Quasi-phase-matched up- and down-conversion in periodically poled layered
semiconductors

Chiara Trovatello1,2,∗, Carino Ferrante3, Birui Yang4, Josip Bajo5, Benjamin
Braun5, Xinyi Xu1, Zhi Hao Peng1, Philipp K. Jenke5, Andrew Ye6, Milan Delor7,
D. N. Basov4, Jiwoong Park6,8,9, Philip Walther5,10, Lee A. Rozema5, Cory
Dean4, Andrea Marini3,11, Giulio Cerullo2,∗ and P. James Schuck1,∗

1Department of Mechanical Engineering, Columbia University, New York, NY
10027, USA
2Dipartimento di Fisica, Politecnico di Milano, Piazza L. da Vinci 32, I-20133
Milano, Italy
3CNR-SPIN, c/o Dip.to di Scienze Fisiche e Chimiche, Via Vetoio, Coppito
(L’Aquila) 67100, Italy
4Department of Physics, Columbia University, New York, NY 10027, USA
5University of Vienna, Faculty of Physics, Vienna Center for Quantum Science
and Technology (VCQ) & Research Network Quantum Aspects of Space Time (TURIS),
1090 Vienna, Austria
6Pritzker School of Molecular Engineering, University of Chicago, Chicago, IL,
USA
7Department of Chemistry, Columbia University, New York, NY 10027, USA
8James Franck Institute, University of Chicago, Chicago, IL, USA
9Department of Chemistry, University of Chicago, Chicago, IL, USA
10Christian Doppler Laboratory for Photonic Quantum Computer, Faculty of
Physics, University of Vienna, 1090 Vienna, Austria
11Department of Physical and Chemical Sciences, University of L’Aquila, Via
Vetoio, 67100 L’Aquila, Italy

∗Corresponding authors. Email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
Nonlinear optics lies at the heart of classical and quantum light
generationBoyd2020 ; Elshaari2020 ; Boes2023 . The invention of periodic
poling revolutionized nonlinear optics and its commercial applications by
enabling robust quasi-phase-matching in crystals such as lithium
niobateBoes2023 ; Fejer1992 . However, reaching useful frequency conversion
efficiencies requires macroscopic dimensions, limiting further technology
development and integration. Here we realize a periodically poled van der
Waals semiconductor (3R-MoS2)Zhao2016 ; Shi2017 ; Xu2022 . Due to its
exceptional nonlinearity, we achieve macroscopic frequency conversion
efficiency over a microscopic thickness of only
$1.2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$, $10-100\times$ thinner
than current systems with similar performancesBoes2023 ; wang2018 ;
suntsov2021watt . Due to unique intrinsic cavity effects, the thickness-
dependent quasi-phase-matched second harmonic signal surpasses the usual
quadratic enhancement by 50%. Further, we report the broadband generation of
photon pairs at telecom wavelengths via quasi-phase-matched spontaneous
parametric down-conversion. This work opens the new and unexplored field of
phase-matched nonlinear optics with microscopic van der Waals crystals,
unlocking applications that require simple, ultra-compact technologies such as
on-chip entangled photon-pair sources for integrated quantum circuitry and
sensing Elshaari2020 ; Boes2023 ; Santiago-Cruz2022 .

The inherent nonlinear response of matter to external electromagnetic stimuli
allows photons of different frequencies incident on a material to interact
with each other, enabling a myriad of photonic applications such as frequency
conversion and the generation of non-classical states of light. Particularly
interesting are second-order nonlinear processes, which occur in non-
centrosymmetric media with $\chi^{(2)}\neq 0$ and are used to produce new
light frequencies e.g., in second harmonic generation (SHG), sum and
difference frequency generation, and entangled photon pairs in spontaneous
parametric down-conversion (SPDC)Boyd2020 ; Shih2016 .

Efficient nonlinear frequency conversion is achieved by fulfilling the so-
called phase matching (PM) condition, which implies a zero wave-vector
mismatch - i.e., momentum conservation - for the interacting waves ($\Delta
k=0$). For both SHG and SPDC, $\Delta
k=k_{2\omega}-2k_{\omega}=\frac{2\omega}{c}(n_{2\omega}-n_{\omega})$ and PM
requires matching the refractive indexes of the fundamental wavelength (FW),
$n_{\omega}$, and of the second harmonic (SH), $n_{2\omega}$. In the absence
of PM, the conversion efficiency reaches its maximum for a propagation
distance corresponding to the coherence length Lc = $\pi$/$\Delta$k and then
oscillates with period 2LcBoyd2020 . Birefringent phase-matching (BPM)
exploits the optical anisotropy of non-centrosymmetric nonlinear crystals by
adjusting the propagation direction inside the crystal so that the fields at
$\omega$ and $2\omega$ with different polarizations experience the same
refractive index. For SHG with BPM, the SH intensity $I_{2\omega}$ grows
quadratically with the thickness $L$ of the nonlinear medium. While BPM is
simple and effective, it can only be applied to a limited number of crystals,
such as the prototypical $\beta$-barium borate (BBO), which display moderate
$\chi^{(2)}$ values of a few pm/VBoyd2020 .

There are many crystals with high nonlinearity but low anisotropy, e.g.,
gallium arsenide (GaAs), for which BPM is not achievable. An alternative to
BPM is quasi-phase-matching (QPM), which introduces periodic phase shifts of
$\pi$ between the fields at the FW and the SH at every coherence length
$L_{c}$, restoring the proper phase relationship and the quadratic growth of
the SH intensity with propagation lengthFejer1992 ; Myers1995 .

Such a phase shift can be obtained by periodic inversion of the
crystallographic orientation during material growth via molecular beam
epitaxyKoh1998 ; Eyres2001 ; Grisard2012 , or by cleaving the nonlinear medium
slabs along the different crystal planes and diffusion-bonding them into
monolithic stacksGordon1993 ; Grisard2012 ; Tanimoto2021 . In ferroelectric
crystals like lithium niobate (LN)Feng1980 ; Feisst1985 , lithium tantalate
Matsumoto1991 or potassium titanyl phosphate VanDerPoel1990 , QPM can be
achieved by periodic poling, i.e., by periodically inverting the sign of the
nonlinear coefficient $\chi^{(2)}$ upon high-voltage switching of
ferroelectric domains following lithographic patterning of the
electrodesHum2007 . The spatial modulation of $\chi^{(2)}$, which follows the
electrode spacing, is the so-called poling period, and it determines the
wavelengths for which a certain nonlinear process can be phase matched. For
LN, the typical values of the poling periods are between 5 and
$50\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ and the modulation can be
extended over crystal thicknesses from millimeters to centimetersHum2007 .

The invention of QPM was a breakthrough in nonlinear optics because it enabled
the use of nonlinear crystals with large $\chi^{(2)}$ of the order of 10-20
pm/V, for which BPM cannot be achievedBoes2023 . Periodically poled LN (PPLN)
crystals provide the highest conversion efficiency for waveguided SHG
($I_{2\omega}/I_{\omega}\approx 70\%$)wang2018 ; suntsov2021watt , optical
parametric amplifiers and oscillatorsMyers1997 ; Yang1999 ; lu2021 ;
Ledezma2023 and entangled photon sourcesSolntsev2018 ; Zhang2021 . Along with
their high conversion efficiencies, standard nonlinear crystals like PPLN can
be directly fabricated on silicon-based optical circuits, though their
millimeter-to-centimeter thickness limits the number of devices that can be
integrated on a single chip. This has spurred recent advances in the
development of nonlinear metasurfacesKrasnok2018 ; Wang2022 ; Fedotova2022 ;
Santiago-Cruz2022 ; Neshev2023 , thin filmsZhu2021 and
nanowaveguidesJankowski2023 , which are now leading a paradigm shift towards
more miniaturized, and more tunable, nonlinear optical device platforms.

The rise of quantum and two-dimensional (2D) material scienceGuo2023 ; Wu2017
; Mueller2018 ; Du2023 ; Ma2023 ; Sheffer2023 has inspired research on less
traditional classes of optical crystals including semiconducting transition
metal dichalcogenides (TMDs), e.g., MoS2, MoSe2, WS2 and WSe2. These are
layered materials comprised of crystalline sheets with strong in-plane
covalent bonds but weak out-of-plane van der Waals interactions. Starting from
the first reports of SHGMalard2013 ; Li2013 ; Wang2015shg , their exceptional
optical nonlinearities ($\chi^{(2)}=100-1000~{}$pm/V), 100-1000x higher than
standard bulk crystals due in part to favourable quantum geometries, have
recently been exploited to demonstrate nonlinear effects at the ultimate thin-
ness limit in single-layer TMDsAutere2018 ; Wen2019 ; Liu2020 ; Dogadov2022 .
However, in such cases, the overall conversion efficiency is still limited by
the sub-nm propagation lengthtrovatello2021optical . In contrast to the most
common 2H crystallographic phase, which is centrosymmetric in the bulk
($\chi^{(2)}=0$), the 3R polytype is non-centrosymmetric ($\chi^{(2)}\neq 0$)
and can be used to boost the conversion efficiency of second-order nonlinear
optical processes by increasing the number of layers $N$ in the TMD sample.
Using thin 3R-MoS2 flakes (1-6 layers), it has been shown that the SH
intensity grows quadratically with the number of layers, i.e., with the
thickness of the nonlinear medium ($I_{\textrm{SHG}}\propto N^{2}$)Zhao2016 .

However, when the number of layers is increased to $\sim 200-300$, the wave
vector mismatch $\Delta k$ results in an inevitable deviation from the $N^{2}$
enhancementShi2017 ; Xu2022 , and the SH oscillates with the sample thickness,
with a semi-period equal to the coherence length $L_{c}$Boyd2020 . Recently,
the coherence length in 3R-MoS2 has been measured at telecom wavelengthsXu2022
. At $L_{c}$ ($\sim$500\text{\,}\mathrm{n}\mathrm{m}$$) the single-pass SHG
efficiency is $1.5\times 10^{4}$ higher than for a monolayer
($I_{2\omega}/I_{\omega}\approx 10^{-6}$), resulting in the highest conversion
efficiency per unit length reported to date for a transparent materialXu2022 .
Further increasing this value to achieve macroscopic nonlinear conversion
efficiencies, while still preserving the micrometer thickness of TMD crystals,
remains an open challenge.

Here we bridge this gap using a non-centrosymmetric van der Waals crystal,
3R-MoS2, to achieve QPM in periodically-poled transition metal dichalcogenides
(PPTMDs). We flip the sign of the optical nonlinearity $\chi^{(2)}$ by
stacking consecutive slabs with opposite dipole orientation. The large
nonlinearity of TMDs combined with QPM unlocks single-pass conversion
efficiencies for SHG higher than $10^{-4}$ at telecom wavelengths and enables
efficient photon-pair generation via SPDC over a microscopic thickness of only
$1.2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ – at least two orders of
magnitude thinner than BBO and PPLN crystals with similar performances. We
observe unique cavity effects prompted by internal reflections of both FW and
SH, which increase the conversion efficiency by an additional 50% over
standard QPM. Finally, we report the broadband generation of photon pairs at
telecom wavelengths via quasi-phase-matched SPDC.

Figure 1: Periodically poled transition metal dichalcogenides (PPTMDs). (a-b)
Micrograph of the 3R-MoS2 flake (a) before and (b) after patterning (electron
beam lithography, etching). (c) Stacking procedure. Slab 1 is first
transferred on a $500\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$-thick
SiO2 transparent substrate. Slab 2 is twisted by
$180\text{\,}\mathrm{\SIUnitSymbolDegree}$ and released on top of slab 1. Slab
3 is transferred on top of slab 2. Finally slab 4 is twisted by
$180\text{\,}\mathrm{\SIUnitSymbolDegree}$ and released on top of the 3
stacked portions. (d) Micrographs of each stacking step. Scale bar
$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$.

The 3R-MoS2 crystals are mechanically exfoliated from a commercial bulk
3R-MoS2 crystal (HQ graphene) grown by chemical vapor transport on a SiO2/Si
substrate. Characterization of the bulk crystal by energy dispersive X-ray
analysis and X-Ray diffraction is provided in Ref.Xu2022 . The thickness of
the exfoliated flakes is measured using atomic force microscopy (Supplementary
Note 4). To fabricate the PPTMD, we select a large flake with lateral size of
$\approx$100\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ and thickness
$\approx$300\text{\,}\mathrm{n}\mathrm{m}$$, shown in Fig. 1a. We pattern the
flake using electron beam lithography and reactive ion etching into smaller
portions (Fig. 1b, Supplementary Note 1 for details). By cutting the different
slabs out of a single flake, we ensure that all areas have identical thickness
and the same macroscopic dipole orientation. Our design and fabrication
methodology thus bypasses the need for angle-resolved SHG for crystal
orientation characterization.

We choose a flake with thickness of $L_{\rm
f}\approx$300\text{\,}\mathrm{n}\mathrm{m}$$, close to the coherence length
$L_{\rm c}$ measured for a FW at 1450 nm (Supplementary Note 7). The largest
portion of the flake (area 1) is transferred onto a
$500\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$-thick transparent fused
silica (SiO2) substrate, and the other portions (area 2, 3, 4) are
individually stacked on top of each other by keeping an interlayer twist angle
of $180^{\circ}$ArxivTwistPM2023 (symmetrically equivalent to $60^{\circ}$ and
$300^{\circ}$ twist angles) to flip the sign of $\chi^{(2)}$ at each coherence
length (see Fig. 1c), resulting in an overall thickness of
$\approx$1.2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$$ (Supplementary
Note 4). The microscope images of the periodically poled crystal at each
stacking step are shown in Fig. 1d.

Figure 2: Pump wavelength-dependent second-harmonic generation from PPTMDs.
(a) Transmission microscope. The PPTMD is excited by the FW from the back side
of the SiO2 substrate with a 40x objective. The SH is collected by a 50x
objective. (b) Pump wavelength-dependent normalized SH maps of the PPTMD,
along with the SH spectra and the SHG enhancement factor with respect to the
bare slab. The error bar represents the standard deviation of the SH emission
calculated over each corresponding slab area. Scale bar
$10\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$.

To characterize the nonlinear response of the PPTMD, we use a custom-built
transmission microscope (Fig. 2a), illuminated by an optical parametric
oscillator (OPO, Coherent Chameleon), emitting
$150\text{\,}\mathrm{f}\mathrm{s}$ pulses at
$80\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$, tunable in the NIR (1000-1600 nm)
wavelength range. We use a long working-distance 40x reflective objective with
numerical aperture (NA) 0.5 to focus the FW on the sample from the backside of
the substrate. The SH is collected by a 50x objective with a 0.95 NA and
directed onto a silicon CCD camera (Supplementary Note 2). The FW is tuned
from 1430 nm to 1580 nm, and the power is kept fixed at
$0.5\text{\,}\mathrm{m}\mathrm{W}$ (peak power
$1.17\text{\,}\mathrm{G}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$). Figure
2b shows the pump wavelength dependent normalized SHG maps, along with the
corresponding SHG spectrum. For each pump wavelength, we extract the SH
enhancement factor, defined as the SH emission from the regions with 2, 3 and
4 slabs normalized to the SH emission of 1 slab, shown in Fig. 2b. The peak of
the SH enhancement is obtained at FW $=$1460\text{\,}\mathrm{n}\mathrm{m}$$,
at approximately the target operation wavelength for this stack. Compared to a
standard first-order QPM ($\chi^{(2)}$ flipped in sign at each coherence
length) that predicts a quadratic enhancement of 4, 9 and 16 for 2, 3 and 4
slabs, respectively, the peak enhancement that we obtain is not monotonically
increasing with the slab number at each pump wavelength, and it is almost 50%
higher in the portion of the stack with 4 slabs. This effect can be explained
by considering the modulation of the optical properties caused by interference
inside the PPTMD. Multiple reflections of FW and SH radiation at the
interfaces between the TMD and the air and glass substrate, acting as optical
microcavities, provide nonlinear cavity enhancement effects.

Figure 3: Theoretical simulations of quasi-phase matching in PPTMDs. (a)
Schematic of simulated PPTMD structure. Multiple reflections of FW and SH
fields are shown in orange and purple, respectively. The inversion of the
electric field at each coherence length is induced by $\chi^{2}$ flipped in
sign via twist-controlled stacking of consecutive slabs. (b) SH enhancement
for different wavelengths and number of slabs, as reported in Fig. 2. (c)
Corresponding SH enhancement extracted from the model. (d) Quantitative
comparison of the wavelength-dependent enhancement between experiment and
model for different number of slabs, i.e., cross sections of panel (b) and
(c), respectively. (e-f) Calculated SH intensity as a function of slab
thickness and number of slabs at FW = 1450 nm and
$5\text{\,}\mathrm{G}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$ intensity,
in two cases: with interference effects of the FW (e), and without
interference (f). The maximum intensity achievable as a function of the total
number of slabs in the PPTMD is shown on the right, showing that interference
boosts SHG, due to intrinsic cavity-induced enhancements.

To better understand the unconventional QPM regime we observe in PPTMDs, we
analytically model our signal by solving the coupled nonlinear equations
considering the interference of FW and SH fields in the slabs, and the SH
electric field sign inversion for the different poling conditions
(Supplementary Note 6). Specifically, we assume that the SH process does not
affect the intensity of the FW (undepleted-pump approximation) and we apply
the boundary conditions at the entrance and the exit of the TMD slabs to
analytically retrieve the interference effects. Calculating the forward and
backward propagating FW electric fields and considering the poling, we are
able to evaluate the SH in the system. We calculate the second order non
linear polarization in the slabs at $2\omega$ induced by FW propagation, and
insert this term into Maxwell’s equations to extract the SH electric and
magnetic fields. Applying again the boundary condition for the SH, we obtain
the forward emitted SH intensity. The analytical calculation is performed
assuming a normal propagation in the different slabs. A sketch of the modeled
structure is shown in Fig. 3a, in which the interference effects of FW and SH
are depicted.

Figure 3b and 3c show the measurements and the theoretical simulations of the
SH enhancement factor (i.e., SH emission from slab 2, 3 and 4 normalized to
the SH from slab 1) as a function of the SH wavelength and the number of
slabs, respectively. For the theoretical simulations, we use the refractive
index reported in Fig. S2,
$|\chi^{(2)}|=~{}$100\text{\,}\mathrm{p}\mathrm{m}\mathrm{/}\mathrm{V}$$ and a
slab thickness equal to 293 nm (Supplementary Note 6). Figure 3d reports the
comparison between experiments and theory for poled structures with 2, 3 and 4
slabs, demonstrating a very good agreement with the data. The theoretical
model also accurately reproduces the nonlinear response of PPTMDs with
different poling periods (Supplementary Note 8). Data and simulations
emphasize the importance of an appropriate choice of the number of slabs for
each FW wavelength and a given slab thickness, to maximize the enhancement of
SH. In particular, for our 300-nm slab thickness, a configuration with 3 slabs
provides a higher enhancement compared to the poled structure with 4 slabs for
SH wavelengths above 750 nm.

Figure 3e shows the intensity profile of SH for different slab thicknesses and
number of slabs, assuming a FW at 1450 nm,
$5\text{\,}\mathrm{G}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$ peak power
and $|\chi^{(2)}|=$100\text{\,}\mathrm{p}\mathrm{m}\mathrm{/}\mathrm{V}$$. The
results show a maximum efficiency for a $L_{\rm
f}\sim$350\text{\,}\mathrm{n}\mathrm{m}$$, with a strong dependence on the
interference effects of the FW. The optimum poling period is strongly
dependent on the FW. For instance, at 1550 nm the maximum efficiency is
achieved for $L_{\rm f}\sim$550\text{\,}\mathrm{n}\mathrm{m}$$ (see
Supplementary Note 6). We also report the same simulation removing the
interference of the FW in the slab (Fig. 3f), which we achieve by setting the
real part of refractive index of air after the slabs equal to that of the TMD.
In this case, the interference pattern is removed, obtaining a more
homogeneous profile. However, the maximum efficiency is notably reduced (more
than $10\times$ lower). This comparison highlights the key role played by the
intrinsic cavity enhancement effects in PPTMDs.

Figure 4: PPTMD second harmonic conversion efficiency. (a) Picture of the SHG
spot at 530 nm, and close-up of the tunable SH from 400 nm to 625 nm. (b)
Broadband SH conversion efficiency with tunable pump wavelength (1020 nm -
1580 nm) and fixed power 40 mW (peak power
$\approx$90\text{\,}\mathrm{G}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$$).
The grey area is the imaginary part of the refractive index $\kappa$. In the
inset we report the nonlinearity of TMDs
($\chi^{(2)}\approx$100\text{\,}\mathrm{p}\mathrm{m}\mathrm{/}\mathrm{V}$$),
extracted from the conversion efficiency measurements, compared with the one
of BBO and LN crystalsBoyd2020 ; Schiek2012 .

We use the frequency tunable OPO to quantitatively measure the wavelength
dependent conversion efficiency of the PPTMD. The photographs of the broadly
tunable SH spots of our
$1.2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$-thick PPTMD are shown in
Fig. 4a. Using a power meter, we measure the SH power emitted by the sample
excited with tunable pump and a constant power of
$40\text{\,}\mathrm{m}\mathrm{W}$ (peak intensity
$\approx$90\text{\,}\mathrm{G}\mathrm{W}\mathrm{/}\mathrm{c}\mathrm{m}^{2}$$).
The measured conversion efficiency for different excitation wavelengths is
shown in Fig. 4b. The peak of the conversion efficiency ($\approx 10^{-4}$) is
recorded at FW $=$1460\text{\,}\mathrm{n}\mathrm{m}$$), i.e., the target
phase-matched wavelength with the poling period we use. We stress the fact
that this macroscopic conversion efficiency is recorded over a sample
thickness of only $1.2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$. PPTMDs
eclipse the previous thickness-conversion efficiency trade-off curves, and now
show macroscopic efficiencies over microscopic thicknesses. At the QPM
resonance the extracted nonlinearity is
$\chi^{(2)}\approx$100\text{\,}\mathrm{p}\mathrm{m}\mathrm{/}\mathrm{V}$$
(Supplementary Note 10). With higher $\chi^{(2)}$ compared to BBO/PPLNBoyd2020
; Schiek2012 (see inset of Fig. 4b), PPTMDs achieve the same efficiency, but
over approximately $10-100\times$ shorter propagation lengths.

Figure 5: Pump wavelength-dependent SPDC from PPTMDs. (a) Experimental setup.
A frequency-doubled beam from the OPO is used to pump the PPTMD in a
transmission geometry. The generated SPDC photons can be directed to a Hanbury
Brown-Twiss interferometer in one arm for photon-pair correlation
measurements, or to an bireferingent Fourier transform interferometer (TWINS)
for spectral characterization. B1 is a pair of birefringent wedges and B2 is a
birefringent plate with optical axis rotated by 90° with respect to the
wedges. (b) The upper panels show the SPDC spectra for pump wavelengths of
710nm and 800nm, from the 4-slabs (left) and 1-slab (right, multiplied $\times
5$) samples. The lower panels plot color map illustrating the SPDC spectrum
intensity for each pump wavelength from 700 nm to 800 nm, for 4-slabs (left)
and 1-slab (right, multiplied $\times 10$) samples. The inset plots the pump
power dependence of the SPDC signal (depicted by orange dots), and its
corresponding linear fit (black line).

Finally, we use PPTMDs to demonstrate SPDC. For these measurements, we employ
a PPTMD stack with peak enhancement at FW
$=$1420\text{\,}\mathrm{n}\mathrm{m}$$ (Supplementary Note 11). We measure
SPDC in this PPTMD using a transmission geometry, as depicted in Fig. 5a. The
tunable pump beam is generated by frequency doubling the signal beam of an OPO
(Stuttgart Instruments, Alpha). This system provides
$170\text{\,}\mathrm{f}\mathrm{s}$ pulses at an
$80\text{\,}\mathrm{M}\mathrm{H}\mathrm{z}$ repetition rate, with a signal
wavelength ranging from $1400\text{\,}\mathrm{n}\mathrm{m}$ to
$1600\text{\,}\mathrm{n}\mathrm{m}$, resulting in doubled wavelengths from 700
to 800 nm. Aspheric lenses before and after the sample (with matched NA=0.68)
are used to focus the laser beam onto the sample and collect the resulting
SPDC signal. After the PPTMD, the pump beam is filtered out with two hard-
coated long-pass filters at a cutoff wavelength of
$1150\text{\,}\mathrm{n}\mathrm{m}$. The SPDC signal can then either be
coupled directly into single-mode fiber (SMF28, Corning), or directed towards
our setup for spectral characterization. The inset of Fig. 5b shows the
expected linear power dependence of the SPDC signal measured with
superconducting nanowire single-photon detectors (PhotonSpot). We measure the
SPDC spectrum with a bireferingent interferometer, the TWINStwins , which is
inserted in the optical path of the SPDC signal after which the photons (both
the signal and idler) are coupled into multi-mode fiber and detected by an
InGaAs single-photon detector (IDQuantique). The setup is described in more
detail in Supplementary Note 3.

We collect SPDC spectra for different excitation wavelengths, ranging from 700
nm to 800 nm in steps of 10 nm, from a 4-slabs and from a 1-slab sample. Fig.
5b plots the resulting broadband SPDC spectrum from slab 4 (left) and slab 1
(right). As the signal is significantly stronger from the 4-slabs sample, we
use an average pump power of $0.5\text{\,}\mathrm{m}\mathrm{W}$ to acquire the
4-slabs data and $2\text{\,}\mathrm{m}\mathrm{W}$ for the 1-slab data to avoid
detector saturation. To account for this, we divide the 4-slabs spectra by
four in order to fairly compare the spectra (this is based on the linear power
dependence confirmed in the inset). In Fig. 5, we further multiply the 1-slab
data by a factor of five and ten in the individual spectra (upper right panel)
and in the integrated SPDC map (lower right panel), respectively, to allow for
a more direct comparison of the spectral features.

In additional to enhanced efficiency, we note several other features in these
spectra. First, we observe a broadband background, which appears to be limited
by our detector response. This is to be expected as the SPDC process works
around the degeneracy condition, for which the group velocities of the signal
and idler waves are matched, resulting in a broad phase-matching
bandwidthjopt2016 . The peak corresponding to the degeneracy condition, which
shifts with pump wavelength, is shown as a dashed line in the lower panel of
the 4-slabs data. Additionally, we plot a vertical dashed line at
$1420\text{\,}\mathrm{n}\mathrm{m}$ which corresponds to the QPM peak. Its
intensity, relative to the broadband background, grows as the pump is brought
into resonance. This is more apparent when comparing the spectra in the upper
panel, taken at the resonant wavelength of $710\text{\,}\mathrm{n}\mathrm{m}$
(purple), and the off-resonant spectrum taken at
$800\text{\,}\mathrm{n}\mathrm{m}$ (orange). In addition to the QPM peak, we
see another peak occurring at the degenerate wavelength (twice the pump
wavelength). Because of the less stringent phase-matching conditions, the
features are not as visible in the single slab data on the right as they are
in the 4-slabs data. We additionally perform correlation measurements on the
SPDC signal, measuring a maximum coincidence-to-accidental ratio of $37.3\pm
2.9$ when pumping at a non-resonant wavelength (Supplementary Note 11).

In conclusion, we introduce periodically poled van der Waals layered materials
as a novel nonlinear platform for classical and quantum light generation. By
periodically stacking 3R-MoS2 slabs with identical thickness and an interlayer
twist angle of $180^{\circ}$, we flip the sign of $\chi^{(2)}$ and achieve
quasi-phase matching, restoring the proper phase relationship between
fundamental and second harmonic, and between fundamental and spontaneously
down-converted fields. Due to the large nonlinearity of 3R-TMDs (100-1000
pm/V) we demonstrate a record single-pass second harmonic conversion
efficiency of $10^{-4}$ at telecom wavelengths, over a thickness of only
$1.2\text{\,}\mathrm{\SIUnitSymbolMicro}\mathrm{m}$ (two poling periods).
Moreover, in the phase-matched interaction, we observe nonlinear enhancement
which surpasses by more than 50% the usual quadratic enhancement typically
observed in standard quasi-phase-matched crystals. This is attributed to
cavity effects, which enhance the field overlap inside the periodically poled
structure. Theoretical simulations accurately reproduce the conditions for
such an unconventional quasi-phase matching, and predict the optimal slab
thickness as a function of the FW/SH wavelength (Supplementary Note 6).
Although the sample has been prepared by manual exfoliation and stacking, the
periodic poling process is potentially scalable, as it can be automatized
using a robotic stacking machinemannixye2022 (Supplementary Note 5). PPTMDs
now provide macroscopic nonlinear conversion efficiencies over microscopic
thicknesses, establishing new routes for designing novel nonlinear optical
devices and quantum nanophotonic circuit elementssilva2017 , such as
entangled-photon generators directly embedded on chip.

## DATA AVAILABILITY

All data generated or analysed during this study, which support the plots
within this paper and other findings of this study, are included in this
published article and its Supplementary Information. Source data are provided
with this paper.

## ACKNOWLEDGEMENTS

We thank Benedikt Ursprung for useful discussions.

This work was supported by Programmable Quantum Materials, an Energy Frontier
Research Center funded by the US Department of Energy, Office of Science,
Basic Energy Sciences, under Award DE-SC0019443. C.T. acknowledges the
European Union’s Horizon Europe research and innovation programme under the
Marie Skłodowska-Curie PIONEER HORIZON-MSCA-2021-PF-GF grant agreement No
101066108. C.T. also acknowledges the Optica Foundation and Coherent Inc. for
supporting this research through the Bernard J. Couillaud prize 2022. G.C
acknowledges support by the Progetti di ricerca di Rilevante Interesse
Nazionale (PRIN) of the Italian Ministry of Research 2022HL9PRP Overcoming the
Classical limits of ultRafast spEctroSCopy with ENtangleD phOtons (CRESCENDO).
A.M. acknowledges funding from the European Union—NextGenerationEU under the
Italian Ministry of University and Research (MUR) National Innovation
Ecosystem Grant No. ECS00000041-VITALITY-CUP E13C22001060006, and Progetti di
ricerca di Rilevante Interesse Nazionale (PRIN) of the Italian Ministry of
Research PHOTO (Photonic Terahertz devices based on topological materials) No.
316 2020RPEPNH. A.Y. acknowledges support from the Department of Defense (DoD)
through the National Defense Science and Engineering Graduate (NDSEG)
Fellowship Program. J.P. acknowledges funding from the Air Force Office of
Scientific Research (FA9550-21-1-0323) and the Office of Naval Research
(N000142212841). P.W. acknowledges support from the Austrian Science Fund
(FWF) through BeyondC (F7113) and from AFOSR via FA8655-20-1-7030 (PhoQuGraph)
and FA8655-23-1-7063 (TIQI); and from the Austrian Federal Ministry for
Digital and Economic Affairs, the National Foundation for Research, Technology
and Development and the Christian Doppler Research Association.

### Author contributions

C.T. conceived the experiment and performed the measurements. C.F. and A.M.
developed the theory model. J.B., B.B., C.T. and P.K.J. performed the
spontaneous parametric down-conversion measurements. B.Y., C.T., Z.P., X.X.
and A.Y. prepared the samples. X.X., C.T. and Z.P. built the experimental
setup and performed the morphological characterization of the samples. M.D.,
D.N.B, J.P., L.A.R., P.W., C.D., G.C. and P.J.S. supervised the study. C.T.
wrote the manuscript with input from all authors.

### Competing interests

The authors declare no competing interests.

### Additional information

## References

  * (1)

## REFERENCES

* (2) Boyd, R. W. Nonlinear Optics. Academic Press. (2020).
* (3) Elshaari, A.W., et al. Hybrid integrated quantum photonic circuits. Nat. Photonics 14, 285–298 (2020).
* (4) Boes, A. et al. Lithium niobate photonics: Unlocking the electromagnetic spectrum. Science. 379, eabj4396 (2023).
* (5) Fejer, M. M., Jundt, D. H., Byer, R. L. & Magelh, G. A. Quasi-phase-matched second harmonic generation: tuning and tolerances. IEEE Journal of Quantum Electronics. 28, 2631 – 2654 (1992).
* (6) Zhao, M. et al. Atomically phase-matched second-harmonic generation in a 2D crystal. Light: Science & Applications. 5, e16131–e16131 (2016).
* (7) Shi, J. et al. 3R MoS2 with Broken Inversion Symmetry: A Promising Ultrathin Nonlinear Optical Device. Adv. Mater. 29, 1701486 (2017).
* (8) Xu, X. et al. Towards compact phase-matched and waveguided nonlinear optics in atomically layered semiconductors. Nat. Photon. 16 698–706 (2022).
* (9) Wang, C. et al. Ultrahigh-efficiency wavelength conversion in nanophotonic periodically poled lithium niobate waveguides. Optica. 5, 1438-1441 (2018).
* (10) Suntsov, S., Rüter, C. E., Brüske, D. & Kip, D. Watt-level 775 nm SHG with 70% conversion efficiency and 97% pump depletion in annealed/reverse proton exchanged diced PPLN ridge waveguides. Opt. Express. 29, 11386–11393 (2021).
* (11) Santiago-Cruz, T. et al. Resonant metasurfaces for generating complex quantum states. Science. 377, 991-995 (2022).
* (12) Shih, Y. An Introduction to Quantum Optics Photon and Biphoton Physics. Taylor & Francis. (2016).
* (13) Myers, L. E. et al. Quasi-phase-matched optical parametric oscillators in bulk periodically poled LiNbO3. Journal of the Optical Society of America B. 12, 2102–2116 (1995).
* (14) Koh, S. et al. Sublattice Reversal in GaAs/Si/GaAs (100) Heterostructures by Molecular Beam Epitaxy. Jpn. J. Appl. Phys. 37, L1493 (1998).
* (15) Eyres, L. A. et al. All-epitaxial fabrication of thick, orientation-patterned GaAs films for nonlinear optical frequency conversion. Appl. Phys. Lett. 79, 904–906 (2001).
* (16) Grisard, A. et al. Fabrication and applications of orientation-patterned gallium arsenide for mid-infrared generation. Physica Status Solidi (C). 9, 1651– 1654 (2012).
* (17) Gordon, L. et al. Diffusion-bonded stacked GaAs for quasiphase-matched second-harmonic generation of a carbon dioxide laser. Electronics Letters. 29, (1993).
* (18) Tanimoto, R., Takahashi, Y. & Shoji, I. Quasi-phase-matching stack of 25 GaAs plates with high transmittance for high-power mid-infrared wavelength conversion fabricated by use of room-temperature bonding. Physica Status Solidi (C). 38, (2021).
* (19) Feng, D. et al. Enhancement of second-harmonic generation in LiNbO3 crystals with periodic laminar ferroelectric domains. Appl. Phys. Lett. 37, (1980).
* (20) Feisst, A. & Koidl, P. Current induced periodic ferroelectric domain structures in LiNbO 3 applied for efficient nonlinear optical frequency mixing. Appl. Phys. Lett. 47, (1985).
* (21) Matsumoto, S., Lim, E. J., Hertz, H. M. & Fejer, M. M. Quasiphase-matched second harmonic generation of blue light in electrically periodically-poled lithium tantalate waveguides. Electronics Letters. 27, (1991).
* (22) Van Der Poel, C. J., Bierlein, J. D., Brown, J. B. & Colak, S. Efficient type I blue second-harmonic generation in periodically segmented KTiOPO4 waveguides. Appl. Phys. Lett. 57, 20 (1990).
* (23) Hum, D. S. & Fejer, M. M. Quasi-phasematching. Comptes Rendus Physique. 8, (2007).
* (24) Myers, L. E. & Bosenberg, W. R. Periodically poled lithium niobate and quasi-phase-matched optical parametric oscillators. IEEE Journal of Quantum Electronics. 33, 10 (1997).
* (25) Yang, S. T. & Velsko, S. P. Frequency-agile kilohertz repetition-rate optical parametric oscillator based on periodically poled lithium niobate. Opt. Lett. 24, 133–135 (1999).
* (26) Lu, J. et al. Ultralow-threshold thin-film lithium niobate optical parametric oscillator. Optica. 8, 539–544 (2021).
* (27) Ledezma, L. et al. Octave-spanning tunable infrared parametric oscillators in nanophotonics. Sci. Adv. 9, eadf9711 (2023).
* (28) Solntsev, A. S. et al. LiNbO3 waveguides for integrated SPDC spectroscopy. APL Photonics. 3, 10 (2018).
* (29) Zhang, C. et al. Spontaneous Parametric Down-Conversion Sources for Multiphoton Experiments. Adv. Quantum Technol. 4, 2000132 (2021).
* (30) Krasnok, A., Tymchenko, M. & Alù, A. Nonlinear metasurfaces: a paradigm shift in nonlinear optics. Materials Today. 21, 8–21 (2018).
* (31) Wang, K., Chekhova, M., & Kivshar, Y. Metasurfaces for quantum technologies. Physics Today. 75, 3745–3763 (2022).
* (32) Fedotova, A. et al. Lithium Niobate Meta-Optics. ACS Photon. 9, 3745–3763 (2022).
* (33) Neshev, D. N. & Miroshnichenko, A. E. Enabling smart vision with metasurfaces. Nat. Photon. 17, 26–35 (2023).
* (34) Zhu, S. et al. Integrated photonics on thin-film lithium niobate. Adv. Opt. Photon. 13, 242–352 (2021).
* (35) Jankowski, M. et al. Supercontinuum generation by saturated second-order nonlinear interactions. APL Photonics. 8 116104 (2023).
* (36) Guo, Q., et al. Ultrathin quantum light source with van der Waals NbOCl2 crystal. Nature 613 53–59 (2023).
* (37) Wu, L., et al. Giant anisotropic nonlinear optical response in transition metal monopnictide Weyl semimetals. Nat. Physics 13 350–355 (2017).
* (38) Mueller, T. & Malic, E. Exciton physics and device application of two-dimensional transition metal dichalcogenide semiconductors. npj 2D Mater. Appl. 2 29 (2018).
* (39) Du, L. et al. Moiré photonics and optoelectronics. Science 379 eadg0014 (2023).
* (40) Ma, Q. et al. Photocurrent as a multiphysics diagnostic of quantum materials. Nat. Rev. Phys. 5 170–184 (2023).
* (41) Sheffer, Y., Queiroz, R. & Stern, A. Symmetries as the Guiding Principle for Flattening Bands of Dirac Fermions Phys. Rev. X 13 021012 (2023).
* (42) Malard, L. M. et al. Observation of intense second harmonic generation from MoS2 atomic crystals. Phys. Rev. B. 87 201401 (2013).
* (43) Li, Y. et al. Probing Symmetry Properties of Few-Layer MoS2 and h-BN by Optical Second-Harmonic Generation. Nano Lett. 13, 3329–3333 (2013).
* (44) Wang, G. et al. Giant enhancement of the optical second-harmonic emission of WSe2 monolayers by laser excitation at exciton resonances. Phys. Rev. Lett. 114 097403 (2015).
* (45) Autere, A. et al. Nonlinear Optics with 2D Layered Materials. Adv. Mater. 30, 1705963 (2018).
* (46) Wen, X., Gong, Z. & Li, D. et al. Nonlinear optics of two-dimensional transition metal dichalcogenides. Wiley Online Library. 1, 317–337 (2019).
* (47) Liu, W. et al. Recent Advances of 2D Materials in Nonlinear Photonics and Fiber Lasers. Adv. Optical Mater. 8, 1901631 (2020).
* (48) Dogadov, O. et al. Parametric Nonlinear Optics with Layered Materials and Related Heterostructures. Laser & Photonics Reviews. 16, 2100726 (2022).
* (49) Trovatello, C. et al. Optical parametric amplification by monolayer transition metal dichalcogenides. Nat. Photonics. 15, 6–10 (2021).
* (50) Hong, H. et al. Twist-phase-matching in two-dimensional materials. arXiv, 2305.11511 (2023).
* (51) Schiek, R. & Pertsch, T. Absolute measurement of the quadratic nonlinear susceptibility of lithium niobate in waveguides. Opt. Mater. Express 2, 126–139 (2012).
* (52) Brida, D. et al. Phase-locked pulses for two-dimensional spectroscopy by a birefringent delay line. Opt. Lett. 37, 3027–3029 (2012).
* (53) Manzoni, C. & Cerullo, G. Design criteria for ultrafast optical parametric amplifiers. J. Opt. 18, 103501 (2016).
* (54) Mannix, A. J., Ye, A. et al. Robotic four-dimensional pixel assembly of van der Waals solids. Nat. Nanotechnol. 17, 361–366 (2022).
* (55) Li, H. et al. Probing dynamical symmetry breaking using quantum-entangled photons. Quantum Science and Technology. 3, 015003 (2017).


# Gauged Skyrme analogue of Chern-Pontryagin

D. H. Tchrakian†∗

†School of Theoretical Physics, Dublin Institute for Advanced Studies,
Burlington Road, Dublin 4, Ireland
∗Department of Computer Science, National University of Ireland Maynooth,
Maynooth, Ireland

###### Abstract

An analogue of the Chern-Pontryagin density for $SO(D)$ gauged $O(D+1)$ Skyrme
systems, referred to as Skyrme–Chern-Pontyagin (SCP) densities iss known for
dimensions $D=2,3,4$. Since these are defined only through a prescription, it
is necessary to extend the realisation to higher $D$, which is carried out
here for $D=5$. It is found that in $D=5$ there is limitation on the group
gauging the Skyrmion, implying that such limitation will persist in higher
dimensions. The construction of SCP densities in $D=2,3,4,5$ is presented here
in a unified notation with the aim of of pointing out to possible
extarpolation to all dimensions.

##  1 Introduction

Chern-Pontryagin (CP) densities are pivotal in the construction of instantons
[1] on ${\rm IR}^{4}$, and their dimensonal descendants play that role for the
magnetic monopoles [2] on ${\rm IR}^{3}$ and the Abelian Higgs vortices [3] on
${\rm IR}^{2}$. CP densities play the same role also in higher dimensions. In
even dimensions, they stabilise Yang-Mills instanton actions and in particular
on ${\rm IR}^{4p}$ they support (anti-)self-dual instantons [4]. Their
dimensional descendants are employed in cosntructing magnetic monopoles [5] in
all odd and even dimensions.

Concerning gauged Skyrmions the situation is entirely different since Skyrme
scalars are not dimensional descendants of Yang-Mills fields in higher (even)
dimensions, and hence CP densities cannot be employed in the same way as for
gauged Higgs fields. The approach taken here is to devise an analogue of the
CP density that accommodates the Skyrme scalar in the given dimension.

The guiding principle for devising a density that can present a lower bound
for the action/energy for a gauged Skyrme system is to find an analogue of the
CP density, which like the latter is both gauge invariant and total
divergence. This then can be exploited to establish Bogomol’nyi like
action/energy lower bounds.

Formally, this density, $\varrho^{(D)}$ in $D$ dimensions is expressed as

$\displaystyle\varrho^{(D)}$ $\displaystyle=$
$\displaystyle\varrho_{G}^{(D)}+W[F,D\phi]$ (1.1) $\displaystyle=$
$\displaystyle\varrho_{0}^{(D)}+\partial_{i}\Omega_{i}[A,\phi]\,,$ (1.2)

where

$\displaystyle\varrho_{0}^{(D)}$ $\displaystyle=$
$\displaystyle\varepsilon_{i_{1}i_{2}\dots i_{D}}\varepsilon^{A_{1}A_{2}\dots
A_{D}A_{D+1}}\partial_{i_{1}}\phi^{A_{1}}\partial_{i_{2}}\phi^{A_{2}}\dots\partial_{i_{D}}\phi^{A_{D}}\phi^{A_{D+1}}$
(1.3) $\displaystyle\varrho_{G}^{(D)}$ $\displaystyle=$
$\displaystyle\varepsilon_{i_{1}i_{2}\dots i_{D}}\varepsilon^{A_{1}A_{2}\dots
A_{D}A_{D+1}}D_{i_{1}}\phi^{A_{1}}D_{i_{2}}\phi^{A_{2}}\dots
D_{i_{D}}\phi^{A_{D}}\phi^{A_{D+1}}$ (1.4)

and where $\varrho_{0}^{(D)}$ defined by (1.3) is the winding number density
of the $O(D+1)$ Skyrme scalar $\phi^{A}\ ,\ A=1,2,\dots,D+1$ in $D$
dimensions, subject to the constraint $|\phi^{A}|^{2}=1$, and
$\varrho_{G}^{(D)}$ is defined by replacing all the partial derivatives in
$\varrho_{0}^{(D)}$ by covariant derivatives. Thus, $\varrho_{0}$ is
effectively a total-divergence but is not gauge-invariant, while
$\varrho_{G}^{(D)}$ is by construction gauge-invariant but is not total-
divergence.

In (1.3) and (1.4), coordinates are labelled by lower indices, $e.g.,$
$x_{i}$, and the indices labelling the $O(D+1)$ scalars by upper indices,
$e.g.,$ $\phi^{A}$. Upper/lower here do not imply contravariant/covariant.
This notation is used throughout.

For $\varrho^{(D)}$ to present an energy lower bound it must be both gauge
invariant and total divergence, which is achieved by constructing the gauge-
invariant quantity $W[F,D\phi]$ and the total-divergence
$\partial_{i}\Omega_{i}[A,\phi]$. In this sense, $\varrho$ is an analogue of
the Chern-Pontryagin (CP), which is the density used in establishing
action/energy lower bounds for instantons/monopoles (and vortices). In the
case of instantons, this is the CP density itself, while in the case of
monopoles (and vortices) it is the appropriate dimensional descendant of a CP
density.

Unlike the CP density however, $\varrho^{(D)}$ in $D$ dimensions, prescribed
by (1.1)-(1.2), is defined for gauge group $SO(D)$, or contractions of
$SO(D)$. To make the prescription (1.1)-(1.2) concrete, it is necessary to
define the gauging prescription and fix the notation.

The $O(D+1)$ Skyrme scalar is $\phi^{A}\ ,\ A=1,2,\dots,D+1$ is subject to

$|\phi^{A}|^{2}=|\phi^{a}|^{2}+(\phi^{D+1})^{2}=1\ ,\quad a=1,2,\dots,D\,.$
(1.5)

The $SO(D)$ gauging prescription is

$\displaystyle D_{i}\phi^{a}$ $\displaystyle=$
$\displaystyle\partial_{i}\phi^{a}+A_{i}\phi^{a}\ ,\quad
A_{i}\phi^{a}=(A_{i}\phi)^{b}=A_{i}^{ab}\phi^{b}$ (1.6) $\displaystyle
D_{i}\phi^{D+1}$ $\displaystyle=$ $\displaystyle\partial_{i}\phi^{D+1}\,,$
(1.7)

where

$A_{i}^{ab}=-A_{i}^{ba}\ \ i=1,2,\dots,D\ ;a=1,2,\dots,D$

is the $SO(D)$ gauge connection, whose curvature is

$\displaystyle F_{ij}^{ab}$ $\displaystyle=$
$\displaystyle\partial_{[i}A_{j]}^{ab}+(A_{[i}A_{j]})^{ab}$
$\displaystyle\equiv$
$\displaystyle\partial_{i}A_{j}^{ab}+A_{i}^{ac}A_{j}^{cb}-(i,j)\,.$

The prescription for constructing the (1.1)-(1.2) is to calculate the
difference

$\varrho_{G}^{(D)}-\varrho_{0}^{(D)}=\partial_{i}\Omega_{i}[A,\phi]-W[F,D\phi]$
(1.9)

such that $W[F,D\phi]$ is gauge invariant.

The relation (1.9) is evaluated explicitly by extracting partial derivatives
unlil a gauge invariant quantity is isolated. In increasing dimensions, there
is some arbitrariness in this splitting. The calculations are carried out
directly, using the Leibnitz rule, tensor and Bianchi identities.

As stated above, the density (1.1)-(1.2),

$\displaystyle\varrho^{(D)}$ $\displaystyle=$
$\displaystyle\varrho_{G}^{(D)}+W[F,D\phi]$ $\displaystyle=$
$\displaystyle\varrho_{0}^{(D)}+\partial_{i}\Omega_{i}[A,\phi]\,,$

is an analogue of the Chern-Pontryagin (CP) density and can therefore be
similarly exploited. It might not be unreasonable to refer to $\varrho^{(D)}$
as a Skyrme–Chern-Pontryagin (SCP) density.

The definition (1.1) of the SCP density is suited for devising Bogomol’nyi
type energy lower bounds. It has been exploited in constructing gauged Skyrme
solitons: on ${\rm IR}^{2}$ in [6], on ${\rm IR}^{3}$ in [7] and on ${\rm
IR}^{4}$ in [8].

The definition (1.2) of the SCP density is useful for calculating the
effective “baryon number” which in the presence of the Chern-Simons term can
cause this to depart from the integer valued topological charge. This was
observed in $2+1$ and $3+1$ dimensional Abelian gauged $O(3)$ and $O(5)$
models, in [9] and [10], respectively. Not less importantly, since (1.2) can
be expressed explicitly as a total divergence, it can be exploited in the
definition of a Chern-Simon like density.

However, unlike the CP density which can be defined for any gauge group in any
even dimension, the gauge group of the SCP density in $D$ dimensions,
prescribed by (1.1)-(1.2), is restricted to be at most $SO(D)$, or some
subgroup of it. But unlike the the CP density which can be defined in even
dimensions only, the definition of the SCP density in not restricted to even
dimensions only and is valid in odd dimensions too.

We are now confronted with the essential question. While the definition of the
CP density for any gauge group and any (even) dimension is unamiguously
stated, the corresponding status of the SCP density depends on explicit
construction in any given dimension, and as such (1.1)-(1.2) is a
prescription. To date, this prescrition was implemented only for $D=2,\ 3$ and
$4$, and was exploited in Refs. [6], [7] and [8] respectively.

To answer this question at least partially, it is necessary to explore the
prescription (1.1)-(1.2) in dimensions $D\geq 5$, and in the present note this
carried out for dimension $D=5$. This is the sole aim of the present work. In
Section 2 below we present the construction of $\varrho^{(D)}$ for $D=2,\ 3,\
4$ and $5$, in successive Subsections. The aim is to give an as transparent as
possible description of these SCP densities in a unified manner. In Section 3,
a summary and outlokk is give, and the further details of the content of
Section 2 are relegated to the Appendix.

##  2 Calculation

The quantity $(\varrho_{G}^{(D)}-\varrho_{0}^{(D)})$ is calculated and cast in
the prescribed form of (1.9), for $D=2,3,4,$ and $5$ in each Subsection below,
respectively.

###  2.1 $D=2$

Using basic tensor identities, $(\varrho_{G}-\varrho_{0})$ given by (1.4)
minus (1.3), is unpacked to give

$\frac{1}{2}(\varrho_{G}-\varrho_{0})=\frac{1}{2}\,\varepsilon_{ij}\,\varepsilon^{fg}\,\partial_{j}\phi^{3}\
A_{i}^{fg}\equiv\varepsilon_{ij}\,\partial_{j}\phi^{3}\ A_{i}\,,$ (2.10)

which by a single application of the Leibniz rule splits in a gauge-invariant
and a total divergence

$(\varrho_{G}-\varrho_{0})=\varepsilon_{ij}\left[2\,\partial_{j}(\phi^{3}\
A_{i})-\phi^{3}\,F_{ij}\right]\,,$ (2.11)

from which the quantities $W$ and $\partial_{i}\Omega_{i}$ in (1.1) and (1.2)
can be read off

$\displaystyle W$ $\displaystyle=$
$\displaystyle\varepsilon_{ij}\,\phi^{3}\,F_{ij}$ (2.12)
$\displaystyle\Omega_{i}$ $\displaystyle=$
$\displaystyle-2\varepsilon_{ij}\,\phi^{3}\,A_{j}$ (2.13)

Of course, in all even $D$ the $(D/2)$-th CP density can be added to both
(1.1) and to (1.2). preserving the CP like properies of $\varrho^{(2)}$. For
$D=2$, this is the $1$-st CP density, with $\varepsilon_{ij}F_{ij}$ being
subtracted from $W$ in (2.12) and $2\varepsilon_{ij}A_{j}$ added to
$\Omega_{i}$ in (2.13).

###  2.2 $D=3$

Using basic tensor identities, $(\varrho_{G}-\varrho_{0})$ given by (1.4)
minus (1.3), is unpacked to give

$\frac{1}{3}(\varrho_{G}-\varrho_{0})=\varepsilon_{ijk}\,\varepsilon^{fga}\
\partial_{k}\phi^{4}\
A_{j}^{fg}\left(\partial_{i}\phi^{a}+\frac{1}{2}A_{i}\phi^{a}\right)\,.$
(2.14)

Applying the Leibniz rule, (2.14) can be written as

$\frac{1}{3}(\varrho_{G}-\varrho_{0})=\partial_{k}\phi^{4}\left(\partial_{i}\Omega_{ik}-\frac{1}{2}\,\varepsilon_{ijk}\varepsilon^{fga}\,\phi^{a}\,F_{ij}^{fg}\right)$
(2.15)

where

$\Omega_{ik}=\varepsilon_{ijk}\varepsilon^{fga}\,(A_{j}^{fg}\,\phi^{a})\,.$
(2.16)

The result (2.15) is of the required form (1.9) as the first term
$\partial_{k}\phi^{4}\partial_{i}\Omega_{ik}$ is total divergence by virtue of
the antisymmetry of $\Omega_{ik}$, and, the second term is gauge invariant as
it stands.

From the viewpoint of the application of Bogomol’nyi inequalities however, it
is expedient to modify this result by extracting the partial derivative from
$\partial_{k}\phi^{4}$ in the gauge invariant term in (2.15). Using the
Bianchi identity, this results in the identity

$\varepsilon_{ijk}\,\varepsilon^{fga}\,\phi^{a}\,F_{ij}^{fg}=\varepsilon_{ijk}\,\varepsilon^{fga}\left(\phi^{4}\,\phi^{a}\,F_{ij}^{fg}--\phi^{4}\,D_{k}\phi^{a}\,F_{ij}^{fg}\right)$
(2.17)

yielding the result in the desired form

$\displaystyle W$ $\displaystyle=$
$\displaystyle\frac{3}{2}\,\varepsilon_{ijk}\varepsilon^{fga}\phi^{4}\,D_{k}\phi^{a}\,F_{ij}^{fg}$
(2.18) $\displaystyle\Omega_{k}$ $\displaystyle=$ $\displaystyle
3\,\phi^{4}\left(\partial_{i}\Omega_{ik}-\frac{1}{2}\,\varepsilon_{ijk}\varepsilon^{fga}\,\phi^{a}\,F_{ij}^{fg}\right)\,.$
(2.19)

Exploiting identities like (2.17) resulting from the extraction of a partial
derivative are a persistent feature in achieving the desired form of $W$ and
$\Omega_{i}$, in all dimensions $D\geq 3$.

###  2.3 $D=4$

Using basic tensor identities, $(\varrho_{G}-\varrho_{0})$ given by (1.4)
minus (1.3), is unpacked to give

$\frac{1}{4}(\varrho_{G}-\varrho_{0})=\varepsilon_{ijkl}\varepsilon^{fgab}\partial_{l}\phi^{5}\,A_{k}^{fg}\left(\partial_{i}\phi^{a}\partial_{j}\phi^{b}+\partial_{i}\phi^{a}A_{j}\phi^{b}+\frac{1}{3}\,A_{i}\phi^{a}A_{j}\phi^{b}\right)\,.$
(2.20)

After multiple applications of the Leibniz rule and tensor identities, this
can be put in the form

$\frac{1}{4}(\varrho_{G}-\varrho_{0})=\partial_{l}\phi^{5}\partial_{i}\Omega_{il}+\frac{1}{2}\,\varepsilon_{ijkl}\varepsilon^{fgab}\partial_{l}\phi^{5}\left\\{F_{ij}^{fg}\,\phi^{a}D_{k}\phi^{b}+\frac{1}{2}|\vec{\phi}|^{2}\Xi_{ijk}^{fgab}\right\\}$
(2.21)

where we have used the notation $|\vec{\phi}|^{2}=|\phi^{a}|^{2}$, and where

$\displaystyle\Omega_{il}$ $\displaystyle=$
$\displaystyle\varepsilon_{ijkl}\varepsilon^{fgab}A_{k}^{fg}\,\phi^{a}\left(\partial_{j}\phi^{b}-\frac{1}{2}A_{j}\phi^{b}\right)\,,$
(2.22)

and

$\Xi_{ijk}^{fgab}=A_{k}^{ab}\left[\partial_{i}A_{j}^{fg}+\frac{2}{3}(A_{i}A_{j})^{fg}\right]\,.$
(2.23)

Note that the $l=4$ component of
$\varepsilon_{ijkl}\varepsilon^{fgab}\,\Xi_{ijk}^{fgab}$, namely

$\varepsilon_{\mu\nu\lambda 4}\varepsilon^{fgab}\,\Xi_{\mu\nu\lambda}^{fgab}\
,\quad\mu=1,2,3\ ,$

is just the (Euler–)Chern-Simons density in three dimensions.

The expressions (2.21) and (2.22) are the analogues of (2.15) and (2.16) above
in the $D=3$ case. Again, the first term in (2.21),
$\partial_{l}\phi^{5}\partial_{i}\Omega_{il}$, is a total divergence by virtue
of the antisymmetry of $\Omega_{il}$, and the second term is manifestly gauge
invariant. But the third term involving $\Xi_{ijk}^{fgab}$ is neither gauge-
invariant nor is it total-divergence. Thus, (2.21) is not of the form (1.9) as
it stands. This is a feature of $(\varrho_{G}-\varrho_{0})$ for all $D\geq 4$.

To render (2.21) to the form (1.9), namely consisting of a gauge invariant
density $W$ and a total divergence, the partial derivative $\partial_{l}$ from
the gauge-variant term in (2.21) must be extracted using the Bianchi
identities. But as above in $D=3$, it is expedient to extract this partial
derivative also from the gauge-invariant term 111This is convenient from the
viewpoint of applying Bogomol’nyi type inequalities..

For the gauge-invariant term this is implemented by

$\displaystyle\varepsilon_{ijkl}\varepsilon^{fgab}\partial_{l}\phi^{5}\,\\{F_{ij}^{fg}\,\phi^{a}D_{k}\phi^{b}\\}$
$\displaystyle=$
$\displaystyle\varepsilon_{ijkl}\varepsilon^{fgab}\bigg{\\{}\partial_{l}\left[\phi^{5}\,F_{ij}^{fg}\,\phi^{a}D_{k}\phi^{b}\right]$
(2.24)
$\displaystyle\qquad\qquad-\phi^{5}\left[F_{ij}^{fg}D_{k}\phi^{a}D_{l}\phi^{b}+\frac{1}{8}|\vec{\phi}|^{2}F_{ij}^{fg}F_{kl}^{ab}\right]\bigg{\\}}$

and for the gauge-variant term by

$\displaystyle\varepsilon_{ijkl}\varepsilon^{fgab}|\vec{\phi}|^{2}\partial_{l}\phi^{5}\,\Xi_{ijk}^{fgab}$
$\displaystyle=$
$\displaystyle\varepsilon_{ijkl}\varepsilon^{fgab}\bigg{\\{}\partial_{l}\left(|\vec{\phi}|^{2}\phi^{5}\,\Xi_{ijk}^{fgab}\right)+\frac{1}{4}(\phi^{5}-\frac{1}{3}(\phi^{5})^{3})F_{ij}^{ab}F_{kl}^{cd}\bigg{\\}}\,.$
(2.25)

Applying the identities (2.24) and (2.25), (2.21) is cast in the required form
(1.9) with

$\displaystyle W$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\varepsilon_{ijkl}\varepsilon^{fgab}\,\phi^{5}\bigg{[}F_{ij}^{fg}D_{k}\phi^{a}D_{l}\phi^{b}+\frac{1}{12}(\phi^{5})^{2}\,F_{ij}^{fg}F_{kl}^{ab}\bigg{]}$
(2.26) $\displaystyle\Omega_{l}$ $\displaystyle=$
$\displaystyle\phi^{5}\bigg{(}\partial_{i}\Omega_{il}+\frac{1}{2}\varepsilon_{ijkl}\varepsilon^{fgab}\bigg{[}F_{ij}^{fg}\phi^{a}D_{k}\phi^{b}+\frac{1}{2}\left(1-\frac{1}{3}(\phi^{5})^{2}\right)\Xi_{ijk}^{fgab}\bigg{]}\bigg{)}\,,$
(2.27)

which are the $D=4$ analogues of $W$ and $\Omega_{i}$, (2.18) and (2.19) in
$D=3$, it being understood that the indices $(i,a)$ run over $(1,2,3)$ in
(2.18)-(2.19) while here they run over $(1,2,3,4)$ in (2.26)-(2.27).

###  2.4 D=5

Using basic tensor identities, $(\varrho_{G}-\varrho_{0})$ given by (1.4)
minus (1.3), is unpacked to give

$\displaystyle\frac{1}{5}(\varrho_{G}-\varrho_{0})$ $\displaystyle=$
$\displaystyle\varepsilon_{ijklm}\,\varepsilon^{fgabc}\ \partial_{m}\phi^{6}\
A_{l}^{fg}\bigg{(}2\partial_{i}\phi^{a}\partial_{j}\phi^{b}\partial_{k}\phi^{c}+3\partial_{i}\phi^{a}\partial_{j}\phi^{b}A_{k}\phi^{c}$
(2.28)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+2\partial_{i}\phi^{a}A_{j}\phi^{b}A_{k}\phi^{c}+\frac{1}{2}\,A_{i}\phi^{a}A_{j}\phi^{b}A_{k}\phi^{c}\bigg{)}$

After multiple applications of the Leibniz rule and tensor identities, (2.28)
can be put in the form analogous to its $D=4$ counterpart (2.21)

$\displaystyle\frac{1}{5}(\varrho_{G}-\varrho_{0})$ $\displaystyle=$
$\displaystyle\partial_{m}\phi^{6}\partial_{i}\Omega_{im}-\varepsilon_{ijklm}\varepsilon^{fgabc}\partial_{m}\phi^{6}\left\\{F_{ij}^{fg}\,\phi^{a}\,D_{k}\phi^{b}D_{l}\phi^{c}+|\vec{\phi}|^{2}\,\Xi_{ijkl}^{fgabc}\right\\}$
(2.29)

where

$\displaystyle\Omega_{im}$ $\displaystyle=$ $\displaystyle
2\varepsilon_{ijklm}\varepsilon^{fgabc}\,A_{l}^{fg}\,\phi^{a}(\partial_{j}\phi^{b}\partial_{k}\phi^{c}+\partial_{j}\phi^{b}A_{k}\phi^{c}+\frac{2}{3}\,A_{j}\phi^{b}A_{k}\phi^{c})$
(2.30)

and

$\displaystyle\Xi_{ijkl}^{fgabc}$ $\displaystyle=$ $\displaystyle
A_{k}^{ab}\left[\left(\partial_{l}\phi^{c}+\frac{2}{3}\,A_{l}\phi^{c}\right)\,\partial_{i}A_{j}^{fg}+\frac{2}{3}\left(\partial_{l}\phi^{c}+\frac{3}{4}\,A_{l}\phi^{c}\right)(A_{i}A_{j})^{fg}\right]\,,$
(2.31)

$\Omega_{im}$ in (2.30) being the $D=5$ analogue of (2.22), and
$\Xi_{ijkl}^{fgabc}$ in (2.31) the $D=5$ analogue of $\Xi_{ijk}^{fgab}$ in
(2.23), in $D=4$.

Again, the first term $\partial_{m}\phi^{6}\partial_{i}\Omega_{im}$ in (2.29)
is a total divergence and the second term is manifestly gauge invariant, while
the third term involving $\Xi_{ijkl}^{fgabc}$ is neither gauge invariant not
total divergence. Thus, to cast (2.29) in the required form consisting of a
gauge invariant part $W$ and a total derivative, the partial derivative
$\partial_{m}$ must be extracted from this term by using the Bianchi
identities. As in $D=3$ and $4$, it is expedient to extract $\partial_{m}$
also from the gauge invariant term in (2.29).

Extracting the partial derivative $\partial_{m}$ from the gauge invariant term
in (2.29) results in the expression

$\displaystyle\varepsilon_{ijklm}\varepsilon^{fgabc}\partial_{m}\phi^{6}\,F_{ij}^{fg}\,\phi^{a}\,D_{k}\phi^{b}D_{l}\phi^{c}=$
$\displaystyle\qquad\qquad=\varepsilon_{ijklm}\varepsilon^{fgabc}\bigg{\\{}\partial_{m}\left[\phi^{6}\,F_{ij}^{fg}\phi^{a}D_{k}\phi^{b}D_{l}\phi^{c}+\frac{1}{12}(\phi^{6})^{3}\,F_{ij}^{fg}\,F_{kl}^{ab}\phi^{c}\right]$
$\displaystyle\qquad\qquad\qquad-\phi^{6}\left[F_{ij}^{fg}D_{k}\phi^{a}D_{l}\phi^{b}D_{m}\phi^{c}-\frac{1}{4}\left(1-\frac{4}{3}(\phi^{6})^{2}\right)F_{ij}^{fg}\,F_{kl}^{ab}D_{m}\phi^{c}\right]\bigg{\\}}\,.$
(2.32)

This expression is the $D=5$ analogue of (2.24) in $D=4$, which likewise
splits up in a gauge invariant and a total divergence part.

Extracting the partial derivative $\partial_{m}$ from the gauge variant term
in (2.29), one has

$\varepsilon_{ijklm}\varepsilon^{fgabc}\partial_{m}\phi^{6}|\vec{\phi}|^{2}\,\Xi_{ijkl}^{fgabc}=\varepsilon_{ijklm}\varepsilon^{fgabc}\left\\{\partial_{m}\left[\left(\phi^{6}-\frac{1}{3}(\phi^{6})^{3}\right)\Xi_{ijkl}^{fgabc}\right]-\left(\phi^{6}-\frac{1}{3}(\phi^{6})^{3}\right)\partial_{m}\Xi_{ijkl}^{fgabc}\right\\}\,,$
(2.33)

which is the $D=5$ analogue of (2.25) in $D=4$.

The question now is, does the divergence in the second term in (2.33) result
in a gauge invariant quantity as was the case in (2.25), in the $D=4$?

The answer is found by calculating the divergence

$\varepsilon_{ijklm}\varepsilon^{fgabc}\,\partial_{m}\Xi_{ijkl}^{fgabc}=\frac{1}{4}\varepsilon_{ijklm}\varepsilon^{fgabc}\left[F_{ij}^{fg}F_{kl}^{ab}D_{m}\phi^{c}-4\,(A_{i}A_{j})^{fg}(A_{k}A_{l})^{ab}A_{m}\phi^{c}\right]\,,$
(2.34)

which consists of a gauge invariant term, and the gauge variant term

$\varepsilon_{ijklm}\varepsilon^{fgabc}(A_{i}A_{j})^{fg}(A_{k}A_{l})^{ab}A_{m}\phi^{c}\,.$
(2.35)

The conclusion is that in unlike in $D\leq 4$, in $D=5$ it is not possible to
express $(\varrho_{G}-\varrho_{0})$ in terms of a gauge invariant quantity $W$
and a total divergence $\partial_{i}\Omega_{i}$ as in (1.9) for $SO(5)$
gauging. One expects that this conclusion holds also in dimensions $D\geq 5$
for $SO(D)$ gauging.

However, this limitation is not an obstacle for exploiting the prescription
(1.3)-(1.4) to contruct solitons in $D$ dimensions, or SCS anomalies in $D-1$
dimensions, as long as the gauge connection is not the full $SO(D)$ connection
but rather a contraction thereof. In that case the gauge variant term (2.35)
vanishes, rendering (2.34) gauge invariant,whence (2.33) becomes

$\displaystyle\varepsilon_{ijklm}\varepsilon^{fgabc}\partial_{m}\phi^{6}|\vec{\phi}|^{2}\,\Xi_{ijkl}^{fgabc}$
$\displaystyle=$
$\displaystyle\varepsilon_{ijklm}\varepsilon^{fgabc}\bigg{\\{}\partial_{m}\left[\phi^{6}\left(1-\frac{1}{3}(\phi^{6})^{2}\right)\Xi_{ijkl}^{fgabc}\right]$
(2.36)
$\displaystyle\qquad\qquad\qquad-\frac{1}{4}\phi^{6}\left(1-\frac{1}{3}(\phi^{6})^{2}\right)F_{ij}^{fg}|_{kl}^{ab}D_{m}\phi^{c}\bigg{\\}}\,,$

Applying (2.33) and (2.36) to (2.29), $(\varrho_{G}-\varrho_{0})$ can be
expressed in terms of $W$ and $\partial_{i}\Omega_{i}$ as prescribed by (1.9)
with

$\displaystyle W$ $\displaystyle=$
$\displaystyle\varepsilon_{ijklm}\varepsilon^{fgabc}\phi^{6}\left[F_{ij}^{fg}D_{k}\phi^{a}D_{l}\phi^{b}D_{m}\phi^{c}+\frac{5}{12}(\phi^{6})^{2}F_{ij}^{fg}F_{kl}^{ab}D_{m}\phi^{c}\right]$
(2.37) $\displaystyle\Omega_{m}$ $\displaystyle=$
$\displaystyle\phi^{6}\partial_{i}\Omega_{im}-\varepsilon_{ijklm}\varepsilon^{fgabc}\phi^{6}\bigg{[}F_{ij}^{fg}\phi^{a}D_{k}\phi^{b}D_{l}\phi^{c}+\frac{1}{12}(\phi^{6})^{2}F_{ij}^{fg}F_{kl}^{ab}\phi^{c}$
(2.38)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad+\left(1-\frac{1}{3}(\phi^{6})^{2}\right)\Xi_{ijkl}^{fgabc}\bigg{]}$

which are the $D=5$ analogues of the $D=4$ expressions (2.26) and (2.27) for
$W$ and $\Omega_{i}$, with however a subtle difference. While in (2.26)-(2.38)
the index $i$ and index $a$ run over $(1,2,3,4)$, here in (2.37)-(2.38) the
index $i$ runs over $(1,2,3,4,5)$, but the index $a$ runs over whatever values
are dictated by the given contraction of $SO(5)$ chosen, except $(1,2,3,4,5)$.

##  3 Summary and outlook

The new result in this study is the implementation of the prescription
(1.1)-(1.2) for the SCP in $D=5$. To give as systematic as possible
presentation, this has been presented in a unified manner together with all so
far known implementations, in $D=2,\ 3$ and $4$.

While in $D=2,\ 3$ and $4$ the prescription is achieved for gauge group
$SO(D)$, it is found that in $D=5$ there is a limitation, to some subgroup of
$SO(D)$. It can be surmised that such a limitation will apply to all higher
$D>5$.

How important is this limitation depends on, to what applications is the SCP
density to be applied. If the application be to the construction of a $SO(D)$
gauged Skyrmion in $D$ dimensions, this limitation is indeed an obstacle. But
such solitons can be construced for an subgroup of $SO(D)$. But this is
perhaps not the most important application of SCP.

By far the most important application SCP comes from the prescription (1.2).
Since (1.2) is essentially total divergence (and explicitly total divergence
in constraint-compliantparametrisation) it can be formally expressed as

$\varrho^{(D)}=\partial_{i}(\omega_{i}^{(D)}+\Omega_{i})\ ,\quad
i=1,2,\dots,D\,.$ (3.39)

Since (3.39) is also gauge invariant, a Chern-Simons like density can be
defined in one dimension lower, in $d=D-1$ “spacetime”,

$\Omega_{\rm SCS}\stackrel{{\scriptstyle\rm
def.}}{{=}}\omega_{i=D}^{(D)}+\Omega_{i=D}$ (3.40)

in which $\omega_{i=D}^{(D)}$ here is the Wess-Zumino term. The density
$\Omega_{\rm SCS}$ in (3.40) referred to as the Skyrme–Chern-Simons (SCS)
density [11, 12] defined in $D_{1}$ “spacetime” dimensions. That the equations
of motion of (3.40) are gauge invariant can be demonstrated exactly as in the
case of the Chern-Simons density, since (1.2) (or (3.39)) is gauge-invariant
and total-derivative like the Chern-Pontryagin (CP) density.

Apart from its intrinsic value, the example $D=5$ is of particular interest in
that it can yield SCS densities in $D-1=4$ spacetime in the same spirit as the
anomalies introduced in Ref. [14], with application to baryon number decay
[15].

## Acknowledgements

The question whether the pescription (1.1)-(1.2) for the SCP density can be
realised beyond $D=4$ was emphasised by V. A. Rubakov (R.I.P.). The present
study is aimed to resolving this question. I am very grateful to my colleague
Eugen Radu for having read the manuscript carefully, and to him and to
Francisco Navarro-Lerida for valuable discussions on these topics.

## References

  * [1] A. A. Belavin, A. M. Polyakov, A. S. Schwartz and Y. S. Tyupkin, Phys. Lett. B 59 (1975), 85-87 doi:10.1016/0370-2693(75)90163-X
  * [2] G. ’t Hooft, ‘Magnetic Monopoles in Unified Gauge Theories,” Nucl. Phys. B 79 (1974), 276-284 doi:10.1016/0550-3213(74)90486-6
A. M. Polyakov, “Particle Spectrum in Quantum Field Theory,” JETP Lett. 20
(1974), 194-195 PRINT-74-1566 (LANDAU-INST).

  * [3] H. B. Nielsen and P. Olesen (1973) “Vortex-line models for dual strings,” Nuclear Physics B (1973), 45–61 doi:10.1016/0550-3213(73)90350-7
  * [4] D. H. Tchrakian, Phys. Lett. B 150 (1985), 360-362 doi:10.1016/0370-2693(85)90994-3
  * [5] T. Tchrakian, J. Phys. A 44 (2011), 343001 doi:10.1088/1751-8113/44/34/343001 [arXiv:1009.3790 [hep-th]].
  * [6] K. Arthur, D. H. Tchrakian and Y. s. Yang, Phys. Rev. D 54 (1996), 5245-5258 doi:10.1103/PhysRevD.54.5245
F. Navarro-Lerida, E. Radu and D. H. Tchrakian, “Effect of Chern-Simons
dynamics on the energy of electrically charged and spinning vortices,” Phys.
Rev. D 95 (2017) no.8, 085016 [arXiv:1612.05835 [hep-th]].
F. Navarro-Lérida and D. H. Tchrakian, “Vortices of $SO(2)$ gauged Skyrmions
in $2+1$ dimensions,” Phys. Rev. D 99 (2019) no.4, 045007 [arXiv:1812.03147
[hep-th]].

  * [7] K. Arthur and D. H. Tchrakian, Phys. Lett. B 378 (1996), 187-193 doi:10.1016/0370-2693(96)00343-7 [arXiv:hep-th/9601053 [hep-th]].
B. M. A. G. Piette and D. H. Tchrakian, “Static solutions in the U(1) gauged
Skyrme model,” Phys. Rev. D 62 (2000), 025020 [arXiv:hep-th/9709189 [hep-th]].
E. Radu and D. H. Tchrakian, “Spinning U(1) gauged skyrmions,” Phys. Lett. B
632 (2006), 109-113 [arXiv:hep-th/0509014 [hep-th]].
Y. Brihaye, B. Kleihaus and D. H. Tchrakian, “Dyon - Skyrmion lumps,” J. Math.
Phys. 40 (1999), 1136-1152 doi:10.1063/1.532793 [arXiv:hep-th/9805059 [hep-
th]].
Y. Brihaye, V. A. Rubakov, D. H. Chrakian and F. Zimmerschied, “A simplified
model for monopole catalysis of nucleon decay,” Theor. Math. Phys. 128 (2001),
1140-1154 [arXiv:hep-ph/0103228 [hep-ph]]. Y. Brihaye, D. Y. Grigoriev, V. A.
Rubakov and D. H. Tchrakian, “An extended model for monopole catalysis of
nucleon decay,” Phys. Rev. D 67 (2003), 034004 [arXiv:hep-th/0211215 [hep-
th]].

  * [8] Y. Brihaye, V. Paturyan, D. H. Tchrakian and B. M. A. G. Piette, J. Math. Phys. 42 (2001), 4669-4683 doi:10.1063/1.1396636 [arXiv:hep-th/0010111 [hep-th]].
F. Navarro-Lerida, E. Radu and D. H. Tchrakian, “$SO(2)$ gauged Skyrmions in
$4+1$ dimensions,” Phys. Rev. D 101 (2020) no.12, 125014 [arXiv:2003.05899
[hep-th]]. F. Navarro-Lerida, E. Radu and D. H. Tchrakian, “On the effects of
the Chern-Simons term in an Abelian gauged Skyrme model in $d=4+1$
dimensions,” Phys. Lett. B 814 (2021), 136083 [arXiv:2010.04093 [hep-th]].

  * [9] F. Navarro-Lérida, E. Radu and D. H. Tchrakian, Phys. Lett. B 791 (2019), 287-292 doi:10.1016/j.physletb.2019.02.011 [arXiv:1811.09535 [hep-th]].

  * [10] F. Navarro-Lerida, E. Radu and D. H. Tchrakian, Phys. Lett. B 814 (2021), 136083 doi:10.1016/j.physletb.2021.136083 [arXiv:2010.04093 [hep-th]].
  * [11] D. H. Tchrakian, “Higgs-and Skyrme–Chern-Simons densities in all dimensions,” J. Phys. A 48 (2015) no.37, 375401 [arXiv:1505.05344 [hep-th]].
  * [12] D. H. Tchrakian, “Some aspects of Skyrme–Chern–Simons densities,” J. Phys. A 55 (2022) no.24, 245401 [arXiv:2111.13097 [hep-th]].
  * [13] D. H. Tchrakian, “Topologically stable lumps in SO(d) gauged O(d+1) sigma models in d dimensions: d = 2,3,4,” Lett. Math. Phys. 40 (1997), 191-201
  * [14] E. Witten, “Global Aspects of Current Algebra,” Nucl. Phys. B 223 (1983), 422-432
  * [15] C. G. Callan, Jr. and E. Witten, “Monopole Catalysis of Skyrmion Decay,” Nucl. Phys. B 239 (1984), 161-176

##  Appendix A Details of the calculation

The $O(D+1)$ Skyrme scalar is $\phi^{A}\ ,\ A=1,2,\dots,D+1$ is subject to

$|\phi^{A}|^{2}=|\phi^{a}|^{2}+(\phi^{D+1})^{2}=1\ ,\quad a=1,2,\dots,D\,.$
(A.1)

The $SO(D)$ gauging prescription is

$\displaystyle D_{i}\phi^{a}$ $\displaystyle=$
$\displaystyle\partial_{i}\phi^{a}+A_{i}\phi^{a}\ ,\quad
A_{i}\phi^{a}=(A_{i}\phi)^{b}=A_{i}^{ab}\phi^{b}$ (A.2) $\displaystyle
D_{i}\phi^{6}$ $\displaystyle=$ $\displaystyle\partial_{i}\phi^{D+1}.$ (A.3)

$A_{i}^{ab}=-A_{i}^{ba}\ \ i=1,2,\dots,D\ ;a=1,2,\dots,D$ is the $SO(D)$ gauge
connection, whose curvature is

$\displaystyle F_{ij}^{ab}$ $\displaystyle=$
$\displaystyle\partial_{[i}A_{j]}^{ab}+(A_{[i}A_{j]})^{ab}$
$\displaystyle\equiv$
$\displaystyle\partial_{i}A_{j}^{ab}+A_{i}^{ac}A_{j}^{cb}-(i,j)\,.$

For $D=5$ the winding number density is

$\varrho_{0}=\varepsilon_{ijklm}\,\varepsilon^{abcdef}\,\partial_{i}\phi^{a}\,\partial_{j}\phi^{b}\,\partial_{k}\phi^{c}\,\partial_{l}\phi^{d}\,\partial_{m}\phi^{e}\
\phi^{d}$ (A.5)

which is total divergence and not gauge invarianct, and replacing the partial
derivatives $\partial_{i}\phi^{a}$ in (A.5) with the covariant derivatives
$D_{i}\phi^{a}$

$\varrho_{G}=\varepsilon_{ijklm}\,\varepsilon^{abcdef}\,D_{i}\phi^{a}\,D_{j}\phi^{b}\,D_{k}\phi^{c}\,D_{l}\phi^{d}\,D_{m}\phi^{e}\
\phi^{d}$ (A.6)

which is gauge invariant but not total divergence but is gauge invarianct 222
For all other $D=4,3,2$, the simpler versions of (A.5) and (A.6) are obvious.

The task is to calculate

$\varrho_{G}-\varrho_{0}=\partial_{i}\Omega_{i}[A,\phi]-W[F,D\phi]$ (A.7)

where $W[F,D\phi]$ is gauge invariant by construction and $\Omega_{i}[A,\phi]$
is gauge variant. (In increasing dimensions, there is some arbitrariness in
the splitting into these two terms.)

The relation (A.7) is evaluated explicitly with gauge group $SO(5)$, that
defines the covariant derivative in (A.2). The calculations are carried out
directly, using the Leibnitz rule and the tensor identities.

Collecting the gauge invariant pieces $\varrho_{G}$ and $W$ in (A.7), and
separately, the individually gauge variant pieces $\varrho_{0}$ and
$\partial_{i}\Omega_{i}$, one has two equivalent definitions of a density

$\displaystyle\varrho$ $\displaystyle=$ $\displaystyle\varrho_{G}+W[F,D\phi]$
(A.8) $\displaystyle=$
$\displaystyle\varrho_{0}+\partial_{i}\Omega_{i}[A,\phi]\,,$ (A.9)

which is adopted as the definition for the density
$\varrho\stackrel{{\scriptstyle\rm def.}}{{=}}\Omega^{(d+1)}_{\rm SCP}$
presenting a lower bound on the “energy” in the same way as does the usual CP
density.

First step

Using the tensor dentity

$V^{b}\,\varepsilon^{{a_{1}}{a_{2}}a_{3}\dots
a_{D}}=V^{a_{1}}\,\varepsilon^{b\,a_{2}a_{3}\dots
a_{D}}+V^{a_{2}}\,\varepsilon^{a_{1}b\,a_{3}\dots
a_{D}}+V^{a_{3}}\,\varepsilon^{a_{1}a_{2}\,b\dots
a_{D}}+V^{a_{D}}\,\varepsilon^{b\,a_{2}a_{3}\dots b}$ (A.10)

in $D$ dimensnions, it can be seen that the quantities
$(\varrho_{G}-\varrho_{0})$ in dimensions $D=2,3,4,5$ reduce to

$\displaystyle\frac{1}{2}(\varrho_{G}-\varrho_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\varepsilon_{ij}\,\varepsilon^{fg}\,\partial_{j}\phi^{3}\
A_{i}^{fg}\equiv\varepsilon_{ij}\,\partial_{j}\phi^{3}\ A_{i}$ (A.11)
$\displaystyle\frac{1}{3}(\varrho_{G}-\varrho_{0})$ $\displaystyle=$
$\displaystyle\varepsilon_{ijk}\,\varepsilon^{fga}\ \partial_{k}\phi^{4}\
A_{j}^{fg}\left(\partial_{i}\phi^{a}+\frac{1}{2}A_{i}\phi^{a}\right)$ (A.12)
$\displaystyle\frac{1}{4}(\varrho_{G}-\varrho_{0})$ $\displaystyle=$
$\displaystyle\varepsilon_{ijkl}\,\varepsilon^{fgab}\ \partial_{l}\phi^{5}\
A_{k}^{fg}\left(\partial_{i}\phi^{a}\partial_{j}\phi^{b}+\partial_{i}\phi^{a}A_{j}\phi^{b}+\frac{1}{3}\,A_{i}\phi^{a}A_{j}\phi^{b}\right)$
(A.13) $\displaystyle\frac{1}{5}(\varrho_{G}-\varrho_{0})$ $\displaystyle=$
$\displaystyle\varepsilon_{ijklm}\,\varepsilon^{fgabc}\ \partial_{m}\phi^{6}\
A_{l}^{fg}\bigg{(}2\partial_{i}\phi^{a}\partial_{j}\phi^{b}\partial_{k}\phi^{c}+3\partial_{i}\phi^{a}\partial_{j}\phi^{b}A_{k}\phi^{c}$
(A.14)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+2\partial_{i}\phi^{a}A_{j}\phi^{b}A_{k}\phi^{c}+\frac{1}{2}\,A_{i}\phi^{a}A_{j}\phi^{b}A_{k}\phi^{c}\bigg{)}$

The second step is to treat each case $D=2,3,4,5$ separately in a unified
manner.

Second step

It is convenient to express (A.12)-(A.14) formally as

$\displaystyle-\frac{1}{D}(\varrho_{G}-\varrho_{0})$ $\displaystyle=$
$\displaystyle\partial_{n}\phi^{D+1}(\omega_{n}^{(1)}+\omega_{n}^{(2)})$
(A.15)

in which $\omega_{n}^{(1)}$ is encoded with terms like $A_{i}\phi^{a}$
$exclusively$, and $\omega_{n}^{(2)}$ by a $mixture$ of terms like
$A_{i}\phi^{a}$ $and$ $\partial_{i}\phi^{a}$.

###  A.1 $D=2$

That (A.11) splits up in a total divergence term and a gauge invariant term is
seen immediately. Applying the Leibnitz rule once, (A.11) can be immediately
be expressed as

$\frac{1}{2}(\varrho_{G}-\varrho_{0})=\varepsilon_{ij}\left[\partial_{j}(\phi^{3}\
A_{i})-\frac{1}{2}\,\phi^{3}\,F_{ij}\right]\,,$ (A.16)

consisting of a total divergence and a gauge invariant term.

###  A.2 $D=3$

In this case

$\displaystyle\omega_{k}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\varepsilon_{ijk}\varepsilon^{fga}\,A_{j}^{fg}\,A_{i}\phi^{a}$
(A.17) $\displaystyle=$
$\displaystyle\frac{1}{2}\,\varepsilon_{ijk}\varepsilon^{fga}(A_{[i}A_{j]})^{fg}\,\phi^{a}$
$\displaystyle\omega_{k}^{(2)}$ $\displaystyle=$
$\displaystyle\varepsilon_{ijk}\varepsilon^{fga}\,A_{j}^{fg}\,\partial_{i}\phi^{a}$
(A.18) $\displaystyle=$
$\displaystyle\partial_{i}\Omega_{ik}-\frac{1}{2}\,\varepsilon_{ijk}\varepsilon^{fga}\,\phi^{a}\,\partial_{[i}A_{j]}^{fg}$

where, (A.17) has resulted from using the identity (A.10) for $D=3$ and
$\Omega_{ik}$ in (A.18) is defined as

$\Omega_{ik}=\varepsilon_{ijk}\varepsilon^{fga}\,(A_{j}^{fg}\,\phi^{a})\,.$
(A.19)

Thus adding $\omega_{k}^{(1)}$ and $\omega_{k}^{(2)}$ as in (A.15), which are
now given by (A.17) and (A.18), results in

$\frac{1}{3}(\varrho_{G}-\varrho_{0})=\partial_{k}\phi^{4}\left[\partial_{i}\Omega_{ik}-\frac{1}{2}\,\varepsilon_{ijk}\varepsilon^{fga}\,\phi^{a}\,F_{ij}^{fg}\right]$
(A.20)

which consists of a part that is gauge invariant, and a total divergence term
which is gauge variant, as seen in

$\frac{1}{3}(\varrho_{G}-\varrho_{0})=\partial_{k}\left[\phi^{4}\partial_{i}\Omega_{ik}\right]-\frac{1}{2}\,\partial_{k}\phi^{4}\,\varepsilon_{ijk}\varepsilon^{fga}\,\phi^{a}\,F_{ij}^{fg}$
(A.21)

or alternatively,

$\displaystyle\frac{1}{3}(\varrho_{G}-\varrho_{0})$ $\displaystyle=$
$\displaystyle\partial_{k}\left[\phi^{4}\partial_{i}\Omega_{ik}-\frac{1}{2}\,\phi^{4}\,\varepsilon_{ijk}\varepsilon^{fga}\,\phi^{a}\,F_{ij}^{fg}\right]+\frac{1}{2}\,\varepsilon_{ijk}\varepsilon^{fga}\phi^{4}\,D_{k}\phi^{a}\,F_{ij}^{fg}$
(A.22) $\displaystyle\equiv$
$\displaystyle\partial_{k}\Omega_{k}+\frac{1}{2}\,\varepsilon_{ijk}\varepsilon^{fga}\phi^{4}\,D_{k}\phi^{a}\,F_{ij}^{fg}\,,$
(A.23)

in the second line of which the vector valued density $\Omega_{k}$ is
preciselythe $D=3$ case of $\Omega_{i}$ appearing in (1.2) .

###  A.3 $D=4$

According to the notation in (A.15), $\omega_{l}^{(1)}$ and $\omega_{l}^{(2)}$
can be read from (A.13) as,

$\displaystyle\omega_{l}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\,\varepsilon_{ijkl}\varepsilon^{fgab}\,A_{k}^{fg}\,A_{i}\phi^{a}A_{j}\phi^{b}$
(A.24) $\displaystyle=$
$\displaystyle\frac{1}{3}\,\varepsilon_{ijkl}\varepsilon^{fgab}(A_{i}A_{j})^{fg}\left(3\phi^{a}A_{k}\phi^{b}+\frac{1}{2}|\vec{\phi}|^{2}A_{k}^{ab}\right)\,,\qquad|\vec{\phi}|^{2}=|\phi^{a}|^{2}$
$\displaystyle\omega_{l}^{(2)}$ $\displaystyle=$
$\displaystyle\varepsilon_{ijkl}\varepsilon^{fgab}\,A_{k}^{fg}\,\partial_{i}\phi^{a}\,D_{j}\phi^{b}$
(A.25) $\displaystyle=$
$\displaystyle\partial_{i}\Omega_{il}^{(1)}+\varepsilon_{ijkl}\varepsilon^{fgab}\,\partial_{i}A_{j}^{fg}\,\phi^{a}D_{j}\phi^{b}-\omega_{l}^{(a)}-\omega_{l}^{(b)}$

where

$\displaystyle\Omega_{il}^{(1)}$ $\displaystyle=$
$\displaystyle\varepsilon_{ijkl}\varepsilon^{fgab}\,A_{k}^{fg}\,\phi^{a}D_{j}\phi^{b}\,.$
(A.26)

The quantities $\omega_{l}^{(a)}$ and $\omega_{l}^{(b)}$ in (A.25) are
calculated to give

$\displaystyle\omega_{l}^{(a)}$ $\displaystyle=$
$\displaystyle\varepsilon_{ijkl}\varepsilon^{fgab}\,A_{k}^{fg}\,\phi^{a}\,\partial_{i}A_{j}\phi^{b}$
(A.27) $\displaystyle=$
$\displaystyle-\varepsilon_{ijkl}\varepsilon^{fgab}\,\partial_{i}A_{j}^{fg}\left(\phi^{a}A_{k}\phi^{b}+\frac{1}{2}|\vec{\phi}|^{2}\,A_{k}^{ab}\right)$

and

$\displaystyle\omega_{l}^{(b)}$ $\displaystyle=$
$\displaystyle\varepsilon_{ijkl}\varepsilon^{fgab}\,A_{k}^{fg}\,\phi^{a}\,A_{j}\partial_{i}\phi^{b}$
(A.28) $\displaystyle=$
$\displaystyle\frac{1}{2}\partial_{i}\Omega_{il}^{(2)}+\varepsilon_{ijkl}\varepsilon^{fgab}\left[\partial_{i}A_{j}^{fg}\left(\phi^{a}A_{k}\phi^{b}+\frac{1}{4}|\vec{\phi}|^{2}\,A_{k}^{ab}\right)-(A_{i}A_{j})^{fg}\phi^{a}\partial_{k}\phi^{b}\right]\,,$

the density $\Omega_{il}^{(2)}$ being

$\displaystyle\Omega_{il}^{(2)}$ $\displaystyle=$
$\displaystyle\varepsilon_{ijkl}\varepsilon^{fgab}\,A_{k}^{fg}\,\phi^{a}A_{j}\phi^{b}\,.$
(A.29)

The analogue of the density $\Omega_{ik}$ (A.19) for $D=3$, can be denoted as

$\displaystyle\Omega_{il}$ $\displaystyle=$
$\displaystyle(\Omega_{il}^{(1)}-\frac{1}{2}\Omega_{il}^{(2)})=\varepsilon_{ijkl}\varepsilon^{fgab}A_{k}^{fg}\,\phi^{a}\left(\partial_{j}\phi^{b}-\frac{1}{2}A_{j}\phi^{b}\right)\,.$
(A.30)

Substituting (A.27) and (A.28) in (A.25), $\omega_{l}^{(2)}$ is

$\displaystyle\omega_{l}^{(2)}$ $\displaystyle=$
$\displaystyle\partial_{i}\Omega_{il}+\varepsilon_{ijkl}\varepsilon^{fgab}\left\\{\partial_{i}A_{j}^{fg}D_{k}\phi^{b}+(A_{i}A_{j})^{fg}\phi^{a}\partial_{k}\phi^{b}+\frac{1}{4}|\vec{\phi}|^{2}\partial_{i}A_{j}^{fg}\,A_{k}^{ab}\right\\}\,.$
(A.31)

Adding (A.24) and (A.31) yields $(\omega_{l}^{(1)}+\omega_{l}^{(2)})$, and
contracting this with $\partial_{l}\phi^{5}$, we have
$(\varrho_{G}-\varrho_{0})$ as $per$ (A.15)

$\frac{1}{4}(\varrho_{G}-\varrho_{0})=\partial_{l}{\phi^{5}\partial_{i}\Omega_{il}}+\frac{1}{2}\,\varepsilon_{ijkl}\varepsilon^{fgab}\partial_{l}\phi^{5}\left\\{F_{ij}^{fg}\,\phi^{a}D_{k}\phi^{b}+\frac{1}{2}|\vec{\phi}|^{2}A_{k}^{ab}\left[\partial_{i}A_{j}^{fg}+\frac{2}{3}(A_{i}A_{j})^{fg}\right]\right\\}$
(A.32)

where in the $4$-th component of the second term inside the chain brackets can
be identfied with the Chern-Simons density in three dimensions.

The expression (A.32) for $(\varrho_{G}-\varrho_{0})$ is at the same stage as
that of its $D=3$ counterpart (A.20), but unlike the former, it is not yet a
sum of a total-divergence and gauge invariant term.

To render it to that form one extracts the partial derivative $\partial_{l}$
in (A.32). The gauge-invariant term yields

$\displaystyle\varepsilon_{ijkl}\varepsilon^{fgab}\partial_{l}\phi^{5}\,\\{F_{ij}^{fg}\,\phi^{a}D_{k}\phi^{b}\\}$
$\displaystyle=$
$\displaystyle\varepsilon_{ijkl}\varepsilon^{fgab}\bigg{\\{}\partial_{l}\left[\phi^{5}\,F_{ij}^{fg}\,\phi^{a}D_{k}\phi^{b}\right]$
(A.33)
$\displaystyle\qquad\qquad-\phi^{5}\left[F_{ij}^{fg}D_{k}\phi^{a}D_{l}\phi^{b}+\frac{1}{8}|\vec{\phi}|^{2}F_{ij}^{fg}F_{kl}^{ab}\right]\bigg{\\}}$

and the gauge-variant term yields

$\displaystyle\varepsilon_{ijkl}\varepsilon^{fgab}|\vec{\phi}|^{2}\partial_{l}\phi^{5}\left\\{A_{k}^{ab}\left[\partial_{i}A_{j}^{fg}+\frac{2}{3}(A_{i}A_{j})^{fg}\right]\right\\}$
$\displaystyle=$
$\displaystyle\varepsilon_{ijkl}\varepsilon^{fgab}\bigg{\\{}\partial_{l}\left[|\vec{\phi}|^{2}\phi^{5}A_{k}^{ab}\left(\partial_{i}A_{j}^{fg}+\frac{2}{3}(A_{i}A_{j})^{fg}\right)\right]$
(A.34)
$\displaystyle\qquad\qquad\quad\quad+\frac{1}{4}(\phi^{5}-\frac{1}{3}(\phi^{5})^{3})F_{ij}^{ab}F_{kl}^{cd}\bigg{\\}}$

one turns up with the final result in the required form

$\frac{1}{4}\,(\varrho_{G}-\varrho_{0})=\partial_{l}\Omega_{l}+\frac{1}{2}\,\varepsilon_{ijkl}\varepsilon^{fgab}\,\phi^{5}\left\\{F_{ij}^{fg}D_{k}\phi^{a}D_{l}\phi^{b}+\frac{1}{12}(\phi^{5})^{2}\,F_{ij}^{fg}F_{kl}^{ab}\right\\}$
(A.35)

where the vector valued density $\Omega_{l}$ is given by

$\Omega_{l}=\phi^{5}\partial_{i}\Omega_{il}+\varepsilon_{ijkl}\varepsilon^{fgab}\left\\{\frac{1}{2}\phi^{5}\,F_{ij}^{fg}\phi^{a}D_{k}\phi^{b}+\frac{1}{4}\left(\phi^{5}-\frac{1}{3}(\phi^{5})^{3}\right)A_{k}^{fg}\left[\partial_{i}A_{j}^{ab}+\frac{2}{3}(A_{i}A_{j})^{ab}\right]\right\\}$
(A.36)

which is precisely the $D=4$ case of $\Omega_{i}$ appearing in (1.2) .

###  A.4 D=5

According to the notation in (A.15), $\omega_{l}^{(1)}$ and $\omega_{l}^{(2)}$
can be read from (A.14) as,

$\displaystyle\omega_{m}^{(1)}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\,\varepsilon_{ijklm}\varepsilon^{fgabc}\,A_{l}^{fg}\,A_{i}\phi^{a}A_{j}\phi^{b}A_{k}\phi^{c}$
(A.37) $\displaystyle=$
$\displaystyle-2\,\varepsilon_{ijklm}\varepsilon^{fgabc}(A_{i}A_{j})^{fg}\,A_{l}\phi^{c}\left(\phi^{a}A_{k}\phi^{b}+\frac{1}{4}|\vec{\phi}|^{2}\,A_{k}^{ab}\right)$

and

$\displaystyle\omega_{m}^{(2)}$ $\displaystyle=$
$\displaystyle\partial_{i}\Omega_{im}^{(1)}-\varepsilon_{ijklm}\varepsilon^{fgabc}\partial_{i}A_{j}^{fg}\phi^{a}(2D_{k}\phi^{b}D_{l}\phi^{c}-\partial_{l}\phi^{c}A_{k}\phi^{b})-\omega_{m}^{(a)}-3\,\omega_{m}^{(b)}-4\,\omega_{m}^{(c)}$
(A.38)

where

$\displaystyle\Omega_{im}^{(1)}$ $\displaystyle=$
$\displaystyle\varepsilon_{ijklm}\varepsilon^{fgabc}A_{l}^{fg}\phi^{a}(2D_{j}\phi^{b}D_{k}\phi^{c}-\partial_{j}\phi^{b}A_{k}\phi^{c})$
(A.39)

and

$\displaystyle\omega_{m}^{(a)}$ $\displaystyle=$
$\displaystyle\varepsilon_{ijklm}\varepsilon^{fgabc}A_{k}^{fg}\phi^{a}(3\partial_{l}\phi^{b}+4A_{l}\phi^{b})\partial_{i}A_{j}\phi^{c}$
(A.40) $\displaystyle\omega_{m}^{(b)}$ $\displaystyle=$
$\displaystyle\varepsilon_{ijklm}\varepsilon^{fgabc}A_{k}^{fg}\phi^{a}\partial_{l}\phi^{b}\,A_{j}\partial_{i}\phi^{c}$
(A.41) $\displaystyle\omega_{m}^{(c)}$ $\displaystyle=$
$\displaystyle\varepsilon_{ijklm}\varepsilon^{fgabc}A_{k}^{fg}\phi^{a}A_{l}\phi^{b}\,A_{j}\partial_{i}\phi^{c}$
(A.42)

Here, $\omega_{m}^{(a)}$, $\omega_{m}^{(b)}$ and $\omega_{m}^{(c)}$ must be
calculated individually, as was done above for in the $D=4$ case above for
$\omega_{l}^{(a)}$ and $\omega_{l}^{(b)}$ in (A.27) and (A.28) respectively,.

The results of these calclations are listed here:

$\omega_{m}^{(a)}=\varepsilon_{ijklm}\varepsilon^{fgabc}\,\partial_{i}A_{j}^{fg}\left(\phi^{a}A_{k}\phi^{b}+\frac{1}{2}|\vec{\phi}|^{2}\,A_{k}^{ab}\right)(3\,\partial_{l}\phi^{c}+4\,A_{l}\phi^{c})$
(A.43) $\displaystyle\omega_{m}^{(b)}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\,\partial_{i}\Omega_{im}^{(2)}+\frac{2}{3}\,\varepsilon_{ijklm}\varepsilon^{fgabc}\bigg{\\{}-\partial_{i}A_{j}^{fg}\,\partial_{l}\phi^{c}\left(\phi^{a}A_{k}\phi^{b}+\frac{1}{4}|\vec{\phi}|^{2}\,A_{k}^{ab}\right)$
(A.44)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+(A_{i}A_{j})^{fg}\,\phi^{a}\partial_{k}\phi^{b}\partial_{l}\phi^{c}\bigg{\\}}$

where

$\Omega_{im}^{(2)}=\varepsilon_{ijklm}\varepsilon^{fgabc}A_{l}^{fg}\phi^{a}\,\partial_{j}\phi^{b}A_{k}\phi^{c}$
(A.45)

and

$\displaystyle\omega_{m}^{(c)}$ $\displaystyle=$
$\displaystyle\frac{1}{3}\partial_{i}\Omega_{im}^{(3)}+\frac{1}{3}\,\varepsilon_{ijklm}\varepsilon^{fgabc}\bigg{\\{}-\partial_{i}A_{j}^{fg}\,A_{l}\phi^{c}\left(3\phi^{a}A_{k}\phi^{b}+|\vec{\phi}|^{2}\,A_{k}^{ab}\right)$
(A.46)
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+(A_{i}A_{j})^{fg}\,\partial_{l}\phi^{c}\left(3\phi^{a}A_{k}\phi^{b}+\frac{1}{2}|\vec{\phi}|^{2}\,A_{k}^{ab}\right)$

where

$\Omega_{im}^{(3)}=\varepsilon_{ijklm}\varepsilon^{fgabc}A_{l}^{fg}\phi^{a}\,A_{j}\phi^{b}A_{k}\phi^{c}$
(A.47)

Substituting (A.43), (A.44) and (A.46) into (A.38)

$\displaystyle\omega^{(2)}_{m}$ $\displaystyle=$
$\displaystyle\partial_{i}\Omega_{im}-2\varepsilon_{ijklm}\varepsilon^{fgabc}\bigg{\\{}\partial_{i}A_{j}^{fg}\left[\phi^{a}\,D_{k}\phi^{b}D_{l}\phi^{c}+\frac{1}{2}\,|\vec{\phi}|^{2}\,A_{k}^{ab}\left(\partial_{l}\phi^{c}+\frac{2}{3}\,A_{l}\phi^{c}\right)\right]$
(A.48)
$\displaystyle\qquad\qquad\qquad\qquad\quad+(A_{i}A_{j})^{fg}\left[\phi^{a}\partial_{k}\phi^{b}(\partial_{l}\phi^{c}+2\,A_{l}\phi^{c})+\frac{1}{3}\,|\vec{\phi}|^{2}\,A_{k}^{ab}\,\partial_{l}\phi^{c}\right]\bigg{\\}}$

where

$\displaystyle\Omega_{im}$ $\displaystyle=$
$\displaystyle\Omega_{im}^{(1)}-\Omega_{im}^{(2)}-\frac{4}{3}\Omega_{im}^{(3)}$
(A.49) $\displaystyle=$ $\displaystyle
2\varepsilon_{ijklm}\varepsilon^{fgabc}\,A_{l}^{fg}\,\phi^{a}(\partial_{j}\phi^{b}\partial_{k}\phi^{c}+\partial_{j}\phi^{b}A_{k}\phi^{c}+\frac{1}{3}\,A_{j}\phi^{b}A_{k}\phi^{c})$

in which $\Omega_{im}^{(1)}$, $\Omega_{im}^{(2)}$ and $\Omega_{im}^{(3)}$ are
given in (A.39), (A.45) and (A.47) respectively, and $\Omega_{im}$ is the
$D=5$ analogue of $\Omega_{il}$ given by (A.30) in $D=4$.

Adding $\omega_{m}^{(1)}$ given in (A.37), to $\omega_{m}^{(2)}$ given in
(A.48),

$\displaystyle\omega_{m}^{(1)}+\omega_{m}^{(2)}$ $\displaystyle=$
$\displaystyle\partial_{i}\Omega_{im}-\varepsilon_{ijklm}\varepsilon^{fgabc}\left\\{F_{ij}^{fg}\,\phi^{a}\,D_{k}\phi^{b}D_{l}\phi^{c}+|\vec{\phi}|^{2}\,\Xi_{ijkl}^{fgabc}\right\\}$
(A.50)

where

$\displaystyle\Xi_{ijkl}^{fgabc}$ $\displaystyle=$ $\displaystyle
A_{k}^{ab}\left[(\partial_{l}\phi^{c}+\frac{2}{3}\,A_{l}\phi^{c})\,\partial_{i}A_{j}^{fg}+\frac{2}{3}(\partial_{l}\phi^{c}+\frac{3}{4}\,A_{l}\phi^{c})(A_{i}A_{j})^{fg}\right]\,,$
(A.51)

so that the term

$\varepsilon_{ijklm}\varepsilon^{fgabc}\,\Xi_{ijkl}^{fgabc}$

in (A.50) is the analogue of the term

$\varepsilon_{ijkl}\varepsilon^{fgab}|\vec{\phi}|^{2}A_{k}^{ab}\left[\partial_{i}A_{j}^{fg}+\frac{2}{3}(A_{i}A_{j})^{fg}\right]$

appearing in the corresponding expression (A.32) in $D=4$, whose component
$l=4$ is the Chern-Simons (CS) density in $3$ dimensions. As such, the $m=5$
component of the density (A.51) is the $4$ dimensional analogue of the Chern-
Simons density in $3$ dimensions.

As $per$ (A.15), (A.50) implies

$\displaystyle\frac{1}{5}(\varrho_{G}-\varrho_{0})$ $\displaystyle=$
$\displaystyle\partial_{m}\phi^{6}\partial_{i}\Omega_{im}-\varepsilon_{ijklm}\varepsilon^{fgabc}\partial_{m}\phi^{6}\left\\{F_{ij}^{fg}\,\phi^{a}\,D_{k}\phi^{b}D_{l}\phi^{c}+|\vec{\phi}|^{2}\,\Xi_{ijkl}^{fgabc}\right\\}$
(A.52)

This expression is the $D=5$ analogue of (A.32) in $D=4$. The first term,
$\partial_{m}(\phi^{6}\partial_{i}\Omega_{im})$, is a total divergence .
Inside the chain bracket the first term is gauge invariant, but the second
term in front of $|\vec{\phi}|^{2}$ is neither gauge invariant nor total
divergence, which is unsatisfactory from the viewpoint of casting this
expression in the form (A.7), in terms of a gauge invariant quantity $W$ and a
total divergence $\partial_{i}\Omega_{i}$. This can be effected by pulling out
333Strictly speaking this does not have to involve the first term inside the
chain bracket of (A.52) since that term is gauge invariant as it stands. But
we opt to do that since that results in a more useful expression for later
application of Bogomol’nyi type inequalities. the partial derivative
$\partial_{m}$ as was done in the $D=4$ case.

But this case being more complex than the $D=4$, it is helpful to treat the
gauge invariant term and the gauge variant term (in front of
$|\vec{\phi}|^{2}$) separately.

Pulling out $\partial_{m}$ from the gauge invariant term in (A.52) results in
the expression

$\displaystyle\varepsilon_{ijklm}\varepsilon^{fgabc}\partial_{m}\phi^{6}\,F_{ij}^{fg}\,\phi^{a}\,D_{k}\phi^{b}D_{l}\phi^{c}=\varepsilon_{ijklm}\varepsilon^{fgabc}\cdot$
$\displaystyle\qquad\qquad\cdot\bigg{\\{}\partial_{m}\left[\phi^{6}\,F_{ij}^{fg}\phi^{a}D_{k}\phi^{b}D_{l}\phi^{c}+\frac{1}{12}(\phi^{6})^{3}\,F_{ij}^{fg}\,F_{kl}^{ab}\phi^{c}\right]$
$\displaystyle\qquad\qquad\qquad-\phi^{6}\left[F_{ij}^{fg}D_{k}\phi^{a}D_{l}\phi^{b}D_{m}\phi^{c}+\frac{1}{4}\left(1-\frac{2}{3}(\phi^{6})^{2}\right)F_{ij}^{fg}\,F_{kl}^{ab}D_{m}\phi^{c}\right]\bigg{\\}}\,.$
(A.53)

This expression is the $D=5$ analogue of (A.33) in $D=4$.

Doing the same with the gauge variant term results in

$\varepsilon_{ijklm}\varepsilon^{fgabc}\partial_{m}\phi^{6}|\vec{\phi}|^{2}\,\Xi_{ijkl}^{fgabc}=\varepsilon_{ijklm}\varepsilon^{fgabc}\left\\{\partial_{m}\left[\left(\phi^{6}-\frac{1}{3}(\phi^{6})^{3}\right)\Xi_{ijkl}^{fgabc}\right]-\left(\phi^{6}-\frac{1}{3}(\phi^{6})^{3}\right)\partial_{m}\Xi_{ijkl}^{fgabc}\right\\}\,,$
(A.54)

in which the divergence term is calculated to be

$\varepsilon_{ijklm}\varepsilon^{fgabc}\,\partial_{m}\Xi_{ijkl}^{fgabc}=\frac{1}{4}\varepsilon_{ijklm}\varepsilon^{fgabc}\left[F_{ij}^{fg}F_{kl}^{ab}D_{m}\phi^{c}-4\,(A_{i}A_{j})^{fg}(A_{k}A_{l})^{ab}A_{m}\phi^{c}\right]\,,$
(A.55)

which is the counterpart of the divergence

$-\varepsilon_{ijkl}\varepsilon^{fgab}\,A_{k}^{ab}\left[\partial_{i}A_{j}^{fg}+\frac{2}{3}(A_{i}A_{j})^{fg}\right]=\frac{1}{4}\varepsilon_{ijkl}\varepsilon^{fgab}\,F_{ij}^{fg}F_{kl}^{ab}\,,$
(A.56)

in $D=4$, which is gauge invariant.

The quantity

$\Xi=\varepsilon_{ijklm}\varepsilon^{fgabc}\,(A_{i}A_{j})^{fg}(A_{k}A_{l})^{ab}A_{m}\phi^{c}$
(A.57)

in (A.55) is manifestly gauge variant. This means that in $D=5$, it is not
possible to cast $(\varrho_{G}-\varrho_{0})$ in terms of $W$ and
$\partial_{i}\Omega_{i}$ as in (A.7), for the $SO(D)$ gauged $O(D+1)$ Skyrme
scalar. It may be reasonable to expect that this will be the case in all
dimensions $D\geq 5$, at least for $D$ odd.

Fortunately, it is possible to cast $(\varrho_{G}-\varrho_{0})$ in terms of
$W$ and $\partial_{i}\Omega_{i}$ as in (A.7), if the $SO(5)$ gauge connection
is suitably 444The contractions to $SO(2)$, $SO(3)$ and $SO(4)$ are possible
but not suitable, since connections to single subgroups do not mix “electric”
and “magnetic” fields. contracted. Two such contractions are

$SO(2)\times SO(2)\quad{\rm and}\quad SO(2)\times SO(3)$

$\displaystyle A_{i}^{ab}$ $\displaystyle=$
$\displaystyle(A_{i}^{\alpha\beta}\in so_{a}(2),A_{i}^{AB}\in
so_{b}(2),A_{i}^{\alpha A},\ A_{i}^{\alpha 5},\ A_{i}^{A5})\,\quad\alpha=1,2\
{\rm and}\ A=3,4$ (A.58) $\displaystyle\equiv$
$\displaystyle(A_{i}^{\alpha\beta}=a_{i}\,\varepsilon^{\alpha\beta}\
,A_{i}^{AB}=\ b_{i}\,\varepsilon^{AB},\ A_{i}^{\alpha A}=0\ ,\ A_{i}^{\alpha
5}=0\ ,\ A_{i}^{A5}=0)$

while in the second case it is

$\displaystyle A_{i}^{ab}$ $\displaystyle=$
$\displaystyle(A_{i}^{\alpha\beta}\in so(2),A_{i}^{AB}\in so(3),A_{i}^{\alpha
A})\,\quad\alpha=1,2\ {\rm and}\ A=3,4,5$ (A.59) $\displaystyle\equiv$
$\displaystyle(A_{i}^{\alpha\beta}=a_{i}\,\varepsilon^{\alpha\beta}\ ,\
A_{i}^{AB}\in so(3)\ ,\ A_{i}^{\alpha A}=0)$

Thus, subject to either one of the contractions (A.58) or (A.59), the
divergence term (A.55) is gauge invariant and hence
$(\varrho_{G}-\varrho_{0})$ in (A.52) can be cast in the form (A.7), namely as
a difference of a total divergence term and a gauge invariant term.