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Dirac Branes for Dirichlet Branes: Supergravity Actions

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Dirac Branes for Dirichlet Branes: Supergravity Actions PHYSICAL REVIEW D 102, 046017 (2020) Dirac branes for Dirichlet branes: Supergravity actions Andrew R. Frey * Department of Physics and Winnipeg Institute for Theoretical Physics, University of Winnipeg, 515 Portage Avenue, Winnipeg, Manitoba R3B 2E9, Canada (Received 26 May 2020; accepted 5 August 2020; published 27 August 2020) Nontrivial Bianchi identities with local magnetic sources are solved by recognizing that gauge potentials are sections rather than globally defined functions, but properly accounting for the source degrees of freedom requires a modification of the field strength. Following work by Teitelboim and by Cariglia and Lechner, we extend Dirac’s string formalism for monopoles to D-branes in type IIA and IIB string theory. We give novel derivations of brane-induced Chern-Simons terms in the supergravity actions, including a prescription for integrating over potentials in the presence of magnetic sources. We give a noncovariant formulation of the IIB theory, keeping only the independent degrees of freedom of the self-dual 4-form potential. Finally, it is well known that D8-branes source the mass parameter of IIA supergravity; we show that the additional couplings of the massive IIA supergravity, including on other D-brane worldvolumes, are a consequence of the corresponding Dirac branes. DOI: 10.1103/PhysRevD.102.046017 I. INTRODUCTION of freedom by introducing dual field strengths and corre- sponding potentials. In this so-called “democratic” formal- In modern mathematical treatments, we recognize the ism, the magnetic sources enter in the equations of motion vector potential of a gauge theory not as a globally defined (EOM) for the dual potentials [3].1 The extra degrees function but as the section of a gauge fiber bundle [1,2].In of freedom are then removed by enforcing duality con- somewhat more pedestrian terms, the vector potential can ditions F − −2 ¼⋆F 2 at the level of the EOM. In the be defined as different vector-valued functions in different D p pþ democratic formalism, the action for magnetic charges coordinate patches of spacetime as long as the distinct includes the same current-potential coupling as for electric vector potentials are related by a gauge transformation charges, so the EOM of the magnetic charges includes the on the overlap of the coordinate patches (we will refer to dual field strengths. Nontrivial Bianchi identities are this as “gauge patching” of the vector potential). Gauge enforced by the duality conditions. As a result, democratic patching allows the description, for example, of a constant formalisms still require gauge patching around magnetic magnetic field strength on a torus (the distinct patches sources. cover different unit cells) or the field of a magnetic Because the gauge transformations in the transition monopole (where the Bianchi identity dF2 ≠ 0 can have regions between gauge patches are part of the definition no globally defined solution). Analogs of both these of the potentials and also depend on the dynamical mag- examples for higher-rank form potentials are important netic currents, the gauge potentials are not independent in string theory as harmonic background flux in compacti- degrees of freedom—they have a hidden dependence on fications and higher-dimensional D-branes (and NS5- the magnetic brane degrees of freedom which should branes) that carry magnetic charges for the fundamental be considered explicit in the language of calculus of potentials. The coupling between the magnetic current and variations. In quantum mechanical language, we need to the potential is implicit in the patching and does not appear separate the brane and gauge degrees of freedom to serve as in the action for the magnetic charge. integration variables in the path integral. An alternative that displays the coupling of magnetic Interestingly, Dirac [6] provided a solution in his early sources explicitly is to double the number of gauge degrees work on magnetic monopoles,2 which Teitelboim [9], *[email protected] 1Including auxiliary fields to enforce duality constraints, the Published by the American Physical Society under the terms of IIA and IIB supergravities are given in a democratic formalism in the Creative Commons Attribution 4.0 International license. [4,5] respectively. Further distribution of this work must maintain attribution to 2In fact, for monopoles, [7,8] showed that Dirac’s formalism is the author(s) and the published article’s title, journal citation, equivalent to defining potentials as sections in part by showing and DOI. Funded by SCOAP3. that some gauge transformations move the Dirac string. 2470-0010=2020=102(4)=046017(19) 046017-1 Published by the American Physical Society ANDREW R. FREY PHYS. REV. D 102, 046017 (2020) Bandos et al. [10], and Lechner and co-workers [11–17] quantization. Further, [17] found a correction to the bulk extended to magnetic p-branes. The key idea is as follows: Chern-Simons (CS) term involving Dirac brane currents. We ˜ Consider a field strength Fpþ2 satisfying Bianchi identity will emphasize how these terms are required for consistency ˜ β⋆ β of the EOM and for gauge invariance. A key point in this dFpþ2 ¼ jD−p−3 with a sign convention. Any con- 3 story is that D-brane currents are not conserved due to the served current can be written as ⋆jD−p−3 ¼ d⋆JD−p−2, so ˜ WZ couplings, but CS terms in the EOM and Bianchi a field strength defined as F 2 ≡ dC 1 þ β⋆J − −2 pþ pþ D p identities cancel the anomaly via an inflow argument satisfies the Bianchi identity; in this way, the magnetic [22,23]; we give a detailed accounting of the anomaly coupling appears in the action. There is actually one inflow in a general theory similar to that of the RR forms. additional subtlety when the branes fill all noncompact ⋆ ⋆ The necessity of reference currents also forces us to explain dimensions; jD−p−3 ¼ d JD−p−2 on a compact manifold what it means to integrate over a gauge-patched potential. is inconsistent when there is net local charge, so we must The plan of this paper is as follows. In Sec. II,we ⋆ − ⋆ à ⋆ define instead jD−p−3 jD−p−3 ¼ d JD−p−2, where demonstrate the anomaly inflow argument of [22,23] for a à jD−p−3 is some specified reference current (see [18] for class of theories of form potentials which includes the RR details in the case of magnetic monopoles). Now, Cpþ1 is potentials of both type II supergravities. We pay particular patched around the reference current, so dynamics of attention to how the inflow argument requires specific jD−p−3 do not affect the potential. The alert reader may relations between various conventional coefficients in the note that a Gauss law constraint means that the net charge EOM and Bianchi identity and confirm the consistency of must vanish on a compact manifold, but in string theory the Dirac brane current with the inflow. Then we consider charge may dissolve in background flux, so the net charge an action principle for the generalized theory of Sec. II and of local objects need not vanish. Reference currents are introduce the modified CS term in Sec. III through a novel necessary to account for this fact. derivation. In Sec. IV, preliminary to our discussion of the There are a variety of choices for the form J for a given supergravity actions, we find a prescription for integrating magnetically charged p-brane with worldvolume M potentials that are gauge patched around magnetic sources, and current jM. As described by [9], we can consider a focusing on RR potentials in the 10D supergravities. We “Dirac (p þ 1)-brane” with worldvolume N of boundary then give a novel derivation of the new terms in the type IIB à ∂N ¼ M − M . Then JN , the current of the Dirac brane, supergravity action that were first described in [17] in satisfies d⋆JN ¼ ⋆ðjM − jMà Þ. As we will see below, both Sec. V. We also eliminate redundant degrees of freedom in the brane and Dirac brane currents are delta-function the self-dual 4-form potential, leading to a noncovariant supported. A less singular option for J used by [13–17] action for the RR fields and discuss gauge invariance. is given by the Chern kernel [19,20], which diverges only Finally, in Sec. VI, we derive the IIA supergravity action as a power law near the current jM. In the following, we including the Romans mass term [24] and D-branes for the will mostly remain agnostic about the nature of J, as our first time. It has long been known that D8-branes source the results are independent of this choice, but we will often use Romans mass, and we show for the first time how the the language of Dirac branes to be concrete and refer to J as corresponding Dirac 9-brane currents reproduce the addi- the Dirac brane current as shorthand. It is also worth noting tional couplings of the massive IIA supergravity. We also that, even fixing to Dirac branes or Chern kernels, J is propose that additional WZ couplings on D-branes in the arbitrary up to its coderivative, but the field strength F˜ is massive theory [25,26] are a consequence of the Dirac invariant. brane currents and conjecture the presence of other new Our goal is to extend this formalism to D-branes in the WZ couplings on type IIA D-branes. We conclude with a IIA and IIB string theories, writing these theories in terms brief discussion of future directions and give our conven- tions and some auxiliary results in the appendixes. A of Ramond-Ramond (RR) potentials C 1 for p ≤ 3.
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