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Toronto Arrows Best Players of All Time
Leave a Comment / Rugby / By John Rizzo
The Toronto Arrows are an exciting new addition to the Major League Rugby landscape. Founded in 2018, they represent a vibrant rugby union community centered around Ontario's provincial team – the Blues. Led by a passionate group of private investors and supporters, this professional side looks to bring top-level competition right into Canada's biggest city. With their innovative style of play and commitment to creating lasting relationships with fans across North America, the Toronto Arrows have quickly become one of MLR's most beloved teams.
1. Lucas Rumball
Lucas Rumball was born on 2 August 1995 in Markham, Ontario. He played rugby union for the Toronto Arrows from 2016-2018. Lucas is a back row player and has represented Canada at U19 level and at the World Rugby Under 20 Championships in 2017.
In 2018, he made his debut for the Maple Leafs' AHL affiliate, the Marlies, playing against Hartford Wolf Pack. Currently (2019), he is contracted to play with Toronto Arrows of Super Rugby league in New Zealand until 2021 Lucas Rumball is a Canadian rugby union player who has played for the Toronto Arrows in both the PRO14 and Heineken Cup.
Born on July 26th, 1992, Lucas Rumball first picked up a rugby ball at just three years old and quickly developed into one of Canada's most promising young players. After impressing with his performances for club side Durham RFC, Rumball made his debut for Canada in 2016 against Georgia. Since then he has gone on to make 35 appearances for his country, scoring 20 points from all over the backline including five tries during qualification for Rugby World Cup 2019 where they were narrowly beaten by New Zealand in the semi-finals.
With an impressive list of accolades under his belt - such as Young Player of The Year (2015), RPA Players' Player Of The Year (2016) and IRB International Sevens Series MVP award (2017) - it is clear that Lucas Rumball is one of the brightest rising stars in world rugby today.
2. Sam Malcolm
Sam Malcolm is a rugby union player who has represented New Zealand at youth level and now plays for the Kamaishi Seawaves in Japan. He started playing rugby as a child, taking up the sport because he enjoyed running with his brothers and cousins.
After high school, Sam decided to pursue a professional career in rugby instead of continuing his studies and took up an opportunity to play for Kamaishi Seawaves in Japan. In 2018, he made his debut for the national side and scored two tries during their win over Samoa later that year.
Sam is currently contracted to stay with the team until 2020, after which he plans on returning home to continue his Rugby Union career with Wanganui High School or another NZ Super Rugby club.. Sam Malcolm is a professional rugby player who plays for the Toronto Arrows in the World Rugby League.
He has played for Manawatu in New Zealand and was part of their successful 2015–2019 campaign. In 2019, he helped guide Toronto to their first-ever trophy by winning the West Harbour Trophy. Malcolm is also an important member of Kamaishi Seawaves, having made 17 appearances since joining them in 2017.
He has been praised for his skillful play on both sides of the ball and his work ethic on and off the pitch - making him one of Canada's most promising players at international level
3. Mike Sheppard
Mike Sheppard is a rugby union player who has played for the Toronto Arrows since 2019. He was born in 1988 and attended McMaster University. Mike has made 17 appearances for the Toronto Arrows so far this season, scoring 25 points.
In addition to his rugby career, Mike also works as a software engineer. Mike is 6' 4" tall and weighs 245 pounds (111 kilograms). Mike Sheppard is a professional rugby player who has played for both the Ontario Blues and Canada A teams.
He started his career with the Ontario Blues in 2013, playing there until 2016 before joining Canada A two years later. In 2018, he helped Canada to qualify for the Rugby World Cup 2019 where they reached the Quarter Finals but were beaten by Fiji.
Mike is an experienced player who will be a key part of Canada's squad at this year's World Cup as they look to win their first ever title.
4. Ross Braude
Ross Braude is a rugby union player who hails from Pretoria, South Africa. He made his debut for the Blue Bulls in 2016 and has since played over 50 matches for them.
Ross is a scrum-half and has been praised by commentators for his skill on the pitch. His biggest achievement to date may be helping the Blue Bulls win their first Super Rugby title in 2018.
Ross is currently studying Business Administration at University of Cape Town and plans to start playing professional rugby again after graduating in 2020. Ross Braude is a professional Canadian soccer player who plays for the Toronto Arrows.
Ross Braude has had an illustrious career thus far, playing for some of the best teams in North America and Europe. Ross Braude has been a regular starter for both Canada and Toronto Arrows, impressing with his ball control and passing ability.
With two years left on his contract with the Arrows, Ross Braude will be looking to continue his good form into future seasons
5. Cole Keith
Cole Keith is a prop for the Toronto Arrows and was born in Saint John, New Brunswick. He made his debut with the Arrows in 2019 and has played 30 games so far this season.
Cole Keith has been playing rugby union since he was just seven years old and has amassed over 100 caps for his country to date. His hard work and determination have seen him achieve great things both on and off the pitch, with many predicting big things from him in future seasons Cole Keith is a professional soccer player who plays for Atlantic Rock in the Canadian Premier League.
He was born on December 12, 1994 in Victoria, Canada and played youth soccer with local clubs before signing his first pro contract with USL side Orange County Blues in 2016. In 2018, he signed with Atlantic Rock and helped them reach the playoffs that year as well as win their first-ever CPL title in 2019. Keith has represented Canada at U20 level but is yet to make an appearance for the senior team.
6. Mason Flesch
Mason Flesch was born on 18 November 1999 in Cobourg, Ontario. He is 1.96 metres tall and weighs 105 kilograms. Mason started playing rugby union at the age of six for his local club and has since played for several provincial teams in Canada including the Toronto Blues, Quebec Bleus and Ottawa Redblacks .
In 2018 he made his debut for the Canadian national team as a flanker during their tour of Europe . Mason is expected to be a key player for Canada at this year's World Rugby U20 Championship where they will face off against France, Italy and Spain in Group A . Mason Flesch is a professional ice hockey player who currently plays for the Toronto Arrows in the American Hockey League (AHL).
Mason was drafted by the Los Angeles Kings in the 4th round of the 2016 NHL Entry Draft. Mason has played with both Team Canada and their U20 team at various international tournaments including World Juniors, IIHF World Championship and Spengler Cup. Mason signed a three-year contract extension with Toronto on August 3rd, 2021 which will keep him with them until 2024.
7. Rob Brouwer
Rob Brouwer was born in Lindsay, Ontario and started playing rugby union at the age of 10. He played for Lindsay Rugby Club before moving to Canada's national team and then onto Glasgow Warriors in Scotland.
Rob has also played for Tonga, France and England A teams during his career. In 2013, he won a Scottish Premiership title with Glasgow Warriors and was named European Player of the Year that same year. Rob retired from professional rugby in 2017 but continues to play for local clubs across North America as well as representing Canada on international level again Rob Brouwer is a Canadian professional ice hockey player who currently plays for the Toronto Arrows in the Ontario Hockey League (OHL).
Rob was drafted by the Toronto Maple Leafs in 2009, but he has spent most of his career with the Arrows. He has played 176 games with Toronto, scoring 71 goals and 143 assists. In 2018-19, he led all OHL players in points and helped lead the Arrows to their first ever Eastern Conference championship. Rob is also a member of Canada's national junior team, having played at both World Juniors Championships and IIHF World Championship tournaments. He is considered one of the top prospects for next year's NHL Entry Draft and looks set to be a key part of any team that selects him
8. Giuseppe du Toit
Giuseppe du Toit is a Canadian professional rugby player who plays for the Toronto Arrows in the PRO Rugby Championship. Du Toit has previously played for Canada's national team, and made his debut in 2015 against Italy.
He was also part of the squad that won silver at the 2017 World Cup. Du Toit attended Brigham Young University where he played American football as well as rugby union. After graduating, he turned to rugby full-time and joined Ontario before signing with Toronto in 2017.
Du Toit is an explosive backrower who excels at running hard across the field and making big plays on defense; both qualities which have seen him become a key member of Toronto's attack over recent seasons.. Giuseppe du Toit is a South African fly-half who currently plays for Toronto Arrows in Major League Rugby.
Du Toit has represented his country at multiple youth levels, and made his senior debut in 2016. He quickly emerged as one of the leading lights in the Pro12, winning both the title and player of the year award in 2017 with Ulster. In 2018, he moved to North America to ply his trade with Toronto Arrows - becoming only the second overseas player ever signed by MLR side San Francisco Deltas (the other being compatriot Handre Pollard).
Du Toit has been an instant success with Toronto; scoring 212 points from 11 tries throughout 18 matches this season – good for sixth place on the league's table overall. A strong finisher around goal, du Toit will be looking to add another trophy to his cabinet when Toronto take on Seattle Storm at KeyBank Stadium tomorrow evening (Friday 7th December).
9. Andrew Ferguson
Andrew Ferguson was born on 1 May 1992 in Mississauga, Ontario. He attended Lorne Park Secondary School and studied at McMasters University where he played rugby union for the university's team.
In 2011, Andrew made his debut with Oakville Crusaders of the Canadian Rugby Union (CRCU). The following year he joined Toronto Arrows and has since been playing regularly for them in the PRO14 competition.
Outside of rugby union, Andrew is involved with a number of charity organisations and is also pursuing a career as a personal trainer. Andrew Ferguson is a Canadian rugby player who currently plays for the Toronto Arrows in the PRO Rugby league.
Previously, he played for both Ontario Blues and Canada u20s and was part of their successful campaigns at both U20 World Cups and the Americas Pacific Challenge. Andrew has also represented his country at senior level, making one appearance off the bench in 2016 against Tonga.
He attended McGill University where he studied business administration before turning to rugby full-time in 2014. In 2018, Andrew was named captain of Toronto Arrows following injuries to other players ahead of their season opener against Seattle Seawolves; they would go on to win that match 34-26 despite losing influential back rower Phil Mackenzie midway through proceedings due to injury himself later on in the game (he would return).
A versatile player capable of playing anywhere across the front line, Andy's ambition is clear – he wants every match to be as competitive as possible so that Toronto can achieve their ultimate goal which is qualifying for Super Rugby competition one day
10. Spencer Jones
Spencer Jones is a 25-year-old Canadian rugby union player who plays as a centre. He has played for Canada at Under 20, Under 23 and Junior levels before making his full international debut in 2018.
In March 2019 he was named in the squad for Canada's Rugby World Cup campaign. Spencer grew up playing football but decided to switch to rugby when he saw how physical the sport was and found that it fit well with his competitive nature.
He currently plays for London Irish RFC in England's Premiership Division 1 where he has made 21 appearances so far this season scoring five tries including two during Ireland's historic victory over New Zealand in November 2018 which sealed their first ever home Six Nations Championship title since 1905.
Spencer is an ambitious footballer who sees potential success both domestically and on the world stage with Canada one day – something which drives him every single day while training and competing. Spencer Jones is a professional soccer player who plays for Toronto Arrows in the Canadian Premier League. He first came to prominence as a youth player, playing for Manchester United's academy and making several appearances for their reserves team before moving on to play for Welsh club Bangor City.
After appearing sparingly with Bangor, Spencer joined Toronto in 2018 where he has since made an impact, scoring 23 goals in 25 apps this season. With Canada set to compete at the international level next year, Spencer will be looking to build on his strong debut campaign with the Arrows and continue his progress towards representing his country at senior level.
11. Andrew Quattrin
Andrew Quattrin is a 26-year-old hooker from Canada who played for the Aurora Barbarians in the Canadian Rugby Union (CRU) Premier Division. Andrew started playing rugby at a young age and was quickly picked up by his local club, the Laurier Golden Hawks Rugby Club.
In 2016, he signed with French side Racing 92 where he made five appearances during their 2016/17 campaign before returning to Canada to rejoin the Laurier Golden Hawks for their 2017 season. He has since been named in Canada's squad for their upcoming tour of Europe which includes games against Scotland, Wales and Ireland this November as well as fixtures against Italy and Romania later in 2018.
With over 100 appearances under his belt across various levels of competition, Andrew is looking forward to continuing his development at home with the Laurier Golden Hawks in 2018/19 while also eyeing opportunities abroad should they arise once more injury setbacks are behind him.. Andrew Quattrin played football for the University of Toronto. He then went on to play professionally in Germany and Austria before coming back to Canada to play with the Edmonton Eskimos.
In 2018, he was signed by the Toronto Arrows as a replacement for Jordan Hamilton who had moved onto Montreal Impact. Andrew Quattrin has been an important part of the team since his arrival and is currently leading all defenders in points scored with 48 total points (20 goals and 28 assists). His contract with the Arrows expires at the end of this season, but there are rumours that he could be joining another Canadian club soon after talks between him and the coaching staff have begun already
12. Gastón Mieres
Gastón Mieres was born in Uruguay on October 5, 1989. He started playing rugby union at the age of 14 and has represented his country at both junior and senior level.
In 2018, Gastón Mieres signed for Italian side Valpolicella as their new full-back. In 2016 he moved to England to play for Coventry Bears where he helped them win promotion from National League 2 North that season.
As of January 2019, Gastón Mieres is still contracted with Coventry Bears for the 2020/21 season Gastón Mieres is a professional rugby player who plays for the Toronto Arrows in the Canadian Rugby Championship. Gastón was born on January 3, 1995 in Montevideo, Uruguay.
He played club rugby with Club Atlético San Martín de los Andes before joining Racing 92 from France as an 18-year-old in 2015. In his first season at Racing 92 he made 15 appearances and scored two tries for their Top 14 side before being signed by Toronto Arrows ahead of the 2017 season where he has since made over 100 appearances across all formats of the game including Premiership Rugby and European competitions such as Champions Cup and Challenge Cup .
He represented Uruguay at Under 20 level but did not make any senior debut until 2018 when he appeared against Denmark scoring a try which secured victory for Uruguay to seal their place at 2019 World Cup after finishing second in Americas qualifying tournament behind Brazil beating Canada 29–19 on aggregate scoreline His father also played professionally for several clubs playing mainly midfield or centre back including Peñarol, Liverpool (England) and River Plate among others while mother is Uruguayan whilst having British passport due to his father's nationality
13. Jamie Mackenzie
Jamie Mackenzie is a Canadian rugby union player who plays as a scrum-half. He played for the University of Victoria before joining Toronto Arrows in 2016.
In his debut season with the Arrows, he scored four tries in thirteen appearances. Jamie also has experience playing club rugby for Oakville RFC and Durham Rugby Club, both in Canada.
Having been born on 28 February 1989, Jamie is just 33 years old and has many years of international rugby ahead of him. Jamie Mackenzie is a rugby player who has represented Canada on several occasions.
He started playing rugby at the age of 12 and quickly progressed through the ranks, becoming a key member of Esher RFC in 2010. In 2013, Jamie Mackenzie was called up to the Canadian national team for their tour of South Africa, where he made his debut against Zimbabwe.
Since then, Jamie Mackenzie has gone on to make 19 appearances for Canada and scored 5 tries during that time span. He also featured prominently for Toronto Arrows in their inaugural season in Major League Rugby (MLR), captaining them until they were relegated at the end of the 2019 season
14. Siaki Vikilani
Siaki Vikilani was born on 7 August 2000 in Canada. Siaki Vikilani is a flanker and number 8 for the Canadian national rugby union team. Siaki Vikilani has played for multiple clubs throughout his career, most notably Toronto Wolfpack and London Irish RFC.
In 2018, Siaki Vikilani helped lead the Canadian side to a win at the Americas Rugby Championship Division I tournament final against Uruguay。 As of 2019, Siaki Vikilani is currently contracted with Premiership side Gloucester Rugby 。 Siaki Vikilani is a professional rugby player who plays for the Toronto Arrows in Major League Rugby.
Siaki has represented Canada at youth level, and made his senior debut for the national team in 2019. In 2021, he will be part of Canada's squad at the World Cup Qualifier tournament. Siaki is a hard-working midfielder who likes to get involved in attacking play; he is an important member of the Arrows' squad.
15. Will Kelly
Will Kelly started playing rugby union in his teenage years and quickly became a standout player. He joined the Toronto Arrows in 2019, and has already made an impact with his accurate passing and powerful running.
Kelly is a graduate of McMaster University where he studied business management. He is also a talented footballer who played for Wales youth teams before making the switch to rugby union full-time at age 20. With impressive statistics thus far, there's no doubt that Will Kelly will continue to make waves on the international stage for many years to come.
Will Kelly is a Canadian soccer player who has been with the Toronto Arrows since 2019. 2. Kelly made his senior international debut for Canada in 2017, and has appeared at youth levels as well. 3. He played college soccer at Clemson University before joining the Arrows full time in 2019. 4. Kelly is an athletic midfielder who can play on both sides of the ball, making him a valuable member of any team he joins up with next season.
16. Tomás de la Vega
Tomás de la Vega was born on September 28, 1990 in Buenos Aires, Argentina. De La Vega began his rugby career with CUBA before moving to Pampas XV in 2013.
In 2018, he joined the Toronto Arrows and has since made 10 appearances for them. De La Vega is a flanker who predominantly plays as a replacement or off the bench but can also play at number 8 if needed.
He is considered one of the most promising young players in Argentine rugby and is likely to continue playing at a high level for some time to come Tomás de la Vega is a football (soccer) player who has played for Argentina and club side Huracán.
He made his international debut in 2009 at the age of 18, playing for the Argentine under-20 team. In 2012 he was transferred to Huracán where he spent two seasons before joining Racing Club on a permanent basis in 2014. A powerful striker with good speed, De La Vega has been described as "an ideal fit" by Racing manager Luis Fernando Tena due to his ability to score goals from distance as well as close range shots.ESPN Stats & Info: Tomás de la Vega Profile De La Vega started all six of Argentina's matches at the 2013 Copa America Centenario tournament which saw them reach the final where they were defeated 2–1 by Chile.
17. Isaac Salmon
Isaac Salmon was born on 5 September 1996 in Christchurch, New Zealand. He played rugby union for Nelson College before moving to play professionally with the Tasman and Toronto Arrows teams.
Isaac is a prop who has been described as "strong and powerful". In May 2018, he re-signed with the Toronto Arrows team until the end of 2020. Isaac Salmon is a professional rugby union player who currently plays for Toronto Arrows in Super Rugby.
He made his senior debut for Tasman in 2017 and has since gone on to score 38 points from 20 appearances for the team. Salmon was also part of the Canada national side, playing two tests during their tour of South Africa in 2018. The 26-year-old is an explosive runner with good passing skills, making him a valuable member of any side he joins next season.
18. Ollie Nott
Ollie Nott was born in Leicester, United Kingdom on 26 May 1995. He attended Shawnigan Lake School and then studied at the University of Victoria where he played rugby union for the UVic Vikes.
Ollie made his debut for England under-20s in March 2016 and has since gone on to play for their senior side. In November 2018, it was announced that Ollie had signed with French club Montauban following his release from Premiership side Wasps RFC..
Ollie is a flanker and has been described as "very athletic". Outside of rugby, Ollie enjoys playing outdoors including cricket and basketball . He is married to Grace Bullock and they have one daughter together named Evangeline Rosemary Nott (born 30 January 2019).
Ollie Nott is a professional rugby player who has played for the Toronto Arrows in the Canadian Rugby League. Nott's senior career started with Canada U20s where he made nine appearances and scored five tries. He followed that up by playing for Canada in 2017, making two appearances but not scoring any points.
After retiring from international play, Nott will now focus on his club career with the Toronto Arrows where he is likely to feature prominently in their 2021 season campaign.
19. Gordon McRorie
Gordon McRorie is a rugby union player who has played for Hong Kong and the Scotland national team. He was born in 1988 in Hong Kong and began playing rugby at an early age.
Gordon made his senior debut for the Scottish national team in 2010, and has since gone on to play at several World Cups, including this year's event in France. Gordon is often referred to as one of the most creative scrum halves in world rugby, due to his ability to get himself into good positions on the field and set up attacks for his teammates.
Gordon is married with two young children, and he enjoys spending time outdoors fishing or skiing when he isn't involved in football or rugby training sessions. Gordon McRorie is a professional rugby player who has played for several teams over the years. Gordon was born in Canada and has represented his country on multiple occasions. He first came to prominence with Rugby Milano, where he spent three seasons playing alongside players such as Tommaso Boni, Matteo Minutoli and Carlo Canna.
After leaving Rugby Milano, Gordon joined Northampton Saints where he made 66 appearances between 2014-2018 before moving onto Montpellier this season. Gordon is currently one of the most experienced players in French Top 14 rugby and is regarded as one of the best scrummers in Europe today. Gordon has been nominated for both European Player of the Year (2019) and World Player of the Year (2019) Awards but has yet to win either award - hopefully he can achieve that goal during his time at Montpellier next season.
20. Kyle Baillie
Kyle Baillie is a Canadian rugby union player who has played for Westshore RFC, Halifax Tars and Castaway Wanderers. He was born on April 7, 1991 in St.
Albert, Alberta and attended Three Oaks High School before turning professional in 2015. Kyle has been a regular member of the Canada national team since making his debut in 2011 and has earned 38 caps to date.
He made his international breakthrough at the 2013 Americas Rugby Championship where he helped Canada finish fourth after scoring two tries in five appearances off the bench as a replacement/substitute.. Kyle also represented Canada at the 2014 World Cup Qualifier where they finished second behind Uruguay but qualified for their first ever Rugby World Cup thanks to being one of only six teams with three points or more from their qualification group stage matches (Canada had won all four games).
At the 2017 Americas Rugby Championship, Kyle started all seven of Canada's matches including their final defeat against Brazil which denied them an automatic place at this year's World Cup tournament held in Japan . Outside of rugbyunion, Kyle enjoys playing ice hockey and football professionally while completing his undergraduate degree at Mount Royal University Kyle Baillie first came to prominence as a rugby player in Canada, playing for the Toronto Arrows. He then moved over to England, where he represented Saracens and London Scottish before joining Ohio Aviators midway through the 2017 season.
In 2019, Kyle signed with New Orleans Gold of the American Rugby Premiership (APR). Kyle is an imposing figure on the field and has been described as one of the most physical players in North America. His determination and commitment have seen him overcome several difficult injuries throughout his career. Kyle is currently contracted with New Orleans until 2021 . Kyle is a hardworking player who always gives 100% during every game he plays; whether it be for Ohio Aviators or any other team he's played for
21. Pat Parfrey
Pat Parfrey is a rugby union fly half who has played for Toronto Arrows in the Canadian Rugby Union. Pat started playing rugby at the age of seven and quickly became one of Canada's most promising young players, representing his country at youth level before making his senior debut in 2019.
His goal is to help Toronto Arrows win titles both domestically and internationally, and he is already looking forward to next season with renewed vigour after injury kept him out of action for much of this year. Pat's natural flair for the game has seen him become a popular figure within the Arrows squad, and he credits his family - including father Terry, mother Julie, sister Shannon and brother Rhett - as major influences throughout his career so far.
Outside of rugby, Pat enjoys cooking healthy meals for himself and friends as well as spending time outdoors hiking or skiing in wintertime Pat Parfrey is a retired Canadian ice hockey player. He played for Canada's national team from 2013 to 2014 and amassed 30 points in 10 apps. After his retirement, he served as an assistant coach with the Calgary Flames of the NHL until 2018-19 season when he was fired after posting a 5-23-0 record in 50 games..
22. Leandro Leivas
Leandro Leivas was born on July 6, 1988 in Brazil. He is 1.80 meters tall and weighs 94 kilograms. After playing rugby union for a number of years in his home country, he moved to Canada to play with the Toronto Arrows club in 2013.
In 2016, Leivas signed with French side Biarritz Olympique where he played until 2018 when he retired due to injury.. Leivas has represented Brazil at various junior levels before making his senior international debut in 2011 against Uruguay.
He has won 24 caps for his country and scored six tries during that time Leandro Leivas is a professional rugby league footballer who plays for the Toronto Arrows in the National Rugby League. He has played for Uruguay national team since 2008, and has made 74 appearances.
In 2019 he was named in their squad for the World Cup in England. Leivas started his career with Uruguayan side Nacional before moving to English club Warrington Wolves in 2014 where he made 52 appearances scoring 10 tries including winning Super League's Man of Steel award that year His move to Toronto came at the end of 2018 season when he signed a 3-year deal worth around $1million
23. Tayler Adams
Tayler Adams is a 29-year-old New Zealand Scrum Half who currently plays for the Toronto Arrows in the Canadian Rugby Union (CRC). Tayler played his first senior rugby union match for the NSW Country Eagles in 2016 and has since then been an integral part of their squad.
In addition to playing at representative level, Tayler also competes regularly with Toronto Arrows in both domestic and international tournaments. Born in Auckland, NZ, Tayler began playing rugby union as a youngster before moving to Sydney to attend Kelston Boys' High School where he joined Southern Institute of Technology's rugby club..
After graduating from college, Tayler returned to play provincial football for The North Harbour Rays before making his debut for the NSW Country Eagles during their 2017 season campaign. Tayler Adams is a professional rugby player who played for the Southland Stags in Super Rugby. Adams made her debut for the Stags in March 2018 and has since appeared in two matches.
Prior to playing rugby, Adams played basketball at university and represented New Zealand at under-16 level before switching to rugby full-time after graduating from college. In 2019, she was named in the New Zealand Women's Sevens team that competed at the World Cup in France as well as making her test debut against Italy later that year. Outside of rugby, Adams works as a personal trainer and also provides commentary for Southland provincial matches on television station Southern Cross Ten Network..
24. Manuel Diana
At 26 years old, Manuel Diana is one of the youngest players in the professional rugby union league. He started playing rugby when he was just six years old and has since progressed through the ranks to become a regular starter for his current team Toronto Arrows.
A powerful number eight, Manuel Diana is known for his accurate passing and strong tackling abilities on both sides of the ball. His impressive achievements have seen him nominated for several international awards, including World Rugby Young Player of The Year in 2016 and 2018 respectively.
Having made over 100 appearances for Uruguay's national side, it seems likely that Manuel Diana will continue his rugby career at some point in the future - we can't wait to see what he achieves next. Manuel Diana is a professional rugby player who currently plays for the Toronto Arrows in the PRO Rugby competition.
He has represented Uruguay on multiple occasions, most notably at the 2015 U20 World Cup and at the 2016 Olympics. Manuel Diana is also a strong defender and can play either flanker or number 8; his international experience suggests he could be an important part of Uruguay's future squads. In March 2020, it was announced that Manuel Diana had signed with Peñarol Rugby Club in Argentina as they prepare to compete in Super Rugby next season - this will give him plenty of opportunity to showcase his skills on a larger stage internationally.
With 35 appearances for Uruguayan national teams so far, there's no doubt that Manuel Diana has achieved a lot both on and off the pitch and we're excited to see what he does next.
25. Juan Cruz González
Juan Cruz González is a rugby union fly-half who played for Cuba and Alcobendas in Spain. He made his debut for CUBA in 2012 and was the team's first choice fly-half until 2019, when he left to play for Toronto Arrows.
Juan Cruz González has represented Argentina at youth level but has never played senior international rugby. His best performance came during the 2018 Americas Rugby Championship where he scored two tries in an eventual victory over Uruguay.
Juan Cruz González married his wife, Natalia, in November 2018 and they are expecting their first child later this year Juan Cruz González is a rugby player from Argentina who represented his nation at the international level. He played for the Argentina XV team in 2016 and 2018, and was also part of their sevens squad that competed at two tournaments during that time frame.
Juan Cruz González is known for his strong running game, which has often proved to be a key component of successful Argentine teams on both sides of the ball. In addition to playing rugby, Juan Cruz González is also an accomplished boxer and track athlete who has won several medals at regional competitions over the years. He announced his retirement from rugby in February 2019 after 18 seasons as a professional player, most notably with Club Atlético de Rosario where he spent 12 seasons before moving to Racing 92 in France earlier this year
26. Gaston Cortes
Gaston Cortes was born on October 6, 1985 in Cordoba, Argentina. He played rugby union for the Argentina national team and represented them at international level from 2007 to 2013.
Gaston Cortes is a tighthead prop who turned out for Bristol Rugby in England during the 2012-13 season. In 2018 he moved back to his home country of Argentina where he currently plays for Club Atlético Colón de Santa Fe in the Top Division of Argentinian rugby union football league system.
Gaston Cortes was born on January 6, 1994 in Cordoba, Argentina. He started playing rugby at the age of 10 and quickly became a fan favorite with his tough play and powerful runs. Cortes made his provincial debut for Córdoba A.C., in 2012, and went on to make 114 appearances for the club over the next three years before joining Bristol Rugby in 2016.
In 2018 he joined Leicester Tigers but left just six months later after failing to break into their first team squad or win any Premiership honours during that time period. In June of 2020 he signed a one-year deal with Glasgow Warriors where he is currently plying his trade as a reserve player behind starters Jonny Gray and Tommy Seymour . He has also represented Argentina at both U20 (2012) and senior level (2013-present).
27. Jason Higgins
Jason Higgins is a rugby union player who has spent most of his career playing for Munster in the Pro14. Jason Higgins first came to prominence when he was part of the Ireland U20 team that won the 2015 Junior World Championship.
In 2017, Higgins joined Munster from Bordeaux-Begles and has since made over 100 appearances for them, scoring 26 try conversions along with 53 penalties. In 2018, Higgins helped Munster reach their first ever Champions Cup Final where they lost to Racing 92 Jason Higgins is a professional rugby player who has represented Canada internationally.
Jason Higgins started playing rugby when he was just 12 years old and quickly became one of the best players in the country. In 2011, Jason Higgins joined the Toronto Arrows and helped them to win the Rugby Championship that year. In 2015, Jason Higgins moved to New York where he joined Rugby United New York (RUNAY) and helped them to win two National Championships in 2016 and 2017.
Jason Higgins will be representing Canada at this year's World Cup which begins on September 18th
28. Manuel Montero
Manuel Montero was born on November 20, 1991 in Buenos Aires, Argentina. He is a rugby union player and has been playing for Toronto Arrows since 2021. Manuel Montero has played for Club Pucará (2011-2015), Olimpia Lions (2012-2015) and Pampas XV (2020-21).
He is 1.94 m tall and weighs 113 kgs which makes him an imposing figure on the pitch. His impressive stats include 180 apps with 55 points scored during his amateur career so far. In terms of international experience, Manuel Montero has represented Argentina at U20 level as well as being part of their senior squad from 2020 onwards Manuel Montero is a professional rugby player who currently plays for the Toronto Arrows.
Manuel Montero began his career with Argentinian side Tucuman before moving to Europe in 2011 to play for Portuguese club Benfica. In 2016, Manuel Montero moved to North America to join the Toronto Arrows and has since been an important part of their team. Manuel Montero has represented Argentina at various youth levels, but he is yet to make his debut for the senior squad. Manuel Montero is a hard-working forward who loves playing physical rugby; this makes him an effective all-round player on the pitch..
29. Ben LeSage
Ben LeSage is a 27-year-old Canadian rugby union player who currently plays for the UBC Thunderbirds. Ben has represented Canada at youth level and was part of the Calgary Canucks side that won the 2016 Amway Cup.
After graduating from University of British Columbia in 2017, Ben joined up with UBC Thunderbirds where he has made 11 appearances so far this season. Ben is known for his powerful running game and strong defence which have seen him become a key member of the team during his time there thusfar.
With several years experience under his belt, Ben looks to continue to develop as both a player and person in order to reach even greater heights in rugby union career Ben LeSage is a dynamic forward who has been playing in the Canadian Premier League for two seasons now. He has scored 25 goals in 18 appearances with the Toronto Arrows and his performances have caught the eye of many teams across North America.
Ben is an excellent finisher, able to score from close range or from further out on the pitch, and he will be a valuable player for whichever team signs him next season. Le Sage is also good at creating chances for others, whether that's through taking on defenders one-on-one or by setting up teammates with accurate passes. His hardworking attitude and willingness to do whatever it takes to help his team win make him a popular member of any squad he joins, and he should continue to develop as a professional footballer over time.
30. Cole Davis
Cole Davis is a Canadian rugby union player who has played for Austin Gilgronis in the past. Cole was born on July 6th, 1997 and stands at 1.88 meters tall (6 feet 2 inches) and weighs 100 kilograms (220 pounds).
He started playing Rugby Union when he was just eight years old and quickly became one of the best players in his age group. In 2020, Cole signed with Austin Gilgronis as their new wing partner and has since made an impact both on and off the pitch, scoring five tries in eight appearances so far this season.
With his quick reflexes, powerful running style and accurate passing abilities, Cole is sure to be a key member of any team he joins next - whether that's in Canada or abroad remains to be seen. Cole Davis is a Canadian rugby union player who has played for the Canada U20 and Canada Sevens teams since 2017. Cole Davis primarily plays as a scrum-half, but can also play at fly-half or in the centres.
He has been praised for his speed, passing ability and kicking skills. In 2018, Cole Davis was named captain of the Canada Sevens team which competed at the World Rugby Sevens Series event in San Francisco. The team finished 7th overall out of 16 teams competing. As of February 2022, Cole Davis had scored 10 points in 7 international appearances for Canada, including 4 apps during their most recent tournament -the Americas Pacific Challenge - where they lost to Chile 36-24 in the final round robin stage
31. Theo Sauder
Theo Sauder was born on April 2, 1996 in Vancouver, British Columbia. He played rugby union for the UBC Thunderbirds and Toronto Arrows during his amateur career.
Sauder is 5' 11" tall and weighs 185 pounds (84 kilograms). The fullback has represented Canada at youth international level and attended the University of British Columbia where he studied business management.
In 2018, Theo Sauder joined the Toronto Arrows professional team as their fullback. Theo Sauder is a professional rugby league footballer who currently plays for the Toronto Arrows in the National Rugby League (NRL).
Theo was born on December 16th, 1995 in Peterborough, Ontario and played his junior football with the London Monarchs before being signed by the Toronto Wolfpack. After making his debut for the Wolfpack during their inaugural season in 2016, he made 11 appearances over two seasons and also represented Canada at international level.
In 2018, Sauder was signed by the Toronto Arrows and has since made five appearances for them as well as representing Canada at international level again.
32. Richie Asiata
Richie Asiata is a prop and hooker who has played for the Auckland Blues, Melbourne Rebels and Newcastle Knights in the NRL. He made his debut for the Auckland Blues in 2013 and has since developed into one of their most reliable players, playing over 100 games during his three-year career with them.
In 2017, he was recruited by the Melbourne Rebels after being given an opportunity to show his skills at Super League side Warrington Wolves where he impressed coaches enough to sign him on a two-year contract. He made his first appearance for the Newcastle Knights in 2018 and is currently their starting prop/hooker.
Richie Asiata is a talented Rugby League player who represents the Queensland Country side. He started playing rugby league in his early teenage years, and quickly rose through the ranks of Queensland's junior teams. In 2016, he was selected to play for Australia U20s, and has since been impressive on the international stage.
Richie Asiata is an exciting player to watch due to his powerful running and accurate passing game. His contract with the Toronto Arrows expires at the end of 2020, but it is likely that he will be looking for another opportunity in 2021 - given his current form and popularity with fans across Australia and Canada.
33. Luke Campbell
Luke Campbell is a rugby union player who has represented Canada at international level. He first played for James Bay AA in 2015 and went on to represent them at senior level in 2019-2020.
Luke Campbell is known for his strong defence and accurate kicking game, which has seen him score many points over the years. In 2020, he signed with Old Glory DC where he will continue playing rugby professionally Luke Campbell is a professional Canadian football player who currently plays for the Toronto Arrows in the Canadian Football League.
Campbell has played for provincial and state sides over the course of his career, including playing for BC Bears in 2016 and 2017 before being drafted to Canada's U20 team in 2018. He made his debut with Canada Selects during their 2019 World Bowl qualifying campaign, amassing 14 points on 5 apps. The 26-year-old Campbell is entering his seventh year with the Toronto Arrows after originally joining them as an undrafted free agent back in 2014
34. Doug Wooldridge
Doug Wooldridge is a prop who has played for England, Scotland and Racing 92. He made his international debut in 2008 and went on to play at the 2011 Rugby World Cup.
Doug was part of the team that won the Six Nations Championship in 2016. Injuries have blighted parts of Doug's career but he still managed to win 36 caps for England before retiring in 2017 due to injury.
After retirement from rugby, Doug started working as a strength and conditioning coach with Bath Rugby Club where he currently works as an assistant head coach/physical performance manager Doug married his wife Katie in May 2018 Doug Wooldridge is a professional rugby player who currently plays for the Toronto Arrows in Major League Rugby.
Doug was born on February 4, 1989 in Ottawa, Ontario and played football at Carleton University before switching to rugby. He made his Canadian national team debut in 2009 and has since gone on to make 23 appearances for his country. Doug has also represented Ontario Blues, Canada A Side and most recently the Vancouver Ravens in the Americas Pacific Challenge Cup competition.
In 2015 he signed with French side Clermont Auvergne where he spent two seasons before joining Toronto Arrows in 2019-20 season of MLR playoffs
35. Tom Dolezel
Tom Dolezel is a prop for Leicester Tigers in the English Premiership. He made his debut in 2009 and has since played over 100 games for the club. He has been part of two Championship-winning sides, as well as reaching the Champions Cup semi-finals in 2015/16.
Tom is married with two children and enjoys spending time outdoors, playing football and golfing. Tom Dolezel is a professional rugby league footballer who currently plays for the Toronto Arrows in the Canadian Rugby League (CRL). Tom was born and raised in Hamilton, Ontario.
He played college football at Lake Erie College before turning his attention to rugby league full-time. In 2018, Tom helped Canada qualify for their first ever World Cup tournament - something they hadn't done since 1987. He has also represented his country at both junior and senior level, making over 50 appearances between them.
Outside of playing rugby league, Tom enjoys spending time with his family and friends - he's got two young boys himself.
36. Liam Murray
Liam Murray is a Canadian prop who has played for the Houston SaberCats in both the American and Canadian rugby union leagues. In his two seasons with Houston, Murray has made 20 appearances, scoring 108 points from 15 conversions and 36 penalties.
He joined Toronto Arrows on loan in February 2020, making six appearances before returning to Houston at the end of that season. With over 100 caps for Canada's U20 side and over 30 appearances for their senior team, Murray is considered one of the country's most experienced players at this level.
A powerful scrummager who can also play as a loose forward or second-rower, Murray will be important to any side he joins next season Liam Murray was born in Edmonton, Alberta, on 10 January 2000. He started playing hockey at the age of four and began competing nationally at the age of twelve. In his first year of junior eligibility (2016-17), he led all Canadian juniors in scoring with 98 points in 58 games played for the WHL's Kelowna Rockets.
The following season (2017-18) saw him lead all rookies with 71 points as a member of Team Canada's U20 squad that won a silver medal at the 2018 World Junior Championships. Liam Murray signed an entry level contract with the Dallas Stars on 25 May 2018 and made his professional debut later that month against the Arizona Coyotes, recording one goal and two assists in six minutes played while earning third star honours in a 4-1 victory.. On 15 November 2018, Liam Murray scored his first NHL goal against Jonathan Quick of the Los Angeles Kings during their home game 7th night/6th game stretch to help seal a 5-4 win for Dallas...
. . The Toronto Arrows are an exciting new addition to the Major League Rugby landscape. Founded in 20.
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Catfish Hunter Hall Of Fame?
The Catfish Hunter Hall of Fame is a prestigious award that honors America's greatest catfish anglers. It was established in and since then, only a handful of individuals have been bestowed with this honor.
How To Make A Batters Box Template Out Of Pvc?
Creating a batting box template out of PVC can be a fun and easy project for anyone looking to improve their batting skills. By following these simple steps, you'll be well on your way to creating the perfect spot for practicing your hits!
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How To Make A Batters Box Template Out Of Pvc
Making a batter's box out of PVC is an easy way to add some fun and excitement to your batting practice.


text:
The top global game advertisers in the US, Japan and South Korea 2019
Where do indies and innovative ideas fit in the current mobile games market?
Comments & Opinions
Aaron Fothergill Co-founder Strange Flavour
Ben Murch Co-Founder Perchang
Dan Menard CEO Double Stallion
Matthew Annal MD Nitrome
Molly Heady-Carroll Co-founder & Lead Artist Arcane Circus
Nathan Fouts Founder Mommy's Best Games
Pierre-Luc Vettier CEO Zero Games Studios
December 7th, 2018 - 12:30pm
By Iain Harris, Staff Writer
Earlier this year we posed the question to our Indie Mavens whether or not they should get in on the battle royale trend, with Epic Games' Fortnite and Tencent's PUBG Mobile locked in a lucrative face-off.
One thing to come from our conversation was the idea that maybe it's not if indies could get on the battle royale trend, but should they?
"I feel it goes against why developers go independent in the first place," said Perchang co-founder Ben Murch.
"We do this to try our own ideas, invent genres, to boldly go - well; you get the idea. We don't do this to copy what's gone before."
His words sum up the indie spirit, but how indomitable can that spirit be in the face of the hard realities of the mobile market?
To delve into that thought a little more, we asked our Indie Mavens how much innovation consumers are really looking for and where indies fit in today's mobile market.
Specifically, we asked:
Does the market want innovation or more of the same?
Where do indies fit in on mobile right now?
The games market is a funny beast these days. It's vast and generally quite quiet.
Moving around like a massive herd of robot cows, gently humming away as they feed upon the latest triple-A release. This peace is occasionally broken by tiny pockets of extremely loud customers who shout for change, cheaper games, better games or Diablo 4.
And because they're so loud, it's easy to think that's what the overall market wants. The robotic bovines (robovines?) raise their heads at the buzz but then go back to their munching.
The quantity of titles being released has grown hugely over the last few years, so it's very easy to get lost.
Ben Murch
Generally speaking, the herd is pretty content playing slightly upgraded versions of their favourite titles. Innovation is welcome for the fringes, but it has to be well measured, tested and integrated slowly to the rest.
Because you really don't want to spook that herd. They might stampede!
The mobile market is like a shoal of robotic fish. Only joking. It can feel like life is slightly tough for indies right now.
The quantity of titles being released has grown hugely over the last few years, so it's very easy to get lost. Couple that with the rise of the free-to-play publisher monopoly and it can be hard for a little Indie to stand out.
Luckily, Apple seems to have recognised this too and are actually pretty great at helping us get exposure. They even have a dedicated section on their store!
Also, the sheer size of the audience plays in our favour, because snagging 0.01 per cent of the estimated 2.1 billion mobile gamers will still see you well on the way to that sequel. And perhaps that indie niche is wide enough to try some innovation.
I agree with Ben on this one. The big money game regarding what gets coverage, marketing budget and sales usually have a number after it (which can sometimes be lower if it's another game by the same publisher but with horses instead of cars).
The thing is, I'm actually kind of in favour of this when it's a game I like. But I can understand players' fury when the new iteration isn't what they thought they wanted, even if I don't agree with how their furious assumption that just because the different version of the game isn't what they want, it shouldn't have been made.
The bit I'd like to see a change in isn't the actual release of the same games with new stuff, but the balance of coverage between these monster iterations and the innovative new stuff.
Blizzard caused a stir when it announced Diablo Immortal for mobile recently
The more the interesting new stuff gets covered and not swamped by yet another story about the same game that's been covered for the last 10 years, the more players will actually get to try them.
I'm always a bit split on this as I'm forever having to go back to our 10-year-old game that scrapes some income because players want to keep playing it. Which is a good thing.
I do, however, like trying new stuff. Sometimes it works (we won an Apple Design Award for innovation back in 2004), sometimes it just vanishes without a trace (usually).
Adam and I are actually writing a whacky new innovative game at the moment. It'll probably get five downloads.
Wherever we can get a handhold and just hang on I think.
As Ben said, Apple is helpful, and they do like to pick up on innovative games, so it's sometimes worth the gamble.
Always the same. Innovation in video games has been a lure for a long time now, but most players absolutely don't care about innovation. They just want to have fun, however it comes.
Indies have been dead on mobile for a few years and innovation in this market is even harder on mobile.
Even if Apple tries to sometimes highlight a few indie games, it hasn't really helped at all.
I think innovation works best in small doses — a tiny tweak here, a small move there.
Everything else has to be perfectly designed. Then you might have a winner.
With our latest Steam release (also planned for mobile later on) Pig Eat Ball, it was probably too much.
It's a polished action-adventure, but so strange, people don't know what to make of it on first glance.
Mommy's Best Games released Finger Derpy back in 2015
And "first glance" is the key on mobile unless you have lots of marketing money. Google's 70/20/10 development rule suggests only 10 per cent on innovation.
So when a game turns almost everything on its head to shake things up, gamers won't know what end is up and skip it.
Our next unannounced mobile game is designed as casual and free-to-play.
Thus, we will be seeking a publisher. I've done free-to-play on my own (Finger Derpy, 2015) and while it had some downloads, it wasn't enough and didn't make much money. Get as much help as you can.
When you are making games, regardless of platform, you are making games for people with their own tastes and needs from their entertainment.
I think indie game development is the most likely space to push the boundaries of what is possible on mobile.
Molly Heady-Carroll
This is why we have both successful games that follow trends and successful games that walk their own creative path. I firmly believe that the variety of players in the world means there is room for both.
Mobile is suffering from somewhat of an image problem (as illustrated by the recent Diablo Immortal controversy), so it is easy for players to not see mobile as a space for creative indie projects.
Despite this, some of my favourite developments in gaming right now are in mobile. Not only is there a whole new audience of players thanks to the accessibility of the mobile platform, but developers also have room to experiment with the relatively new and unexplored platform.
I'm not sure if this is the same thing but the market wants F2P and sadly that does have a big impact on innovation.
More and more at Nitrome we find ourselves throwing away ideas because they just don't fit well with currently successful business models.
The business model does tend to suit certain sorts of games so as a result innovation is hindered.
I don't think that means the market doesn't want innovation. It's probably always a good thing to make something that stands out, but finding something that does so without negatively impacting the models that convert can be challenging.
I think the days of making very small games based around an original idea and turning a profit purely of getting featured are behind us.
To be successful now, I think a stronger monetisation model is needed and that means making a bit bigger games with more depth. That means more risk which probably makes the idea of using a publisher more appealing than it was a couple of years back.
Chiming back in on this one as the current project was still under wraps when I first replied. I'm kind of hoping the market still wants a bit of innovation as we're about to release the first ever nose and tongue controlled game!
Face Candy is about picking up candy with your nose and sticking your tongue out to slurp it down. It's very silly.
If all goes well, it'll be on the App Store within a couple of weeks of the article going live, so we'll know by the next one whether the market wants it or not.
The market says it wants innovation, but it also tends to reward games that feel familiar and stick to genres and tropes. Often innovating means sacrificing something to go out into the unknown.
Usually, it's expensive and you won't have the best iteration of that new mechanic/idea because it's new to you, the developer, as well.
You can find more indie games insights live at Pocket Gamer Connects London 2019 on January 21st to 22nd. We'll be hosting a special 'Indie Track' dedicated to all things - you guessed it - indie at the show.
Iain Harris
Iain is a freelance writer based in Scotland with a penchant for indies and all things Nintendo. Alongside PocketGamer.Biz, he has also appeared in Kotaku, Rock Paper Shotgun, PCGamesN and VG24/7.
Mobile Mavens Mar 6th, 2018
Is there any room for indie games developers to get in on the battle royale trend?
Mobile Mavens Jan 11th, 2018
Should aspiring indie games developers strike out on their own straight away or get experience at a studio first?
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Are indie developers interested in releasing games on the Nintendo Switch?
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redpajama_set_name: RedPajamaCommonCrawl


"What's the difference between Pilates and Yoga" is a question that I am asked probably more than any other, so I am sure all teachers have been asked this many times as well. It is clearly something that many people do not understand.
It's been a warm weekend in Barcelona. The weather has turned a little so whilst it's still hot, there is a slight freshness that is reminding everyone that the heady days of summer are passing and that autumn is not so far away now. The children are back at school and there is the usual surge of people wanting to get back into their fitness and health before the next holidays at Christmas and New Year. I like this time of year. The oppressive heat is dying down, and many of the younger people have returned to their colleges or work, so the demographic of visitors here in Barcelona is older and calmer.
During the weekend I was working with some clients who have travelled to Barcelona from Germany and Russia for one of my wellness and fitness retreats, and yet again, the Pilates or yoga question was asked. I like to let the Universe direct me to what my blogs should be about, so the message at the weekend was clear: what is the best way to explain to people the difference between Pilates and yoga?
In my experience, it's quite typical that many yoga and Pilates devotees are passionate about their chosen discipline, but it's clear to me that many members of the public do not understand the differences and are therefore not in a good position to decide which is best for them (or of course decide to do both as I do). As people are so passionate about their chosen discipline, if you ask one of them you are unlikely to get an objective and unbiased opinion.
Some Pilates teachers incorporate elements of their yoga practise into their Pilates teachings and vice versa, so sometimes it could be argued that the lines get blurred (can't resist a musical MTV reference to Miley Cyrus, our twerking Pilates fan!) I use some Kundalini in my Pilates system Pilates EVO, primarily as stretches and transition movements.
So what really and simply is the difference between yoga and Pilates? How do you describe it to your clients who ask?
Yoga is an ancient practice that focuses the body, mind and spirit. The yoga lifestyle upholds more than just a healthy physical lifestyle by embracing good moral values such as honesty and tolerance. Its intention was to join the individual spirit with the universal spirit. Pilates is a modern school of exercise developed by Joseph Pilates as a way to improve core strength and spinal alignment. Pilates combines flexibility, core strength and body awareness to support physical rehabilitation.
Ancient yoga practices were passed down individually between teacher and student, so there are many schools and types of yoga. Each school of yoga combines a series of postures, breathing exercises and meditation to form different routines. The repetitive movement of yoga can be very therapeutic while also building flexibility and strength. Yoga has also developed popularity as a great way to relieve stress. Exercises are completed on a floor mat and need no special equipment to begin.
Pilates improves your body from the centre outward by strengthening your core and improving your posture. Pilates is considered a resistance exercise to build muscle and support for the body while aiding flexibility. The core and spinal cord are the central and most important part of whole-body strength and awareness. The cornerstones of Pilates exercises include concentration, control, centring, breathing, flow and precision. A floor mat and specialized machines may be used in Pilates practice.
Pilates reigns supreme as the key to a strong core and improved posture. Practitioners experience stronger abdominal muscles and those hard-to-target obliques from holding positions and using resistance. Pilates exercises will make your body appear taller and leaner. You will also see improved flexibility in your hamstrings and hips. Yoga will provide the most improvement in flexibility. Holding poses while focusing on breathing stretches the body and relaxes muscles, and certain poses will target certain muscles.
So do we all agree? If not, can we all agree on a definitive explanation to clearly explain the differences, or in your experience do you not think we really need one?
Working as a fitness exercise trainer in Barcelona, I am often asked the same type of questions. In my blog I will try to answer some of these questions. Today I talk about losing weight without losing performance.
Many athletes and fitness participants wish to lose weight, either for health or performance reasons.
However, rapid weight loss can have serious health consequences leading to a reduction in performance.
Since 95% of dieters fail to maintain their weight loss within a five-year period, lifestyle management is the key to long-term weight management. Two methods that should not be entertained are crash dieting and dehydration.
It has been proven that rapid weight loss results in diminished aerobic capacity. Fat is more fattening calorie for calorie than carbohydrate. This is supported by a number of well-controlled studies. It is the over- consumption of fat, rather than carbohydrate or protein, that results in body fat gain. To lose body fat, you simply have to expand more energy (calories) than you consume.
Avoid a significant reduction in resting metabolic rate (this is the number of calories your body burns if you did nothing all day).
The key to successful fat loss is to cut your calories mostly from fat and to reduce carbohydrates by only a modest amount. If you reduce carbohydrate intake below the daily average requirement, glycogen stores become depleted which results in lean tissue loss, so is not a good idea.
Set realistic goals, and write them down.
Monitor body composition changes by measuring chest, waist, hips, arms and legs, and obtain skin-fold thickness measurements by using callipers.
Aim to lose no more than 0.5 kg each week.
Keep a food diary. Most people eat far more than they realise or admit to.
Trim the saturated fat by using less butter, deep-fried foods, fatty meats, pastry, cakes, biscuits and chocolate.


text:
\section{Introduction}\label{intro}

Topological field theories have been the subject of a thorough investigation in theoretical physics \cite{Witten:1982im,Witten:1988ze,Atiyah:1989vu}. The initial aim was to unveil to what extent they could give hints in better understanding gravity without matter \cite{Witten:1988hc}, but it was soon recognized that they also had and still have a different role if one adds a boundary \cite{Moore:1989yh,Cattaneo:2012zs}. Indeed it is the boundary which  plays a physical role and it is on the boundary that the local observables of a new and different physics live.
The introduction of a boundary in a field theory was first proposed by  Symanzik who introduced a separability ansatz in order to study the Casimir effect of two parallel plates \cite{Symanzik:1981wd}. The method proposed by Symanzik concerns a space divided into a left and a right hand side, and it has been applied to topological field theories of different types \cite{Amoretti:2012kb}, obtaining results particularly relevant for the theory of the  fractional quantum Hall effect \cite{Blasi:2008gt} and of the topological insulators in three and four spacetime dimensions \cite{Amoretti:2014iza}.  Later on  field theories with a single-sided boundary have also been considered, which also lead to interesting results.
The first and most studied model is the Chern-Simons theory in three dimensions \cite{Chern:1974ft,Blasi:1990pf}, soon after appeared the BF models \cite{Horowitz:1989ng}, the generalizations of the Chern-Simons model which can only live in an odd spacetime \cite{Karlhede:1989hz,Blasi:2005vf} .
The topological nature of the Chern-Simons  action is that it does not depend on the metric tensor and hence the energy momentum tensor vanishes; later on Witten  proposed another kind of topological theory where the action is not in the cohomology of the BRS operator and hence the theory has no physical observables \cite{Birmingham:1991ty}.
The common denominator to all these models is that they are gauge field theories with the gauge field $A_\mu$ as the main actor and where the Maxwell term does not appear since it breaks the topological nature of the model, which seems to be at the basis of duality relations characterizing the boundary degrees of freedom of these models, whose bulk theories are purely topological \cite{Amoretti:2014kba}. Duality relations of this type are known \cite{Aratyn:1983bg,Aratyn:1984jz} to allow to extract fermionic degrees of freedom out of bosonic ones, in a way compatible with the existence of Hall or quantum spin states on the edge of higher dimensional bulk theories.
It is natural to ask the question to what extent this is true, investigating whether fermionizing duality relations hold also on the boundary of non topological field theories, which would broaden the possible candidates for the theories of fractional quantum Hall effect and of topological insulators.
Moreover, the Maxwell coupling is expected to be quite relevant in whatever physics may arise on the boundary and that is why we are studying models where the Maxwell term is included in the bulk action. This has been done for Chern-Simons theory with both double \cite{Blasi:2010gw} and single-sided \cite{Maggiore:2018bxr} boundary, with significantly different results. We note also that the introduction of a Maxwell term in the Chern-Simons action gives rise to topologically massive theories \cite{Deser:1981wh,Deser:1982vy} by means of a mechanism which cannot to be replicated in spacetime dimensions other than three. The question also arises in what sense a topological theory with a Maxwell coupling is still topological. We propose here the Maxwell-BF model  as a new kind of topological theory in four spacetime dimensions. The model does not fit into the known ``topological'' classes of quantum field theories since it does depend explicitly on the metric and the Maxwell term makes it cohomologically non trivial. Nevertheless the bulk theory has  no local observables due to the equations of motion which enforce the field strength $F_{\mu\nu}$ to vanish, and hence the gauge field $A_\mu$ is pure gauge. This model has never been considered before with a boundary and the question is not irrelevant. The fact that the physics of the model live on the boundary is neither intuitive nor immediately deducible as a simple exercise.\\

The paper is organized as follows. In Section 2 the model with planar boundary is introduced. The boundary conditions and the Ward identities, broken by the presence of the boundary, are derived. A kind of electromagnetic structure is found on the boundary, with Maxwell equations solved by potentials, which will play the role of degrees of freedom for the 3D theory. The identification of electric and magnetic fields makes possible a  physical interpretation of the role of the Maxwell term, as deformation of the magnetic field. In Section 3, following standard methods of quantum field theory, the algebra formed by the conserved electromagnetic currents is found, which heavily depends on the Maxwell term in the bulk action. The algebra, written in terms of the 3D potentials, allows for the construction, in Section 4, of the 3D theory, whose symmetries are identified. The holographic contact is realized by means of the equations of motion of the 3D theory, which are required to be compatible with the boundary conditions of the bulk 4D theory. The resulting equation is recognized to be the duality relation which characterizes the existence of fermionic degrees of freedom on the boundary and which therefore turns out not to be peculiar of purely topological bulk field theories only. Or, putting it in another way, the fact that the physical properties are the same in  the holographic theory whether the Maxwell term is present or not, clarifies the meaning of topological quantum field theories when a boundary is introduced.

\section{The model: action, boundary conditions and Ward identities}\label{sect2}

The action of the 4D Maxwell-BF theory in euclidean spacetime
is
\begin{equation}
S_{bulk}=\int d^4x\, \theta(x_3) \left(
k_1\,\epsilon_{\mu\nu\rho\sigma}F_{\mu\nu}B_{\rho\sigma}
+ k_2\, F_{\mu\nu}F_{\mu\nu}\right)\ ,
\label{Sbulk}\end{equation}
where the presence of the step-function $\theta(x_3)$ restricts the model on the half-space $x_3\geq 0$, with planar boundary at $x_3=0$, $F_{\mu\nu}(x)$ is the electromagnetic tensor for the gauge field $A_\mu(x)$ and $B_{\mu\nu}(x)=-B_{\nu\mu}(x)$ is the rank-2 antisymmetric tensor of the 4D topological BF theory \cite{Blau:1989dh,Blau:1989bq}, and the canonical mass dimensions of the quantum fields are
\begin{equation}
[A]=1 \ \ ;\ \  [B]=2\ .
\label{dimensions}\end{equation}
Finally, $k_1$ and $k_2$ are constants which could be reabsorbed by a redefinition of the fields, but which we prefer to keep, in order to be able to trace the contributions of the topological $BF$ and Maxwell $F^2$ term, respectively.

In absence of boundary, $i.e.$ without the $\theta$-function in \eqref{Sbulk}, the action of the Maxwell-BF theory is invariant under the following two symmetries:
\begin{equation}
\begin{split}
\delta^{(1)} A_\mu &= \partial_\mu\Lambda \\
\delta^{(1)}B_{\mu\nu} &=0
\end{split}
\label{delta1}\end{equation}
and
\begin{equation}
\begin{split}
\delta^{(2)} A_\mu &= 0 \\
\delta^{(2)}B_{\mu\nu} &=\partial_\mu\zeta_\nu-\partial_\nu\zeta_\mu\ ,
\end{split}
\label{delta2}\end{equation}
where $\Lambda(x)$ and $\zeta_\mu(x)$ are gauge parameters. The presence of the boundary in $S_{bulk}$ breaks the $\delta^{(2)}$- invariance, preserving the usual gauge symmetry $\delta^{(1)}$:
\begin{equation}
\delta^{(1)}S_{bulk}=0\ \ ;\ \ \delta^{(2)}S_{bulk}=-4k_2\int d^4x \delta(x_3)\zeta_i\epsilon_{ijk}\partial_jA_k\ ,
\label{brokensym}\end{equation}
where latin indices run from 0 to 2: $i,j,...=\{0,1,2\}$.

The total action $S_{tot}$ is composed by four terms
\begin{equation}
S_{tot}=S_{bulk}+S_{gf}+S_{ext}+S_{bd}\ ,
\label{Stot}\end{equation}
where $S_{bulk}$ is given by \eqref{Sbulk}, $S_{gf}$ is the gauge fixing term
\begin{equation}
S_{gf}=\int d^4x\, \theta(x_3)\left(bA_3+d_iB_{3i}\right)\ ,
\label{Sgf}\end{equation}
and $b(x)$ and $d_i(x)$ are Lagrange multipliers implementing the gauge conditions
\begin{equation}
A_3=B_{3i}=0\ .
\label{gaugecond}\end{equation}
In $S_{ext}$ external sources $J_i(x)$ and $J_{ij}(x)$ are introduced
\begin{equation}
S_{ext}=\int d^4x\, \theta(x_3)\left(J_iA_i+\frac{1}{2}J_{ij}B_{ij}\right)\ ,
\label{Sext}\end{equation}
by means of which the quantum fields surviving the gauge conditions \eqref{gaugecond} can be defined.
Finally, $S_{bd}$ is the most general boundary  term defined on $x_3=0$ compatible with power counting
\begin{equation}
S_{bd}=\int d^4x\, \delta(x_3)\left(
a_1\epsilon_{ijk}A_i\partial_jA_k + a_2A_i\partial_3A_i +a_3\epsilon_{ijk}A_iB_{jk} + \frac{m}{2}A_iA_i
\right)\ ,
\label{Sbd}\end{equation}
where $a_i$ and $m$ are constant parameters and the canonical mass assignments \eqref{dimensions} and $[\delta]=1$ have been used.
Notice that
\begin{equation}
\begin{split}
\int d^4x\, \theta(x_3) \epsilon_{\mu\nu\rho\sigma}F_{\mu\nu}F_{\rho\sigma}
&=
2\int d^4x\, \theta(x_3)\partial_\mu (\epsilon_{\mu\nu\rho\sigma}A_\nu F_{\rho\sigma})\\
&=-4\int d^4x\, \delta(x_3)\epsilon_{ijk}A_i\partial_jA_k\ ,
\end{split}\label{FFdual}\end{equation}
which justifies the fact that we did not introduce in $S_{bulk}$ \eqref{Sbulk} a term $F\tilde{F}$ in favor of its Chern-Simons-like boundary counterpart, identified by the constant $a_1$ in \eqref{Sbd}. In addition, we chose to keep 3D covariance on the boundary. A more general, non covariant boundary term could have been written \cite{Maggiore:2017vjf,Blasi:2015lrg,Blasi:2017pkk}.  \\

The equations of motion of the fields $A_i$ and $B_{ij}$ are
\begin{equation}
\begin{split}
\frac{\delta S_{tot}}{\delta A_i} &=
\theta(x_3)
[-2k_1\epsilon_{ijk}\partial_3B_{jk} - 4k_2\partial_3^2A_i-4k_2\partial_j^2A_i + 4k_2\partial_i\partial_jA_j +J_i]
\\
&+\delta(x_3)
[(a_3-2k_1)\epsilon_{ijk}B_{jk} + (a_2-4k_2) \partial_3A_i +2a_1\epsilon_{ijk}\partial_jA_k+mA_i]
=0
\end{split}
\label{eomA}\end{equation}
and
\begin{equation}
\frac{\delta S_{tot}}{\delta B_{ij}} =
\theta(x_3)
[4k_1\epsilon_{ijk}\partial_3A_k+ J_{ij}]
+\delta(x_3)
[2a_3\epsilon_{ijk}A_k]
=0\ .\label{eomB}
\end{equation}
From the equations of motion \eqref{eomA} and \eqref{eomB} and performing $\lim_{\epsilon\rightarrow 0} \int_{-\epsilon}^{+\epsilon} dx_3$, we get the boundary conditions
\begin{equation}
\left.(a_3-2k_1)\epsilon_{ijk}B_{jk} + (a_2-4k_2) \partial_3A_i +2a_1\epsilon_{ijk}\partial_jA_k+mA_i \right|_{x_3=0}= 0
\label{bcA}
\end{equation}
\begin{equation}
\left.a_3A_i\right|_{x_3=0} = 0\ .
\label{bcB}
\end{equation}
The equations of motion lead also to the following Ward identities, broken by terms on the r.h.s. due to the presence of the boundary
\begin{equation}
\int_0^\infty dx_3\, \partial_iJ_i =
-\partial_i\left.\left(2k_1\epsilon_{ijk}B_{jk}+4k_2\partial_3A_i\right)\right|_{x_3=0}
\label{wardA}\end{equation}
\begin{equation}
\int_0^\infty dx_3\, \partial_jJ_{ij}=
(4k_1-2a_3)\left. \epsilon_{ijk}\partial_jA_k\right|_{x_3=0}\ ,
\label{wardB}\end{equation}
where, to obtain \eqref{wardA}, the boundary conditions \eqref{bcA} have been used. On the boundary $x_3=0$ and at vanishing external sources, $i.e.$ on the mass shell, the above Ward identities imply
\begin{equation}
\left.\partial_i\left(2k_1\epsilon_{ijk}B_{jk}+4k_2\partial_3A_i\right)\right|_{x_3=0} = 0
\label{cond1}\end{equation}
and
\begin{equation}
\left.\epsilon_{ijk}\partial_jA_k\right|_{x_3=0} = 0\ .
\label{cond2}\end{equation}
Notice that, due to \eqref{cond2}, the $a_1$-term in the boundary action $S_{bd}$ \eqref{Sbd} vanishes. Therefore, without loss of generality, we may rule it out
\begin{equation}
a_1=0\ .
\label{a1out}\end{equation}
The equations \eqref{cond1} and \eqref{cond2} reveal an electromagnetic structure on the boundary $x_3=0$, since they suggest to define an  ``electric'' and a ``magnetic'' field:
\begin{eqnarray}
{\cal E}_i &\equiv& A_i \label{3Delectric} \\
{\cal H}_i &\equiv& k_1\epsilon_{ijk}B_{jk}+2k_2\partial_3A_i\ , \label{3Dmagnetic}
\end{eqnarray}
which allow to identify the degrees of freedom on the boundary $x_3=0$ as the corresponding electromagnetic potentials. Indeed equations \eqref{cond1} and \eqref{cond2} are solved by introducing a 3D vector field $\xi_i(X)$ and a 3D scalar field $\Phi(X)$, respectively:
\begin{equation}
\sqrt{M}\epsilon_{ijk}\partial_j\xi_k(X)\equiv\left.2k_1\epsilon_{ijk}B_{jk}(x)+4k_2\partial_3A_i(x)\right|_{x_3=0}
\label{defxi}\end{equation}
\begin{equation}
\frac{1}{\sqrt{M}}\partial_i\Phi(X)\left.\equiv A_i(x)\right|_{x_3=0}\ ,
\label{defphi}\end{equation}
where a massive scaling parameter $M$ has been introduced in order to make compatible the mass dimensions \eqref{dimensions} of the 4D fields $A_\mu$ and $B_{\mu\nu}$ with those of their 3D boundary counterparts $\Phi$ and $\xi_i$ \cite{Blasi:2005bk}. In 3D spacetime dimensions, in fact, vector fields and scalar fields should have the following mass dimensions
\begin{equation}
[\xi]=[\Phi]=\frac{1}{2}\ .
\label{3Ddimensions}\end{equation}\\

A comment is in order. It turns out that a particular solution of the general boundary conditions \eqref{bcA} and \eqref{bcB}  exists, which makes the physics independent of the Maxwell term. In fact, the choice
\begin{equation}
\ a_2 \neq 4k_2\ ; a_3=2k_1\ ; m= \mbox{any}
\label{particularchoicebc}\end{equation}
implies, from \eqref{bcA} and \eqref{bcB}, (Dirichlet and) Neumann boundary conditions for the gauge field $A_i$ on $x_3=0$
\begin{equation}
\left.\partial_3A_i=0\right|_{x_3=0}\ .
\label{neumann}\end{equation}
Consequently, the Ward identities do not depend on the coupling $k_2$ of the Maxwell term in the action \eqref{Sbulk}, and the theory is indistinguishable, under any respect, from the pure topological BF theory with planar boundary. The fact that  Neumann condition for the gauge field $A_i$, which is a solution of the general boundary conditions \eqref{bcA} and \eqref{bcB}, make the Maxwell term transparent and the non-topological theory equivalent to a topological one, is the first nontrivial result of this paper. Since the aim of this paper is to study if and how the non-topological Maxwell term has an impact on the physics on the boundary, we proceed from now disregarding the solution \eqref{particularchoicebc}. In particular the boundary condition \eqref{bcB} is solved by
\begin{equation}
a_3=0\ .
\label{a3out}\end{equation}\\

The $k_2$-Maxwell term manifests itself on the r.h.s. of the Ward identity \eqref{wardA} by means of  $\left.\partial_3 A_i\right|_{x_3=0}$. We observe that, on the boundary $x_3=0$, the fields $\left.A_i\right|_{x_3=0}$ and $\left.\partial_3 A_i\right|_{x_3=0}$ must be treated as  independent dynamical fields \cite{Karabali:2015epa}. Consequently, we need to couple, on the boundary $x_3=0$, an external source $\widehat{J}_i$  to $\left.\partial_3 A_i\right|_{x_3=0}$, as done  in \cite{Maggiore:2019wie}, where the 3D Maxwell theory with boundary has been studied:
\begin{equation}
S_{ext}\rightarrow \hat{S}_{ext}=\int d^4x\,
\left[\theta(x_3)\left(J_iA_i+\frac{1}{2}J_{ij}B_{ij}\right)
+ \delta(x_3)\widehat{J}_i\partial_3A_i\right]\ .
\label{Sexthat}\end{equation}

\section{The boundary algebra}\label{sect3}

Differentiating the two Ward identities \eqref{wardA} and \eqref{wardB} with respect to the external sources $J_i(x)$, $J_{ij}(x)$ and $\hat{J}_i(x)$ and then going at $J=0$ lead to six algebraic relations. We consider the subalgebra obtained as follows. \\

Differentiating the Ward identity \eqref{wardA} with respect to $J_m(x')$, and then going at vanishing external source, we get
\begin{equation}
\partial_m^X\delta^{(3)}(X-X') =
\partial_i^X \left(
-2k_1\epsilon_{ijk}\Delta_{A_mB_{jk}}(X',X) -4k_2 \Delta_{A_m\partial_3A_i}(X',X)
\right).
\label{3.1}\end{equation}
In \eqref{3.1} $\partial_i^X\equiv\frac{\partial}{\partial X_i}$, and the time-ordered propagator between two generic fields $\Phi(X)$ and $\Phi(X)$ is defined on the generating functional of the connected Green functions $Z_c[J]$, as usual, as
\begin{equation}
\begin{split}
\Delta_{\Phi\Psi}(X,X')
&\equiv
\left.
\frac{\delta^{(2)} Z_c}{\delta J_\Phi(X)\delta J_\Psi(X')}
\right|_{J_\Phi=J_\Psi=0}\\
&= \theta(x_0-x'_0)\langle \Phi(X)\Psi(X')\rangle +  \theta(x'_0-x_0)\langle \Psi(X')\Phi(X)\rangle.
\end{split}\label{defprop}\end{equation}
Hence, from \eqref{3.1} we have
\begin{equation}
\begin{split}
\partial^X_m\delta^{(3)}(X-X') =\ &
\delta(x_0-x'_0) \langle[A_m(X'),2k_1\widetilde{B}(X)+4k_2\partial_3A_0(X)]\rangle\\
&-2\theta(x_0-x_0')\langle\left[\partial_i^X\left(k_1\epsilon_{ijk}B_{jk}+2k_2\partial_3A_i\right)(X) A_m(X')\right]\rangle \\
&-2\theta(x_0'-x_0)\langle\left[A_m(X')\partial_i^X\left(k_1\epsilon_{ijk}B_{jk}+2k_2\partial_3A_i\right)(X) \right]\rangle,
\end{split}
\label{3.3}\end{equation}
where we defined
\begin{equation}
\widetilde{B}(X)\equiv\epsilon_{\alpha\beta}B_{\alpha\beta}(X)\ .
\label{Bdual}\end{equation}
The last two terms on the r.h.s. of \eqref{3.3} vanish on-shell due to \eqref{cond1} so that, at vanishing external sources, we get
\begin{equation}
\partial_m^X\delta^{(3)}(X-X') = \delta(x_0-x'_0)\langle[A_m(X'),2k_1\widetilde{B}(X)+4k_2\partial_3A_0(X)]\rangle\ .
\label{prealgebra1}\end{equation}
Following the same steps, differentiating the Ward identity \eqref{wardA} with respect to the external sources $J_{mn}(x')$ and $\widehat{J}_m(x')$ , we get, respectively
\begin{equation}
\delta(x_0-x'_0)\langle[B_{mn}(X'),k_1\widetilde{B}(X)+2k_2\partial_3A_0(X)]\rangle=0
\label{prealgebra2}\end{equation}
and
\begin{equation}
\delta(x_0-x'_0)\langle[k_1\widetilde{B}(X)+2k_2\partial_3A_0(X),\partial_3A_m(X')]\rangle=0\ .
\label{prealgebra3}\end{equation}
On the other hand, deriving the Ward identity \eqref{wardB} with respect to $J_{mn}(x')$, we obtain
\begin{equation}
\delta(x_0-x'_0)\langle[B_{mn}(X'),k_1\widetilde{B}(X)+2k_2\partial_3A_0(X)]\rangle=0\ .
\label{prealgebra4}\end{equation}
From the above algebraic relations, we get the following subalgebra formed by equal-time commutators:
\begin{equation}
\langle[A_\alpha(X),2k_1\widetilde{B}(X')+4k_2\partial_3A_0(X')]\rangle_{x_0=x'_0} = \partial_\alpha^X\delta^{(2)}(X-X')
\label{algebra1}\end{equation}
\begin{equation}
\langle\left[A_\alpha(X),A_\beta(X')\right]\rangle_{x_0=x'_0} = 0
\label{algebra2} \end{equation}
\begin{equation}
\left\langle\left[2k_1\widetilde{B}(X)+4k_2\partial_3A_0(X),2k_1\widetilde{B}(X')+4k_2\partial_3A_0(X')\right]\right\rangle_{x_0=x'_0} = 0\ ,
\label{algebra3}\end{equation}
which, written in terms of the 3D  boundary fields $\xi_i(X)$ \eqref{defxi} and $\Phi(X)$ \eqref{defphi}, implies
\begin{eqnarray}
\langle[\Phi(X),\epsilon_{\alpha\beta}\partial_\alpha\xi_\beta(X')]\rangle_{x_0=x'_0} &=& \delta^{(2)}(X-X') \label{cancomm1}\\
\langle\left[\Phi(X),\Phi(X')\right]\rangle_{x_0=x'_0} &=& 0 \label{cancomm2}\\
\langle\left[\epsilon_{\alpha\beta}\partial_\alpha\xi_\beta(X),\epsilon_{\alpha\beta}\partial_\alpha\xi_\beta(X')\right]\rangle_{x_0=x'_0} &=& 0\ . \label{cancomm3}
\end{eqnarray}

\section{The action induced on the 3D boundary}\label{sect4}

The commutators \eqref{cancomm1}, \eqref{cancomm2} and \eqref{cancomm3} can be interpreted as equal-time canonical commutation relations for the 3D canonically conjugate variables
\begin{equation}
\begin{split}
q(X) &\equiv \frac{1}{\sqrt{M}}\Phi(X)\\
p(X) &\equiv \sqrt{M}\epsilon_{\alpha\beta}\partial_\alpha\xi_\beta(X)\ ,
\end{split}
\label{qp}\end{equation}
and this allows us to identify the most general action $S_{3D}[\Phi,\xi]$ induced on the planar boundary $x_3=0$ of the 4D Maxwell-BF theory, which must display the following features:
\begin{enumerate}
\item
The 3D action $S_{3D}[\Phi,\xi]$ must be a local integrated functional of the 3D fields \eqref{defxi} and \eqref{defphi}, with canonical mass dimension equal to three.
\item
The 3D Lagrangian ${\cal L}_{3D}[\Phi,\xi]$ must be such that the relation between the canonically conjugate variables \eqref{qp} holds true:
\begin{equation}
p(X)=\frac{\partial{\cal L}_{3D}}{\partial\dot{q}(X)}\ .
\label{pqdot}\end{equation}
This implies that the Lagrangian ${\cal L}_{3D}[\Phi,\xi]$ must contain time derivatives only in the term
$p\dot{q}$.
\item
The action $S_{3D}[\Phi,\xi]$ must display the two symmetries which leave invariant the definitions  \eqref{defxi} and \eqref{defphi}:
\begin{enumerate}
\item gauge symmetry
\begin{equation}
\delta_{gauge}\xi_i=\partial_i\Lambda
\label{3Dgauge}\end{equation}
\item shift symmetry
\begin{equation}
\delta_{shift}\Phi=\mbox{constant}\ .
\label{3Dshift}\end{equation}
\end{enumerate}
\end{enumerate}
The most general action satisfying the above requests is
\begin{equation}
S_{3D}[\Phi,\xi] = \int d^3X \left [
c_1(\epsilon_{\alpha\beta}F_{\alpha_\beta})(\partial_0\Phi)
+c_2 F_{\alpha\beta}F_{\alpha\beta}
+c_3 \partial_\alpha\Phi\partial_\alpha\Phi
\right],
\label{3Daction}\end{equation}
where $F_{\alpha\beta}=\partial_\alpha\xi_\beta-\partial_\beta\xi_\alpha$ and $c_i,\ i=1,2,3$ are constants. From the action $S_{3D}[\Phi,\xi]$, we get the equations of motion
\begin{eqnarray}
\frac{\delta S_{3D}}{\delta\Phi} &=&
-2\partial_\alpha (
c_1\epsilon_{\alpha\beta}\partial_0\xi_\beta
+c_3\partial_\alpha\Phi) =0 \label{eomphi} \\
\frac{\delta S_{3D}}{\delta\xi_\alpha} &=&
2\partial_\beta (
c_1\epsilon_{\alpha\beta}\partial_0\Phi
+2c_2F_{\alpha\beta}) =0\ . \label{eomxi}
\end{eqnarray}

\section{Holographic constraint and duality}\label{sect5}

The equations of motion \eqref{eomphi} and \eqref{eomxi} of the scalar-vector 3D action $S_{3D}[\Phi,\xi]$ \eqref{3Daction} must be compatible with the boundary conditions \eqref{bcA} and \eqref{bcB} of the 4D Maxwell-BF theory  on the planar boundary $x_3=0$. In order to make this holographic contact \cite{Amoretti:2017xto}, the boundary condition \eqref{bcA} written in terms of the boundary degrees of freedom $\xi_i$ \eqref{defxi}  and $\Phi$ \eqref{defphi}  is
\begin{equation}
\epsilon_{ijk}\partial_j\xi_k-\kappa\partial_i\Phi =0\ ,
\label{duality}\end{equation}
where, besides \eqref{a1out} and \eqref{a3out}, we chose
\begin{equation}
a_2=0\ ,
\label{a2out}\end{equation}
 and we defined the dimensionless normalized mass parameter
\begin{equation}
\kappa\equiv\frac{m}{M}\ .
\label{defkappa}\end{equation}
We recognize in \eqref{duality} the duality relation found in \cite{Amoretti:2013nv} which extracts fermionic degrees of freedom from bosonic ones \cite{Aratyn:1983bg,Aratyn:1984jz,Amoretti:2013xya}. We shall come back to this point later. Here, it appears as the unique boundary condition which relates the 4D Maxwell-BF theory with boundary and  its holographic 3D counterpart(s), as we shall explicitly show.

Notice that the three components $i=\{0,\alpha\}$ of \eqref{duality} are
\begin{eqnarray}
\epsilon_{\alpha\beta}F_{\alpha\beta}-2\kappa\partial_0\Phi &=& 0 \label{duality0} \\
\epsilon_{\alpha\beta}\partial_0\xi_\beta-\epsilon_{\alpha\beta}\partial_\beta\xi_0+\kappa\partial_\alpha\Phi &=& 0\ , \label{dualityalpha}
\end{eqnarray}
which are compatible with \eqref{eomphi} and \eqref{eomxi} if
\begin{equation}
c_2=-\frac{1}{2\kappa}c_1\ \ ;
c_3=\kappa c_1
\label{detci}\end{equation}
and if the temporal gauge choice for the 3D gauge field $\xi_i$ is imposed
\begin{equation}
\xi_0=0\ .
\label{gaugechoicexi}\end{equation}

\section{Summary of results and discussion}\label{sect6}

When a boundary is introduced in a quantum field theory, a crucial role is played by the boundary term, which in the case studied in this paper is represented by $S_{bd}$ \eqref{Sbd}, which depends by a number of constant parameters which need to be fine tuned in order to determine the holographic theory induced on the boundary. For the 4D Maxwell-BF theory, the boundary term finally reduces to
\begin{equation}
S_{bd}=\frac{m}{2}\int d^4x\, \delta(x_3)
A_iA_i\ ,
\label{finalSbd}\end{equation}
which depends on one massive parameter $m$ only. \\

In presence of a boundary, the question naturally arises of which boundary conditions should be imposed. The procedure described in this paper leads to the following boundary condition compatible with the holographic construction:
 \begin{equation}
\left. 2k_1\epsilon_{ijk}B_{jk} + 4k_2 \partial_3A_i  - mA_i\right|_{x_3=0} = 0\ ,
\label{finalbc} \end{equation}
which involves both the $k_1$-BF and the $k_2$-Maxwell terms. Notice that it depends on one parameter only $(m)$. It is of a nonstandard type, since it does not fall into the usual Dirichlet, Neumann or Robin boundary conditions on each field, separately. On the contrary, it mixes both the fields, and the effect of the Maxwell term is to introduce a dependence on the transverse component of the gauge field with respect to the planar boundary, which is independent of the longitudinal components, and consequently must be treated as an independent dynamical variable on the boundary. As we remarked, an unexpected consequence of the boundary conditions \eqref{bcA} and \eqref{bcB}, is that they can be solved by the set of parameters \eqref{particularchoicebc}, which implies, in particular, Neumann boundary condition for the gauge field $A_\mu$. This corresponds to eliminating any dependence from $k_2$, $i.e.$ from the Maxwell term, in the physics on the boundary, represented by the current algebra and by the boundary conditions themselves.   \\

The boundary breaks all the invariances of the unbounded theory: translations, parity and gauge symmetries. The first consequence concerns the choice of the gauge conditions, implemented by the gauge fixing term $S_{gf}$ \eqref{Sgf}, which does not need to be covariant. With deeper consequences, the Ward identities describing the Ward identities are broken by boundary terms:
\begin{equation}
\int_0^\infty dx_3\, \partial_iJ_i =
-\partial_i\left.\left(2k_1\epsilon_{ijk}B_{jk}+4k_2\partial_3A_i\right)\right|_{x_3=0}
\label{wardA.1}\end{equation}
\begin{equation}
\int_0^\infty dx_3\, \partial_jJ_{ij}=
4k_1\left. \epsilon_{ijk}\partial_jA_k\right|_{x_3=0}\ .
\label{wardB.1}\end{equation}
At $J=0$, $i.e.$ going on-shell, the vanishing of the boundary breakings lead to define on $x_3=0$ an ``electric'' and a ``magnetic'' field, and the corresponding potentials:
\begin{eqnarray}
{\cal E}_i &\equiv& A_i \propto \partial_i\Phi \label{defphi.1} \\
{\cal H}_i &\equiv& k_1\epsilon_{ijk}B_{jk}+2k_2\partial_3A_i \propto \epsilon_{ijk}\partial_j\xi_k\ , \label{defxi.1}
\end{eqnarray}
which allow to physically interpret the contribution of the Maxwell term as a deformation of the magnetic field on the boundary.  Without Maxwell term, $i.e.$ at $k_2=0$, the fields $A_i$ and $B_{ij}$ are  interpreted as electric and magnetic fields on the boundary, respectively. Their transverse component do not enter in the game and could safely be eliminated by Neumann boundary conditions. In presence of the Maxwell term, this is no longer true for the gauge field $A$, for which both the longitudinal and transverse components are physically important: the former as electric field, the latter as deformation of the main magnetic field represented by the dual of the $B$-field. The potentials corresponding to the boundary electric and magnetic fields  \eqref{defphi.1} and \eqref{defxi.1} are the degrees of freedom by means of which the holographic 3D theory is constructed: a vector field $\xi_i$ and a scalar field $\Phi$.
\\

The algebra obtained from the broken Ward identities \eqref{wardA.1} and \eqref{wardB.1} by differentiating them with respect to the external sources $J$, written in terms of the boundary degrees of freedom is
\begin{eqnarray}
\langle[\Phi(X),\epsilon_{\alpha\beta}\partial_\alpha\xi_\beta(X')]\rangle_{x_0=x'_0} &=& \delta^{(2)}(X-X') \label{cancomm1.1}\\
\langle\left[\Phi(X),\Phi(X')\right]\rangle_{x_0=x'_0} &=& 0 \label{cancomm2.1}\\
\langle\left[\epsilon_{\alpha\beta}\partial_\alpha\xi_\beta(X),\epsilon_{\alpha\beta}\partial_\alpha\xi_\beta(X')\right]\rangle_{x_0=x'_0} &=& 0\ , \label{cancomm3.1}
\end{eqnarray}
which can be seen as equal-time canonical commutation relations between canonical variables $q(X)$ and $p(X)$. \\

Once identified the canonical variables, the corresponding action is found to be
\begin{equation}
S_{3D}[\Phi,\xi] = c_1\int d^3X \left [
(\epsilon_{\alpha\beta}F_{\alpha\beta})(\partial_0\Phi)
-\frac{1}{2\kappa} F_{\alpha\beta}F_{\alpha\beta}
+\kappa \partial_\alpha\Phi\partial_\alpha\Phi
\right],
\label{final3Daction}\end{equation}
which is the most general 3D local integrated functional built with a scalar and a vector field, respecting power counting and invariant under the gauge \eqref{3Dgauge} and shift \eqref{3Dshift} symmetries, and, most important, whose equations of motion are compatible with the boundary conditions \eqref{finalbc}:
\begin{equation}
\epsilon_{ijk}F_{jk}-2\kappa\partial_i\Phi =0\ .
\label{duality1}\end{equation}
The equation \eqref{duality1} coincides with the duality relation between a scalar and a vector field which has been invoked in \cite{Aratyn:1983bg,Aratyn:1984jz,Amoretti:2013nv} as the main tool for the mechanism of fermionization of bosonic degrees of freedom. In other words, the effective dynamical variables living on the boundary of the 4D Maxwell-BF are fermionic. This feature is crucial for the interpretation of the boundary degrees of freedom as the edge states of the 3D topological insulators \cite{Cho:2010rk,Blasi:2011pf}. The new fact that we are recovering here, is that this property, which has been related to the topological character of the bulk theory \cite{Amoretti:2014kba}, holds indeed also for a non-topological theory like Maxwell-BF theory. We shall comment in more detail this point later. The duality relation \eqref{duality1} has also more field theoretical consequences. Thanks to \eqref{duality1}, indeed, the action \eqref{final3Daction} is covariant, despite the appearances. In fact,  the scalar field $\Phi$ can be eliminated from the action through \eqref{duality1}, and we find the 3D Maxwell theory
\begin{equation}
S_{Max}[\xi]= \frac{c_1}{2\kappa} \int d^3X\ F_{ij}F_{ij}\ .
\label{3Dmax}\end{equation}
Alternatively, the duality relation \eqref{duality1} allows us to trade the vector for the scalar field
\begin{equation}
S_{KK}[\Phi]= \kappa c_1\int d^3X\ \partial_i\Phi\partial_i\Phi\ .
\label{3Dkk}\end{equation}
The scalar-vector 3D theory described by \eqref{final3Daction}, the Maxwell action \eqref{3Dmax} and the massless Klein-Gordon  action \eqref{3Dkk} are all holographic counterparts of the 4D Maxwell-BF theory with planar boundary.
The two actions \eqref{3Dmax} and \eqref{3Dkk} are related by the duality relation \eqref{duality}, which may be written in a way to emphasize its strong-weak coupling aspect:
\begin{equation}
\partial_i\Phi\leftrightarrow\epsilon_{ijk}F_{jk}\ \
\cup
\ \  \kappa\leftrightarrow\frac{1}{\kappa}\ .
\label{kappaduality}\end{equation}
In this form, it is apparent that the coupling $\kappa$ \eqref{defkappa}  governs the regimes where one type of action dominates with respect to the other: at strong coupling (very large $\kappa$) the dominating  term is the Maxwell one, while at weak coupling (very small $\kappa$), it is the massless scalar action which dominates. At intermediate regimes, both the degrees of freedom are present, and the relevant action is \eqref{final3Daction}, whose degrees of freedom are fermionic, due to \eqref{duality1}, as we already remarked. It is this intermediate regime which is relevant for the topological insulators, for which the same action \eqref{final3Daction} has been proposed in \cite{Cho:2010rk}. Surprisingly, the order parameter $\kappa$  which distinguishes the various regimes is directly related through \eqref{defkappa} to the only effective parameter $m$ which survives in the boundary term \eqref{finalSbd}, which hence plays a much more crucial role than one might expect at first sight. \\

We conclude this paper with a remark concerning the effect of the presence of the Maxwell term in the bulk action \eqref{Sbulk} together with a general consideration on the topological nature of quantum field theories. The main feature of topological quantum field theories is the lack of local observables, the only observables being global geometrical properties of the manifolds on which they are built. Technically, this means that the local cohomology of the BRS operator is empty. Examples of topological quantum field theories are the 3D Chern-Simons theory and BF theories, which may be defined in any spacetime dimensions. The 4D Maxwell-BF theory studied in this paper is not topological, since its action depends on the metric through the Maxwell term, and its cohomological structure is nontrivial. Nevertheless the bulk action has no local observables since the equations of motion enforce the vanishing of the $F_{\mu\nu}$ tensor; thus we are looking at a different class of ``topological'' theories where the boundary is expected to carry all the physical information.  If a boundary is introduced like we did, this nontriviality  reflects in the algebra derived from the Ward identities \eqref{wardA.1} and \eqref{wardB.1}, which is rather complicated. We wrote down the four relations (out of six) \eqref{prealgebra1}, \eqref{prealgebra2}, \eqref{prealgebra3} and \eqref{prealgebra4}, where the dependence on the Maxwell term is highlighted by the coefficient $k_2$. In addition, the boundary condition \eqref{finalbc} explicitly depends on the Maxwell term, which physically results in a perturbation of the magnetic field ${\cal H}$ \eqref{defxi.1}. On the other hand, when constructing the holographic scalar-vector 3D theory, the presence of the Maxwell term is buried in the definition of the vector potential $\xi_i$ \eqref{defxi.1}, and both the 3D actions (in their duality-related representations \eqref{final3Daction}, \eqref{3Dmax} and \eqref{3Dkk} and the duality relation \eqref{duality1} are the same as in the case of pure BF with boundary. In other words, from the holographic point of view, Maxwell-BF theory with boundary is indistinguishable from the pure topological BF theory, like if holography would protect the topological character of bulk theories.

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