{'fraction_non_alphanumeric': 0.06699368364467141, 'fraction_numerical': 0.01988979975809703, 'mean_word_length': 4.1462655601659755, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Constructing Doubly Self-Dual Chiral p-Form Actions in D=2(p+1) Spacetime Dimensions

A Siegel-type chiral p-form action is proposed in D=2(p+1) spacetime dimensions. The approach we adopt is to realize the symmetric second-rank Lagrange-multiplier field, introduced in Siegel's action, in terms of a normalized multiplication of two (q+1)-form fields with q indices of each field contracted in the even p case, or of two pairs of (q+1)-form fields with q indices of each pair of fields contracted in the odd p case, where the (q+1)-form fields are of external derivatives of one auxiliary q-form field for the former, or a pair of auxiliary q-form fields for the latter. Using this action, it is straightforward to deduce the recently constructed PST action for q equal to zero. It is found that the Siegel-type chiral p-form action with a fixed p (even or odd) is doubly self-dual in D=2(p+1) spacetime dimensions when the auxiliary field(s) is/are also chosen to be of p-form. This result includes PST's as a special case where only the chiral 0-form action is doubly self-dual in D=2 dimensions.

Chiral p-forms exist in the D = 2(p + 1) dimensional spacetime since their external derivatives, i.e., the field strengths, are (p+1)-forms and the Hodge duals as well, which requires the spacetime dimensions to be twice big the form number of the field strengths. The chirality usually means that the field strengths satisfy a self-duality condition. In spacetime with Lorentzian metric signature, self-duality requires the chiral p-forms to be complex if p is odd.
However, the self-duality requires the chiral p-forms to be real if p is even. For this case one p-form potential, A (p) (x), is enough 2 , and its field strength, F (p+1) (A) = dA (p) (x), of course, satisfies the usual self-duality condition, For the sake of conciseness, we combine these two (self-)duality conditions into one general form 3 for both even p and odd p cases, For the main reason why chiral p-forms have received so much attention, we may summarize that they appear in various theoretical models that relate to superstring theories, and reflect especially the existence of a variety of important dualities that connect these theories among one another. 1 If A a(p) (x) are treated as an O(2) doublet and an extended dualization is defined, the extended duals of F a(p+1) (A), if we choose −ǫ ab * F b(p+1) (A), satisfy a self-duality condition. See, for example, Ref. [2] for details. 2 We do not introduce two p-form potentials that constitute a doublet adopted in Ref. [2] because that treatment is not economical for the even p case. 3 For the definition of Γ ab , see eq.(10).
One has to envisage the two basic problems in a Lagrangian description of chiral pforms: one is the consistent quantization and the other is the harmonic combination of manifest duality and spacetime covariance, since the equation of motion of a chiral p-form, i.e., the self-duality condition, is first order with respect to the derivatives of space and time. In order to solve these problems, some non-manifestly spacetime covariant [1,[3][4][5][6][7][8] and manifestly spacetime covariant [9][10][11][12] models have been proposed. It is remarkable that these chiral p-form models have close relationships among one another, especially various dualities that have been demonstrated in detail from the points of view of both configuration [12][13][14][15] and momentum [16,17] spaces.
The recently constructed PST action [12], motivated by the SS action [8] together with the idea [18] of spacetime covariance in it, has become a successful model in putting both the manifest spacetime and duality symmetries into a chiral p-form action. The characteristic of this model is to introduce a Lagrangian-multiplier term in a non-polynomial way, or more precisely, in terms of a normalized multiplication of two vector (1-form) fields that are "external" derivatives of one auxiliary scalar (0-form) field. Though there is the merit of connecting the Zwanziger [1] with SS actions or with others through various dualities, it is quite unsatisfactory that the model lacks of [13] double self-duality in higher (than two) dimensions. The double self-duality means that the action of chiral p-forms is self-dual with respect not only to dualization of chiral fields, but also to dualization of auxiliary fields.
We may ask whether the double self-duality does or does not appear in higher (than two) dimensions.
Analysing in detail Siegel's action 4 [9] and its generalization [17] in D = 2(p + 1) (p even) dimensions, we immediately find the relation between the Siegel and the PST actions.
That is, the former becomes the latter if the symmetric second-rank Lagrangian-multiplier is realized in terms of the approach mentioned above. In fact, this discovery relates closely to our question. As we have proven, if the Lagrangian-multiplier is realized in terms of higher than zero form auxiliary fields, the double self-duality can occur in higher (than two) dimensions. In this way, we generalize PST's result deduced by the assumption of introducing one and the same auxiliary scalar field in any even dimensions.
It is the aim of this paper to construct a doubly self-dual chiral p-form action in D = 2(p + 1) spacetime dimensions. The paper is arranged as follows. In the next section a Siegel-type chiral p-form action is proposed in D = 2(p + 1) dimensions. The method we utilize is to realize the symmetric second-rank Lagrange-multiplier field of Siegel's action in terms of a normalized multiplication of two (q+1)-form fields with q indices of each field contracted for the even p case 5 , or of two pairs of (q+1)-form fields with q indices of each pair of fields contracted for the odd p case, where the (q+1)-form fields are of external derivatives of one auxiliary q-form field for the former, or of one pair of auxiliary q-form fields for the latter. In section 3 we investigate duality properties of this action with respect to both the dualization of chiral p-form fields and the dualization of auxiliary q-form fields, and make search for the condition when the double self-duality appears. Finally section 4 is devoted to a conclusion.

Siegel-type chiral p-form action in D=2(p+1) dimensions
We begin with Siegel's action in D = 2(p + 1) (p even) dimensions [17], which is called the self-dual tensor in Refs. [12,13], is the difference of the field strength and its Hodge dual. Note that the Lagrange-multiplier, λ µν (x), is in general a symmetric second-rank tensor field. We may introduce a real auxiliary q-form field, Y µ 1 ···µq (x), and 5 Zero is included in the even p case throughout this paper.
realize λ µν (x) in some sense in terms of a normalized multiplication of two (q+1)-form fields with q indices of each field contracted, where the (q+1)-form field, T µ 1 ···µ q+1 (x), is defined as the external derivative of the q-form field, i.e., where Here we mention that q takes numbers 0, 1, · · · , D−2 = 2p for a fixed p (even or odd) because dimensions. For details, we refer to the next section. Substituting eq.(4) into eq. (3), one obtains a Siegel-type action for chiral p-forms in D = 2(p + 1) (p even) dimensions, When p is odd, we may introduce a pair of real p-form fields, A a µ 1 ···µp (x), a = 1, 2, as explained in section 1. Correspondingly, we also introduce a pair of real auxiliary q- Therefore, in accordance with eq.(6), we conclude that the Siegel-type action that combines both even p and odd p cases takes the form, where and ∆ = 1, for even p, 2, for odd p, for even p, ǫ ab , a, b = 1, 2, and ǫ 12 = −ǫ 21 = 1, for odd p.
In addition, one obtains from eq.(8) that for the both cases the field strength differences, or the so-called self-dual tensors, satisfy the relations which will be useful in calculations in the next section. These relations are the generalization of the special forms discussed in D = 2, 4, 6 dimensions [12,13].
We point out that in the even p case the Siegel-type action, eq.(7) or eq. (6)

Dualization of the chiral p-form fields
By introducing two (p even) or two pairs of (p odd) independent (p+1)-form fields, F a µ 1 ···µ p+1 and G b µ 1 ···µ p+1 , we construct a new action to replace S ST , where F aµ 1 ···µ p+1 has the same definition as that of F aµ 1 ···µ p+1 (A) in eq. (8), but here it is not dealt with as a functional of A a µ 1 ···µp (x). Variation of S ′ ST with respect to G aµ 1 ···µ p+1 gives which, when substituted into S ′ ST , yields the classical equivalence between the two actions, S ST and S ′ ST . Furthermore, variation of S ′ ST with respect to F a µ 1 ···µ p+1 leads to the expression of G aµ 1 ···µ p+1 in terms of F aµ 1 ···µ p+1 and other fields, If one defines another (p even) or another pair of (p odd) field strength difference(s) that is/are relevant to G aµ 1 ···µ p+1 , and substitutes eq.(14) into eq.(15), one then establishes the relation(s) between the two (p even) or two pairs of (p odd) field strength differences, One may note that such a relationship exists in various chiral p-form actions, which has been pointed out in both configuration [14] and momentum [17] spaces. With the aid of eq.(16), one can easily obtain from eq.(14) F aµ 1 ···µ p+1 expressed in terms of G aµ 1 ···µ p+1 and other fields, We can check from eq.(14) that when the self-duality condition is satisfied, i.e., F aµ 1 ···µ p+1 = 0, which is also called an "on mass shell" condition, F aµ 1 ···µ p+1 and G bµ 1 ···µ p+1 relate with a duality, Substituting eq.(17) into eq.(12) and dropping a total derivative term, we obtain the dual Variation of eq.(19) with respect to A a µ 1 ···µp (x) gives ∂ µ G aµµ 1 ···µp = 0, whose solution has to be where B a µ 1 ···µp (x) is one (p even) or are a pair of (p odd) arbitrary p-form field(s) we introduce. Substituting eq.(20) into eq. (19), one finally obtains the dual action in terms of B a µ 1 ···µp (x) and other fields, This action, eq.(21), has the same form as the original one, eq.(7), only with the replacement of A a µ 1 ···µp (x) by B a µ 1 ···µp (x). As analysed above, A a µ 1 ···µp (x) and B b µ 1 ···µp (x) coincide with each other up to a constant when the self-duality condition is imposed. Consequently, the Siegel-type chiral p-form action is self-dual with respect to A a µ 1 ···µp (x) − B b µ 1 ···µp (x) dualization given by eqs. (13), (18) and (20).
In order for the dual of eq.(23) to possess self-duality, or putting it differently, in order for eq.(22) to be self-dual with respect to dualization of the auxiliary fields, Y a µ 1 ···µq (x) and That is, the auxiliary fields have to be p-form. This is the condition that the double self-duality happens.
With eq.(29) one obtains from eq.(26) the dual relation between V aµ 1 ···µ p+1 (x) and where the proportionality coefficient, which is not explicitly shown, is in general a functional of U a µ 1 ···µ p+1 (x) and F b µ 1 ···µ p+1 (A), but with no indices left free. By using eq.(30), one can prove the crucial relation, where the identity, which can be deduced from eq.(11), was used. We note that it is a hard job to determine the coefficient in eq.(30) even for the special cases of D = 2, 4, 6 dimensions as can be seen in Ref. [13]. However, it is interesting that we are not obliged to determine the concrete form of the coefficient by following our line of analyses shown from eq.(25) to eq.(32).
Substituting eqs.(26), (29), (31) and the special case of eq.(28) with q = p into eq.(23) and using eq.(32) again, one finally obtains the dual action in terms of the dual fields, As expected, the Siegel-type action with auxiliary p-form fields, Y a µ 1 ···µp (x), possesses selfduality with respect to dualization of the auxiliary fields given by eqs.(24), (28), (29) and (30). Note that the dualization of the auxiliary fields is "off mass shell," which is different from the dualization of the chiral p-form fields. This property is obviously shown in eq.(25). We mention that the Siegel-type action is not self-dual with respect to dualization of auxiliary fields if the condition, eq.(29), is not satisfied.

Conclusion
We have constructed the Siegel-type chiral p-form action by realizing the symmetric secondrank Lagrange-multiplier field in terms of a normalized multiplication of two (q+1)-form fields with q indices of each field contracted in the even p case, or of two pairs of (q+1)-form fields with q indices of each pair of fields contracted in the odd p case, where the (q+1)-form fields are of external derivatives of one auxiliary q-form field of the former, or of a pair of auxiliary q-form fields of the latter. From this action, one can deduce the PST action in D = 2 (p = 0) and D = 6 (p = 2) dimensions simply by letting q equal to zero, or the PST action in D = 4 (p = 1) dimensions by introducing only one auxiliary 0-form field. This means that the PST action can be treated as a special example of the Siegel-type action.
It is known [12][13][14][15] that the PST action originates from the SS action that is dual to the Zwanziger action and that it has close relations with others. For instance, on the one hand, it reduces to the non-manifestly covariant FJ action [5,6] provided appropriate gauge fixing conditions are chosen, on the other hand, it becomes the MWY action [11] if one gets rid of the PST action's non-polynomiality and eliminates its scalar auxiliary field at the price of introducing polynomially auxiliary (p+1)-forms, or, vice versa, if one consistently truncates the MWY action's infinite tail and puts at its end the auxiliary scalar field. Consequently, our result shows that Siegel's action is in some sense the source of chiral p-form actions.
We have shown that the Siegel-type chiral p-form action is self-dual with respect to dualization of chiral p-form fields when q assumes any of the numbers 0, 1, · · · , 2p, but it is self-dual with respect to dualization of auxiliary q-form fields only when q equals p, for a fixed p in D = 2(p + 1) spacetime dimensions. As a consequence, q = p is the condition under which the double self-duality appears. This result includes PST's as a special case where only the chiral 0-form action is doubly self-dual in D = 2 dimensions.
Finally, comment on the odd p case.
(i) The Siegel-type action is still self-dual with respect to dualization of chiral p-form fields even if only one auxiliary q-form field is introduced. In fact, this self-duality exists without any relation to concrete realization of the Lagrange-multiplier field.
(ii) Regarding the self-duality with respect to dualization of auxiliary fields, we emphasize that it is not enough to merely introduce one auxiliary p-form field. The reason is that no duality relation between V µ 1 ···µ p+1 (x) and U µ 1 ···µ p+1 (x), like eq.(30), could be derived from eq.(26) with q = p if there were only one auxiliary p-form field.