Sailing, Yachts and Sailboats News and Info- 07-15-2005, 00:05.
"While we realize that another IPO as lucrative Google's is a long shot, we have noticed that several names new to the market are boasting solid debuts. The end of June saw four impressive initial offerings from apparel designer Volcom (VLCM), telecom-services firm NeuStar (NSR), regional bank Western Alliance Bancorp (WAL) and shoe retailer DSW (DSW). All of the stocks priced above their projected ranges and continue to trade in the black. The string of successes got us to wondering whether investors are once again hungry for IPOs. According to David Menlow, president of IPO Financial Network, a Millburn, N.J."
"FISHING REDFISH: One of the largest redfish tournaments in the country takes place July 22 and 23 in Inglewood. The FLW Redfish Series holds its third event in the Eastern Division off Charlotte Harbor. There will be 150 teams. Anglers, competing in two-man teams, will fish for a top award of $25,000. Also at stake are points that earn teams a berth into the FLW Series Championship in October in Mobile, Ala. Mid July is the perfect time for redfish in the Venice/Inglewood area, as the fish tend to school around grass flats and oyster bars. There will be daily weigh-ins at 3 p.m. at Stump Pass Marina. For more information, check out www."
"BOULDER, Colorado - Following its smashing success earlier this month with comet Tempel 1, the Deep Impact Flyby spacecraft is being readied for potential retargeting to yet another scientific destination. The two-part Deep Impact craft consisted of the destroyed-on-purpose, battery-powered Impactor probe that was literally run over by Tempel 1, and a still-healthy Flyby spacecraft that monitored the event from a safe distance."
"In the extreme north-east corner of Oxfordshire, tucked in between Warwickshire and Northamptonshire, quite near the site of the Civil War battle of Edgehill in 1642 and nearer still to the 1644 engagement at Cropredy Bridge, is the village of Wardington, lying on high ground between 400 and 500 feet (120-150 metres). The manor house of the same name, built on the site of an ancient nunnery in 1665, contained, until last year, a notable private library of rare antiquarian books comprising early English bibles, incunabula (i.e."
"SINGAPORE - Loss-making semiconductor test and packaging vendor STATS ChipPAC said it expects to report 13-percent sales growth for the quarter ending June 30. Analysts said the results were largely in line with expectations but unlikely to be sustainable. The company said in a statement that based on preliminary results it expects revenues of $264 million for the second quarter, driven by strong demand for commodity PBGA packaging used in chip sets and advanced packages. The revenue increase is solidly in the middle of the 10 to 15 percent growth over the first quarter of 2005 that STATS ChipPAC forecast in April."
"Sweeney's injury takes luster off Royals' win KANSAS CITY, Mo. -- Even when the Royals win, as they did Wednesday in beating the Seattle Mariners 5-1, it comes with an asterisk. No question over who's wearing yellow; After officials rejected his gesture of sportsmanship, Armstrong safely negotiates fifth stage to maintain lead > TOURDE FRANCE MONTARGIS, France -- No yellow jersey Wednesday. No race today. Sonics lose McMillan to closest rival; Blazers lure coach away from franchise where he spent his playing career PORTLAND, Ore. -- Nate McMillan is leaving the Seattle SuperSonics to coach the Portland Trail Blazers."
"MIAMI – AT&T announced today it has won a $15 million networking contract from Carnival Cruise Lines, the most popular cruise line in the world. The three-year contract for an extensive portfolio of data and voice services renews and expands a long-standing relationship between the companies. "Our call centers and Web site have to stay up and running non-stop," said Ken Eberhardt, Director of IT and Administrative Services at Carnival. "We need to be available 24 hours a day, 7 days a week. With the reliability of AT&T, we know we'll be up and running through hurricanes, power failures or floods."
"For some folks, it's treasure: the first signed U.S. copy of the new Harry Potter book, Harry Potter and the Half-Blood Prince. Appropriate, then, that's it was transported in a five-by-seven-foot treasure chest (aka steamer trunk) aboard a Royal British Mail Ship (aka the Queen Mary 2) on a grand adventure (aka a transatlantic crossing) before reaching these shores. That's the saga of one little book, which was signed by author J. K. Rowling in Edinburgh, Scotland, brought to Southampton by a Scholastic Books executive, and loaded onto Queen Mary 2 bound for New York."
"Speeds measured by GPS in brisk, windy conditions on Biscayne Bay near Dinner Key, Fla., w/three people on board, half fuel and half water. Sound levels measured at the helm in dB-A. Standard Equipment (major items): Bimini top; anchor windlass; 7.3-kW gas genset; stereo remote; fender storage compartment; overhead lighting; freshwater and raw-water washdowns; hot and cold hand shower; cockpit door screen; pressurized freshwater; Bennett hydraulic trim tabs; 11-gallon water heater. Optional Equipment (major items): Two-burner propane cooktop; air conditioning; central vacuum system; flybridge icemaker and cocktail table; full canvas; 9."


The crazy thing about Unused Ending II is ONLY available on the rip of the Mega Drive port. As far as I know, it doesn't exist on any other version.
That track is so, so very jam. Thanks for sharing!
Utopia Nemo now holds the record for most VGM podcasts playing a single recommendation!!
At his behest, I played it in my own little program. It's an amazing jam.
To be precise, I merely shared an amazing jam with friends. The track itself did the behesting.
Incidentally, I also shared it with Rob F, so we may yet hear more of Unused Track II in the near future.
Lastly, the ending comment on my testimonial was rather snarky (PS: You're WELCOME). I immediately regretted typing it; a redacted version would have read "PS: THANK YOU".
....and today it's being played on LMH. In the words of the great the Diad, HAT TRICK!
Well, I'm a liar......Rob played Town from the game, not Unused Ending II. BUT, he also played Prairie from Bad Omen, which was also my suggestion, which Pixeltunes played when they did the crossover with VGMJB.
I'm on my second re-listen to A Better Beginning after hearing it here; it draws me in deeper every time. Slow and atmospheric it may be, but there is emotional force in it. And for all its breadth it is unified by that two-note melodic idea recurring and recurring.
I enjoyed Fight Theme 1 thoroughly too -- an opposite kind of track at the opposite end of the show. I love a good short-loop, tightly composed, intensely energetic NES piece.
And Jupiter Jazz's testimonial for the Printer Music was fantastic; this needs to be said.
Like the hosts, I am not sure what a Dragon Slayer is: one time, catalysed by an earlier episode, I was listening to Dragon Slayer: Legend of Heroes 1 and I skipped ahead to a track called Dragon; it was not what I expected.
But enough of frivolity. Soulblighter walks the earth again, and if that fact doesn't send a cold shiver down your spine then you are a fool.
The Myth games were already on my mind as I glanced over the track list; one of my favorite non-VGM podcasts just released an episode about them. Visit Three Moves Ahead (www.idlethumbs.net/3ma/episodes/myth) if you want to hear an erudite and well spoken discussion of what made Myth unique and good.
But they did leave the topic of music and sound design mostly untouched and Myth was exceptional there too: I haven't played Myth II for about eighteen years but I still cannot think of it without thinking of the Victory Screen music and the powerful sense of hard-won triumph it evoked. And Limbs, Heads, and Smoking Craters lent Josh's NPC story, a moment from work, from ordinary life, a kind of mono no aware.
Now, Myth was made by Bungie, who went on to become much better known for the Halo games. This is interesting because almost the only thing Myth and Halo have in common is music. They have a very similar sound in places and both series leave long sections without music so that when tracks do finally play they have an incredible impact. And Bungie acknowledged this connection by hiding a track from Myth I in Halo.
My mind is buzzing with Myth. There is more to say. On another occasion.
Spritz, your last sentence gave me chills.
I'm always so honored to have my testimonials read on air and I feel doubly grateful for Josh's and Emily's kind words about what I said.
This is my favorite VGM podcast, not just because of the music, not just because of the excellent quality of the hosts and discussion, but especially because of the amazing sense of community that exists around it.


text:
Q: The property and event halves of the two way binding are not bound at the same target I want to make a two way binding with reactive forms in Angular. I want to transfer the data from the child component to the parent component and back.
My child component HTML:
<input type="text" #name class="form-control"
            (keydown)="oss.test(); addNewItem(name.value)"  formControlName="name">

My child component TypeScript:
  @Input()  namex: string=  "";
  @Output() additem = new EventEmitter<string>();

The parent component HTML:
 <app-general [(count)]="count" [(namex)]="currentItem" (value)="this.generalInformation=$event; this.print()"
              (isValid)="this.generalInformationIsValid=$event">
            </app-general>

The parent component TypeScript:
currentItem: string = "";

I get following error message:
The property and event halves of the two-way binding 'namex' are not bound to the same target.
            Find more at https://angular.io/guide/two-way-binding#how-two-way-binding-works

Can someone please help me why I get this error?
I want to get the data from the child component to the parent component and back.

A: you have @Input() namex: string = ""; without @Output() namexChange = new EventEmitter(), try adding the namexChange event

redpajama_set_name: RedPajamaStackExchange


text:
OLF-8 to come before Escambia County Commission with final design this week
Plans to transform a roughly 600-acre swath of land in Beulah into a mixed-use development encompassing a school, green space and residential, retail and industrial space are moving forward with a final conceptual design that will come before the Escambia County Commission this week.
Marina Khoury, the lead spokeswoman for the OLF-8 design for firm DPZ, spoke Tuesday morning before the county's planning board and presented a more detailed breakdown of the land's potential future use than has been given in the past few months of contentious and sometimes heated county meetings.
The property, which fronts Nine Mile Road near Navy Federal Credit Union's campus, was intended as a commercial space or industrial park when discussions first began for a land swap with the Navy decades ago. But as Beulah began to boom and become suburbanized, residents became more outspoken about wanting it to become a downtown-type hub with a school, retail options and amenities.
More:OLF-8 hybrid plan overcomes major hurdle, proceeds to zoning phase
More:Beulah is booming, but how much? School enrollment, EMS calls, home sales offer clues
The struggle between how much land to allocate to industry versus residential or community facilities has been at the forefront of the contention over the OLF-8 property, and one Khoury said has been addressed in the latest design.
The DPZ team has created a series of zoning maps that will "sunset" over the next 16 years, which will give the county government the power to change the zoning laws based on future needs of commercial or residential development.
The initial design is zoned heavily industrial, but if by 2027, when the second zoning map goes into effect, there isn't the need or demand for so much industrial space, a portion of the land can be easily rezoned and converted to allow for commercial and higher density residential development like townhomes and duplexes.
The same concept comes into play in 2032, when there is the option to convert another portion of the OLF-8 land from industrial into neighborhood developments with single-family homes if there is a need at that time.
"It's intended to activate the master plan. It's what we heard the community say they wanted, it's what we heard the commissioners say they wanted," Khoury said at the planning meeting.
The planning board voted unanimously Tuesday to recommend a change in the county's future land use map for the OLF-8 project to move ahead. DPZ will present a similar design presentation Thursday to the County Commission, which will have the final say on moving forward.
If approved, the design will go through further analysis of such elements as stormwater and transportation needs before a final iteration of the plan comes back in the fall.
Emma Kennedy can be reached at ekennedy@pnj.com or 850-480-6979.

redpajama_set_name: RedPajamaCommonCrawl


Much like the shepherds and wise men during the first Christmas, people around the world had their attention turned toward the heavens 35 years ago this week, when the Apollo 8 crew delivered its famous Christmas Eve broadcast from lunar orbit.
To mark the occasion, they sent Christmas greetings and live images back to their home planet and read from the Book of Genesis. Borman closed the message with the words "good night, good luck, a Merry Christmas, and God bless all of you -- all of you on the good Earth."
This is an example of one of the breathtaking images of the Moon and the Earth taken by the Apollo 8 crew.
A Christmas tree made out of cans by the Skylab 4 crew.
The first Christmas aboard the International Space Station occurred in 2000 with the Expedition One crew. Astronaut Bill Shepherd and Cosmonauts Yuri Gidzenko and Sergei Krikalev spent a quiet Christmas Day opening gifts and talking to their families.


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Sell The Blueprint Drawings 8
The Blueprint Drawings 8 Keith Haring
You are seeing this page because we currently don't have this work for sale, but we do have an active buyer seeking this work. Please click the link below if you have The Blueprint Drawings 8 by Keith Haring to sell.
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Medium: Screen Print
Format: Signed Print
Size: H 108cm x W 132cm
Framing details: Not framed
This signed screen print from 1990 is a limited edition of 33 from Keith Haring's The Blueprint Drawings series. Set against a dotted landscape and monumental pyramid on the horizon, Haring depicts a number of anonymous male figures in this image in his simplified linear style.
All works are inspected prior to delivery, work will be sent out tracked and insured at the buyers cost. If you'd like to make specific arrangements or discuss collection then please contact us directly.
KEITH HARING, THE BLUEPRINT DRAWINGS 8, THE BLUEPRINT DRAWINGS SERIES, SIGNED SCREEN PRINT, EDITION OF 33, 1990
Created in the artist's immediately recognisable linear style, The Blueprint Drawings 8 is a print from Keith Haring's The Blueprint Drawings series from 1990. Haring depicts a number of anonymous male figures in the foreground holding their erect penises, set against a dotted landscape and monumental pyramid on the horizon. One of Haring's most iconic pictograms, the UFO, can be seen to the right of image, striking one of the figures with its radiating beams.
Completed the year of Haring's tragic death by AIDS in 1990, this image cannot be considered outside of the context of sexuality, HIV/AIDS and death. Haring's faceless figures in this print, characterised only by their erect penises, plays into the idea of anonymous sex as taboo in the homosexual community. As the HIV/AIDS epidemic devastated the gay community in the 1980s, Haring spoke of the horrors that could result from extreme promiscuity at the time. The Blueprint Drawings 3, shows these figures crying for help as they are struck by the UFO one by one, symbolising this fear of dying from the virus.
The inclusion of both the pyramid and the UFO symbolises the presence of unknown forces, creating a tension between ancient civilisation, eternity and an extra-terrestrial future. In particular the UFO represents otherness and those who lie outside of social norms, notably those figures who live with HIV/AIDS and face the stigma attached to the virus.
Read more about The Blueprint Drawings by Keith Haring.
ABOUT KEITH HARING
Known for his bold graphic style and playful sense of humour, Keith Haring is one of the most influential and adored artists of the 20th century.
Born in Pennsylvania, in 1958, Haring was a talented draughtsman as a child and developed his cartoonish style at the hands of his father and the work of Walt Disney and Dr Seuss. However it would take some time before he realised he could marry this kind of drawing with being a fine artist. Upon graduating from high school he enrolled in a commercial art school before realising he had little interest in pursuing a career as an illustrator or graphic designer. After dropping out of college he joined the hippie movement and hitchhiked across the country where he made anti-Nixon t-shirts to pay for food and Grateful Dead tickets. Learn more about Keith Haring.
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Here's the view of three of our guests so far this summer.
"Thank you Ian and Mayberry Kayaking! We had 2 great days out in our kayaks guided by Ian; an excellent and pleasant coach who clearly loves his job! Ian listened and observed and gave great feedback, and a good program was set up for the two days in a nice small group of 3. We were challenged, but all fun and safe.
First day the water was calm, and day two it was windy and more challenging with waves. We learned a lot; planning (tide/map/weather), instructions in kayak and paddling techniques(launching, landing, different strokes, edging, rescuing, one guy in the group, Ieuan, even tried eskimo rolls). Been kayaking inside exiting caves, through narrow passages and around rocks. Also got info about wildlife along the coast and saw some amazing birds up close. We had lots of fun in the water trying to maintain balance whilst fooling around on top of kayak. Packed lunch tasted great outdoors on pebbled beaches with pleasant people. We can really recommend this!
"Dear Mike, Just a line to thank you for another great week. I congratulate you on recruiting Ian and Amy. Ian's paddling skills are evident. I appreciated his sensitive, good humoured leadership. He is un-macho -and quietly encouraging. His calmness, which influenced all the group, is underpinned by his competence. At first I was uncertain that he would be able to assert himself and maintain his authority in a strong group -but was delighted to see him do so when he needed to. And, when a real rescue in exciting conditions was needed (me!) he showed his worth -well, I am sure you would expect nothing else.


text:
Jennifer Lawrence Nude Pics Leak, Rep Confirms They Are Real
The likes of Kim Kardashian, Ariana Grande and more celebs are also said to have been hacked! Read more here...
Monday, September 1, 2014 - 08:57
Naked images of Jennifer Lawrence have emerged online, sparking fears that more A-list stars have been targeted in a huge celebrity hacking breach.
Nude snaps of The Hunger Games actress hit the internet on Sunday (August 31), with intimate snaps of J-Law posted on website 4Chan.
According to The Mirror, the hacker is thought to have broken into 101 famous people's iCloud accounts, with the likes of Kim Kardashian, Rihanna and Ariana Grande also said to have been targeted.
Brit babes Michelle Keegan, Cara Delevingne and Kelly Brook are also among those named on the hacker's list.
The source of the leak is said to have obtained as many as 60 private snaps of Lawrence, with some of the photos showing the 24-year old fully naked, while others see her posing in swimwear.
A rep for Jennifer has since confirmed that the racy images are real, but also threatened legal action against anyone who posts or publishes the stolen pics.
They told Buzzfeed: "This is a flagrant violation of privacy. The authorities have been contacted and will prosecute anyone who posts the stolen photos of Jennifer Lawrence."
However, a spokesperson for Grande has insisted the nude images of the Problem singer are "completely fake", while US actress Victoria Justice has stated that the alleged nude snaps of her are also fake.
She tweeted: "These so-called nudes of me are FAKE people. Let me nip this in the bud right now. *pun intended*."
These so called nudes of me are FAKE people. Let me nip this in the bud right now. *pun intended*
— Victoria Justice (@VictoriaJustice) August 31, 2014
Swimwear model Kate Upton also apparently had snaps stolen, with her reps saying they are "looking into" the claims.
Lawrence is yet to directly comment on the leak, but a Twitter account attributed to the actress posted: "It's so weird and hard how people take your privacy away from you... But, that's the price to have a successful career…"
She allegedly added: "But, just because i am "famous" i can't have a privacy life too? I really don't understand, it's too complicated."
The likes of Scarlett Johansson and Mila Kunis were also hit by a nude photo leak in 2012, with a man later sentenced to 10 years in prison for hacking their private online accounts.
Keep it MTV News UK for more updates on this story…
More From Jennifer Lawrence
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Jennifer Lawrence And The Red Sparrow Cast Go Speed Dating | MTV Movies
Jennifer Lawrence Explains Why She Won't Be Running For President | MTV Movies
MTV Movies Screening Room: Red Sparrow, Wonder Wheel & Gringo!
Jennifer Lawrence Shuts Down Reports She Had An Affair With Chris Pratt


text:
\section{Equivariant branes}\label{S:SectEqSh} In this
section, we introduce the category $\mathfrak{Mod}^G$ of
$G$-equivariant ${\mathcal O}$-modules, define a representation on
the cohomology groups of the branes which are objects of
$D(\mathfrak{Coh}^G)$, and characterize the space of strings stretching between some equivariant branes on a flag manifold of $G$.

\subsection{Equivariant sheaves}\label{SubSectEquivSheav}  By ${\mathcal O}_X$, or simply by ${\mathcal O}$,  we denote the
sheaf of regular functions on $X$.
 Let  $\mu:G\times X\to X$  be an {\em analytic} action of a reductive Lie group $G$
on $X$. Essentially,
 a $G$-equivariant structure on the ${\mathcal O}$-module  ${\mathcal H}$ is given by a family
 $\{\lambda_{g,x}\}$ of isomorphisms between the stalks
 \begin{equation}\label{asterisco1}
 \lambda_{g,x}:{\mathcal H}_x  \to{\mathcal H}_{\mu(g,x)},\;\;\;\hbox{for all}\;\;g\in G,\;x\in X,
  \end{equation}
 compatible with the multiplication in $G$ (i. e., satisfying the cocycle condition).

 To formulate the cocycle condition,

 we introduce the following notations
$$m:  G\times G\to  G,\;\;\;m(g_1,\,g_2)=g_1g_2.\;\;\;\;
 b: G\times X\mapsto x\in X,\;\;\; b(g,x)=x.$$
$$p:G\times G\times X\to G\times X,\;\;\; p(g_1,\,g_2\,,x)=(g_2\,,x).$$
Thus, one has the maps $p,$ $ m\times 1_X$ and $1_G\times \mu$
from $G\times G\times X$ to $G\times X$ and the corresponding
functors
\begin{equation}\label{functors0}
\xymatrix{\mathfrak{Mod}({\mathcal O}_X)\ar@/^/[r]^{b^*}
\ar@/_/[r]_{\mu^*} & \mathfrak{Mod}({\mathcal O}_{G\times X})
\ar[r]^{p^*} \ar@/_1pc/[r]_{(1_G\times \mu)^*}
\ar@/^2pc/[r]^{(m\times 1_X)^*} & \mathfrak{Mod}({\mathcal
O}_{G\times G\times X})\,, }
\end{equation}
where an asterisk as superscript is used for denoting  the inverse
image functor between the corresponding  categories, and
$\mathfrak{Mod}({\mathcal O}_Z)$ stands for the category of
${\mathcal O}_Z$-modules. The equalities
$$b\circ(m\times 1_X)=b\circ p,\,\; b\circ(1_G\times\mu)=\mu\circ p,\,\; \mu\circ(1_G\times\mu)=\mu\circ(m\times 1_X)$$
give rise to equalities between the respective compositions of the
functors in (\ref{functors0}).

Let ${\mathcal H}$ be an ${\mathcal O}$-module and $\lambda$ an
isomorphism $\lambda: b^{*}{\mathcal H}\to\mu^{*}{\mathcal H}$,
which satisfies the cocycle condition
 \begin{equation}\label{cocycle2}
(m\times 1_X)^*(\lambda)=(1_G\times\mu)^*(\lambda)\circ
p^*(\lambda).
\end{equation}
 We say that  the pair $({\mathcal H},\,\lambda)$
    is a $G$-equivariant sheaf of ${\mathcal O}$-modules, or simply a $G$-equivariant ${\mathcal O}$-module.

One  defines the category $\mathfrak{Mod}^G$, whose objects
    are the $G$-equivariant  ${\mathcal O}$-modules.  If $({\mathcal H}',\,\lambda')$
    and $({\mathcal H},\,\lambda)$ are objects in this category, a morphism in $\mathfrak{Mod}^G$
    from $({\mathcal H}',\,\lambda')$ to $({\mathcal H},\,\lambda)$ is a morphism  of ${\mathcal O}$-modules $f:{\mathcal H}'\to{\mathcal H}$,
   such that $\lambda b^*(f)=\mu^*(f)\lambda'$.

Given $g\in G$, we  put $L_g$ for the map defined by
  $x\in X\mapsto \mu(g,x)=gx\in X$.  Since the $G$-action on $X$ is analytic,  the composition with $\mu$ defines a morphism of sheaves
 of rings ${\mathcal O}_X\to{\mathcal O}_{G\times X}$. In particular,  given an open subset $U\subset X$ and $g\in G$,
  the map
  $$h\in{\mathcal O}(U)\mapsto h\circ L_{g^{-1}}\in {\mathcal O}(gU)$$
   determines a ring isomorphism ${\mathcal O}(U)\to {\mathcal O}(gU).$

In other terms, we have an isomorphism of sheaves of rings
 \begin{equation}\label{O-LgO}
 {\mathcal O}\to (L_{g^{-1}})_*{\mathcal O}.
 \end{equation}

We have the following proposition.

\begin{Prop}\label{P:Propiso}
 Let $({\mathcal H},\,\lambda)$ be a $G$-equivariant ${\mathcal O}$-module and $g$ an element of $G$,
 then $\lambda$ determines  an isomorphism  of ${\mathcal O}$-modules
 $$\lambda_g:{\mathcal H}\to (L_{g^{-1}})_*{\mathcal H}, $$

 where the ${\mathcal O}$-structure of $(L_{g^{-1}})_*{\mathcal H}$ is defined through the isomorphism (\ref{O-LgO}).
  \end{Prop}

The image of
$\sigma_U\in{\mathcal H}(U)$ by the isomorphism
${\mathcal H}(U) \stackrel{\sim}{\longrightarrow} {\mathcal
 H}(gU)$
 will be denoted
$g\cdot\sigma_U$.

A consequence of the cocycle condition is  the following proposition.
\begin{Prop}\label{enumerate}
For each $g\in G$,
\begin{enumerate}
\item $\lambda_h\circ\lambda_g=\lambda_{hg}.$ \item The map
$g\mapsto \Hat{\lambda}_g:=\lambda_g(X)$ is a group homomorphism
from $G$ to the group of automorphisms of the complex vector space
${\mathcal H}(X)$.
\end{enumerate}
\end{Prop}

Although the results stated in Propositions \ref{P:Propiso} and  \ref{enumerate} are easy to understand, we give detailed proofs of these propositions   in Appendix.

In summary,  the cocycle condition for $({\mathcal H},\,\lambda)$
gives rise to a representation of $G$ in the space
$H^0(X,\,{\mathcal H})$.

The following step will be to define a representation in
the cohomologies $H^i(X,\,{\mathcal G})$, when ${\mathcal G}$ is an object of   $D(\mathfrak{Coh}^G)$.

\begin{Prop}\label{Proprephcoho}
 If ${\mathcal G}$ is a brane of the category $D(\mathfrak{Coh}^G)$,

  then for
 each $i$ the cohomology group $H^i(X,\,{\mathcal G})$ supports a
 representation of $G$ induced by the $G$-structure of ${\mathcal G}$. When ${\mathcal G}$ is an object of $\mathfrak{Coh}^G$  the
 representation on $H^0(X,\,{\mathcal G})$ is the one of
 Proposition
 \ref{enumerate} (2).
  \end{Prop}
{\it Proof.} As the category $\mathfrak{Mod}^G$ has enough injectives \cite{Tohoku},   following the well-known procedure, it is possible to construct an
equivariant Cartan-Eilenberg resolution ${\mathcal
J}^{\bullet\bullet}$ of ${\mathcal G}^{\bullet}$ in
$\mathfrak{Mod}^G$ \cite[Thm. 10.45]{R}.  Then
${\mathcal G}^{\bullet}$ is quasi-isomorphic to the total complex
${\mathcal I}^{\bullet}=({\rm Tot}({\mathcal J}^{\bullet}),\,\partial^{\bullet})$, a complex of in $\mathfrak{Mod}^G$ consisting of injective objects.

 By the second item in Proposition  \ref{enumerate}, the space the
${\mathcal I}^{i}(X)$ carries the representation $\rho^i$ of $G$.
 Since the diagrams
\begin{equation}\label{DiagExact2}
    \xymatrix{  {\mathcal I^i}(X) \ar[d]_{\rho^i_g}\ar[r]^{{\partial}^i(X)} & {\mathcal I}^{i+1}(X)\ar[d]^{\rho_g^{i+1}} \\
   {\mathcal I}^i(X)\ar[r]^{{\partial}^i(X)} & {\mathcal I}^{i+1}(X) }
 \end{equation}
   are commutative, one has a representation of $G$ on each cohomology group $h^i({\mathcal I}^{\bullet }(X))$ of
 the complex ${\mathcal I}^{\bullet}(X)$. That is, a representation on the cohomology $H^i(X,\,{\mathcal G})$. Thus, we have the proposition.
  \qed

\smallskip

If in the statement of Proposition \ref{Proprephcoho}  ${\mathcal G}$ is a locally free ${\mathcal O}$-module, the representation on $H^i(X,\,{\mathcal G})$ can be constructed by means
of the Dolbeault resolution.
Let ${\mathcal G}={\mathcal O}(V)$ be the sheaf   of germs of sections of the holomorphic vector bundle $V$. We put ${\mathcal A}^{0,q}$ for the sheaf of germs of holomorphic differential forms on $X$ of type $(0,q)$. Since ${\mathcal O}(V)$ is a flat ${\mathcal O}$-module, the tensor product of this module by the Dolbeault resolution of ${\mathcal O}$ gives
 rise to the following fine resolution of ${\mathcal O}(V)$
  \begin{equation}\label{resolutionOmega(V)}
0\to{\mathcal O} (V){\longrightarrow}{\mathcal
A}^{0,0}(V)\overset{1\otimes\bar\partial}
{\longrightarrow}{\mathcal
A}^{0,1}(V)\overset{1\otimes\bar\partial}
{\longrightarrow}{\mathcal A}^{0,2}(V){\longrightarrow}\dots
\end{equation}
 where ${\mathcal A}^{0,q}(V):={\mathcal O}(V)\otimes_{\mathcal O}{\mathcal A}^{0,q}.$
   Thus,
\begin{equation}\label{Hq(X}
 H^q(X,\,{\mathcal O}(V))=h^q(A^{0,\bullet}(V)),
\end{equation}
 where $A^{0,q}=\Gamma(X,\,{\mathcal A}^{0,q}(V))$.

Now, we use the fact that $V$ is a $G$-equivariant vector bundle. If  $\sigma$ a holomorphic section of   $V$ and $\omega$ a $(0,q)$-form on $X$, we put
\begin{equation}\label{gcdot(}
g\cdot (\sigma\otimes \omega):=(g\cdot\sigma)\otimes L_{g^{-1}}^*\omega,
 \end{equation}
  $L_{g^{-1}}^*\omega$ being the pullback of $\omega$ by the diffeomorphism
  $L_{g^{-1}}$ and $g\cdot\sigma$ the section defined as just after
  Proposition \ref{P:Propiso}.

  If $f$ is a holomorphic function on $X$, then
 $$g\cdot(f\sigma\otimes\omega)=g\cdot(\sigma\otimes f\omega).$$
  On the other hand, since the $G$ acts analytically on $X$,
  $1\otimes\bar\partial$  commutes with the $G$-action (\ref{gcdot(}). So, we have  a representation of $G$
  on $H^q(X,\,{\mathcal O}(V))$, which is equivalent to the one of Proposition \ref{Proprephcoho}.

\smallskip

\subsection{Vertex operators.} In this subsection, we consider the spaces of vertex operators  for
 strings stretching between equivariant branes.

We put ${\mathcal Hom}(\,.\,,\,.\,)$ for the sheaf functor Hom of the category $\mathfrak{Mod}$
(see \cite[page 87]{Kas-Sch})
$${\mathcal Hom}(\,.\,,\,.\,): \mathfrak{Mod}^{\rm op}\times\mathfrak{Mod}\to \mathfrak{Sh},$$
where $\mathfrak{Sh}$ is the category of sheaves of abelian groups over $X$.

Let $({\mathcal F},\,\gamma)$, $({\mathcal G},\,\beta)$ be
$G$-equivariant ${\mathcal O}$-modules.
We set ${\mathcal K}:={\mathcal Hom}({\mathcal F},\,{\mathcal G})$ for the sheaf of homomorphisms
from ${\mathcal F}$ to ${\mathcal G}$.
 Given an open  subset $U\subset X$, a section $\Phi\in {\mathcal K}(U)={\rm
Hom}_{{\mathcal O}(U)}({\mathcal F}(U),\,{\mathcal G}(U))$ and
$g\in G$, we define $g\cdot \Phi\in {\mathcal K}(gU)$ as follows:

Let $\tau\in {\mathcal F}(gU)$ be a section of ${\mathcal F}$ on
$gU$, then $g^{-1}\cdot\tau\in {\mathcal F}(U)$,  with the
notation introduced below   Proposition \ref{P:Propiso}. Thus,
$\Phi(g^{-1}\cdot\tau)\in {\mathcal G}(U)$ and
$g\cdot\Phi(g^{-1}\cdot\tau)\in {\mathcal G}(gU)$.
 We put
 \begin{equation}\label{gcdotP}
 (g\cdot\Phi)(\tau):=g\cdot(\Phi(g^{-1}\cdot\tau)).
 \end{equation}

 So, we have constructed an isomorphism

 \begin{equation}\label{Etau}
   \eta_g(U)

   :{\mathcal K}(U)\longrightarrow
{\mathcal K}(gU),\;\; \;\;  \Phi\mapsto g\cdot \Phi.
\end{equation}
Moreover,
\begin{equation}\label{EquatioEta}

 \eta_h(gU)\circ \eta_g(U)=\eta_{hg}(U).
 \end{equation}
 Hence, the isomorphisms

 $\{\eta_g(X)\}_g$
 define a representation of $G$ on the space ${\mathcal K}(X)$. Thus, one has the following proposition.

\begin{Prop}\label{PropEta}
Let $({\mathcal F},\,\gamma)$, $({\mathcal G},\,\beta)$ be
$G$-equivariant ${\mathcal O}$-modules,  then
\begin{enumerate}
\item The isomorphisms (\ref{Etau}) convert the ${\mathcal O}$-module
$${\mathcal K}={\mathcal Hom}({\mathcal F},\,{\mathcal G})$$
in a $G$-equivariant ${\mathcal O}$-module.
\item For $g\in G$ and $\Psi\in{\mathcal K}(X)={\rm Hom}_{\mathfrak{Mod}}({\mathcal
F},\,{\mathcal G})$, we put
$g\cdot\Psi$ for the element of ${\mathcal K}(X)$ defined by
$$(g\cdot\Psi)_U(\sigma):= g\cdot(\Psi_{g^{-1}U}(g^{-1}\cdot\sigma)),$$
where $U$ is an open subset of $X$ and $\sigma\in{\mathcal F}(U)$. The correspondence $\Psi\to g\cdot\Psi$
  defines a representation of $G$ in vector space ${\mathcal K}(X)$.
\end{enumerate}
\end{Prop}

Given ${\mathcal G}^{\bullet}$ a complex in the category $\mathfrak{Coh}^G$, we consider ${\mathcal I}^{\bullet}$ the complex consting of injective objects in $\mathfrak{Mod}^G$ described  in the proof of Proposition \ref{Proprephcoho}.
If ${\mathcal F}^{\bullet}$ is another complex in
$\mathfrak{Mod}^G$, we put
$${\mathcal C}^n:=\prod_a{\mathcal Hom}({\mathcal F^a,\,{\mathcal
I}^{a+n}}).$$ By Proposition \ref{PropEta}, each ${\mathcal
Hom}({\mathcal F^a,\,{\mathcal I}^{a+n}})$ is an object of
$\mathfrak{Mod}^G$.

On the other hand,

 the coboundary operator $\delta^n:{\mathcal C}^n\to{\mathcal C}^{n+1}$ is defined  by (\cite[page 17]{Iversen})
$$\delta^n(\Psi_a)=\big(\partial^{a+n}\circ\Psi_a+(-1)^{n+1}\Psi_{n+1}\circ\partial ^a  \big),$$
where
$$\Psi_a\in {\mathcal Hom}({\mathcal F^a,\,{\mathcal
I}^{a+n}}).$$
Since the   $\partial$'s are $G$-operators, so is
 $\delta^n$. Thus, $({\mathcal
 C}^{\bullet},\,\delta^{\bullet})$ is a complex in the category
 $\mathfrak{Mod}^G$.  The complex  ${\mathcal
 C}^{\bullet}$ is   usually denoted by ${\mathcal
 Hom}^{\bullet}\big({\mathcal F}^{\bullet},\,{\mathcal I}^{\bullet}
 \big).$ By definition, ${\mathcal Ext}^p({\mathcal F}^{\bullet},\,{\mathcal
 G}^{\bullet})$ is the cohomology object
 $h^p({\mathcal
 Hom}^{\bullet}\big({\mathcal F}^{\bullet},\,{\mathcal
 I}^{\bullet})).$

 As ${\mathcal
 Hom}^{\bullet}\big({\mathcal F}^{\bullet},\,{\mathcal I}^{\bullet}
 \big)$ is a complex in the abelian category $\mathfrak{Mod}^G$,
 its cohomologies are also in $\mathfrak{Mod}^G$; i. e. are
 $G$-equivariant sheaves. Hence, from Proposition
 \ref{Proprephcoho}, it follows the following proposition.
 \begin{Prop}\label{PropArbitraryBranes}  Let ${\mathcal F}$ and ${\mathcal G}$ be branes in the derived category
 $D(\mathfrak{Coh}^G)$.
Then the $G$-structures of ${\mathcal F}$ and ${\mathcal G}$ induce on the space of vertex operators
  $$H^q(X,\,{\mathcal Ext}^p({\mathcal F},\,{\mathcal G}))$$
a representation of $G$.
 \end{Prop}

The spectral sequence of the double complex,
  $$E^{p,q}=H^p(X,\,{\mathcal Ext}^q({\mathcal F},\,{\mathcal G})),$$
  converges to the space of strings ${\mathit Ext}^{p+q}({\mathcal F},\,{\mathcal G})$. Then from Proposition \ref{PropArbitraryBranes}, we conclude:

  \begin{Prop}\label{Repres_Strings}
   If ${\mathcal F}$ and ${\mathcal G}$ are branes of the derived category
  $D(\mathfrak{Coh}^G)$,

   then the space ${\mathit Ext}^k ({\mathcal F},\,{\mathcal G})$ of strings between ${\mathcal F}$ and ${\mathcal G}$ with ghost number $k$
   supports a representation of $G$ induced by the $G$-structures of ${\mathcal F}$ and ${\mathcal G}$.
  \end{Prop}

When ${\mathcal F}$ and ${\mathcal G}$ are locally free ${\mathcal
O}$-modules the representation stated in the previous proposition can
be formulated in terms of differential forms.

First of all, one has the following resolution for ${\mathcal F}$
consisting of locally free ${\mathcal O}$-modules
 $$\dots\to 0\to 0\to 0\to{\mathcal F}\overset{1}{\to}{\mathcal F}\to
 0.$$
 By \cite[Proposition 6.5, page 234]{Hart}, ${\mathcal Ext}^k({\mathcal
 F},\,{\mathcal G})=0$, for $k\ne 0$. Hence, when ${\mathcal F}$ is a locally free module we consider only the
 spaces of vertex operators $H^q(X,\,{\mathcal Hom}({\mathcal
 F},\,{\mathcal G}))$. Furthermore, in this case,
\begin{equation}\label{cuerdasVectorbundles}
{\mathit Ext}^k({\mathcal F},\,{\mathcal G})=H^k(X,\,{\mathcal Hom}({\mathcal F},\,{\mathcal G})).
  \end{equation}

Let  $V_1$ and $V_2$ be two   holomorphic $G$-equivariant vector
bundles on $X$. We denote by $V$
  the holomorphic  vector bundle   ${\rm Hom} (V_1,\,V_2)$. Given $\psi$ is an element
  of $\Gamma(X,\,{\mathcal O}(V))$, putting
\begin{equation}\label{gcdotAux}
(g\cdot\psi)(-)=g\cdot(\psi(g^{-1}\cdot (-)))
 \end{equation}
 we
define a representation of $G$ on $\Gamma(X,\,{\mathcal O}(V))$; it
is the one stated in item 2 of Proposition \ref{PropEta}. More
explicit, given $x\in X$, the homomorphism $(g\cdot
\psi)_x:V_{1x}\to V_{2x}$ is the composition
\begin{equation}\label{composition}
\lambda^2_{g,g^{-1}x}\circ\psi_{g^{-1}x}\circ\lambda^1_{g^{-1},x},
 \end{equation}
where the $\lambda $'s are the isomorphisms (\ref{asterisco1}); that is, $\lambda^i_{g,y}:(V_i)_y\to (V_i)_{gy}$.

As   ${\mathcal Hom}({\mathcal O}(V_1),\,{\mathcal
O}(V_2))={\mathcal O}({\rm Hom}(V_1,\,V_2))$,  then (\ref{Hq(X})
applied to $V={\rm Hom}(V_1,\,V_2)$ gives
 \begin{equation}\label{Hq(X1}
H^q(X,\,{\mathcal Hom}({\mathcal O}(V_1),\,{\mathcal O}(V_2)))
= h^{q}(A^{0,\bullet}({\rm Hom}(V_1,\,V_2))).
  \end{equation}
 As in (\ref{gcdot(}), from the $G$-action
 (\ref{gcdotAux}),  we can construct the corresponding representation
 on the space (\ref{Hq(X1}). Thus, given
 $$\psi\in\Gamma(X,\,{\mathcal Hom}({\mathcal
 O}(V_1),\,{\mathcal O}(V_2)))$$
  and the differential form $\omega$ of type
 $(0,q)$, we have
 \begin{equation}\label{gcdotLOCALLY}
 g\cdot(\psi\otimes\omega)=g\cdot
 \psi\otimes L^*_{g^{-1}}\omega.
  \end{equation}
Thus, Proposition \ref{PropArbitraryBranes} adopts the following
form when the branes are locally free sheaves.
\begin{Prop}\label{P:equivariant-Llocally} Let $V_1$, $V_2$ be $G$-equivariant holomorphic vector
bundles on $X$. Then the $G$-action (\ref{gcdotLOCALLY}) induces a
representation of $G$ on the space of vertex operators
$$H^q(X,\,{\mathcal Hom}({\mathcal O}(V_1),\,{\mathcal
O}(V_2))).$$
\end{Prop}

\smallskip

\noindent
{\bf Vertex operators on flag varieties.} When $X$ is a flag manifold, the dimension of the spaces of vertex operators mentioned in
 Proposition \ref{P:equivariant-Llocally} can be determined, in some particular cases, by means of the Borel-Weil-Bott theorem and the Weyl's dimension formula.

Let $G_{\mathbb C}$   be a connected complex semi-simple Lie group and $Q\subset G_{\mathbb C}$  a parabolic subgroup of $G_{\mathbb C}$. Then the flag manifold  $X=G_{\mathbb C}/Q$ is a compact homogeneous simply connected K\"ahler
  algebraic variety (see for example \cite{Borel, W1}).  Moreover, every  compact homogeneous simply connected K\"ahler manifold is isomorphic to such a quotient.  In \cite{Grant}, Grantcharov showed several examples
  flag manifolds which are also Calabi-Yau.

The holomorphic equivariant vector bundles over $X$ are related with the holomorphic representations of $G_{\mathbb C}$.
 Let $V\to X$ be a holomorphic vector bundle endowed with a holomorphic $G_{\mathbb C}$-action by bundle maps, that lies over the action on $X$. Denoting with $x_0\in X$ the class $eQ$,  then  stabilizer of $x_0$ acts on the the fibre $V_{x_0}$. Hence, there is a holomorphic representation
$\xi$
of $Q$ on ${\mathsf V}$, the standard fibre of $V$.
   Moreover, one has an isomorphism of holomorphic bundles $V\simeq  G_{\mathbb C}\times_{\xi} {\mathsf V}$.
Conversely, equivariant holomorphic bundles on $X$ can be constructed from representations.

 We denote by $G$ a compact real form of $G_{\mathbb C}$.
 As it is  known, the finite dimensional representations of $G$ are in bijective correspondence with the holomorphic representations of finite dimension of $G_{\mathbb C}$.

 We put $L:=G\cap Q$,   then one can identify $G/L$ with  $X$. Moreover, each irreducible representation of  $L$ on a complex space induces a unique extension to one holomorphic representation of $Q$. So, an irreducible representation $\alpha$ of $L$ on the complex vector space ${\mathsf V}$ gives rise to one homogeneous holomorphic vector bundle
$$V(\alpha):=G_{\mathbb C}\times_Q{\mathsf V},$$
on $X$.
It is very easy to check that $V(\alpha\oplus\beta)=V(\alpha)\oplus V(\beta),$ and $V(\alpha^*)=(V(\alpha))^*.$

Given $\alpha$, $\beta$ irreducible representations of $L$ the representation tensor product $\alpha^*\otimes\beta$ can be written as direct sum of irreducible representations
\begin{equation}\label{Littlewood-Richard}
\alpha\otimes\beta=\bigoplus_{\nu} m^{\nu}\nu,
 \end{equation}
where $\nu$ is an irreducible representation of $L$ and the $m^{\nu}\in{\mathbb Z}_{\geq 0}$ are the corresponding Littlewood-Richardson coefficients.

Hence, the bundle of homomorphisms (over the identity) from $V(\alpha)$ to $V(\beta)$ can be written as
\begin{equation}\label{HomValpha}
 \mathit{Hom}(V(\alpha),\,V(\beta))= V(\alpha)^*\otimes V(\beta)= V(\alpha^*\otimes \beta)=\oplus_{\nu}m^{\nu}V(\nu).
 \end{equation}
 Since ${\mathcal O}(V(\alpha))$ is a locally free ${\mathcal O}$-module,   it follows from (\ref{HomValpha}) together with (\ref{cuerdasVectorbundles}) that  the space of string with ghost number $k$ stretching between ${\mathcal O}(V(\alpha))$ and ${\mathcal O}(V(\beta))$
$$\mathit{Ext}^k\big({\mathcal O}(V(\alpha)),\,{\mathcal O}(V(\beta))\big)=\bigoplus_{\nu}m^{\nu}H^k(X,\,{\mathcal O}(V(\nu))).$$

 \smallskip
 {\it Notations.}
We put
${\mathfrak g}:={\rm Lie}(G_{\mathbb C})$ and ${\mathfrak g}_0$ for ${\rm Lie}(G)$ for the Lie algebras of the corresponding groups. Let $c$ be the complex conjugation in ${ \mathfrak g }$ with respect to  ${ \mathfrak g }_0.$
 We fix a maximal abelian subalgebra ${\mathfrak h}_0$ of ${\mathfrak g}_0$ and   denote by ${\mathfrak h}$ its complexification with respect to $c$. Let $\Delta$ be the set of roots of the Cartan subalgebra ${\mathfrak h}$ in ${\mathfrak g}$.

  The system  of positive roots $\Delta^{+}$ for the pair $({\mathfrak g},\,{\mathfrak h})$ is chosen so that
   $${\rm Lie}(Q)={\mathfrak q}={\mathfrak h}\oplus\bigoplus_{\gamma\in\Phi}{\mathfrak g}^{-\gamma},$$
   where $\Phi$  is a subset of $\Delta$ containing $\Delta^+$ and   ${\mathfrak g}^{-\gamma}$ being the root space associated to $\gamma$.
   We denote by  $\rho$   the half sum of positive roots,

  write $\Lambda$ for the weight lattice and put $\Lambda^+$ for the set of dominant weights.

\smallskip

$\zeta(\alpha)$ denotes the highest weight of the irreducible representation $\alpha$ of $L$.
  By the Borel-Weil-Bott theorem  \cite{Gr-Sch},   $H^k(X,\,{\mathcal O}(V(\nu))=0,$ if $\zeta(\nu)+\rho$ is singular. When $\zeta(\nu)+\rho$ is regular, let $w_{\nu}$ the unique element of the Weyl group    ${\mathbf W}$ such that  $w_{\nu}(\zeta(\nu)+\rho)$ is dominant and define
 $$l({\nu}):=\#\{\alpha\in\Delta^+\,|\, (\zeta(\nu)+\rho,\,\alpha)<0  \},$$
where the bilinear product $(\,.\,,\,.\,)$ is defined by the Killing form.
 The following proposition is consequence of Borel-Weil-Bott theorem.

 \begin{Prop}\label{P:Bore-Weil-Bott}
If $\alpha$ and $\beta$ are irreducible representations of $L$, then the dimension of the space of strings $\mathit{Ext}^k\big({\mathcal O}(V(\alpha)),\,{\mathcal O}(V(\beta))\big)$ is
$$\sum_{\tau}m^{\tau}\delta_{k,l(\tau)}\,{\rm dim}\,\xi_{\tau},$$
where $\tau$ runs over the irreducible representations of $L$, such that $\zeta(\tau)+\rho$ is a regular weight,  $\xi_{\tau}$ being the irreducible representation of $L$ with highest weight $w_{\tau}(\zeta(\tau)+\rho)-\rho$ and the $m^{\tau}$'s the Littlewood-Richardson coefficients defined in (\ref{Littlewood-Richard}).
 \end{Prop}

When the representations $\alpha$ and $\beta$ are unidimensional the statement of Proposition \ref{P:Bore-Weil-Bott} admits a simpler formulation.

Given $\upsilon\in \Lambda$, it determines a holomorphic character ${\rm e}^{\upsilon}:Q\to {\mathbb C}^{\times}$. So, one can construct the fiber product $L_{\upsilon}=G_{\mathbb C}\times_{Q}{\mathbb C}$, which is a holomorphic $G_{\mathbb C}$-equivariant line bundle over $X$.
If $\lambda,\mu\in \Lambda$, then
$$L_{-\mu+\lambda}=L_{\mu}^*\otimes L_{\lambda}= \mathit{Hom}(L_{\mu},\,L_{\lambda}).$$
Hence, we have isomorphisms  among the following sheaves of holomorphic sections
$${\mathcal Hom}({\mathcal O}(L_{\mu}),\, {\mathcal O}(L_{\lambda}))= {\mathcal O}\big(\mathit{Hom}(L_{\mu},\,L_{\lambda}) \big)
= {\mathcal O}(L_{-\mu+\lambda}).$$

 We put   $\xi:=-\mu +\lambda$ for the weight that determines the line bundle $L_{\mu}^*\otimes L_{\lambda}$, and
 $$p(\xi):=\#\{\alpha\in\Delta^+\,|\, (\xi+\rho,\,\alpha)<0  \}.$$

 A direct application of  the Borel-Weil-Bott theorem  to the weight  $\xi$  proves the following proposition, which characterizes some spaces of vertex operators relative to the flag manifold $X=G_{\mathbb C}/Q$.

\begin{Prop}\label{Flag}
If $\xi+\rho$ is either singular or $p\ne p(\xi)$, then the space of vertex operators
$H^p(X,\, {\mathcal Hom}({\mathcal O}(L_{\mu}),\, {\mathcal O}(L_{\lambda})) )$

is zero. Otherwise, the   representation of $G$ in that space
 is the irreducible representation with highest weight is  $w(\xi+\rho)-\rho$, where $w$ is the element of the Weyl group such
 that $w(\xi+\rho)$ is dominant.
\end{Prop}

 As ${\mathcal O}(L_{\mu})$ is a locally free ${\mathcal O}$-module, the space of strings from the brane ${\mathcal O}(L_{\mu})$ to
${\mathcal O}(L_{\lambda})$ with ghost number $k$ is
 $$ {\mathit Ext}^k({\mathcal O}(L_{\mu}),\, {\mathcal O}(L_{\lambda}))=
 H^k(X,\, {\mathcal Hom}({\mathcal O}(L_{\mu}),\, {\mathcal O}(L_{\lambda}))).$$

 Thus, taking into account the proposition,
the Weyl dimension formula permits us to determine the dimension of these spaces.

\smallskip

\subsection{Correlation functions.} Let us assume that $X$ is a
projective Calabi-Yau variety of dimension $n$. For $j=0,\dots,
k$, let ${\mathcal F}_j$ be
a general brane. Given  $a_j\in
H^{q_j}(X,\,{\mathcal Ext}^{p_j}({\mathcal F}_{j-1},\,{\mathcal
F}_j))$,
 when ${\mathcal F}_k={\mathcal F}_0$ and $\sum(q_j+p_j)=n$, one
 can define the correlation function $\langle a_1\dots a_k\rangle$
 of the vertex operators $a_j$'s (see \cite{Vina}).

 If the ${\mathcal F}_j$ are locally free sheaves, i.e. ${\mathcal F}_j={\mathcal
 O}(V_j)$, with $V_j$ holomorphic vector bundle, the correlation
 function  can be calculated as follows.
 Given
 $$a_j\in H^{q_j}(X,\,{\mathcal Hom}({\mathcal
 O}(V_{j-1}),\,{\mathcal O}(V_j)))=H^{0.q_j}_{\bar\partial}(X, {\rm Hom}(V_{j-1},\,V_j)),$$
 $a_j$ will be the class of a form $\psi^j\otimes\omega^j$, with $\psi^j$
 a holomorphic section of ${\rm Hom}(V_{j-1},\,V_j)$ and
 $\omega^j$ a differential  form on $X$ of type $(0,\,q_j)$. Then
 \cite{Aspin, Vina}
\begin{equation}\label{CorrelationLocally}
\langle a_1\dots a_k\rangle=\int_X{\rm
tr}(\psi^k\circ\dots\circ\psi^1)\,\omega^k\wedge\dots\wedge\omega^1\wedge\Omega,
 \end{equation}
where $\Omega$ is  an fixed $n$-holomorphic form on $X$ which vanishes
nowhere.

Next, let us assume that the ${\mathcal O}$-modules ${\mathcal
F}_j$ are $G$-equivariant, and we denote  by $\lambda^j$ the isomorphism that defines the $G$-structure on ${\mathcal F}_j$. Given $g\in G$, by (\ref{gcdotLOCALLY})
\begin{equation}\label{Correlationg}
\langle g\cdot a_1\dots g\cdot a_k\rangle=\int_X{\rm
tr}(g\cdot\psi^k\circ\dots\circ
g\cdot\psi^1)\,L^*_{g^{-1}}\omega^k\wedge\dots\wedge
L^*_{g^{-1}}\omega^1\wedge\Omega.
 \end{equation}
 From (\ref{composition}) together with the cocycle condition, it follows
 $$(g\cdot\psi^j)_x\circ(g\cdot\psi^{j-1})_x=\lambda^j_{g,g^{-1}x}\circ\psi^j_{g^{-1}x}\circ \psi^{j-1}_{g^{-1}x}\circ\lambda^{j-2}_{g^{-1},x}.$$
 As $\lambda^0=\lambda^k$
\begin{align}
{\rm tr}(g\cdot\psi^k\circ\dots\circ g\cdot\psi^1)(x)&={\rm
tr}(\lambda^0_{g,g^{-1}x}\circ\psi^k_{g^{-1}x}\circ\dots\circ\psi^1_{g^{-1}x}\circ\lambda^0_{g^{-1},x})
 \notag \\
&={\rm tr}(\psi^k\circ\dots\circ \psi^1)(g^{-1}\cdot x). \notag
\end{align}

Since the $G$ action on $X$ is analytic,
$L^*_{g^{-1}}\Omega=\Omega$; so, the integrals
(\ref{CorrelationLocally}) and  (\ref{Correlationg}) are equal;
that is,
\begin{equation}\label{langecdota}
\langle g\cdot a_1\dots g\cdot a_k\rangle=\langle a_1\dots
a_k\rangle.
\end{equation}
Thus, we have the following proposition.

\begin{Prop}\label{Thm:CorrelationInvar}
Let ${\mathcal F}$ and ${\mathcal G}$ be   $G$-equivariant locally free ${\mathcal O}$-modules, then the correlation functions associated to vertex operators for strings between ${\mathcal F}$ to ${\mathcal G} $ are $G$-invariant.
\end{Prop}

\section{Cohomology of equivariant branes}\label{S:equivariant}

Because of the equivariance, the branes in  $D(\mathfrak{Coh}^G)$ admit a lift to  branes on the homotopy quotient of $X$, and so one can   define the corresponding equivariant cohomology for these objects. In this section, we will consider that equivariant cohomology.

Let $G$ be a compact Lie group. By the Peter-Weyl theorem $G$ is a
closed subgroup of $GL(N,\,{\mathbb R})$ for some $N$. The limit
as $n\to\infty$ of  Stiefel manifold of $N$-frames in ${\mathbb
R}^{N+n}$ is a smooth model for the universal $G$-bundle.
 We will adopt this model and denote by

  $EG\overset{q}{\to} BG:=EG/G$ the corresponding $G$-fibration.

 We put $\bar X:=(EG\times X)/G$ the homotopy quotient
of $X$ by $G$ and set $\tau$ and $\pi$ for the projections
 $X\overset{\tau}{\longleftarrow} EG\times X\overset{\pi}{\longrightarrow} \bar X.$
 Let $X\to\bar X=EG\times_G X\overset{\nu}{\to}BG$ be the fibration
constructed from the action of $G$ on $X$, and let $p$ denote the composition
$EG\times X \overset{proj.} {\longrightarrow} EG\overset{q}{\to} BG$. One has the following commutative diagram
$$ \xymatrix{ \bar X \ar[rd]_{\nu} & EG\times X \ar[l]_{\pi} \ar[d]^{p} \ar[r]^{\tau} & X\\
 {}&  BG & {}
 }$$

Let ${\mathcal H}$ be a $G$-equivariant coherent sheaf  on $X$, then the inverse image $\tilde{\mathcal H}:=\tau^*{\mathcal H}$ is $G$-equivariant  coherent sheaf on $EG\times X$. Denoting  by $\tilde L_g$ the obvious multiplication by $g\in G$ in $EG\times X$, we put  $\tilde\lambda_g$ by the isomorphism $\tilde\lambda_g:\tilde{\mathcal H}\to(\tilde L_{g^{-1}})_*\tilde{\mathcal H}$ determined by the equivariance of $\tilde{\mathcal H}$ (see Proposition \ref{P:Propiso}).

 On the other hand, if $V$ is an open set of $\bar X$, $\pi_*\tilde{\mathcal H}(V)=\tilde{\mathcal H}(W)$, where $W$ is the $G$-invariant subset $\pi^{-1}(V)$.  Thus, there is a $G$-action on the sheaf $\pi_*\tilde{\mathcal H}$ defined on
$\pi_*\tilde{\mathcal H}(V)$ by the isomorphism $\tilde\lambda_{g,W}$.
 We put $\bar{\mathcal H}$ for denoting the subsheaf of $\pi_*\tilde{\mathcal H}$  defined by $G$-invariant sections of $\pi_*\tilde{\mathcal H}$.
Then the sheaves $\tau^*{\mathcal H}$ and $\pi^*\bar{\mathcal H}$ are isomorphic (see \cite[page 3]{Be-Lu}) and denote by $h$ the isomorphism
 $h:\tau^*{\mathcal H}\to\pi^*\bar{\mathcal H}$.

For any open not empty subset $ U\subset BG$, one has $\tau p^{-1}(U)=X$  and $\nu^{-1}(U)=\pi p^{-1}(U)$.

 \begin{align} \label{H0nu}
\Gamma(\nu^{-1}(U),\,\bar{\mathcal H}) &=
\bar{\mathcal H}(\pi p^{-1}(U))=\pi^*\bar{\mathcal H}(p^{-1}(U)) \\
&\simeq\tau^*{\mathcal H}(p^{-1}(U))=\Gamma(X,\,{\mathcal H}), \notag
 \end{align}

where the isomorphism is determined by $h$. We have proved the following proposition.
\begin{Prop}\label{P:AUXforH}
Given ${\mathcal H}$   a $G$-equivariant coherent sheaf  on $X$,   it determines a sheaf $\bar{\mathcal H}$ on $\bar X$ and an isomorphism
$\tau^*{\mathcal H}\simeq\pi^*\bar{\mathcal H}.$ Furthermore, $\nu_*\bar{\mathcal H}$ is the constant sheaf $\Gamma(X,\,{\mathcal H})$.
\end{Prop}

Given   ${\mathcal F}$ an object of $D(\mathfrak{Coh}^G)$, i.e., a $G$-equivariant brane, according to the preceding remarks, it defines a object
$\bar{\mathcal F}$ of the derived category $D(\mathfrak{Sh}(\bar X))$, of abelian sheaves on $\bar X$, and an isomorphism $f$ of   the category of $D(\mathfrak{Sh}
  ({EG\times X}))$   from $\tau^*({\mathcal F})$ to
  $\pi^*(\bar{\mathcal F})$. Then the triple ${\mathbf F}=({\mathcal F},\,\bar{\mathcal F},\,f)$ is an object of the equivariant derived category
	$D_G(X)$, defined in \cite{Be-Lu}.

If  ${\mathbf F}$ is an object as above,
let $\bar
{\mathcal I}^{\bullet}$ and $
{\mathcal I}^{\bullet}$ be complexes of injectives on
 $\bar X$
 and on ${ X}$ that represent $\bar{\mathcal F}$ and ${\mathcal F}$, respectively.  By Proposition \ref{P:AUXforH},

 $$R^q\nu_*\bar{\mathcal F}=H^q(\nu_*(\bar{\mathcal I}^{\bullet}))=H^q(\Gamma(X,\,{\mathcal I}^{\bullet}))=H^q(X,\,{\mathcal F}).$$

Hence, $R^q\nu_*\bar{\mathcal F}$ is the constant sheaf on $BG$
defined by $H^q(X,\,{\mathcal F})$.

The equivariant cohomology $H_G(X,\,{\mathbf F})$ of ${\mathbf
F}\in { D}_G(X)$, is by definition the cohomology $H(\bar
X,\,\bar{\mathcal F})$; that is, the cohomology  $H(BG,\, R\nu_*\bar{\mathcal F})$.

\begin{Prop}\label{ThmSpectral}
The spectral sequence $E_2^{pq}=H^p(BG)\otimes H^q(X,\,{\mathcal
F})$

  abuts to $H^{p+q}_G(X,\,{\mathbf F})$.
\end{Prop}
 {\it Proof.} Since $R^q\nu_*\bar{\mathcal F}$ is the constant sheaf $H^q(X,\,{\mathcal F})$,

  the $E_2$ term of the Leray-Serre spectral sequence associated to the fibration
$X\to\bar X\to BG$ is
 $$E_2^{pq}=H^p(BG,\,R^q\nu_*(\bar{\mathcal F}))=H^p(BG)\otimes
H^q(X,\,{\mathcal F}),$$
 and the sequence abuts to $H^{p+q}(\bar X,\,\bar{\mathcal F})= H^{p+q}_G(X,\,{\mathbf F})$.
 \qed

 \begin{Cor} If  $H ^q(X,\, {\mathcal F})=0$ for $q$ odd, then

$$H^{p+q}_G(X,\,{\mathbf F})\simeq H^p(BG)\otimes
H^q(X,\,{\mathcal F}).$$
\end{Cor}

{\it Proof.}
As the cohomology $H^p(BG)$ vanishes when $p$ is odd,   then the differential operators of  spectral sequence  $E_2^{pq}=H^p(BG)\otimes H^q(X,\,{\mathcal
F})$ vanish and  spectral sequence collapses.

\qed

 We assume that $G$ is
the {\em torus} $T=(U(1))^k$.
The $T$-equivariant cohomology with complex coefficients of a
point $H_T({\rm pt};\,{\mathbb C})=H(BT;\,{\mathbb C})$ can be identified to the
algebra ${\mathbb C}[{\mathfrak t}^*_{\mathbb C}]$,  of polynomials on the complexification ${\mathfrak
t}_{\mathbb C}$ of the Lie algebra of $T$.

We   denote by $\Xi$ the multiplicative subset of ${\mathbb
C}[{\mathfrak t}^*_{\mathbb C}]$ consisting of the non-zero
polynomials, and let $S$   denote the fixed point set of the
$T$-action.
 From the localization theorem
\cite[Sect. 6]{G-K-M} one deduces that the restriction map defines
an isomorphism
 $$H_T(X;\,{\mathbf F})_{\Xi}\to H_T(S;\,{\mathbf F})_{\Xi}$$
between the the corresponding localization modules.

As  the action of $T$ on $S$ is trivial, $ET\times_T S=BT\times S$. Thus, for the particular
case of complex coefficients
\begin{equation}\label{F-free}
H_T(S;\,{\mathbf F})\simeq H(BT)\otimes_{\mathbb C} H(S;\,{\mathcal F}).
 \end{equation}
 Hence,
 \begin{equation}\label{Sigma}
 H_T(X; \,{\mathbf F})_{\Xi}\simeq{\mathbb C}(t^*_{\mathbb
 C})\otimes_{\mathbb C} H(S; \,{\mathcal F}),
  \end{equation}
   ${\mathbb C}(t^*_{\mathbb C})$ being the field of rational
 functions on ${\mathfrak t}_{\mathbb C}$. In other words,
 $ H_T(X;\,{\mathbf F})_{\Xi}$ is the result of the extension of scalars in $H(S;\,{\mathcal F})$ from ${\mathbb C}$
 to ${\mathbb C}(t^*_{\mathbb C})$.

\begin{Prop}\label{P:EquivFiniteFixed}
Given ${\mathcal F}$ a brane which belongs to $\mathfrak{Coh}^T$, if the fixed point set for the $T$ action is $\{x_1,\dots,x_r\}$, then
$$H_T(X,\,{\mathbf F})\simeq\bigoplus_{i=1}^r\big({\mathbb C}({\mathfrak t}^*_{\mathbb C})  \otimes{\mathcal F}_{x_i} \big).$$
\end{Prop}

\begin{Cor}\label{CorFiniteFixed} Under the hypotheses of Proposition \ref{P:EquivFiniteFixed}, if ${\mathcal G}$ is other object of $\mathfrak{Coh}^T$ such that $\bigoplus_{i=1}^r\big({\mathbb C}({\mathfrak t}^*_{\mathbb C})  \otimes{\mathcal F}_{x_i} \big)$ and
 $\bigoplus_{i=1}^r\big({\mathbb C}({\mathfrak t}^*_{\mathbb C})  \otimes{\mathcal G}_{x_i} \big)$ are not isomorphic as ${\mathbb C}({\mathfrak t}^*_{\mathbb C})$-vector spaces, then ${\mathcal F}$ and ${\mathcal G}$ are inequivalent $T$-equivariant branes.
\end{Cor}

\smallskip

 \subsection{Equivariant charges.}\label{SubsectEqch}  The charge of a brane  ${\mathcal F}$ which is
 a locally free ${\mathcal O}$-module is an element
  of the cohomology of $X$ defined from certain characteristic clases of $X$ and ${\mathcal
  F}$  \cite{Aspin, Harvey, M-M}. When ${\mathcal F}$ is a coherent sheaf, to define the
  charge it is necessary to pass to a locally free resolution of ${\mathcal
  F}$; the existence of such resolutions is a well-known fact  if $X$
  is a smooth variety  \cite{Fulton1}. This property permits to extent the definition

  to objects of the Grothendieck group  of $X$, when $X$ is a variety.

   In this section, we study the equivariant versions of that process.

  The triangulated category $D(\mathfrak{Coh}^G)$, of $G$-equivariant branes on $X$, has associated the corresponding Grothendieck group
  $K(D(\mathfrak{Coh}^G))$. By $K^G(X)$ we denote the Grothendieck group of the abelian category $\mathfrak{Coh}^G$. The map
  \begin{equation}\label{Grothen-Groups}
  [{\mathcal F}]\in K( D(\mathfrak{Coh}^G))\mapsto \sum_i(-1)^i[H^i({\mathcal F})]\in K^G(X),
   \end{equation}
   where $[Z]$ the equivalence class of the object $Z$, is an isomorphism of abelian groups.

   Let  $K'^G(X)$ be the Grothendieck group
of $G$-equivariant locally free ${\mathcal O}$-modules on $X$
\cite{Segal,Thomason}. In particular, $K'^G({\rm pt})$ is the ring $R(G)$ of virtual
 representations of $G$. The tensor product defines a ring structure
on $K'^G(X)$, and  the tensor product of locally free sheaves by coherent sheaves gives to $K^G(X)$ the structure of module on the ring  $K'^G(X)$.

  Given a $G$-equivariant brane, it seems  natural to assign it a $G$-equivariant charge. When the brane is a locally free sheaf, that assignation can be carried out   by means of the respective $G$-equivariant  characteristic classes.   The resulting charge will be an element of the equivariant cohomology
  $H_G(X)$.

   As in the non equivariant setting, an appropriate choice of the characteristic classes will permit to extent the definition to the objects of $K'^G(X)$.    To define an equivariant charge for branes in $\mathfrak{Coh}^G$,

   it is necessary to consider the cases for which the Grothendieck groups $K'^G(X)$ and $K^G(X)$ are isomorphic. In this situation, the equivariant charges are defined for arbitrary $G$-equivariant branes through  the isomorphism (\ref{Grothen-Groups}).

 One says that   $X$ has the $G$-equivariant resolution property if any $G$-equivariant
 coherent ${\mathcal O}$-module is the quotient of a locally free $G$-module.
 In this case, the natural homomorphism $K'^G(X)\to K^G(X)$ between the

 Grothendieck groups is an  isomorphism.

 Thomason \cite{Thomason1}
  proved the $G$-equivariant resolution property
  for actions of linear algebraic groups which act on smooth varieties.

According with this result,
from now on in this subsection, we assume that:
  \begin{itemize}
  \item The K\"ahler manifold $X$ is an algebraic manifold; that is, $X$ admits a complex analytic embedding as a closed
  submanifold of ${\mathbb C}P^N$, for some $N$. \\
  \item  $G$ is a linear algebraic group. \\
  \item The action of $G$ on  $X$ is algebraic.
   \end{itemize}
 Under these assumptions the smooth algebraic variety $X$ has the $G$-resolution
  property and,  hence,

  the  homomorphism $K'^G(X)\to K^G(X)$ between the algebraic Grothendieck groups is an  isomorphism.

  By the GAGA principle, given a $G$-equivariant coherent ${\mathcal O}$-module ${\mathcal F}$, it is algebraic and  coherent with
  respect the algebraic structure.  So,   ${\mathcal F}$  is the quotient of an algebraic locally free $G$-equivariant sheaf  ${\mathcal E}_0$ on $X$.
    Consequently, it is possible to  construct a resolution
 \begin{equation} \label{resolution}
  0\to{\mathcal E}_m\to{ \mathcal E}_{m-1}\to\dots\to{\mathcal E }_0\to {\mathcal F}\to 0,
   \end{equation}
  consisting of $G$-equivariant  locally free sheaves and $m\leq {\rm
  dim}\,X$ \cite{Thomason1}.

Given an algebraic $G$-equivariant vector bundle $V$ on $X$ of
rank $r$, by the equivariant splitting principle \cite[page
315]{G-G-K}, $V$ has $r$ Chern roots: $x_1\dots,x_r$, such that
the equivariant Chern class $c^G_j(V)=\sigma_j(x_1,\dots,x_r)$,
with $\sigma_j$ the $j$th elementary symmetric function.

One defines the equivariant Chern character of $V$ by the
following element of the equivariant cohomology of $X$
$${\rm ch}^G(V)=\sum_1^re^{x_i}\in H_G(X,\,{\mathbb Q}).$$

The equivariant Chern character is ``additive" with respect the
exact sequences of equivariant vector bundles; that is, if $0\to
V'\to V\to V''\to 0$ is an exact sequence,

 then
${\rm ch}^G({V})={\rm ch}^G({ V}')+{\rm ch}^G({ V}'')$.
 Hence, it
admits an extension to the algebraic Grothendieck group
$K'^G(X)\simeq K^G(X)$
 of $G$ equivariant coherent sheaves.

 In particular,
given a $G$-equivariant brane ${\mathcal
 F}\in\mathfrak{Coh}^G$, the complex ${\mathcal F}$
 is quasi-isomorphic to a complex ${\mathcal E}_{\bullet}$,
 where ${\mathcal E}_i$ is the sheaf of sections of a $G$-equivariant holomorphic vector bundle
 $V_i$. Then we define

$$ {\rm ch}^G({\mathcal F}):= \sum_i (-1)^i{\rm ch}^G({
V}_i).$$

The constant map  $p:X\to{\rm pt}$ gives rise to a pushforward
homomorphism $p_*:K(D(\mathfrak{Coh}^G))\to R(G)$ which maps the class $[{\mathcal F}]$
  to the virtual representation
$$\sum_i(-1)^iR^ip_*{\mathcal F}= \sum_i(-1)^iH^i(X,\,{\mathcal F}),$$
  where the representations
 on the cohomologies are the ones of Proposition
 \ref{Proprephcoho}.

The Chern character does not commutes with the pushforward by
proper maps \cite[page 280]{Fulton1}, \cite{Borel-Serre} and the
equivariant one neither \cite{E-G}.
If $V\to X$ is
 an equivariant vector bundle on $X$,

 then the non commutativity of the Chern character with the pushforward $p_*$ is
expressed by the statement of the equivariant
Hirzebruch-Riemann-Roch theorem:
$${\rm ch(p_*(V}))=p_*\big({\rm ch}^G(
V)\,{\rm td}^G(X)\big),$$
 where ${\rm td}^G(X)$ is the
$G$-equivariant Todd  of the bundle $TX$, that  can be defined by
means of  the equivariant splitting principle \cite[page
317]{G-G-K}.
  The equivariant Hirzebruch-Riemann-Roch
 allows us to express the pushforward ${\rm ch(p_*(V}))$  in terms of equivariant characteristic classes.

\smallskip

Let ${\mathcal F}$ be a general  brane of $D(\mathfrak{Coh}^G)$. The
previous remarks lead us to associate with ${\mathcal F}$ the
following
equivariant cohomology class
  \begin{equation}\label{EquiCharge0}
 Q^G({\mathcal F}):= {\rm ch}^G({\mathcal F})\,{\rm td}^G(X),
  \end{equation}
 that can be considered as an {\it equivariant charge} of the brane ${\mathcal F}$.

 Different equivariant charges can be defined by means of other  $G$-equivariant forms on $X$, for example $\sqrt{{\rm td}^G(X)}$, etc.

\smallskip

\noindent
{\bf Brane on an equivariant subvariety.}
We will consider the natural  brane defined by a subvariety of
the algebraic variety $X$. Let ${\mathcal I}$ be an ideal sheaf of
${\mathcal O}$. We put
$$Z={\rm supp}({\mathcal O}/{\mathcal I}).$$
$Z$ is an analytic subvariety of $X$ and ${\mathcal F}:={\mathcal
O}/{\mathcal I}$ is a coherent sheaf on $X$, that can be
considered as the structure sheaf of $Z$.

Locally, on an open $U$, the sheaf ${\mathcal I}$ will be generated
by the holomorphic functions $f_1,\dots,f_r$. We will assume that
each function is $G$-invariant. Hence, $Z$ is a $G$-invariant
subvariety of $X$ and ${\mathcal O}/{\mathcal I}$ is an
equivariant coherent sheaf.

Let us assume that $f_1,\dots,f_r$ is a regular sequence of
functions and let $e_1,\dots,e_r$ be the canonical basis of ${\mathbb
C}^r$. We put
$${\mathcal E}_k:={\mathcal O}\otimes_{\mathbb C}\big( \wedge^k{{\mathbb
C}^r} \big).$$
 Then ${\mathcal E}_k\simeq{\mathcal O}^{\oplus \tilde k},$ where
 $\tilde k:=\binom {r} {k}$. The known Koszul complex \cite[page 687]{G-H} is
 the following
 $G$-equivariant locally free resolution of ${\mathcal O}/{\mathcal
 I}$
 \begin{equation}\label{Koszul}
 0\to{\mathcal E}_r\overset{\partial}{\rightarrow}{\mathcal
 E}_{r-1}\to\dots\to{\mathcal E}_1\overset{\partial}{\rightarrow}{\mathcal E}_0={\mathcal O}\overset{\rm proj.}\to{\mathcal F}
 \to 0,
  \end{equation}
 where
$$\partial(h\otimes e_{j_1}\wedge\dots\wedge
e_{j_k})=\sum_i(-1)^ihf_i  e_{j_1}\wedge\dots \hat{e}_{j_i}
\dots\wedge e_{j_k}.$$
 Since the $f_i$ are $G$-invariant, the operator $\partial $ is
 equivariant.

 We can use the equivariant locally free resolution (\ref{Koszul}) to define ${\rm ch}^G({\mathcal O}/{\mathcal I})$
 $${\rm ch}^G({\mathcal O}/{\mathcal I})=\sum_{k=0}^r(-1)^k{\rm ch}^G({\mathcal E}_k)=\big(\sum_{k=0}^r(-1)^k\tilde k\big)\,{\rm ch}^G({\mathcal O}).$$
Since $\sum_{k=0}^r(-1)^k\tilde k=(-1+1)^r=0$, we have the following proposition
\begin{Prop}\label{PropchO/I}
The $G$-equivariant charge of the the brane ${\mathcal O}/{\mathcal I}$ is zero.
\end{Prop}

\smallskip

\noindent
{\bf Index of the Dirac operator.}
In some particular cases, the   charge has a natural
interpretation
 in terms of the index  of an elliptic operator.
 The exterior bundle $\Lambda^*T^*X$  of $X$ with the connection
 induced by the Levi-Civita connection  and the standard
 Clifford multiplication  is a Dirac bundle (see \cite[page 114]{L-M}). This  bundle
 has associated the corresponding Dirac operator ${\mathsf D}$. If $G$ acts as a group of isometries
  of $X$, then ${\mathsf D}$ is a $G$-operator \cite[page 211]{L-M}; i.e. ${\mathsf D}$ is $G$-equivariant.

 Let us assume that $G$ is compact and that

  ${ V}$ is a $G$-brane consisting of a locally free

  sheaf. The compactness of $G$ allows us to average over the group
 for obtaining $G$-invariant metrics on $X$ and $G$-invariant connections on ${ V}$. On the other hand,
 the tensor product of $(\Lambda^*T^*X) \otimes{ V}$ is a Dirac bundle (see \cite[page 122]{L-M}) and the corresponding Dirac operator $D$ is also $G$-equivariant,  by the $G$-invariance of the metric and the connection.
  As $D$ is elliptic, ${\rm ker} \, D$ and ${\rm coker} \, D$ are representations
  of $G$ of finite dimension,  since $X$ is compact.

   For $g\in G$ the virtual character $\chi(D)(g)$ of $D$ at $g$ is defined by
  $$\chi(D)(g)={\rm trace}(g|_{{\rm ker}\,D})-{\rm trace}(g|_{{\rm coker}\,D}).$$

By the equivariant index theorem  \cite[Chapter 8]{B-G-V}, we have the proposition.
\begin{Prop}\label{P:DiracVirtual}
 In a neighborhood of
   $0\in{\mathfrak g}:={\rm Lie}(G)$
  \begin{equation}\label{chi(D)}
   \chi(D)\circ{\rm exp}=\int_XQ^G({ V}).
   \end{equation}
\end{Prop}

 The value of $\chi(D)({\rm exp}(\xi))$, for $\xi\in{\mathfrak g}$,  can be calculated by the localization formula in
 equivariant cohomology. The result is the Atiyah-Segal-Singer
 fixed point formula \cite{B-G-V,L-M}.
Next, we will evaluate  the integral (\ref{chi(D)}) when $X$ is a toric variety.

\smallskip
\noindent
{\it Toric varieties.}
Let $\Sigma$ be a fan in $N={\mathbb Z}^r$, such that each cone in $\Sigma$ is generated by a subset of  a basis of ${\mathbb Z}^r$.  We will denote by $X$
the smooth toric variety defined by $\Sigma$  \cite{C-L-S,Fulton,Oda}.
We put $M:={\rm Hom}_{\mathbb Z}(N,\,{\mathbb Z})$

 and $T$ for the torus
$$T=N\otimes{\mathbb C}^{\times}={\rm Hom}_{\mathbb Z}(M,\,{\mathbb
C}^{\times}).$$

We denote by $X^T$ the   fixed point set  for the $T$-action. For $x\in X^T$,

 let $\nu_{i,x}\in 2\pi({\mathbb Z})^n$,$\;i=1,\dots,n$, be the weights of the isotropy representation
 of $T$ on the tangent space $T_xX$. The fixed points of the $T$-action are in bijective correspondence
with the $n$-dimensional cones in $\Sigma$ \cite[\textsection
3.2]{C-L-S}. If the point $x$ is associated with the cone
$\sigma$, then
  \begin{equation}\label{omegaip}
  \omega_{i,x}=\frac{\nu_{i,x}}{2\pi}
  \end{equation}
  are the generators of $\sigma^{\vee}\cap M$, where $\sigma^{\vee}$ is the dual cone of $\sigma$.

Let $V$ be a $T$-equivariant holomorphic vector bundle on $X$ of range $m$. The
equivariant splitting principle together with the fact that  $T$ acts trivially on $X^T$, permit us to express the restriction    of $V$ to $X^T$ as
 a direct sum of $T$-equivariant line bundles
 $$V|_{X^T}=\bigoplus_{j=1}^m{ L}_j,$$
 where action of $T$ on $V|_{X^T}$ will be defined   by $m$ weights $\varphi_j$.
 Thus,    the $T$-equivariant
 Chern class of ${ L}_j$ is given by (see \cite{G-G-K})
  \begin{equation}\label{cT1}
  c^T_1({ L_j})=c_1({ L}_j)+\frac{1}{2\pi}\varphi_j.
   \end{equation}
 The $T$-equivariant Chern character of $V|_{X^T}$ is

 \begin{equation}\label{chT}
 {\rm ch}^T(V|_{X^T})=\sum_{j=1}^m{\rm exp}(c_1^T( L_j)).
  \end{equation}

\begin{Prop}\label{PropFixedPoint}
Let $X$ be a toric manifold and
${ V}$  be a holomorphic $T$-equivariant vector bundle on $X$.
Denoting by $\{\varphi_{j,x}\}_{j=1,\dots,m}$ the weights of the
representation of $T$ on the fibre of ${ V}$ at a fixed
point $x$, then  (\ref{chi(D)}) is equal to
 \begin{equation}\label{2-n}
\sum_{x\in X^T}\Big(\sum_{j=1}^m{\rm
e}^{\frac{1}{2\pi}\varphi_{j,x}}
\Big)\prod_{i=1}^n\Big(1-{\rm e}^{-\omega_{i,x}}     \Big)^{-1},
 \end{equation}
 where $X^T$ is the set of fixed points of $X$ for the $T$-action and the $\omega_{i,x}$ are defined in (\ref{omegaip}).
\end{Prop}
{\it Proof.}
The localization theorem  in equivariant cohomology
\cite[\textsection 10.9]{G-S} allows us to calculate the value
(\ref{chi(D)}) as a sum of contributions of the connected
components of $X^T$. As $X^T$ is discrete, the localization formula
adopts the following form
\begin{equation}\label{chi(D)1}
 \chi(D)\circ{\rm exp}=(2\pi)^n\sum_{x\in X^T}\frac{Q^T(V)(x)}{\prod_i\nu_{i,x}},
 \end{equation}
where the $\nu_{i,x}$ are the weights of the isotropy representation of $T$ at the fixed point $x$.

From (\ref{cT1}) and (\ref{chT}), it follows
$${\rm ch}^T( V){\big|_x}=\sum_{j=1}^m{\rm e}^{\frac{1}{2\pi}\varphi_{j,x}}.$$
Similarly (see \cite[page 230]{L-M}),
$${\rm td}^T(TX){\big|_x}=\prod_{i=1}^n \omega_{i,x}\Big( 1-{\rm e}^{-\omega_{i,x}}   \Big)^{-1}.$$
The proposition follows from (\ref{EquiCharge0}) together with
(\ref{omegaip}). \qed

Note that
 the contribution of the manifold  $X$ to (\ref{2-n}) is encoded in the $n$-dimensional
cones of the fan $\Sigma$.

\section{Appendix}\label{S:Appendix}
 In this section, we will prove Propositions \ref{P:Propiso} and \ref{enumerate}.

 Let $({\mathcal H},\,\lambda)$ be an object of  $\mathfrak{Mod}^G$ and $g\in G$.

With the notations introduced in Subsection \ref{SubSectEquivSheav},  $b^*{\mathcal H}$ is the sheaf associated with the presheaf
 $$W\mapsto {\mathcal P}(W):={\mathcal O}_{G\times X}(W)\otimes_{b^{-1}{\mathcal O}(W)}b^{-1}{\mathcal H}(W),$$
 $W$ being an open set of $G\times X$. Given an open subset $U\subset X$ and $g\in G$, we put
  $U_g:=\{g\}\times U\subset G\times X.$ Identifying ${\mathcal O}_{G\times X}(U_g)$ with ${\mathcal O}(U)$,
   one has the following isomorphism of ${\mathcal O}(U)$-modules
   \begin{equation}\label{identif1}
   {\mathcal P}(U_g)= {\mathcal O}_{G\times X}(U_g)\otimes_{{\mathcal O}(U)}{\mathcal H}(U)\to{\mathcal
   H}(U),\;\;\;\tilde f\otimes\tau\mapsto f\tau,
  \end{equation}
  with $f(x)=\tilde f(g,\,x)$.

 Similarly, the sheaf $\mu^*{\mathcal H}$ is associated to the
 presheaf ${\mathcal M}$, with
$$W\mapsto {\mathcal M}(W):={\mathcal O}_{G\times X}(W)\otimes_{\mu^{-1}{\mathcal O}(W)}b\mu^{-1}{\mathcal H}(W).$$
One has the isomorphism of ${\mathcal O}(gU)$-modules
   \begin{equation}\label{identif2}
  {\mathcal M}(U_g)={\mathcal O}_{G\times X}(U_g)\otimes_{{\mathcal O}(gU)}{\mathcal H}(gU)\to{\mathcal
  H}(gU),\;\;\;\Hat h\otimes\tau\mapsto h\tau
   \end{equation}
    with $h(gx)=\Hat h(g,\,x)$.

 The restriction to $U_g$ of the morphism of sheaves
$\lambda$ is denoted  $\lambda|_{U_g}$
\begin{equation}\label{alpha(U}
\lambda|_{U_g}:b^{*}{\mathcal H}(U_g)
\longrightarrow \mu^{*}{\mathcal H}(U_g).
\end{equation}
By the above identifications (\ref{identif1}) and
(\ref{identif2}), $\lambda|_{U_g}$ determines an isomorphism of
${\mathcal O}(U)$-modules

 \begin{equation}\label{alpha(U}

 \lambda|_{U_g}: {\mathcal H}(U) \stackrel{\sim}{\longrightarrow} {\mathcal
 H}(gU),
 \end{equation}

where the ${\mathcal O}(U)$-module structure of ${\mathcal
 H}(gU)$ is defined through the isomorphism (\ref{O-LgO}).

  Thus,  we have proved   Proposition \ref{P:Propiso}.

As a consequence of  Proposition \ref{P:Propiso}, there is an isomorphism of ${\mathcal O}_x$-modules
 \begin{equation}\label{mathcalHx}
  \lambda_{g,x}:{\mathcal H}_x\to {\mathcal H}_{gx},
 \end{equation}
 where the  ${\mathcal O}_x$-structure of ${\mathcal H}_{gx}$ is induced by the isomorphism (\ref{O-LgO}).

 In this way, the isomorphisms (\ref{asterisco1}) are recovered.

\smallskip

Until now,in this appendix, we have only used the   existence of    the isomorphism
$\lambda$. Next, we will exploit the cocycle condition. Both sides
of equation (\ref{cocycle2}) are isomorphisms between two objects
on the category of ${\mathcal O}_{G\times G\times X}$-modules.
 The cocycle condition means  the commutativity of
the following triangle
 \begin{equation}\label{triangleZ}
  \xymatrix{
 {\mathcal Z}_1\ar[dr]_{(m\times 1_X)^*(\lambda)}\ar[rr]^{p^*(\lambda)} &&{\mathcal Z}_{2}\ar[dl]^{(1_G\times \mu)^*(\lambda)}\\
 & {\mathcal Z}_{3}, }
  \end{equation}
where
$$\begin{aligned}\notag
&{\mathcal Z}_1:=p^*b^*({\mathcal H})=(m\times 1_X)^*b^*({\mathcal
H}),\;\;{\mathcal Z}_2:=p^*\mu^*({\mathcal
H})=(1_G\times\mu)^*b^*({\mathcal H}) \\ &{\mathcal Z}_3:=(m\times
1_X)^*\mu^*({\mathcal H})=(1_G\times\mu)^*\mu^*({\mathcal H}).
\end{aligned}$$

As a consequence of the cocycle condition, we have the
following lemma.

\begin{Lem}\label{Lema}
Let $g,h$ be elements of $G$ and $U$ an open set of $X$, then
$$\lambda|_{(gU)_h}\circ\lambda|_{U_g}=\lambda|_{U_{hg}}.$$
In particular,
$\lambda|_{X_h}\circ\lambda|_{X_g}=\lambda|_{X_{hg}}$.
\end{Lem}

{\it Proof.} We consider   the commutative diagram
(\ref{triangleZ})
and  restrict this triangle to $\{h\}\times U_g\subset G\times
G\times X$. The restriction of $(m\times 1_X)^*(\lambda)$ is  the
morphism
$${\mathcal Z}_1(\{h\}\times U_g )={\mathcal H}(U)\longrightarrow {\mathcal Z}_3(\{h\}\times U_g )={\mathcal
H}((hg)U)$$
induced by $\lambda$. Thus,  by (\ref{alpha(U}), the mentioned restriction is
 $\lambda|_{U_{hg}}$.

 The restriction of $p^{*}(\lambda)$ to  $\{h\}\times U_g$
 $${\mathcal Z}_1(\{h\}\times U_g )={\mathcal H}(U)\longrightarrow {\mathcal Z}_2(\{h\}\times U_g )={\mathcal
H}(gU)$$
is (\ref{alpha(U}).

Finally, we consider the restriction of
$(1_G\times\mu)^*(\lambda)$. It is the morphism
 $${\mathcal Z}_2(\{h\}\times U_g )={\mathcal H}(gU)\longrightarrow {\mathcal Z}_3(\{h\}\times U_g )={\mathcal
H}(h(gU))$$
 induced by $\lambda$, and according to (\ref{alpha(U}) it is $\lambda|_{(gU)_h}$.
Then the lemma follows from   the commutativity of (\ref{triangleZ}).
 \qed

\smallskip

Proposition \ref{enumerate} is a direct consequence the Lemma \ref{Lema}.

redpajama_set_name: RedPajamaArXiv


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The Webby Awards is the leading international award honoring excellence on the Internet. The Webby Awards presents two honors in every category — The Webby Award and The Webby People's Voice Award. Members of the International Academy of Digital Arts and Sciences (IADAS) select the Nominees for both awards in each category, as well as the Winners of The Webby Awards. In the spirit of the open Web, The Webby People's Voice is awarded by the voting public. Each year, The Webby People's Voice Awards garners millions of votes from all over the world.


Looking for a great deal on cheap flights to Chicago? Vayama compares all destinations, airlines and itineraries to offer you the cheapest flights to Chicago. Vayama is a travel agency specialized in cheap international flights. We continuously look for ways to make it easier for you to find the cheap flight you are looking for. That's why we show the lowest possible flight found by our customers in the last 48 hours on all continents, countries, cities and airports. Look for our low priced flights from major US airports like John F Kennedy Airport, San Francisco, LAX and Chicago O'Hare.Whether you are visiting family across states, or abroad, Vayama finds the flight that's right for you. International travellers will find Vayama has a limitless selection of cheap one-way, return or multi-destination flights to North America, Europe, Asia, South America, the Middle East, Africa and the Caribbean. As a travel agent Vayama offers cheap flights on a range of regular and low cost carriers from the United States (United Airlines, American Airlines, Delta Airlines, US Airways) and abroad (AerLingus, British Airways, Air France, KLM, Etihad Airways, Emirates, Air Canada, Lufthansa, Turkish Airlines and Thai Airways and more).
Chicago, the third largest city in the United States is served by two airports: O'Hare International Airport and Midway Airport. O'Hare International Airport is located at about 29 kms from the city center and consists of four terminals and Midway Airport is located at about 16 kms from the city center and consists of one terminal. O'Hare International Airport has its own automated 24-hour ATS train system connecting all four terminals.Various buses, taxis and trains services are provided from the airport to help passengers travel to the city center. Car rental and limo agencies are also located at both the airport to help passengers willing to travel on their own. Information desks are located at the airport to help passengers with their queries.
Passengers willing to have a limo ride can opt for the limousines by doing a pre-booking before arriving at the airport. A number of local agencies offer this service and there is a car rental booth located within the airport terminal. Services are available even on hourly basis.
Chicago Area Regional Transportation Authority operates bus services between the two Airports. Information desks are located at the airport to help passengers with their queries.The Chicago Transit Authority Orange Line serves the southwest side of Chicago and connects the O Hare International Airport with Midway Airport. The Chicago Transit Authority runs the 24-hour Metro service to downtown Chicago from O'Hare International Airport. The CTA station is located on the lowest level of the main parking garage at Terminals 1, 2 and 3.
Taxis ply from O' Hare International Airport to help passengers travel to the city center.Taxis at the airport can be availed outside the terminal and takes a journey time of about 30 minutes to travel to the city center. Car hire companies also provide rental services to help passengers willing to travel on their own.
Transportation from OHare Intl. Arpt. to Midway Intl. Arpt.
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Common Core is not 'federal intrusion'
The results of last year's international Programme for International Student Assessment exam, which tested the 15-year-olds of 60 of the world's industrialized nations, show that education in the U.S. has flat-lined.
PISA reports from last month indicate the U.S. fell to No. 17, in the world in reading, No. 21 in science and No. 26 in math, falling behind China, Canada, Germany and other countries.
On a national level, Florida ranks near the bottom of those sinking scores.
Since 2010, the federal government has been pushing for states to adopt the Common Core State Standards, a set of national standards designed to ensure that students graduating from high school meet a basic level of competency.
But recently, Gov. Rick Scott was quoted as telling the state GOP members "We're not going to have the federal government telling us how to do our education system."
But if any state should argue about federal intrusion in education, it shouldn't be the state with scores lower than the country's national average.
Clearly someone needs to tell Florida how to fix its education system, and if Scott refuses to listen to the federal government, he better start asking the European and Asian nations who seem to be doing something better than us.
The Common Core shouldn't be considered as a "federal intrusion," but more as a national benchmark in which all states should work toward.
The purpose of Common Core is to at least ensure a common ground for children in every state, ideally boosting those in lower performing states — and Florida definitely ranks as one of those.
While curriculum should be considered a local responsibility, much the same way as academic freedom and curriculum are the responsibility of individual professors at universities, having a basic set of standards to meet and set by a body that can hold feet to fire is something that is essential for the success of future Floridians.
Though educational reforms by the federal government are always been far from perfect, the governing bodies of Florida have never done education right either.


So here we are, homeschoolers amid a flurry of conventions, conferences, webinars, and workshops (and in case you couldn't tell by the title, my homeschool is amid alliteration in literature)! Y'all know that I love all the different aspects of a homeschool "shin-dig" (my professional vocabulary word for all of the above), because I've written about it in an earlier blog post.
I personally love meeting new people, soaking up all the wisdom from talented speakers, seeing all the new ways we can include the Gospel in teaching our children, and all of the other amazing things that technology and new generations have been drumming up to improve the already overwhelming success of homeschoolers like you! I would attend one of these "shindigs" every week if I really could!
On the flip side, many homeschoolers aren't regulars on the conference scene or have never been to one. When asked, here were the top 3 reasons folks reported not attending. And before you agree and walk away, know that I've also got a positive solution for you (if you'll read on)!
So, I admit that those factors can really put a stick in the spoke of your book-crate-on-wheels because I've been there! I often think it's really not an inadequacy on the part of the attendee when it comes to implementation and choosing what to buy—it's really more of a problem of organization! So…let me introduce you to The 2017 Homeschool Convention Planner & Journal and Lesli over at This Crazy Homeschool Life! She's offered a pretty simple solution to the above flaws.
I want to share with you my experience using this tool as a full-time working, homeschooling mom of two elementary age students. I've been using this throughout the season, and let me tell you how organized and ready to use all of my conference notes and tips were almost immediately upon return home! This has been priceless. No more fumbling around on notebook paper trying to relive the sessions to make sense of my notes or pull them from the shopping fliers from around the convention hall. I simply printed the downloadable resource and loaded it into a three-ring binder for each separate conference I planned on attending. I was instantly hooked (of course, if you're like me, I had you at planner, or binder, or well…resource)!
This planner is not just a notebook that you can pick up anywhere. Lesli has put together a well thought out list of what you need to do to not only survive but revive your homeschool through convention attendance! This is how you make the convention work for you! You no longer have to feel overwhelmed or live in fear of what lies in the bottom remains of your free tote bag.
She addresses everything from the basics, budget forms, session plans, meal planners, and shopping guides all the way to packing snacks and entertainment for the children. Then she divides the convention up by day and segments it into schedules, plans, and notes. A separate section is laid out for any recorded sessions that you might take home so you can incorporate those notes into your plan as well.
My favorite part of this compilation is the very last section: the Post- Convention Journaling! It really was revolutionary to how I utilized my information from the sessions. This section includes Thoughts, Changes and Goals, Change Action Plans, Goal Action Plansefinitely what had been missing for me. I had always been able to gather and enjoy the information, but the implementation was more of a challenge—especially after I collapsed at home with all of my goodies! Never again! I will have it all noted and organized so that when I catch my breath, I'll be able to flip to the "time to take action" section and get started!
Now the LAST TWO STEPS!
Download this amazing tool! To make this an even sweeter deal for you, HSLDA readers are able to take advantage of The 2017 Homeschool Convention Planner & Journal for only $4.75!
Find a homeschool "shindig" near you and take action this year!
I want to leave you with a few tips for planning your next adventure from Lesli!
Check the convention schedule before attending and plan out your classes, but be willing to be flexible when you get there.
Plan your shopping list before entering the vendor hall.
Don't buy anything on your first trip through the vendor hall, but simply enjoy the displays and then return to shop later.
Don't feel like you've got to absorb everything at your first convention. You will likely be homeschooling for many years, so concentrate on what you need for the stage you're in.
Bring some kind of rolling cart. Books are heavy!!
*If you think these might be helpful, you can read even more from Lesli and home school conventions on This Crazy Homeschool Life.
Excellent tips. I thoroughly enjoyed the one convention I attended. Unfortunately, nobody checks my calendar before scheduling these things, and they tend to conflict with sports tournaments for which I am the transportation. I wish conventions were the last week of February or first week of March (or last week of July, but that's probably too late for a lot of people).
Sometimes, the Home School concept is very difficult to handle.


Internationally renowned high school mathematics teacher and internet sensation Eddie Woo delivered an inspirational and thought provoking Australia Day Address.
Arguably Australia's most famous mathematics teacher, Eddie Woo makes maths fun. The head mathematics teacher at Cherrybrook Technology High School, the largest secondary school in New South Wales, Eddie started posting videos online in 2012 for a student who was sick with cancer and missing a lot of school. Before long, he was sharing the videos across the country and beyond. Wootube now boasts more than 100,000 subscribers and has attracted more than eight million views worldwide and counting.
With infectious enthusiasm, the father-of-three's unique and caring approach to teaching destigmatises mathematics as an inaccessible and difficult subject. Outside his high school classroom, Eddie is a volunteer facilitator with the University of Sydney's Widening Participation and Outreach program and has motivated more than 1,400 students from disadvantaged backgrounds.
A brilliant student, Eddie could have chosen any field, but in defiance of social convention and his parents opted for teaching. Today, he is using his vocation to "pay it forward" and make education equitable for all.
Eddie Woo is changing the face of teaching mathematics and is the national recipient of the 2018 Australia's Local Hero Award.


text:
Konstantīns Pēkšēns (born 3 March 1859, Mazsalaca parish, Russian Empire — died 23 June 1928, Bad Kissingen, Weimar Republic) is one of the most prominent Latvian architects of all times. After Jānis Baumanis he is the epitome of the second generation of Latvian architects. Many Latvian cities and towns take pride in buildings designed by Pēkšēns, but Riga alone can boast more than 250 multi-storey brick buildings and a great number of wooden houses erected following his designs.

Career
Pēkšēns was born in the Nuķi estate near Mazsalaca but in 1896 his family moved to Riga. 1875, Pēkšēns began studies at Riga Polytechnical Institute — at first in the Engineering Department, but from 1880 in the Department of Architecture. He took an active part in social life of students, was founding member of Selonija, a fraternity of Latvian students, participated much in sports. He graduated from the institute in 1885 and spent some time working for the construction office of Jānis Baumanis, in 1886 Pēkšēns opened his own practice. In 1889 he was among those who re-established Riga Architects Society that had disintegrated earlier. Pēkšēns also sat in the councils of a number of credit institutions and banks, and acted as a spokesman for the Riga Latvian Society. Since 1909 he was a member of the Riga City Council, but after World War I was actively involved in several technical commissions under the Riga City Executive Board. Pēkšēns also participated in the publishing of several Latvian newspapers. His broad professional experience allowed him to make a considerable contribution at the debate on how to restore the damages of World War I. Pēkšēns company of plumbing works was well known as the largest local enterprise dealing with the assembly of central heating systems in Riga prior to the war. This company remained in operation until 1940.

At the beginning of the 20th century Pēkšēns also acted as a jury member in several large-scale competitions, while in some other competitions his designs received the highest evaluation. For example, he won the competition for a residential house for workers in 1907, together with Eižens Laube, the Riga Merchants Credit Society Bank at 14, Tērbatas Street (1909), together with Arthur Moedlinger, and Ozoliņš' apartment building at 88, Brīvības Street (1910) together with Ernest Pole. All those associates of Pēkšēns, just like Aleksandrs Vanags, August Malvess, and others who later became well known in Riga in their own right, accumulated their professional experience while working in the construction and technical office of the master.

In 1928 in an attempt to improve his seriously deteriorating health Pēkšēns went to Bad Kissingen in Germany, but the trip was to no avail. The sad news of his death reached his native country on the night of the Līgo festival. The master was buried at the Forest Cemetery in Riga.

Art Nouveau
During the golden age of Art Nouveau of the early 20th century several buildings with a fascinating abundance and variety of decorative motifs typical for this new style were designed by Pēkšēns. Examples are the houses at 6, Strēlnieku Street, 13, Kaļķu Street and 2, Smilšu Street built in 1902. The peculiarity of the latter lies in the fact that an every single window decoration is different, and under the bay windows are hermas—supports as sculpted upper bodies. One of these sculptures was given the title Miss Riga from among the stone females displayed on the facades of the buildings in Riga.

The most characteristic feature of Pēkšēns creative work, is the respective reserve and deep logic of architectural forms arising out of the Art Nouveau artistic principle, namely, the beauty of a building should not depend on outside applications, but derive from a practical and utilitarian layout. Most of Pēkšēns buildings are defined by a clear arrangement of volumes corresponding to the layout of interior spaces, careful choice of building materials and decor governed by the basic architectural form.

Such rational Art Nouveau evidence is present in the buildings at 46, Brīvības Street (1907) and 14, Tērbatas Street (1909), apartment houses at 14, Ausekļa Street (1909), 1 and 44, Avotu Street (both 1904) and 66, Avotu Street (1912), 5, Krišjāņa Barona Street (1909), 45, Baznīcas Street (1909), 148 and 172, Brīvības Street (1912 and 1911), 5, Hospitāļu Street (1904), 6, Marijas Street (1904), 5, Noliktavas Street (1904), 3, 9 and 13, Rūpniecības Street (1908, 1910 and 1909), 32, Skolas Street (1904), 31, Slokas Street (1908), 9/11, Tērbatas Street (1912), 4 and 10, Vīlandes Street (both 1908), 12 and 14, Vīlandes Street (both 1909) and 16, Vīlandes Street (1910), — it is impossible to list them all. Some designs by Pēkšēns contain romanticised reminiscences of historical motifs, others stand out by a vertical compositional arrangement that became especially characteristic of Riga's Art Nouveau movement around 1910. Nevertheless, all the buildings are united by strong forms, elegance, and overall reserve in decorations perceivable only in close-up.

National Romanticism
The National Romantic style had a special role within Riga's Art Nouveau style, and Pēkšēns was one of the originators of this stylistic trend. One of the first National Romantic buildings in Riga was the apartment house at 4, Lāčplēša Street (1905). It was followed by apartment buildings at 40, Krišjāņa Barona Street, 192, Brīvības Street, 1a, Sapieru Street (1907) and 46, Ģertrūdes Street (1908) designed in this idiom as interpreted by Pēkšēns. Today it is difficult to say how much of the architecture of those buildings was created by the master himself and what was contributed by his young associates. It is known with certainty that six very interesting buildings at 23, Tallinas Street, 12, Alberta Street, 26, Aleksandra Čaka Street, 10, Kronvalda Boulevard, as well as 15/17, Tērbatas Street and 33/35, Tērbatas Street, are the result of creative cooperation with Eižens Laube. Although the construction designs bear the signature by Pēkšēns, the author of the artistic ideas here was Laube. At the same time, the facade of the house at 192, Brīvības Street, contains patterns of expressions characteristic to buildings designed by August Malvess.

Outside Riga
The most outstanding houses designed by Pēkšēns outside Riga are the District Council buildings in Ādaži and Dole, the church in Lejasciems (1895), his own summer house in Dzintari, the temporary concert hall in Jelgava built for the IV General Latvian Song Festival (1895), and Nicholas' Church of the Latvian parish in Jelgava (1904—09, demolished after World War II).

See also
 Art Nouveau architecture in Riga

References

Sources

1859 births
1928 deaths
Architects from Riga
Riga Technical University alumni
Art Nouveau architects
People from Mazsalaca
Burials at Forest Cemetery, Riga

redpajama_set_name: RedPajamaWikipedia


This show examines the current boom in robotics. It comprises more than 200 exhibits from the fields of design and art and includes robots used in the home, in nursing care, and industry, as well as computer games, media installations, and examples of films and literature featuring robots. It shows the wide variety of forms that robotics takes today while broadening our awareness of the associated ethical, social and political issues. Hello, Robot introduces us to robots in four steps: the first traces the fascination that robots have long exerted over people and looks at how popular culture has shaped our perception of robots. The second is devoted to the field in which robotics first made a breakthrough: industry and the working environment. The third shows how we are gradually coming face-to-face with this new technology – as a friend and helper in our everyday lives. The last section looks at the increasing blurring of boundaries between humans and robots. For while robotics has become more approachable, more personal, more essential even, the question remains: does it make our world any better?