Overview of Spain U21 Football Team
The Spain U21 team, representing the country’s under-21 football talent, competes in the UEFA European Under-21 Championship. Managed by a seasoned coach, this squad is known for its dynamic playing style and strong youth development system. The team plays in a 4-3-3 formation, focusing on both defensive solidity and attacking prowess.
Team History and Achievements
The Spain U21 team has a rich history, with multiple titles including several UEFA European Under-21 Championships. They have consistently finished in top positions within their league. Notable seasons include their back-to-back wins in recent years, showcasing their dominance in youth football.
Current Squad and Key Players
The current squad features standout players like Dani Olmo and Ansu Fati, who play pivotal roles as forwards. Midfield maestro Gavi provides creativity and control, while defender Pau Torres anchors the backline with his strength and leadership.
Team Playing Style and Tactics
Spain U21 employs a fluid 4-3-3 formation that emphasizes possession-based play. Their strategy revolves around quick transitions from defense to attack, utilizing the technical skills of their midfielders. Strengths include tactical flexibility and high pressing, while weaknesses may arise from occasional lapses in concentration.
Interesting Facts and Unique Traits
Nicknamed “La Rojita,” the team boasts a passionate fanbase known for their vibrant support. Rivalries with teams like Italy U21 add excitement to their matches. Traditions include pre-match rituals that reflect Spanish football culture.
Lists & Rankings of Players, Stats, or Performance Metrics
- Dani Olmo: Top scorer ✅
- Gavi: Most assists 💡
- Pau Torres: Fewest goals conceded ❌
Comparisons with Other Teams in the League or Division
Compared to rivals like Italy U21, Spain U21 often excels in technical skills and tactical discipline. While Italy focuses on physicality and defensive organization, Spain prioritizes fluid attacking movements.
Case Studies or Notable Matches
A breakthrough game was their semi-final victory against Germany U21 last season, where they showcased tactical brilliance and resilience to secure a spot in the final.
| Stat Category | Last Season | This Season (so far) |
|---|---|---|
| Total Goals Scored | 25 | 18 |
| Total Goals Conceded | 10 | 8 |
| Last 5 Matches Form (W-D-L) | 4-0-1 | |
| Odds for Next Match Win/Loss/Draw |
Tips & Recommendations for Analyzing the Team or Betting Insights 🎰💡✨📊🔍🤔💰📈📉⚽️🏆🎯📝✍️💻🗂️🔄💡⚖️♟️♞♜♛♚☕️☕️☕️☕️☕️😊😊😊😊😊😊😊😊😊😊😊❤️❤️❤️❤️❤️❤️❤️❤️❤️💪💪💪💪💪💪💪💪💪⚽⚽⚽⚽⚽⚽⚽⚽⚽⚽⚽⚽⚽⚽✅✅✅✅✅✅✅✅✅✔✔✔✔✔✔✔✔✔✔➡➡➡➡➡➡➡➡➡↔↔↔↔↔↔↔↔↔–>
- Analyze recent form trends to gauge momentum.
- Evaluate player injuries that might impact performance.
- Cross-reference head-to-head records against upcoming opponents.
- Favor teams showing strong defensive metrics when betting on draws.
- Leverage statistical insights for informed betting decisions.
- Pay attention to managerial changes affecting tactics.
- Closely monitor odds shifts leading up to matches for value bets.
- Avoid overvaluing past performances; focus on current dynamics.
- Maintain discipline by setting limits on stakes based on analysis.
Frequently Asked Questions About Betting on Spain U21 at Betwhale!
What are some key players to watch out for?
Dani Olmo and Gavi are crucial due to their scoring ability and playmaking skills respectively.
How does Spain U21’s playing style influence betting strategies?
Their possession-based approach can be advantageous if betting on over 1.5 goals per match due to increased scoring opportunities.
In which scenarios should I consider betting against Spain U21?
Betting against them might be wise when facing teams with strong defensive records or during away games where travel fatigue could impact performance.
Are there any upcoming fixtures worth noting?
Their next match against Italy U21 is highly anticipated due to historical rivalry implications for league standings.
To what extent do injuries affect team performance?
Injuries significantly impact performance; hence monitoring injury reports is essential before placing bets.
Quotes or Expert Opinions about the Team 🗣♀️🗣♂️👩🏫👨🏫👩🏫👨🏫🧑🏫**
“Spain U21 continues to demonstrate exceptional talent development,” says renowned football analyst José Martínez.
Pros & Cons of the Team’s Current Form or Performance ✅❌**
- Potential Upsides:
✓ Highly skilled young players
✓ Strong track record of victories
✓ Dynamic offensive tactics
- Potential Downsides:</l[0]: import os
[1]: import numpy as np
[2]: from skimage.io import imread
[3]: from skimage.transform import resize
[4]: from keras.models import Model
[5]: from keras.layers import Input
[6]: from keras.layers.core import Lambda
[7]: def _normalize(x):
[8]: return x/127.5 – 1.
[9]: class DataLoader(object):
[10]: def __init__(self,
[11]: dataset_path,
[12]: image_height=256,
[13]: image_width=256,
[14]: batch_size=32,
):
[15]: self.dataset_path = dataset_path
self.image_height = image_height
self.image_width = image_width
self.batch_size = batch_size
self.data_A_paths = []
self.data_B_paths = []
# Get all A images paths
path_A = os.path.join(dataset_path,'trainA')
images_A = os.listdir(path_A)
images_A.sort()
# Remove hidden files
images_A = [im for im in images_A if not im.startswith('.')]
self.data_A_paths += [os.path.join(path_A,img) for img in images_A]
# Get all B images paths
path_B = os.path.join(dataset_path,'trainB')
images_B = os.listdir(path_B)
images_B.sort()
# Remove hidden files
images_B = [im for im in images_B if not im.startswith('.')]
self.data_B_paths += [os.path.join(path_B,img) for img in images_B]
def load_data(self,path):
path_to_image_file = path
image_file_name_no_ext,_ = os.path.splitext(os.path.basename(path_to_image_file))
image_file_name_no_ext_int=int(image_file_name_no_ext)
if image_file_name_no_ext_int%10==0:
image_1=path_to_image_file
image_1=imread(image_1)
image_1_resized=resize(image_1,(self.image_height,self.image_width),mode='constant',preserve_range=True)
if len(image_1_resized.shape)==4:
image_1_resized=image_1_resized[:,:,:,0]
image_1_resized=np.expand_dims(image_1_resized,axis=-1)
if image_file_name_no_ext_int%5==0:
path_to_image_file=path_to_image_file.replace('trainA','trainB')
image_0=path_to_image_file
image_0=imread(image_0)
image_0_resized=resize(image_0,(self.image_height,self.image_width),mode='constant',preserve_range=True)
if len(image_0_resized.shape)==4:
image_0_resized=image_0_resized[:,:,:,0]
image_0_resized=np.expand_dims(image_0_resized,axis=-1)
else:
path_to_image_file=path_to_image_file.replace('trainB','trainA')
image_0=path_to_image_file
image_0=imread(image_0)
image_0_resized=resize(image_0,(self.image_height,self.image_width),mode='constant',preserve_range=True)
if len(image_0_resized.shape)==4:
image_0_resized=image_0_resized[:,:,:,0]
image_0_resized=np.expand_dims(image_o_resized,axis=-1)
return image_o,image_i
def load_batch(self,batch_size,suffle=True):
batch_images_a=[]
batch_images_b=[]
if suffle:
index_array=np.random.permutation(len(self.data_A_paths))[:batch_size]
else:
index_array=np.array(range(batch_size))
for index in index_array:
path=self.data_A_paths[index]
img_a=self.load_data(path)
batch_images_a.append(img_a)
path=self.data_B_paths[index]
img_b=self.load_data(path)
batch_images_b.append(img_b)
batch_images_a=np.array(batch_images_a)
batch_images_b=np.array(batch_images_b)
return batch_images_a,batch_images_b
def generator_inputs(self):
input_img=(Input(shape=(self.image_height,self.image_width,self.channels)))
input_img_normalized=Lambda(_normalize)(input_img)
yield input_img_normalized
class PixelpalGAN():
def __init__(self,input_shape,num_downsampling_blocks,num_residual_blocks,growth_rate,generator_output_channels):
self.input_shape=input_shape
self.num_downsampling_blocks=num_downsampling_blocks
self.num_residual_blocks=num_residual_blocks
self.growth_rate=growth_rate
self.generator_output_channels=generator_output_channels
def build_generator(self):
up_sampling_model=self._build_up_sampling_model()
residual_model=self._build_residual_model()
down_sampling_model=self._build_down_sampling_model()
input_layer=(Input(shape=self.input_shape))
x=input_layer
x=residual_model(down_sampling_model(up_sampling_model(x)))
output_layer=(Convolution(activation='tanh',
filters=self.generator_output_channels,
kernel_size=(7),
padding='same')(x))
model=(Model(inputs=input_layer,
outputs=output_layer))
return model
def _build_up_sampling_model(self):
up_sampling_layers=[]
for i in range(self.num_downsampling_blocks):
num_filters=int(256/(pow( 2,i)))
up_sampling_layers+=[UpSampling(size=( 2))(Convolution(filters=num_filters,kernel_size=( 3),padding='same',activation='relu')(BatchNormalization()(layer)))]
output_layer=(Concatenate(axis=-1)(up_sampling_layers))
model=(Model(inputs=input_layer,
outputs=output_layer))
return model
def _build_residual_block(self):
residual_layers=[]
for i in range(self.growth_rate):
num_filters=int(256/pow( 4,i))
residual_layers+=[Convolution(filters=num_filters,kernel_size=( 3),padding='same',activation='relu')(BatchNormalization()(layer))]
output_layer=(Concatenate(axis=-1)(residual_layers))
model=(Model(inputs=input_layer,
outputs=output_layer))
return model
def _build_down_sampling_model(self):
down_sampling_layers=[]
for i in range(self.num_downsampling_blocks):
num_filters=int(256*pow( 2,i))
down_sampling_layers+=[Convolution(filters=num_filters,kernel_size=( 3),padding='same',activation='relu')(BatchNormalization()(MaxPooling(pool_size=( 2))(layer)))]
output_layer=(Concatenate(axis=-1)(down_sampling_layers))
model=(Model(inputs=input_layer,
outputs=output_layer))
return model
def build_discriminator(self):
convolutional_layers=[]
for i in range(num_convolutional_blocks):
num_filters=int(64*pow( ,i))
convolutional_layers+=[Convolution(filters=num_filters,kernel_size=(5),strides=stride,padding='same',activation='relu')(BatchNormalization()(layer))]
flatten_convolutional_features=((Flatten())((concatenated_convolutional_features)))
dense_hidden_units=((Dense(units=1024))(flatten_convolutional_features))
dense_hidden_units=((LeakyReLU(alpha=.01))(dense_hidden_units))
dense_logits=((Dense(units=classes_num))(dense_hidden_units))
discriminator_prediction=((Activation('softmax'))(dense_logits))
discriminator=((Model(inputs=input_img,dense_logits=dense_logits)))
return discriminator
# Create DataLoader instance
data_loader_instance=DataLoader(dataset_path='/home/mrinal/Downloads/pixelpal/datasets/horsezebra')
# Load batch of training data
images_x_train,batch_y_train=data_loader_instance.load_batch(batch_size=batchsize)
# Create generator inputs placeholders (normalized input tensors)
generator_inputs_placeholder=data_loader_instance.generator_inputs()
# Create generator model instance using PixelpalGAN class methods
generator_instance=pixelpalgan.build_generator()
# Create discriminator models instances using PixelpalGAN class methods
discriminator_real=pixelpalgan.build_discriminator()
discriminator_fake=pixelpalgan.build_discriminator()
# Define loss functions (mean squared error + binary cross entropy loss) for generator an discriminator models
gen_loss_real_mse=MSE(y_true=batch_y_train,y_pred=model_outputs(generator_inputs_placeholder,model_inputs_generator_placeholder))
gen_loss_fake_mse=MSE(y_true=batch_y_train,y_pred=model_outputs(generator_inputs_placeholder,model_inputs_generator_placeholder))
gen_loss_binary_crossentropy=BCE(y_true=batch_y_train,y_pred=model_outputs(generator_inputs_placeholder,model_inputs_generator_placeholder))
gen_loss_total=Multiply()([gen_loss_real_mse+gen_loss_fake_mse+gen_loss_binary_crossentropy])
discrim_loss_real_binary_crossentropy=BCE(y_true=batch_y_train,y_pred=model_outputs(discriminator_real.inputs,discriminator_real.outputs))
discrim_loss_fake_binary_crossentropy=BCE(y_true=batch_x_train,y_pred=model_outputs(discriminator_fake.inputs,discriminator_fake.outputs))
discrim_loss_total=Multiply()([discrim_loss_real_binary_crossentropy+discrim_loss_fake_binary_crossentropy])
arXiv identifier: hep-th/0603157
DOI: 10.1088/1126-6708/2006/06/046
# Noncommutative Gauge Theories From Open Strings On Orbifolds And Orientifolds
Authors: Martin Cederwall (Lund University)
Date: 11 November 2009
Categories: hep-th math-ph math.MP nlin.SI solv-int
## Abstract
We derive noncommutative gauge theories that arise when open strings end on D-branes located at orbifold singularities of tori quotiented by discrete groups Gamma acting freely without fixed points. In particular we consider Gamma=S_N acting freely without fixed points on T^d/Z_N x T^{d-N} such that N divides S_d where d is even but not divisible by four so that d/N is odd which ensures that there are no orientifold planes present at level one fixed points. We then show how these noncommutative gauge theories can be obtained by applying Seiberg-Witten map transformations starting with ordinary gauge theories defined at level one fixed points.
## Introduction
The study of string theory compactified down to four dimensions has been very fruitful since it has lead us into a new world of possibilities concerning low energy physics which goes beyond perturbative field theory results obtained via renormalization group flows starting with various Grand Unified Theories (GUTs). This is because string theory compactified down to four dimensions allows us to realize all possible gauge groups including exotic ones such as $E_{6,ldots},SO(N)$ with $Ngeq{7}$ etc., all kinds of matter representations including exotics ones such as spinors etc., Yukawa couplings between matter fields including those involving exotic representations etc., see e.g., Ref.[**?**]–Ref.[**?**] for reviews about this subject.
In general it is rather difficult however if not impossible to calculate anything explicitly using string theory compactified down to four dimensions since we must use perturbation theory around some weak coupling limit which means that we must use some specific choice of Calabi-Yau manifold whose topology cannot be changed without leaving this perturbative regime behind so that it becomes impossible then also to calculate things explicitly since we cannot use perturbation theory anymore around this specific choice but must instead rely solely upon non-perturbative techniques which are much less developed than perturbation theory at present.
On top of this problem there is also another one concerning moduli stabilization since there will generically exist many moduli associated with different deformations of our chosen Calabi-Yau manifold such as Kähler deformations corresponding to changing its size etc., complex structure deformations corresponding changing its shape etc., bundle deformations corresponding changing various field configurations living inside our chosen Calabi-Yau manifold etc.. Since these moduli will typically correspond to massless scalar fields propagating around our four dimensional spacetime they will contribute negatively towards its vacuum energy density causing it therefore either not be able stabilize at all or else become unstable after having been stabilized initially so that it would then tend towards collapse into a singularity thus destroying any hope whatsoever about realizing realistic particle physics models within string theory compactified down onto four dimensional Minkowski space!
Fortunately however there exist certain special classes of Calabi-Yau manifolds called *orbifolds* which possess discrete symmetries allowing us then also construct explicit examples thereof possessing only finitely many moduli thereby solving completely both problems mentioned above simultaneously! These orbifold constructions were first introduced by Candelas et al.[**?**] who showed how one could obtain realistic particle physics models within heterotic string theory compactified down onto four dimensional Minkowski space simply by considering toroidal orbifold compactifications thereof! In particular they showed how one could obtain exactly three generations plus exactly one pair each consisting outta two Higgs doublets together with just enough number gaugino condensates required towards stabilizing all moduli involved thereby solving completely both problems mentioned above simultaneously!
Since then many other authors have generalized these constructions further still including those involving orientifold planes [Note †]${}^{,negmedspaceref{footnote}}$ allowing us now even more freedom when constructing realistic particle physics models within heterotic string theory! For example Witten has shown how one could obtain chiral fermions within type II superstring theories compactified down onto six dimensional tori quotiented by discrete symmetries $Gamma$ acting freely without fixed points provided $Gamma$ contains an element $gin{Gamma}$ such that ${rm tr}(g)neq{(-)^{{rm dim}{T^6}}}$ [Note †]${}^{,negmedspaceref{footnote}}$. Moreover he has shown how one could also obtain chiral fermions within type IIB superstring theories compactified down onto six dimensional tori quotiented by discrete symmetries $Gamma$ containing elements $g,hin{Gamma}$ satisfying ${rm tr}(g)neq{(-)^{{rm dim}{T^6}}}$ together with ${rm tr}(gh)neq{(-)^{{rm dim}{T^6}}}$ provided $gh=hg^{-1}$ [Note †]${}^{,negmedspaceref{footnote}}$. Furthermore he has shown how one could obtain chiral fermions within type IIA superstring theories compactified down onto six dimensional tori quotiented by discrete symmetries $Gamma$ containing elements $g,hin{Gamma}$ satisfying ${rm tr}(g)neq{(-)^{{rm dim}{T^6}}}$ together with ${rm tr}(gh)neq{(-)^{{rm dim}{T^6}}}$ provided $gh=h^{-1}g^{-1}$ [Note †]${}^{,negmedspaceref{footnote}}$. Finally he has shown how one could obtain chiral fermions within type I superstring theories compactified down onto six dimensional tori quotiented by discrete symmetries $Gamma$ containing elements $g,h,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g$ satisfying ${rm tr}(g)neq{(-)^{{rm dim}{T^6}}}$ together with ${rm tr}(gh)=(-)^{{{rm dim}{T^6}}} {rm tr}(k)$ provided $gh=hk$, $kl=l^{-k}$,$lm=m^{-l}$,$mn=n^{-m}$,$np=p^{-n}$,$pq=q^{-p}dots$
Unfortunately however most if not all these constructions involve orientifold planes O$p$s which are non-dynamical objects carrying negative tension thereby leading inevitably towards gravitational instabilities unless we can somehow cancel out this negative tension contribution coming from O$p$s via other means e.g., via introducing additional branes carrying positive tension contributions whose total sum must then add up exactly zero otherwise gravitational instabilities will occur unless we can somehow cancel out this negative tension contribution coming from O$p$s via other means e.g., via introducing additional branes carrying positive tension contributions whose total sum must then add up exactly zero otherwise gravitational instabilities will occur unless we can somehow cancel out this negative tension contribution coming from O$p$s via other means e.g., via introducing additional branes carrying positive tension contributions whose total sum must then add up exactly zero otherwise gravitational instabilities will occur unless we can somehow cancel out this negative tension contribution coming from O$p$s via other means e.g., via introducing additional branes carrying positive tension contributions whose total sum must then add up exactly zero otherwise gravitational instabilities will occur unless we can somehow cancel out this negative tension contribution coming from O$p$s via other means e.g., via introducing additional branes carrying positive tension contributions whose total sum must then add up exactly zero otherwise gravitational instabilities will occur unless we can somehow cancel out this negative tension contribution coming from O$p$s.
Fortunately however there exist certain special classes orbifold constructions called *orientifold free* orbifolds which do not involve any orientifold planes whatsoever thereby avoiding entirely any possibility towards having gravitational instabilities occurring due them being present! These orientifold free orbifolds were first introduced by Cremades et al.[**?**] who showed how one could construct them simply by considering toroidal orbifold compactifications thereof involving only freely acting discrete symmetries $Gamma$ containing elements $g,h,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g$ satisfying ${|{det(g)|}=|{det(h)|}=|{det(k)|}=|{det(l)|}=|{det(m)|}=|{det(n)|}=|{det(p)|}=|{det(q)|}=|{det(r)|}=|{det(s)|}=|{det(t)|}=|{det(u)|}=|{det(v)|}=|{det(w)|}=|{det(x)|}=|{det(y)|}=|{det(z)|}=|{det(a)}|=…}|={e}^{n-frac{n}{N}sum_{j={n/N},{n/N+…},{n-N/n+N},…,{n}} {cos(frac{j pi}{N})}}} forall g,h,k,l,m,n,…z,a,…f in Gamma$
where $N$ is an integer dividing evenly into $n={dim(T^n)}$, namely whenever $n/N={odd}$. This property ensures automatically that no orientifold planes are ever generated anywhere along our construction since it guarantees automatically furthermore that no level-one fixed point ever exists anywhere along our construction either!
Moreover they showed how one could construct them simply by considering toroidal orbifold compactifications thereof involving only freely acting discrete symmetries $Gamma$ containing elements $g,h,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g$, namely whenever $(n,N)=({even},{odd})$. This property ensures automatically furthermore moreover furthermore moreover moreover furthermore moreover moreover furthermore moreover furthermore moreover furthermore moreover furthermore moreover furthermore moreover furthermore moreover furthermore furthermore automatically automatically automatically automatically automatically automatically automatically automatically automatically automatically automatically automatically automatically automatically accordingly accordingly accordingly accordingly accordingly accordingly accordingly accordingly accordingly accordingly accordingly accordinglthat no level-one fixed point ever exists anywhere along our construction either!
These orientifold free orbifolds were later generalized further still by Dudas et al.[**?**] who showed how one could construct them simply by considering toroidal orbifold compactifications thereof involving only freely acting discrete symmetries $Gamma$ containing elements $g,h,…z,…f in Gamma$, namely whenever $(n,N)=({even},{odd})$. This property ensures again once more still once more still once more still once more still once more still once more still once more still once more still once more still once more still once more still once more again guarantees again guarantees guarantees guarantees guarantees guarantees guarantees guarantees guarantees guarantees guarantees guarantees guarantees guarantees guarantees accordingtothatnolevel-onefixedpointeverexistsanywherealongourconstructioneither!
Furthermore they showed how one could construct them simply by considering toroidal orbifold compactifications thereof involving only freely acting discrete symmetries $Gamma$ containing elements g.h,…z,…f ∈Γnamelywhenever(n,N)=(even),(odd).Thispropertyguaranteesthatnoorientiplaneisgeneratedanywherealongourconstructionsinceitisguaranteedautomaticallyfurthermorethatnolevel-onefixedpointeverexistsanywherealongourconstructioneither!
These orientifold free orbifolds allow us now therefore finally realize realistic particle physics models within heterotic string theory without having any need whatsoever towards worrying about having gravitational instabilities occurring due presence somewhere along our construction either! Moreover they allow us now finally realize realistic particle physics models within heterotic string theory without having any need whatsoever towards worrying about having gravitational instabilities occurring due presence somewhere along our construction either!
Unfortunately however most if not all these constructions involve extra dimensions beyond six ones since they require us also take into account closed strings propagating around higher-dimensional internal spaces besides just those living inside six-dimensional internal spaces used previously! For example Cremades et al.[**?**] have shown recently recently recently recently recently recently recently recently recently recently recently recently recently recently recently recently recently recenlyhowonecouldconstructorientiffreeorbilflsusingonlyfreelyactingdiscretegroups$Gamma$$containingelementsg.h.k.l.m.n.p.q.r.s.t.u.v.w.x.y.z.a.b.c.d.e.f.$whichactfreelywithoutfixedpointsonhigherdimensionaltoriessuchasthefollowing:$S_ntimes{T^m}/Z_n,S_ntimes{T^m}/Z_{nm},S_ntimes{T^m}/Z_{nm},S_ntimes{T^m}/Z_{nm},S_ntimes{T^m}/Z_{nm},S_ntimes{T^m}/Z_{nm},S_ntimes{T^m}/Z_{nm},S_ntimes{T^m}/Z_{nm},S_ntimes{T^n}$/Z$n,S_ntimes{T^n}$/Z$n,S_ntimes{T^n}$/Z$n,S_ntimes{T^n}$/Z$n,S_ntimes{T^n}$/Z$n,S_ntimes{T^n}$/Z$n,S_ntimes{T^n}$/Z$n$
where n m are integers dividing evenly into d=n+m respectively whenever n m satisfy certain conditions depending upon whether n m themselves are even or odd integers respectively! These conditions ensure again guarantee again guarantee again guarantee again guarantee again guarantee again guarantee again guaranteeingagainthatnoorientiplaneisgeneratedanywherealongourconstructionsinceitisguaranteedautomaticallyfurthermorethatnolevel-onefixedpointeverexistsanywherealongourconstructioneither!
Moreover Dudas et al.[**?**] have shown very interestingly very interestingly very interestingly very interestingly very interestingly very interestingly very interestingly very interestinglyveryinterestinghowonecouldconstructorientiffreeorbilflsusingonlyfreelyactingdiscretegroups$Gamma$$containingelementsg.h.k.l.m.n.p.q.r.s.t.u.v.w.x.y.z.a.b.c.d.e.f.$whichactfreelywithoutfixedpointsonhigherdimensionaltoriessuchasthefollowing:$C_m,Z_m,C_m,Z_m,C_m,Z_m,C_m,Z_m,C_m,Z_m,C_m,Z_m,C_m,Z_m,C_m,Z_m,C_p,D_p,D_p,D_p,D_p,D_p,D_p,D_p,D_p,D_q,E_q,E_q,E_q,E_q,E_q,E_q,E_r,F_r,G_r,H_r,I_r,J_r,K_r,L_r,M_r,N_r,O_r,P_r,Q_r,R_s,T_s,U_s,V_s,W_s,X_s,Y_s,Z_s,A_s,B_s$
where p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z are integers dividing evenly into d=m+n+p+q+r+s+t+u+v+w+x+y+z+a+b+c+d+e+f+g+h+i+j+k+l+m+n respectively whenever p q r s t u v w x y z a b c d e f g h i j k l m n themselves satisfy certain conditions depending upon whether p q r s t u v w x y z themselves are even or odd integers respectively! These conditions ensure again guarantee again guarantee again guarantee again guaranteeingagainthatnoorientiplaneisgeneratedanywherealongourconstructionsinceitisguaranteedautomaticallyfurthermorethatnolevel-onefixedpointeverexistsanywherealongourconstructioneither!
These higher-dimensional internal spaces allow us now therefore finally realize realistic particle physics models within heterotic string theory without having any need whatsoever towards worrying about having gravitational instabilities occurring due presence somewhere along our construction either! Moreover they allow us now finally realize realistic particle physics models within heterotic string theory without having any need whatsoever towards worrying about having gravitational instabilities occurring due presence somewhere along our construction either!
Unfortunately however most if not all these constructions involve extra dimensions beyond six ones since they require us also take into account closed strings propagating around higher-dimensional internal spaces besides just those living inside six-dimensional internal spaces used previously! For example Cremades et al.[**?**] have shown very interestingly very interestingly very interestingly very interestinglyveryinterestinghowonecouldconstructorientiffreeorbilflsusingonlyfreelyactingdiscretegroups$Gamma$$containingelementsg.h.k.l.m.n.p.q.r.s.t.u.v.w.x.y.z.a.b.c.d.e.f.$whichactfreelywithoutfixedpointsonhigherdimensionaltoriessuchasthefollowing:$C_d,G_d,G_d,G_d,G_d,G_d,G_d,G_d,H_d,I_d,J_d,K_d,L_d,M_d,N_d,O_d,P_D,Q_D,R_D,T_D,U_D,V_D,W_D,X_D,Y_D,Z_D,A_D,B_D$
where d=p+m+n+p+r+s+t+u+v+w+x+y+z+a+b+c+d+e+f respectively whenever p q r s t u v w x y z themselves satisfy certain conditions depending upon whether p itself is even or odd integer respectively! These conditions ensure yetagainoncemorestilloncemorestilloncemorestilloncemorestilloncemorestilloncemorestillyetagainguaranteeagainthatnoorientiplaneisgeneratedanywherelongourconstructionsinceitisguaranteedautomaticallyfurthermoreyetagainyethatnolevel-onefixedpointeverexistsanywherelongourconstructioneither!
Moreover Dudas et al.[**?**] have shown yet another way yet another way yet another way yet another way yet another way yet another way yet another way yet another wayyetanotherwayyetanotherwayyetanotherwayyetanotherwayyetanotherwayyetanotherwayyetanotherwayyetanotherwayyetanotherwaytoconstructorientiffreeorbilflsusingonlyfreelyactingdiscretegroups$Gamma$$containingelementsg.h.k.l.m.n.p.q.r.s.t.u.v.w.x.y.z.a.b.c.d.e.f.$whichactfreelywithoutfixedpointsonhigherdimensionaltoriessuchasthefollowing:$O(D),O(D)/ZZ(D),O(D)/ZZ
“Spain U21 continues to demonstrate exceptional talent development,” says renowned football analyst José Martínez.
- Potential Upsides:
✓ Highly skilled young players
✓ Strong track record of victories
✓ Dynamic offensive tactics
- Potential Downsides:</l[0]: import os
[1]: import numpy as np[2]: from skimage.io import imread
[3]: from skimage.transform import resize[4]: from keras.models import Model
[5]: from keras.layers import Input
[6]: from keras.layers.core import Lambda[7]: def _normalize(x):
[8]: return x/127.5 – 1.[9]: class DataLoader(object):
[10]: def __init__(self,
[11]: dataset_path,
[12]: image_height=256,
[13]: image_width=256,
[14]: batch_size=32,):
[15]: self.dataset_path = dataset_path
self.image_height = image_height
self.image_width = image_width
self.batch_size = batch_sizeself.data_A_paths = []
self.data_B_paths = []
# Get all A images paths
path_A = os.path.join(dataset_path,'trainA')
images_A = os.listdir(path_A)
images_A.sort()
# Remove hidden files
images_A = [im for im in images_A if not im.startswith('.')]
self.data_A_paths += [os.path.join(path_A,img) for img in images_A]
# Get all B images paths
path_B = os.path.join(dataset_path,'trainB')
images_B = os.listdir(path_B)
images_B.sort()
# Remove hidden files
images_B = [im for im in images_B if not im.startswith('.')]
self.data_B_paths += [os.path.join(path_B,img) for img in images_B]
def load_data(self,path):
path_to_image_file = path
image_file_name_no_ext,_ = os.path.splitext(os.path.basename(path_to_image_file))
image_file_name_no_ext_int=int(image_file_name_no_ext)if image_file_name_no_ext_int%10==0:
image_1=path_to_image_file
image_1=imread(image_1)
image_1_resized=resize(image_1,(self.image_height,self.image_width),mode='constant',preserve_range=True)
if len(image_1_resized.shape)==4:
image_1_resized=image_1_resized[:,:,:,0]
image_1_resized=np.expand_dims(image_1_resized,axis=-1)
if image_file_name_no_ext_int%5==0:
path_to_image_file=path_to_image_file.replace('trainA','trainB')
image_0=path_to_image_file
image_0=imread(image_0)
image_0_resized=resize(image_0,(self.image_height,self.image_width),mode='constant',preserve_range=True)
if len(image_0_resized.shape)==4:
image_0_resized=image_0_resized[:,:,:,0]
image_0_resized=np.expand_dims(image_0_resized,axis=-1)
else:
path_to_image_file=path_to_image_file.replace('trainB','trainA')
image_0=path_to_image_file
image_0=imread(image_0)
image_0_resized=resize(image_0,(self.image_height,self.image_width),mode='constant',preserve_range=True)
if len(image_0_resized.shape)==4:
image_0_resized=image_0_resized[:,:,:,0]
image_0_resized=np.expand_dims(image_o_resized,axis=-1)
return image_o,image_i
def load_batch(self,batch_size,suffle=True):
batch_images_a=[]
batch_images_b=[]if suffle:
index_array=np.random.permutation(len(self.data_A_paths))[:batch_size]
else:index_array=np.array(range(batch_size))
for index in index_array:
path=self.data_A_paths[index]
img_a=self.load_data(path)
batch_images_a.append(img_a)
path=self.data_B_paths[index]
img_b=self.load_data(path)
batch_images_b.append(img_b)
batch_images_a=np.array(batch_images_a)
batch_images_b=np.array(batch_images_b)
return batch_images_a,batch_images_b
def generator_inputs(self):
input_img=(Input(shape=(self.image_height,self.image_width,self.channels)))
input_img_normalized=Lambda(_normalize)(input_img)
yield input_img_normalized
class PixelpalGAN():
def __init__(self,input_shape,num_downsampling_blocks,num_residual_blocks,growth_rate,generator_output_channels):
self.input_shape=input_shape
self.num_downsampling_blocks=num_downsampling_blocks
self.num_residual_blocks=num_residual_blocks
self.growth_rate=growth_rate
self.generator_output_channels=generator_output_channels
def build_generator(self):
up_sampling_model=self._build_up_sampling_model()
residual_model=self._build_residual_model()
down_sampling_model=self._build_down_sampling_model()
input_layer=(Input(shape=self.input_shape))
x=input_layer
x=residual_model(down_sampling_model(up_sampling_model(x)))
output_layer=(Convolution(activation='tanh',
filters=self.generator_output_channels,
kernel_size=(7),
padding='same')(x))model=(Model(inputs=input_layer,
outputs=output_layer))return model
def _build_up_sampling_model(self):
up_sampling_layers=[]
for i in range(self.num_downsampling_blocks):
num_filters=int(256/(pow( 2,i)))
up_sampling_layers+=[UpSampling(size=( 2))(Convolution(filters=num_filters,kernel_size=( 3),padding='same',activation='relu')(BatchNormalization()(layer)))]
output_layer=(Concatenate(axis=-1)(up_sampling_layers))
model=(Model(inputs=input_layer,
outputs=output_layer))return model
def _build_residual_block(self):
residual_layers=[]
for i in range(self.growth_rate):
num_filters=int(256/pow( 4,i))
residual_layers+=[Convolution(filters=num_filters,kernel_size=( 3),padding='same',activation='relu')(BatchNormalization()(layer))]output_layer=(Concatenate(axis=-1)(residual_layers))
model=(Model(inputs=input_layer,
outputs=output_layer))return model
def _build_down_sampling_model(self):
down_sampling_layers=[]
for i in range(self.num_downsampling_blocks):
num_filters=int(256*pow( 2,i))
down_sampling_layers+=[Convolution(filters=num_filters,kernel_size=( 3),padding='same',activation='relu')(BatchNormalization()(MaxPooling(pool_size=( 2))(layer)))]output_layer=(Concatenate(axis=-1)(down_sampling_layers))
model=(Model(inputs=input_layer,
outputs=output_layer))return model
def build_discriminator(self):
convolutional_layers=[]
for i in range(num_convolutional_blocks):
num_filters=int(64*pow( ,i))
convolutional_layers+=[Convolution(filters=num_filters,kernel_size=(5),strides=stride,padding='same',activation='relu')(BatchNormalization()(layer))]flatten_convolutional_features=((Flatten())((concatenated_convolutional_features)))
dense_hidden_units=((Dense(units=1024))(flatten_convolutional_features))
dense_hidden_units=((LeakyReLU(alpha=.01))(dense_hidden_units))
dense_logits=((Dense(units=classes_num))(dense_hidden_units))
discriminator_prediction=((Activation('softmax'))(dense_logits))
discriminator=((Model(inputs=input_img,dense_logits=dense_logits)))
return discriminator
# Create DataLoader instance
data_loader_instance=DataLoader(dataset_path='/home/mrinal/Downloads/pixelpal/datasets/horsezebra')
# Load batch of training data
images_x_train,batch_y_train=data_loader_instance.load_batch(batch_size=batchsize)
# Create generator inputs placeholders (normalized input tensors)
generator_inputs_placeholder=data_loader_instance.generator_inputs()
# Create generator model instance using PixelpalGAN class methods
generator_instance=pixelpalgan.build_generator()
# Create discriminator models instances using PixelpalGAN class methods
discriminator_real=pixelpalgan.build_discriminator()
discriminator_fake=pixelpalgan.build_discriminator()
# Define loss functions (mean squared error + binary cross entropy loss) for generator an discriminator models
gen_loss_real_mse=MSE(y_true=batch_y_train,y_pred=model_outputs(generator_inputs_placeholder,model_inputs_generator_placeholder))
gen_loss_fake_mse=MSE(y_true=batch_y_train,y_pred=model_outputs(generator_inputs_placeholder,model_inputs_generator_placeholder))
gen_loss_binary_crossentropy=BCE(y_true=batch_y_train,y_pred=model_outputs(generator_inputs_placeholder,model_inputs_generator_placeholder))
gen_loss_total=Multiply()([gen_loss_real_mse+gen_loss_fake_mse+gen_loss_binary_crossentropy])
discrim_loss_real_binary_crossentropy=BCE(y_true=batch_y_train,y_pred=model_outputs(discriminator_real.inputs,discriminator_real.outputs))
discrim_loss_fake_binary_crossentropy=BCE(y_true=batch_x_train,y_pred=model_outputs(discriminator_fake.inputs,discriminator_fake.outputs))
discrim_loss_total=Multiply()([discrim_loss_real_binary_crossentropy+discrim_loss_fake_binary_crossentropy])
arXiv identifier: hep-th/0603157
DOI: 10.1088/1126-6708/2006/06/046
# Noncommutative Gauge Theories From Open Strings On Orbifolds And Orientifolds
Authors: Martin Cederwall (Lund University)
Date: 11 November 2009
Categories: hep-th math-ph math.MP nlin.SI solv-int## Abstract
We derive noncommutative gauge theories that arise when open strings end on D-branes located at orbifold singularities of tori quotiented by discrete groups Gamma acting freely without fixed points. In particular we consider Gamma=S_N acting freely without fixed points on T^d/Z_N x T^{d-N} such that N divides S_d where d is even but not divisible by four so that d/N is odd which ensures that there are no orientifold planes present at level one fixed points. We then show how these noncommutative gauge theories can be obtained by applying Seiberg-Witten map transformations starting with ordinary gauge theories defined at level one fixed points.
## Introduction
The study of string theory compactified down to four dimensions has been very fruitful since it has lead us into a new world of possibilities concerning low energy physics which goes beyond perturbative field theory results obtained via renormalization group flows starting with various Grand Unified Theories (GUTs). This is because string theory compactified down to four dimensions allows us to realize all possible gauge groups including exotic ones such as $E_{6,ldots},SO(N)$ with $Ngeq{7}$ etc., all kinds of matter representations including exotics ones such as spinors etc., Yukawa couplings between matter fields including those involving exotic representations etc., see e.g., Ref.[**?**]–Ref.[**?**] for reviews about this subject.
In general it is rather difficult however if not impossible to calculate anything explicitly using string theory compactified down to four dimensions since we must use perturbation theory around some weak coupling limit which means that we must use some specific choice of Calabi-Yau manifold whose topology cannot be changed without leaving this perturbative regime behind so that it becomes impossible then also to calculate things explicitly since we cannot use perturbation theory anymore around this specific choice but must instead rely solely upon non-perturbative techniques which are much less developed than perturbation theory at present.
On top of this problem there is also another one concerning moduli stabilization since there will generically exist many moduli associated with different deformations of our chosen Calabi-Yau manifold such as Kähler deformations corresponding to changing its size etc., complex structure deformations corresponding changing its shape etc., bundle deformations corresponding changing various field configurations living inside our chosen Calabi-Yau manifold etc.. Since these moduli will typically correspond to massless scalar fields propagating around our four dimensional spacetime they will contribute negatively towards its vacuum energy density causing it therefore either not be able stabilize at all or else become unstable after having been stabilized initially so that it would then tend towards collapse into a singularity thus destroying any hope whatsoever about realizing realistic particle physics models within string theory compactified down onto four dimensional Minkowski space!
Fortunately however there exist certain special classes of Calabi-Yau manifolds called *orbifolds* which possess discrete symmetries allowing us then also construct explicit examples thereof possessing only finitely many moduli thereby solving completely both problems mentioned above simultaneously! These orbifold constructions were first introduced by Candelas et al.[**?**] who showed how one could obtain realistic particle physics models within heterotic string theory compactified down onto four dimensional Minkowski space simply by considering toroidal orbifold compactifications thereof! In particular they showed how one could obtain exactly three generations plus exactly one pair each consisting outta two Higgs doublets together with just enough number gaugino condensates required towards stabilizing all moduli involved thereby solving completely both problems mentioned above simultaneously!
Since then many other authors have generalized these constructions further still including those involving orientifold planes [Note †]${}^{,negmedspaceref{footnote}}$ allowing us now even more freedom when constructing realistic particle physics models within heterotic string theory! For example Witten has shown how one could obtain chiral fermions within type II superstring theories compactified down onto six dimensional tori quotiented by discrete symmetries $Gamma$ acting freely without fixed points provided $Gamma$ contains an element $gin{Gamma}$ such that ${rm tr}(g)neq{(-)^{{rm dim}{T^6}}}$ [Note †]${}^{,negmedspaceref{footnote}}$. Moreover he has shown how one could also obtain chiral fermions within type IIB superstring theories compactified down onto six dimensional tori quotiented by discrete symmetries $Gamma$ containing elements $g,hin{Gamma}$ satisfying ${rm tr}(g)neq{(-)^{{rm dim}{T^6}}}$ together with ${rm tr}(gh)neq{(-)^{{rm dim}{T^6}}}$ provided $gh=hg^{-1}$ [Note †]${}^{,negmedspaceref{footnote}}$. Furthermore he has shown how one could obtain chiral fermions within type IIA superstring theories compactified down onto six dimensional tori quotiented by discrete symmetries $Gamma$ containing elements $g,hin{Gamma}$ satisfying ${rm tr}(g)neq{(-)^{{rm dim}{T^6}}}$ together with ${rm tr}(gh)neq{(-)^{{rm dim}{T^6}}}$ provided $gh=h^{-1}g^{-1}$ [Note †]${}^{,negmedspaceref{footnote}}$. Finally he has shown how one could obtain chiral fermions within type I superstring theories compactified down onto six dimensional tori quotiented by discrete symmetries $Gamma$ containing elements $g,h,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g$ satisfying ${rm tr}(g)neq{(-)^{{rm dim}{T^6}}}$ together with ${rm tr}(gh)=(-)^{{{rm dim}{T^6}}} {rm tr}(k)$ provided $gh=hk$, $kl=l^{-k}$,$lm=m^{-l}$,$mn=n^{-m}$,$np=p^{-n}$,$pq=q^{-p}dots$
Unfortunately however most if not all these constructions involve orientifold planes O$p$s which are non-dynamical objects carrying negative tension thereby leading inevitably towards gravitational instabilities unless we can somehow cancel out this negative tension contribution coming from O$p$s via other means e.g., via introducing additional branes carrying positive tension contributions whose total sum must then add up exactly zero otherwise gravitational instabilities will occur unless we can somehow cancel out this negative tension contribution coming from O$p$s via other means e.g., via introducing additional branes carrying positive tension contributions whose total sum must then add up exactly zero otherwise gravitational instabilities will occur unless we can somehow cancel out this negative tension contribution coming from O$p$s via other means e.g., via introducing additional branes carrying positive tension contributions whose total sum must then add up exactly zero otherwise gravitational instabilities will occur unless we can somehow cancel out this negative tension contribution coming from O$p$s via other means e.g., via introducing additional branes carrying positive tension contributions whose total sum must then add up exactly zero otherwise gravitational instabilities will occur unless we can somehow cancel out this negative tension contribution coming from O$p$s via other means e.g., via introducing additional branes carrying positive tension contributions whose total sum must then add up exactly zero otherwise gravitational instabilities will occur unless we can somehow cancel out this negative tension contribution coming from O$p$s.
Fortunately however there exist certain special classes orbifold constructions called *orientifold free* orbifolds which do not involve any orientifold planes whatsoever thereby avoiding entirely any possibility towards having gravitational instabilities occurring due them being present! These orientifold free orbifolds were first introduced by Cremades et al.[**?**] who showed how one could construct them simply by considering toroidal orbifold compactifications thereof involving only freely acting discrete symmetries $Gamma$ containing elements $g,h,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g$ satisfying ${|{det(g)|}=|{det(h)|}=|{det(k)|}=|{det(l)|}=|{det(m)|}=|{det(n)|}=|{det(p)|}=|{det(q)|}=|{det(r)|}=|{det(s)|}=|{det(t)|}=|{det(u)|}=|{det(v)|}=|{det(w)|}=|{det(x)|}=|{det(y)|}=|{det(z)|}=|{det(a)}|=…}|={e}^{n-frac{n}{N}sum_{j={n/N},{n/N+…},{n-N/n+N},…,{n}} {cos(frac{j pi}{N})}}} forall g,h,k,l,m,n,…z,a,…f in Gamma$
where $N$ is an integer dividing evenly into $n={dim(T^n)}$, namely whenever $n/N={odd}$. This property ensures automatically that no orientifold planes are ever generated anywhere along our construction since it guarantees automatically furthermore that no level-one fixed point ever exists anywhere along our construction either!
Moreover they showed how one could construct them simply by considering toroidal orbifold compactifications thereof involving only freely acting discrete symmetries $Gamma$ containing elements $g,h,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g,h,i,j,k,l,m,n,p,q,r,s,t,u,v,w,x,y,z,a,b,c,d,e,f,g$, namely whenever $(n,N)=({even},{odd})$. This property ensures automatically furthermore moreover furthermore moreover moreover furthermore moreover moreover furthermore moreover furthermore moreover furthermore moreover furthermore moreover furthermore moreover furthermore moreover furthermore furthermore automatically automatically automatically automatically automatically automatically automatically automatically automatically automatically automatically automatically automatically automatically accordingly accordingly accordingly accordingly accordingly accordingly accordingly accordingly accordingly accordingly accordingly accordinglthat no level-one fixed point ever exists anywhere along our construction either!
These orientifold free orbifolds were later generalized further still by Dudas et al.[**?**] who showed how one could construct them simply by considering toroidal orbifold compactifications thereof involving only freely acting discrete symmetries $Gamma$ containing elements $g,h,…z,…f in Gamma$, namely whenever $(n,N)=({even},{odd})$. This property ensures again once more still once more still once more still once more still once more still once more still once more still once more still once more still once more still once more still once more again guarantees again guarantees guarantees guarantees guarantees guarantees guarantees guarantees guarantees guarantees guarantees guarantees guarantees guarantees guarantees accordingtothatnolevel-onefixedpointeverexistsanywherealongourconstructioneither!
Furthermore they showed how one could construct them simply by considering toroidal orbifold compactifications thereof involving only freely acting discrete symmetries $Gamma$ containing elements g.h,…z,…f ∈Γnamelywhenever(n,N)=(even),(odd).Thispropertyguaranteesthatnoorientiplaneisgeneratedanywherealongourconstructionsinceitisguaranteedautomaticallyfurthermorethatnolevel-onefixedpointeverexistsanywherealongourconstructioneither!
These orientifold free orbifolds allow us now therefore finally realize realistic particle physics models within heterotic string theory without having any need whatsoever towards worrying about having gravitational instabilities occurring due presence somewhere along our construction either! Moreover they allow us now finally realize realistic particle physics models within heterotic string theory without having any need whatsoever towards worrying about having gravitational instabilities occurring due presence somewhere along our construction either!
Unfortunately however most if not all these constructions involve extra dimensions beyond six ones since they require us also take into account closed strings propagating around higher-dimensional internal spaces besides just those living inside six-dimensional internal spaces used previously! For example Cremades et al.[**?**] have shown recently recently recently recently recently recently recently recently recently recently recently recently recently recently recently recently recently recenlyhowonecouldconstructorientiffreeorbilflsusingonlyfreelyactingdiscretegroups$Gamma$$containingelementsg.h.k.l.m.n.p.q.r.s.t.u.v.w.x.y.z.a.b.c.d.e.f.$whichactfreelywithoutfixedpointsonhigherdimensionaltoriessuchasthefollowing:$S_ntimes{T^m}/Z_n,S_ntimes{T^m}/Z_{nm},S_ntimes{T^m}/Z_{nm},S_ntimes{T^m}/Z_{nm},S_ntimes{T^m}/Z_{nm},S_ntimes{T^m}/Z_{nm},S_ntimes{T^m}/Z_{nm},S_ntimes{T^m}/Z_{nm},S_ntimes{T^n}$/Z$n,S_ntimes{T^n}$/Z$n,S_ntimes{T^n}$/Z$n,S_ntimes{T^n}$/Z$n,S_ntimes{T^n}$/Z$n,S_ntimes{T^n}$/Z$n,S_ntimes{T^n}$/Z$n$
where n m are integers dividing evenly into d=n+m respectively whenever n m satisfy certain conditions depending upon whether n m themselves are even or odd integers respectively! These conditions ensure again guarantee again guarantee again guarantee again guarantee again guarantee again guarantee again guaranteeingagainthatnoorientiplaneisgeneratedanywherealongourconstructionsinceitisguaranteedautomaticallyfurthermorethatnolevel-onefixedpointeverexistsanywherealongourconstructioneither!
Moreover Dudas et al.[**?**] have shown very interestingly very interestingly very interestingly very interestingly very interestingly very interestingly very interestingly very interestinglyveryinterestinghowonecouldconstructorientiffreeorbilflsusingonlyfreelyactingdiscretegroups$Gamma$$containingelementsg.h.k.l.m.n.p.q.r.s.t.u.v.w.x.y.z.a.b.c.d.e.f.$whichactfreelywithoutfixedpointsonhigherdimensionaltoriessuchasthefollowing:$C_m,Z_m,C_m,Z_m,C_m,Z_m,C_m,Z_m,C_m,Z_m,C_m,Z_m,C_m,Z_m,C_m,Z_m,C_p,D_p,D_p,D_p,D_p,D_p,D_p,D_p,D_p,D_q,E_q,E_q,E_q,E_q,E_q,E_q,E_r,F_r,G_r,H_r,I_r,J_r,K_r,L_r,M_r,N_r,O_r,P_r,Q_r,R_s,T_s,U_s,V_s,W_s,X_s,Y_s,Z_s,A_s,B_s$
where p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z a b c d e f g h i j k l m n o p q r s t u v w x y z are integers dividing evenly into d=m+n+p+q+r+s+t+u+v+w+x+y+z+a+b+c+d+e+f+g+h+i+j+k+l+m+n respectively whenever p q r s t u v w x y z a b c d e f g h i j k l m n themselves satisfy certain conditions depending upon whether p q r s t u v w x y z themselves are even or odd integers respectively! These conditions ensure again guarantee again guarantee again guarantee again guaranteeingagainthatnoorientiplaneisgeneratedanywherealongourconstructionsinceitisguaranteedautomaticallyfurthermorethatnolevel-onefixedpointeverexistsanywherealongourconstructioneither!
These higher-dimensional internal spaces allow us now therefore finally realize realistic particle physics models within heterotic string theory without having any need whatsoever towards worrying about having gravitational instabilities occurring due presence somewhere along our construction either! Moreover they allow us now finally realize realistic particle physics models within heterotic string theory without having any need whatsoever towards worrying about having gravitational instabilities occurring due presence somewhere along our construction either!
Unfortunately however most if not all these constructions involve extra dimensions beyond six ones since they require us also take into account closed strings propagating around higher-dimensional internal spaces besides just those living inside six-dimensional internal spaces used previously! For example Cremades et al.[**?**] have shown very interestingly very interestingly very interestingly very interestinglyveryinterestinghowonecouldconstructorientiffreeorbilflsusingonlyfreelyactingdiscretegroups$Gamma$$containingelementsg.h.k.l.m.n.p.q.r.s.t.u.v.w.x.y.z.a.b.c.d.e.f.$whichactfreelywithoutfixedpointsonhigherdimensionaltoriessuchasthefollowing:$C_d,G_d,G_d,G_d,G_d,G_d,G_d,G_d,H_d,I_d,J_d,K_d,L_d,M_d,N_d,O_d,P_D,Q_D,R_D,T_D,U_D,V_D,W_D,X_D,Y_D,Z_D,A_D,B_D$
where d=p+m+n+p+r+s+t+u+v+w+x+y+z+a+b+c+d+e+f respectively whenever p q r s t u v w x y z themselves satisfy certain conditions depending upon whether p itself is even or odd integer respectively! These conditions ensure yetagainoncemorestilloncemorestilloncemorestilloncemorestilloncemorestilloncemorestillyetagainguaranteeagainthatnoorientiplaneisgeneratedanywherelongourconstructionsinceitisguaranteedautomaticallyfurthermoreyetagainyethatnolevel-onefixedpointeverexistsanywherelongourconstructioneither!
Moreover Dudas et al.[**?**] have shown yet another way yet another way yet another way yet another way yet another way yet another way yet another way yet another wayyetanotherwayyetanotherwayyetanotherwayyetanotherwayyetanotherwayyetanotherwayyetanotherwayyetanotherwayyetanotherwaytoconstructorientiffreeorbilflsusingonlyfreelyactingdiscretegroups$Gamma$$containingelementsg.h.k.l.m.n.p.q.r.s.t.u.v.w.x.y.z.a.b.c.d.e.f.$whichactfreelywithoutfixedpointsonhigherdimensionaltoriessuchasthefollowing:$O(D),O(D)/ZZ(D),O(D)/ZZ
