator ∆ mn ij in (2.15) defined on 6d field B ij (x µ ) is, in general, non-degenerate and the gauge condition B i6 = 0 does not lead to a non-trivial (A-dependent) ghost determinant (cf. (2.11)). Note that the gauge-fixed action (2.14) is still invariant under the following 5d local gauge transformations (U (x i ) ∈ G): Provided the regularization preserves this symmetry, the effective action (2.16) should thus be built out of gauge-invariant combinations of F ij and D i .

Self-dual B-field model
Let us now consider the analog of the non-abelian action (2.14) in the case of the B-field with a self-dual field strength. Let us first review the free-field case of a single self-dual field. In Minkowski signature the real 6d self-duality condition reads As is well known, one way to find the action corresponding to (2.18) is to relax the manifest Lorentz symmetry. A systematic approach is to start with the phase-space path integral for the non-chiral H 2 µνλ theory, impose the standard "time-like" gauge B i0 = 0, trade the momenta corresponding to B ij for another 2-form field and then impose the self-duality truncation ending up with the "EB − BB" type action (E is "electric" and B is "magnetic") [17] (see also [18][19][20][21]). 9 Switching to the Euclidean notation that we shall use below (x 0 → ix 6 , with the gauge condition B i6 = 0) the resulting action is (i, j, . . . = 1, . . . , 5) It formally has a residual gauge invariance δB ij = ∂ i ξ j − ∂ j ξ i . 10 Taking the variation of (2.19) over B ij we obtain the equation of motion which may be written as 11 (2.20) 8 We ignore the Ai-independent factors in the partition function Z = e −Γ that should agree with (A.4) in the free limit. See appendix A for a discussion of the free B-field partition function in the "axial" gauge Bi6 = 0. 9 To get the right count of degrees of freedom at the level of path integral one should also keep track of appropriate Jacobians in the path integral measure. 10 This a 6d symmetry; the action is invariant up to a surface term. This residual symmetry is an artifact of the action (2.19) -it is absent in the required self-duality equation O+Bij = 0. 11 As in (2.15), we adopt the standard convention δ mn ij = 1 2 (δ m i δ n j − δ n i δ m j ).

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It is solved by for a 5d function q i (x i ) and a function f ij (x 6 ) that does not depend on 5d coordinates. Absorbing q i part into a formal redefinition of B ij in ∂ 6 B ij term in O + B ij and imposing the boundary condition that the self-duality condition O + B ij = 0 is satisfied at "spatial infinity" |x i | = ∞ we conclude that f ij = 0 and thus O + B ij = 0 is satisfied everywhere.
Integrating over B ij in the path integral defined by the action (2.19) and taking into account the necessary determinant factors in the measure one finds that the resulting partition function is where O ⊥ + acts on transverse B ⊥ ij field and thus describes 3 dynamical degrees of freedom as expected. 12 The equivalent results can be obtained by starting with an alternative ("EB − EE") action Here the equations of motion The B ij path integral measure here should have an extra factor of (det ∂ 6 ) 1/2 that ensures 6d Lorentz invariance; as a result one finds the same chiral partition function (2.22) (cf. appendix A). Note that the free limit of the non-chiral action (2.14) is equivalent to a combination of the self-dual and anti self-dual models: in the free limit the kinetic term in (2.14), (2.15) may be written as and thus the corresponding partition function is given by The above discussion has a straightforward generalization to the non-abelian case. Namely, let us require that the self-duality condition (2.18) or its Euclidean counterpart in the B i6 = 0 gauge should be satisfied for the non-abelian field strength (2.6) or (2.10), i.e.

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One may expect that this condition should follow (under the corresponding boundary conditions discussed above) from the direct analogs of (2.23) and (2.19): This is indeed obvious for (2.27) but is not immediately so for the second action (2.28). The equations of motion following from (2.28), D [iÔ+ B jk] = 0, may be solved asÔ One may then attempt to absorb the F -term by a (non-local in to arrive at the self-duality condition. However, the quantum equivalence of the theories based on (2.27) and (2.28) becomes unclear as an extra determinant of the operator 1 2 i ijrpq D r coming from (2.28) will now have a non-trivial dependence on A i .
In what follows we shall use the simplest action (2.27) as defining the non-abelian selfdual B-field model. 13 Since the ∂ 6 operator factorizes in (2.27), the corresponding partition function is given by the direct analog of (2.22) with O + →Ô + (A). 14 It is straightforward to check that the operator ∆(A) in the non-chiral action (2.14), (2.15) is given again by the product of the self-dual and anti self-dual operators in (2.26): As a result, the non-chiral B-field quantum effective action (2.16) may be written as While Γ should be parity (P) even, Γ + and Γ − may a priori contain imaginary P-odd parts that cancel in their sum in (2.26) (as, e.g., in the case of an external gravitational field [14]). However, it is easy to see that this does not happen in the present case when the external field A i does not depend on x 6 . Indeed, ∂ 6 → −∂ 6 combined with 5 → − 5 is a symmetry of the classical action (2.27) and thus should be present also in the effective action. As the P-odd part of Γ ± should contain an odd number of 5 = ( ijkmn ) factors it should thus have an odd number of p 6 factors (in momentum representation) but then the integral over p 6 vanishes. The absence of P-odd part implies also the absence of an anomalous (5d gauge symmetry breaking) part of Γ ± . Thus both the effective action Γ of the full non-chiral theory and Γ + of the self-dual theory should be invariant under the residual gauge symmetry of the A-field in (2.17).
To conclude, we have

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3 Structure of divergent part of effective action Before describing the details of the computation of the divergent part Γ ∞ of the effective action (2.16) corresponding to non-chiral non-abelian B-field action (2.14) and the self-dual model (2.27) and verifying their relation in (2.31), let us first discuss the general structure of Γ ∞ in a background gauge vector field. Let us consider the 1-loop effective action for a 6d model containing standard 2derivative Yang-Mills vectors, scalars and spinors coupled to a background gauge field. To prepare for the discussion of the models in the previous section we will specify to the case when the background field is chosen to be the 5-dimensional one as in (2.8) (i.e. use indices m, n, . . . = 1, . . . , 5). Using, e.g., the heat kernel representation and proper-time cutoff = Λ −2 → 0 one finds [12] from the general expression for the corresponding heat kernel coefficient [22] (see also [23,24]) β 2 and β 3 are numerical coefficients of the two independent dimension-6 invariants built out of the background field. 15 Note that in dimensional regularization one gets Γ ∞ = 1 d−6 B 6 where 1 d−6 corresponds to − log Λ in (3.1). Here tr is over the matrix indices of the particular representation to which the quantum field belongs: if it is in the adjoint representation (with hermitian generators (t a ) bc = −if a bc ) one has the gauge field as a matrix A ab n = f acb A c n and For generic representation R with generators satisfying tr(t a t b ) = T R δ ab one is to replace Both coefficients vanish in the case of the maximally (1,1) supersymmetric YM theory (SYM) in 6d which can be obtained by dimensional reduction from the 10d SYM giving 17 Note that the expression for the coefficient β 3 of the F 3 invariant in (3.5) happens to coincide with the number of effective degrees of freedom 15 The other two invariants of the same dimension are related by use of Bianchi identities: tr(DmF kn DmF kn ) = 2 tr(DmFmn D k F kn ) − 4 tr(FmnF nk F km ) + total derivative , tr(DmF kn D k Fmn) = 1 2 tr(DmF kn DmF kn ). 16 We use this opportunity to correct two unfortunate misprints in [12]: d − 1 2 → d − 42 in eq. (3.9) (here d = 6) and − 1 72 → + 1 90 in eq. (3.5) (results in eq. (3.6) there are correct). 17 Equivalently, if we consider the (1,0) SYM coupled to one adjoint hypermultiplet we get the same 1-loop finite theory (cf. [27][28][29]).
As we shall find below, in the case of the self-dual B-field the divergent part of the effective action Γ + in (2.30) is given by (3.1), (3.2) with β 2 = −27, β 3 = −57. In the case of the non-chiral B-field with the effective action in (2.16) these coefficients are doubled, in agreement with (2.31). Thus, in the presence of N T self-dual tensors, the coefficients in (3.5) become (3.6) Here all fields are assumed to be in the adjoint representation; otherwise N T , N 0 , N 1 2 are to be rescaled by the corresponding factors T R /C 2 .

Calculation of one-loop divergences
Let us now compute the coefficients in the logarithmically divergent part of the one-loop effective actions Γ + and Γ for the self-dual (2.27) and the non-chiral (2.14) 2-form models.
We adopt dimensional regularization, by continuing the theory to d = 6 − 2ε dimensions. Since the sixth direction is treated separately in the classical action and in the gauge fixing condition (B i6 = 0), it is natural to keep it one-dimensional, while continuing the remaining 5 directions, setting 6 = 1 + 5 → 1 + d, d = 5 − 2ε. Within the dimensional regularization we consider an analog of the four-dimensional helicity scheme, where all the momentum numerator algebra is first done in an integer number of dimensions and then the scalar integrals are continued to d dimensions. This guarantees that the number of physical states in loops is unchanged by the regulator.
To find the coefficients in the divergent part of the effective action one may compute, e.g., the terms quadratic and cubic in the vector field A and compare them with (3.2). This is what we will do below. Alternatively, one may find the terms with six powers of A which appear in (3.2) without derivatives and thus can be isolated by taking the non-abelian field A to be constant. This will be done in appendix C on the example of the self-dual model (2.27).

Self-dual B-field model
The effective action corresponding to the classical action (2.27) is 18

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The operator ∆ + is thus linear in the background field A i , i.e. 19 Expanding the non-trivial part of Γ + in powers of A, we have Since the background field A i is independent of x 6 , the trace projects out all terms containing an odd number of ∂ 6 factors and also produces an overall factor of length L 6 = dx 6 . As was already mentioned in section 2, together with the symmetry of the gauge-fixed action (2.27) under ∂ 6 → −∂ 6 combined with 5 → − 5 , this implies the effective action Γ + is parity-even. The evaluation of traces is standard, by using momentum space basis of states and assuming that the background field is A a i (x k ) = The matrix element of (∆ (0) ) −1 in momentum representation is the free B-field propagator

A 2 term
Inserting complete sets of momentum eigenstates between any two operators in (4.4) and using (4.5), (4.6) and momentum conservation, we have 20 where d = 5 − 2 and L 6 = dx 6 . Since the external field does not depend on x 6 here all the factors have the same 6-th component of momentum p 6 . 19 Here a, b, c are Lie algebra indices. We assume that B is in adjoint representation; otherwise t a bc = −if a bc is to be replaced by the corresponding hermitian generators. 20 The same expression may be obtained by computing the two-point function of A and promoting it to a term in the effective action. In this approach, the numerical factors are symmetry factors and the signs related to resummation of one-loop corrections to the A-field two-point function.

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The background-field gauge invariance requires that (4.7) should vanish for constant A i , i.e. for s i = 0. Setting s i = 0 and carrying out index contractions we get where we used the identity (B.4). Thus, for a constant external field, the A 2 contribution vanishes in d = 5 even before performing the integration over the p 6 momentum. Contracting group indices using eq. (3.3), introducing Feynman parameter y in the momentum integral, doing tensor reduction with the help of (B.2), and finally using the identity (B.4) gives the following expression for (4.7): (4.10) The d-dimensional integral here is standard (cf. eq. (B.1)); while it is finite for d → 5, taking this limit before the p 6 integral makes the latter divergent. To carry out the p 6 integral it is convenient to change the variable p 6 → µ as p 6 = µ y(1 − y)s 2 1/2 ; then the remaining µ-integral can be computed using (B.5). As a result, we find that the divergent part of the A 2 term in the effective action is (4.12) As was already mentioned below (3.4), in the case of the B-field being in generic representation R the coefficient C 2 is to be replaced by the corresponding index T R .

A 3 term
To find β 3 in (3.2) we need to compute the A 3 part of the effective action. The evaluation of Γ 3 in (4.4) follows the same steps as that of Γ 2 . For a B-field in an arbitrary representation (4.4) becomes × P j 3 j 4 i 3 i 4 (p i + s 2i , p 6 )V i 5 i 6 j 3 j 4 (s 3i , p 6 ) P j 5 j 6 i 5 i 6 (p i + s 2i + s 3i , p 6 ) . (4.14)

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Here V ij mn is the vertex in (4.6) with f acb replaced by −it c where t c is hermitian generator in some representation R (coming from the covariant derivative . To compute the trace over the group indices we use that where A R is the anomaly coefficient of a given representation. In adjoint representation T R = C 2 , A R = 0. The momentum-dependent coefficient of the symmetric d abc tensor part is P-odd (containing one power of 5 ) and should thus vanish identically as discussed above.
After carrying out the index contraction, Feynman parametrization and momentum integration, the divergent part of G 3 may be written as (in the adjoint representation) a 1 a 2 a 3 K a 1 a 2 a 3 (s 1 , s 2 , s 3 ) , (4.16) where K a 1 a 2 a 3 is a 5d invariant constructed from 3 powers of the background fieldÃ i (s k ) and the corresponding momenta (s 2 r ≡ s r · s r ): It simplifies in the transverse background gauge s iÃ a i (s) = 0: Comparing this to the two terms in (3.2) (which both contribute to the A 3 term) and using that β 2 was already determined in (4.12) we conclude that (cf. (3.7)) We have confirmed this result independently by computing constant A 6 term in the effective action in appendix C.

A 2 term
The first term in Γ 2 in (4.22) is a tadpole which vanishes in dimensional regularization; the second term gives (using the same notation as in (4.14))

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Following similar steps as in the self-dual model in section 4.1.1 we find (p 2 = p i p i ) s ·Ã a (s) s ·Ã a (−s).
It is useful to use eq. (B.4) to relate the integrals with loop momenta in the numerators to scalar bubble integrals. Unlike the self-dual theory case, here the A 2 term takes a gauge-invariant form only after one carries out all the integrals; its divergent part is found to be This is twice the value in the self-dual case (4.11), i.e. the corresponding β 2 coefficient in (3.2) is (cf. (4.12)) (4.31)

A 3 term
Unlike the case of Γ 2 , the matrix element of ∆ (2) in (4.25) contributes nontrivially to Γ 3 in (4.22) (cf. (4.13)) Here we used (4.15) to do the group index contraction (with the momentum-dependent coefficient of d abc again vanishing in general) and considered the adjoint representation. Introducing Feynman parameters and shifting the integration variable in such a way that the denominator becomes a symmetric function, we use the SO(d) symmetry to express the tensor momentum integrals in terms of the scalar ones. Further using eq. (B.4) the integrals with various powers of the d-momentum can be reduced to scalar triangle and/or bubble integrals. Evaluating first the d-dimensional integral and then appropriately changing

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the variable of the p 6 integral one can decouple the latter from that over the Feynman parameters. As before, the logarithmic UV divergence we are interested in emerges after the last (p 6 ) integral is evaluated using (B.5). The contribution of the first structure in (4.33) written in coordinate representation gives a term proportional to 1 which matches the cubic term in tr(∂ m F mn ) 2 in (3.2). Gauge invariance and consistency with the A 2 term calculation (4.30) are restored by the inclusion of the second term in (4.33).
The full computation is straightforward but tedious so we simply state that the final result is consistent with (3.2) with β 2 found above in (4.31) and with β 3 being again twice the value in the self-dual case (4.19): (4.34)

Concluding remarks
In this paper we have studied a model of 6d 2-form B-fields in some representation of an internal symmetry group G coupled consistently to a non-abelian gauge field A which "lives" only in a 5d subspace. We computed one-loop logarithmic divergences in such a theory by integrating out the B-field and treating the gauge field A as a background. The resulting divergent part of the effective action (3.1), (3.2) contains the terms ∼ tr[3β 2 (D m F mn ) 2 − 2β 3 F mn F nk F km ] with the coefficients β 2 , β 3 given by (3.6), (3.7) or (4.12), (4.19) in the case of the self-dual theory (and twice these values in the case of the non-chiral B-field model given in (4.31), (4.34)). The presence of these divergences suggests that in the full theory where A-field (or its 6d extension) is also quantized the higher-derivative c 1 (DF ) 2 + c 2 F 3 terms should be added to the bare 6d action. 21 One may hope that these divergences could be cancelled by adding some other fields to the model and imposing, e.g., supersymmetry constraint. For a collection of N T self-dual tensors, N 1 standard 2-derivative YM vectors, N 0 real scalars and N 1 2 Weyl fermions in 6d coupled to background vector, we conclude that 22 Thus the self-dual B-field coupled minimally to a 5d gauge field contributes to the β 2 in the logarithmic divergent part of the effective action with the same sign as the YM vector. A somewhat unexpected feature is that its contribution to β 3 turns out to be opposite in sign compared to other standard 2-derivative bosonic fields. A naive expectation could be 21 (DF ) 2 + F 3 theory is of course classically conformally invariant but this symmetry will be anomalous at loop level. Let us note that the presence of similar (DF ) 2 terms in 6d theory was suggested by requiring dual conformal symmetry in six dimensions in [30], though precise connection of this to the present work is not clear to us. 22 Note that quantizing the 5d gauge field with the action L6 d 5 x[c1(DiFij) 2 + c2F 3 ij ] will not produce extra one-loop logarithmic UV divergences as this theory is effectively 5-dimensional one. It would be interesting to add also the one-loop contribution of the genuine 6d non-abelian vector model. Choosing the background gauge field to be a 5d one we would then get additional contributions to β2 and β3.

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that each field should contribute to β 3 proportionally to its number of dynamical degrees of freedom: (5.2) One may formally consider fields that form 6d supermultiplets containing self-dual B-field and couple them to a background 5d gauge field. In the case of (1,0) tensor multiplet with N T = 1, N 1 = 0, N 0 = 1, N 1 2 = 1 a natural coupling is to (1,0) SYM (N 1 = 1, N 0 = 0, N 1 2 = 1); in this case we would expect to get β 3 = 0 as the F 3 -invariant should be prohibited by supersymmetry. However, while ν in (5.2) indeed vanishes in this case, from (5.1) we get β 3 = −60 = 2β 2 . This suggests that the model considered in this paper (with A i treated as a 5d background field) does not admit a (1,0) supersymmetric extension. This may not be surprising as the model lacks 6d Lorentz symmetry. 23 A possible role or application of the non-abelian coupled (B, A) model discussed in this paper remains an open question. It might be related to some intersecting brane configuration where a 5d gauge field lives on a 5d brane "defect". Another option is that the 5d A-field may happen to play an auxiliary role and eliminating it one may end up with an effective interacting theory of a set of B-field only. Yet another possibility is the existence of a generalization where the A gauge field becomes fully 6-dimensional, Lorentz invariance is formally restored but the resulting classical action might become effectively non-local.
Analogous considerations in the self-dual case described by the classical actions (2.19) or (2.23) lead to the partition function (2.22) or the "square root" of (A.4) (cf. (2.25)).

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C A 6 term in effective action of self-dual B-field To obtain the β 2 and β 3 coefficients in the divergent part of the effective action (3.2) one may either compute the derivative-dependent A 2 and A 3 terms as was done in section 4, or consider a constant non-abelian A a i -field and find the coefficients of the independent A 6 terms in Γ. For constant potential one has F a ij = f abc A b i A c j and thus

D Non-abelian chiral 4d vector model
Here we consider a 4d analog of the 6d self-dual B-field model (2.27) where the tensor field B ij is replaced by a vector B i (with i = 1, 2, 3) which is coupled to a gauge field living in a 3d subspace (or a "defect"): We assume that B i is in adjoint representation and depends on all 4 coordinates while A a i depends only on 3 coordinates x i . Here we can not have ∂ 4 in the ijk term as otherwise this term vanishes, so we need to introduce a mass parameter m to balance the dimensions. The action is invariant under the local symmetry (with U = U (x i ), cf. (2.17)) Integrating out the B i -field one should then get a gauge-invariant effective action Γ(A) depending on the 3d field A i . Since the classical action is not parity-invariant, Γ may contain a non-local P-odd part. The analogs of the propagator (4.5) and the vertex (4.6) linear in A here are (p 2 = p i p i ) P ab ij = δ ab P ij (p s , p 4 ) = The one-loop two-point function of the external A a i -field appearing in the A 2 term of the effective action may be written as (k s is an external 3-momentum) iΠ ab ij (k s ) = dp 4 2π d d p (2π) d V cda mni P de nr (p s , p 4 ) V ef b rqj P f c qm (p s + k s , p 4 ) = iδ ab Π ij (k s ) , iΠ ij (k s ) = −m 2 C 2 imn jpr dp 4 2π d d p (2π) d P nr (p i , p 4 )P qm (p i + k i , p 4 ) . In contrast to the 6d case in (4.7)-(4.11), here the A 2 term in the effective action is UV finite and contains two 3d gauge-invariant non-local structures: P-even one F ij ( m 2 ∂ 2 ) 1/4 F ij and P-odd one kij A k (m 2 ∂ 2 ) 1/4 F ij .
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